/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


MAYBE
proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


H-Termination with start terms of the given HASKELL could not be shown:

(0) HASKELL
(1) IFR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 22 ms]
(6) HASKELL
(7) LetRed [EQUIVALENT, 0 ms]
(8) HASKELL
(9) NumRed [SOUND, 0 ms]
(10) HASKELL
(11) Narrow [SOUND, 0 ms]
(12) AND
    (13) QDP
        (14) DependencyGraphProof [EQUIVALENT, 0 ms]
        (15) AND
            (16) QDP
                (17) QDPOrderProof [EQUIVALENT, 43 ms]
                (18) QDP
                (19) DependencyGraphProof [EQUIVALENT, 0 ms]
                (20) QDP
                (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (22) YES
            (23) QDP
                (24) MRRProof [EQUIVALENT, 0 ms]
                (25) QDP
                (26) PisEmptyProof [EQUIVALENT, 0 ms]
                (27) YES
    (28) QDP
        (29) MNOCProof [EQUIVALENT, 0 ms]
        (30) QDP
        (31) InductionCalculusProof [EQUIVALENT, 0 ms]
        (32) QDP
        (33) TransformationProof [EQUIVALENT, 0 ms]
        (34) QDP
            (35) TransformationProof [EQUIVALENT, 0 ms]
            (36) QDP
            (37) UsableRulesProof [EQUIVALENT, 0 ms]
            (38) QDP
            (39) QReductionProof [EQUIVALENT, 0 ms]
            (40) QDP
            (41) TransformationProof [EQUIVALENT, 0 ms]
            (42) QDP
            (43) UsableRulesProof [EQUIVALENT, 0 ms]
            (44) QDP
            (45) QReductionProof [EQUIVALENT, 0 ms]
            (46) QDP
            (47) TransformationProof [EQUIVALENT, 0 ms]
            (48) QDP
            (49) TransformationProof [EQUIVALENT, 0 ms]
            (50) QDP
            (51) DependencyGraphProof [EQUIVALENT, 0 ms]
            (52) QDP
            (53) TransformationProof [EQUIVALENT, 0 ms]
            (54) QDP
            (55) DependencyGraphProof [EQUIVALENT, 0 ms]
            (56) QDP
            (57) TransformationProof [EQUIVALENT, 0 ms]
            (58) QDP
            (59) DependencyGraphProof [EQUIVALENT, 0 ms]
            (60) QDP
            (61) TransformationProof [EQUIVALENT, 0 ms]
            (62) QDP
            (63) DependencyGraphProof [EQUIVALENT, 0 ms]
            (64) QDP
            (65) TransformationProof [EQUIVALENT, 0 ms]
            (66) QDP
            (67) DependencyGraphProof [EQUIVALENT, 0 ms]
            (68) AND
                (69) QDP
                    (70) UsableRulesProof [EQUIVALENT, 0 ms]
                    (71) QDP
                    (72) QReductionProof [EQUIVALENT, 0 ms]
                    (73) QDP
                    (74) TransformationProof [EQUIVALENT, 0 ms]
                    (75) QDP
                    (76) DependencyGraphProof [EQUIVALENT, 0 ms]
                    (77) QDP
                    (78) TransformationProof [EQUIVALENT, 0 ms]
                    (79) QDP
                    (80) DependencyGraphProof [EQUIVALENT, 0 ms]
                    (81) QDP
                    (82) TransformationProof [EQUIVALENT, 0 ms]
                    (83) QDP
                    (84) TransformationProof [EQUIVALENT, 0 ms]
                    (85) QDP
                    (86) DependencyGraphProof [EQUIVALENT, 0 ms]
                    (87) AND
                        (88) QDP
                            (89) UsableRulesProof [EQUIVALENT, 0 ms]
                            (90) QDP
                            (91) QReductionProof [EQUIVALENT, 0 ms]
                            (92) QDP
                            (93) TransformationProof [EQUIVALENT, 0 ms]
                            (94) QDP
                            (95) TransformationProof [EQUIVALENT, 0 ms]
                            (96) QDP
                            (97) TransformationProof [EQUIVALENT, 0 ms]
                            (98) QDP
                            (99) TransformationProof [EQUIVALENT, 0 ms]
                            (100) QDP
                            (101) TransformationProof [EQUIVALENT, 0 ms]
                            (102) QDP
                            (103) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (104) QDP
                            (105) TransformationProof [EQUIVALENT, 0 ms]
                            (106) QDP
                            (107) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (108) QDP
                            (109) TransformationProof [EQUIVALENT, 0 ms]
                            (110) QDP
                            (111) QDPSizeChangeProof [EQUIVALENT, 2 ms]
                            (112) YES
                        (113) QDP
                            (114) UsableRulesProof [EQUIVALENT, 0 ms]
                            (115) QDP
                            (116) QReductionProof [EQUIVALENT, 0 ms]
                            (117) QDP
                            (118) TransformationProof [EQUIVALENT, 0 ms]
                            (119) QDP
                            (120) TransformationProof [EQUIVALENT, 0 ms]
                            (121) QDP
                            (122) TransformationProof [EQUIVALENT, 0 ms]
                            (123) QDP
                            (124) TransformationProof [EQUIVALENT, 0 ms]
                            (125) QDP
                            (126) TransformationProof [EQUIVALENT, 0 ms]
                            (127) QDP
                            (128) TransformationProof [EQUIVALENT, 0 ms]
                            (129) QDP
                            (130) TransformationProof [EQUIVALENT, 0 ms]
                            (131) QDP
                            (132) TransformationProof [EQUIVALENT, 0 ms]
                            (133) QDP
                            (134) TransformationProof [EQUIVALENT, 0 ms]
                            (135) QDP
                            (136) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (137) QDP
                            (138) TransformationProof [EQUIVALENT, 3 ms]
                            (139) QDP
                            (140) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (141) AND
                                (142) QDP
                                    (143) TransformationProof [EQUIVALENT, 0 ms]
                                    (144) QDP
                                    (145) TransformationProof [EQUIVALENT, 0 ms]
                                    (146) QDP
                                    (147) TransformationProof [EQUIVALENT, 0 ms]
                                    (148) QDP
                                    (149) QDPOrderProof [EQUIVALENT, 1293 ms]
                                    (150) QDP
                                    (151) DependencyGraphProof [EQUIVALENT, 0 ms]
                                    (152) TRUE
                                (153) QDP
                                    (154) QReductionProof [EQUIVALENT, 0 ms]
                                    (155) QDP
                                    (156) InductionCalculusProof [EQUIVALENT, 0 ms]
                                    (157) QDP
                (158) QDP
                    (159) UsableRulesProof [EQUIVALENT, 0 ms]
                    (160) QDP
                    (161) QReductionProof [EQUIVALENT, 0 ms]
                    (162) QDP
                    (163) TransformationProof [EQUIVALENT, 0 ms]
                    (164) QDP
                    (165) DependencyGraphProof [EQUIVALENT, 0 ms]
                    (166) QDP
                    (167) TransformationProof [EQUIVALENT, 0 ms]
                    (168) QDP
                    (169) DependencyGraphProof [EQUIVALENT, 0 ms]
                    (170) QDP
                    (171) TransformationProof [EQUIVALENT, 0 ms]
                    (172) QDP
                    (173) TransformationProof [EQUIVALENT, 0 ms]
                    (174) QDP
                    (175) DependencyGraphProof [EQUIVALENT, 0 ms]
                    (176) AND
                        (177) QDP
                            (178) UsableRulesProof [EQUIVALENT, 0 ms]
                            (179) QDP
                            (180) QReductionProof [EQUIVALENT, 0 ms]
                            (181) QDP
                            (182) TransformationProof [EQUIVALENT, 0 ms]
                            (183) QDP
                            (184) TransformationProof [EQUIVALENT, 0 ms]
                            (185) QDP
                            (186) TransformationProof [EQUIVALENT, 0 ms]
                            (187) QDP
                            (188) TransformationProof [EQUIVALENT, 0 ms]
                            (189) QDP
                            (190) TransformationProof [EQUIVALENT, 0 ms]
                            (191) QDP
                            (192) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (193) QDP
                            (194) TransformationProof [EQUIVALENT, 0 ms]
                            (195) QDP
                            (196) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (197) QDP
                            (198) TransformationProof [EQUIVALENT, 0 ms]
                            (199) QDP
                            (200) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                            (201) YES
                        (202) QDP
                            (203) UsableRulesProof [EQUIVALENT, 0 ms]
                            (204) QDP
                            (205) QReductionProof [EQUIVALENT, 0 ms]
                            (206) QDP
                            (207) TransformationProof [EQUIVALENT, 0 ms]
                            (208) QDP
                            (209) TransformationProof [EQUIVALENT, 0 ms]
                            (210) QDP
                            (211) TransformationProof [EQUIVALENT, 0 ms]
                            (212) QDP
                            (213) TransformationProof [EQUIVALENT, 0 ms]
                            (214) QDP
                            (215) TransformationProof [EQUIVALENT, 0 ms]
                            (216) QDP
                            (217) TransformationProof [EQUIVALENT, 0 ms]
                            (218) QDP
                            (219) TransformationProof [EQUIVALENT, 0 ms]
                            (220) QDP
                            (221) TransformationProof [EQUIVALENT, 0 ms]
                            (222) QDP
                            (223) TransformationProof [EQUIVALENT, 0 ms]
                            (224) QDP
                            (225) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (226) QDP
                            (227) TransformationProof [EQUIVALENT, 2 ms]
                            (228) QDP
                            (229) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (230) AND
                                (231) QDP
                                    (232) TransformationProof [EQUIVALENT, 0 ms]
                                    (233) QDP
                                    (234) TransformationProof [EQUIVALENT, 0 ms]
                                    (235) QDP
                                    (236) TransformationProof [EQUIVALENT, 0 ms]
                                    (237) QDP
                                    (238) QReductionProof [EQUIVALENT, 0 ms]
                                    (239) QDP
                                    (240) InductionCalculusProof [EQUIVALENT, 0 ms]
                                    (241) QDP
                                (242) QDP
                                    (243) QReductionProof [EQUIVALENT, 0 ms]
                                    (244) QDP
                                    (245) InductionCalculusProof [EQUIVALENT, 0 ms]
                                    (246) QDP
                (247) QDP
                    (248) UsableRulesProof [EQUIVALENT, 0 ms]
                    (249) QDP
                    (250) QReductionProof [EQUIVALENT, 0 ms]
                    (251) QDP
                    (252) TransformationProof [EQUIVALENT, 0 ms]
                    (253) QDP
                    (254) DependencyGraphProof [EQUIVALENT, 0 ms]
                    (255) QDP
                    (256) TransformationProof [EQUIVALENT, 0 ms]
                    (257) QDP
                    (258) DependencyGraphProof [EQUIVALENT, 0 ms]
                    (259) QDP
                    (260) TransformationProof [EQUIVALENT, 0 ms]
                    (261) QDP
                    (262) TransformationProof [EQUIVALENT, 0 ms]
                    (263) QDP
                    (264) DependencyGraphProof [EQUIVALENT, 0 ms]
                    (265) QDP
                    (266) TransformationProof [EQUIVALENT, 0 ms]
                    (267) QDP
                    (268) TransformationProof [EQUIVALENT, 0 ms]
                    (269) QDP
                    (270) TransformationProof [EQUIVALENT, 0 ms]
                    (271) QDP
                    (272) DependencyGraphProof [EQUIVALENT, 0 ms]
                    (273) QDP
                    (274) TransformationProof [EQUIVALENT, 0 ms]
                    (275) QDP
                    (276) TransformationProof [EQUIVALENT, 0 ms]
                    (277) QDP
                    (278) TransformationProof [EQUIVALENT, 0 ms]
                    (279) QDP
                    (280) TransformationProof [EQUIVALENT, 0 ms]
                    (281) QDP
                    (282) DependencyGraphProof [EQUIVALENT, 0 ms]
                    (283) QDP
                    (284) TransformationProof [EQUIVALENT, 0 ms]
                    (285) QDP
                    (286) TransformationProof [EQUIVALENT, 0 ms]
                    (287) QDP
                    (288) TransformationProof [EQUIVALENT, 0 ms]
                    (289) QDP
                    (290) TransformationProof [EQUIVALENT, 0 ms]
                    (291) QDP
                    (292) TransformationProof [EQUIVALENT, 0 ms]
                    (293) QDP
                    (294) TransformationProof [EQUIVALENT, 0 ms]
                    (295) QDP
                    (296) TransformationProof [EQUIVALENT, 0 ms]
                    (297) QDP
                    (298) DependencyGraphProof [EQUIVALENT, 0 ms]
                    (299) QDP
                    (300) TransformationProof [EQUIVALENT, 0 ms]
                    (301) QDP
                    (302) DependencyGraphProof [EQUIVALENT, 0 ms]
                    (303) AND
                        (304) QDP
                            (305) UsableRulesProof [EQUIVALENT, 0 ms]
                            (306) QDP
                            (307) QReductionProof [EQUIVALENT, 0 ms]
                            (308) QDP
                            (309) TransformationProof [EQUIVALENT, 0 ms]
                            (310) QDP
                            (311) TransformationProof [EQUIVALENT, 0 ms]
                            (312) QDP
                            (313) TransformationProof [EQUIVALENT, 0 ms]
                            (314) QDP
                            (315) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (316) QDP
                            (317) TransformationProof [EQUIVALENT, 0 ms]
                            (318) QDP
                            (319) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (320) QDP
                            (321) TransformationProof [EQUIVALENT, 0 ms]
                            (322) QDP
                            (323) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                            (324) YES
                        (325) QDP
                            (326) UsableRulesProof [EQUIVALENT, 0 ms]
                            (327) QDP
                            (328) QReductionProof [EQUIVALENT, 0 ms]
                            (329) QDP
                            (330) TransformationProof [EQUIVALENT, 0 ms]
                            (331) QDP
                            (332) TransformationProof [EQUIVALENT, 0 ms]
                            (333) QDP
                            (334) TransformationProof [EQUIVALENT, 0 ms]
                            (335) QDP
                            (336) TransformationProof [EQUIVALENT, 0 ms]
                            (337) QDP
                            (338) TransformationProof [EQUIVALENT, 0 ms]
                            (339) QDP
                            (340) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (341) QDP
                            (342) TransformationProof [EQUIVALENT, 0 ms]
                            (343) QDP
                            (344) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (345) QDP
                            (346) TransformationProof [EQUIVALENT, 0 ms]
                            (347) QDP
                            (348) QDPSizeChangeProof [EQUIVALENT, 13 ms]
                            (349) YES
                        (350) QDP
                            (351) UsableRulesProof [EQUIVALENT, 0 ms]
                            (352) QDP
                            (353) QReductionProof [EQUIVALENT, 0 ms]
                            (354) QDP
                            (355) TransformationProof [EQUIVALENT, 0 ms]
                            (356) QDP
                            (357) TransformationProof [EQUIVALENT, 0 ms]
                            (358) QDP
                            (359) TransformationProof [EQUIVALENT, 0 ms]
                            (360) QDP
                            (361) TransformationProof [EQUIVALENT, 0 ms]
                            (362) QDP
                            (363) TransformationProof [EQUIVALENT, 0 ms]
                            (364) QDP
                            (365) TransformationProof [EQUIVALENT, 0 ms]
                            (366) QDP
                            (367) TransformationProof [EQUIVALENT, 0 ms]
                            (368) QDP
                            (369) TransformationProof [EQUIVALENT, 0 ms]
                            (370) QDP
                            (371) TransformationProof [EQUIVALENT, 0 ms]
                            (372) QDP
                            (373) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (374) QDP
                            (375) TransformationProof [EQUIVALENT, 0 ms]
                            (376) QDP
                            (377) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (378) QDP
                            (379) TransformationProof [EQUIVALENT, 0 ms]
                            (380) QDP
                            (381) TransformationProof [EQUIVALENT, 0 ms]
                            (382) QDP
                            (383) TransformationProof [EQUIVALENT, 0 ms]
                            (384) QDP
                            (385) TransformationProof [EQUIVALENT, 0 ms]
                            (386) QDP
                            (387) DependencyGraphProof [EQUIVALENT, 0 ms]
                            (388) AND
                                (389) QDP
                                    (390) UsableRulesProof [EQUIVALENT, 0 ms]
                                    (391) QDP
                                    (392) TransformationProof [EQUIVALENT, 0 ms]
                                    (393) QDP
                                    (394) TransformationProof [EQUIVALENT, 0 ms]
                                    (395) QDP
                                    (396) TransformationProof [EQUIVALENT, 0 ms]
                                    (397) QDP
                                    (398) InductionCalculusProof [EQUIVALENT, 0 ms]
                                    (399) QDP
                                (400) QDP
                                    (401) UsableRulesProof [EQUIVALENT, 0 ms]
                                    (402) QDP
                                    (403) TransformationProof [EQUIVALENT, 0 ms]
                                    (404) QDP
                                    (405) UsableRulesProof [EQUIVALENT, 0 ms]
                                    (406) QDP
                                    (407) InductionCalculusProof [EQUIVALENT, 0 ms]
                                    (408) QDP
                                (409) QDP
                                    (410) InductionCalculusProof [EQUIVALENT, 0 ms]
                                    (411) QDP
    (412) QDP
        (413) DependencyGraphProof [EQUIVALENT, 0 ms]
        (414) AND
            (415) QDP
                (416) MRRProof [EQUIVALENT, 0 ms]
                (417) QDP
                (418) PisEmptyProof [EQUIVALENT, 0 ms]
                (419) YES
            (420) QDP
                (421) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (422) YES
    (423) QDP
        (424) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (425) YES
    (426) QDP
        (427) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (428) YES
    (429) QDP
        (430) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (431) YES
    (432) QDP
        (433) TransformationProof [EQUIVALENT, 0 ms]
        (434) QDP
        (435) DependencyGraphProof [EQUIVALENT, 0 ms]
        (436) QDP
        (437) UsableRulesProof [EQUIVALENT, 0 ms]
        (438) QDP
        (439) QReductionProof [EQUIVALENT, 0 ms]
        (440) QDP
        (441) TransformationProof [EQUIVALENT, 0 ms]
        (442) QDP
        (443) UsableRulesProof [EQUIVALENT, 0 ms]
        (444) QDP
        (445) TransformationProof [EQUIVALENT, 0 ms]
        (446) QDP
        (447) UsableRulesProof [EQUIVALENT, 0 ms]
        (448) QDP
        (449) QReductionProof [EQUIVALENT, 0 ms]
        (450) QDP
        (451) MNOCProof [EQUIVALENT, 0 ms]
        (452) QDP
        (453) InductionCalculusProof [EQUIVALENT, 0 ms]
        (454) QDP
        (455) TransformationProof [EQUIVALENT, 0 ms]
        (456) QDP
            (457) DependencyGraphProof [EQUIVALENT, 0 ms]
            (458) QDP
            (459) UsableRulesProof [EQUIVALENT, 0 ms]
            (460) QDP
            (461) QReductionProof [EQUIVALENT, 0 ms]
            (462) QDP
            (463) TransformationProof [EQUIVALENT, 0 ms]
            (464) QDP
            (465) DependencyGraphProof [EQUIVALENT, 0 ms]
            (466) QDP
            (467) TransformationProof [EQUIVALENT, 0 ms]
            (468) QDP
            (469) DependencyGraphProof [EQUIVALENT, 0 ms]
            (470) QDP
            (471) TransformationProof [EQUIVALENT, 0 ms]
            (472) QDP
            (473) TransformationProof [EQUIVALENT, 0 ms]
            (474) QDP
            (475) DependencyGraphProof [EQUIVALENT, 0 ms]
            (476) QDP
            (477) TransformationProof [EQUIVALENT, 0 ms]
            (478) QDP
            (479) DependencyGraphProof [EQUIVALENT, 0 ms]
            (480) QDP
            (481) TransformationProof [EQUIVALENT, 0 ms]
            (482) QDP
            (483) TransformationProof [EQUIVALENT, 0 ms]
            (484) QDP
            (485) DependencyGraphProof [EQUIVALENT, 0 ms]
            (486) QDP
            (487) TransformationProof [EQUIVALENT, 0 ms]
            (488) QDP
            (489) DependencyGraphProof [EQUIVALENT, 0 ms]
            (490) QDP
            (491) TransformationProof [EQUIVALENT, 0 ms]
            (492) QDP
            (493) TransformationProof [EQUIVALENT, 0 ms]
            (494) QDP
            (495) DependencyGraphProof [EQUIVALENT, 0 ms]
            (496) QDP
            (497) TransformationProof [EQUIVALENT, 0 ms]
            (498) QDP
            (499) DependencyGraphProof [EQUIVALENT, 0 ms]
            (500) QDP
            (501) TransformationProof [EQUIVALENT, 0 ms]
            (502) QDP
            (503) TransformationProof [EQUIVALENT, 0 ms]
            (504) QDP
            (505) TransformationProof [EQUIVALENT, 0 ms]
            (506) QDP
            (507) TransformationProof [EQUIVALENT, 0 ms]
            (508) QDP
            (509) TransformationProof [EQUIVALENT, 0 ms]
            (510) QDP
            (511) TransformationProof [EQUIVALENT, 0 ms]
            (512) QDP
            (513) TransformationProof [EQUIVALENT, 0 ms]
            (514) QDP
            (515) TransformationProof [EQUIVALENT, 0 ms]
            (516) QDP
            (517) DependencyGraphProof [EQUIVALENT, 0 ms]
            (518) QDP
            (519) TransformationProof [EQUIVALENT, 0 ms]
            (520) QDP
            (521) TransformationProof [EQUIVALENT, 0 ms]
            (522) QDP
            (523) DependencyGraphProof [EQUIVALENT, 0 ms]
            (524) QDP
            (525) TransformationProof [EQUIVALENT, 0 ms]
            (526) QDP
            (527) TransformationProof [EQUIVALENT, 0 ms]
            (528) QDP
            (529) TransformationProof [EQUIVALENT, 0 ms]
            (530) QDP
            (531) TransformationProof [EQUIVALENT, 0 ms]
            (532) QDP
            (533) TransformationProof [EQUIVALENT, 0 ms]
            (534) QDP
            (535) TransformationProof [EQUIVALENT, 0 ms]
            (536) QDP
            (537) TransformationProof [EQUIVALENT, 0 ms]
            (538) QDP
            (539) TransformationProof [EQUIVALENT, 0 ms]
            (540) QDP
            (541) DependencyGraphProof [EQUIVALENT, 0 ms]
            (542) QDP
            (543) TransformationProof [EQUIVALENT, 0 ms]
            (544) QDP
            (545) TransformationProof [EQUIVALENT, 0 ms]
            (546) QDP
            (547) DependencyGraphProof [EQUIVALENT, 0 ms]
            (548) QDP
            (549) TransformationProof [EQUIVALENT, 0 ms]
            (550) QDP
            (551) TransformationProof [EQUIVALENT, 0 ms]
            (552) QDP
            (553) TransformationProof [EQUIVALENT, 0 ms]
            (554) QDP
            (555) TransformationProof [EQUIVALENT, 0 ms]
            (556) QDP
            (557) TransformationProof [EQUIVALENT, 0 ms]
            (558) QDP
            (559) TransformationProof [EQUIVALENT, 0 ms]
            (560) QDP
            (561) TransformationProof [EQUIVALENT, 0 ms]
            (562) QDP
            (563) TransformationProof [EQUIVALENT, 0 ms]
            (564) QDP
            (565) DependencyGraphProof [EQUIVALENT, 0 ms]
            (566) QDP
            (567) TransformationProof [EQUIVALENT, 0 ms]
            (568) QDP
            (569) TransformationProof [EQUIVALENT, 0 ms]
            (570) QDP
            (571) DependencyGraphProof [EQUIVALENT, 0 ms]
            (572) QDP
            (573) TransformationProof [EQUIVALENT, 0 ms]
            (574) QDP
            (575) TransformationProof [EQUIVALENT, 0 ms]
            (576) QDP
            (577) TransformationProof [EQUIVALENT, 0 ms]
            (578) QDP
            (579) TransformationProof [EQUIVALENT, 0 ms]
            (580) QDP
            (581) TransformationProof [EQUIVALENT, 0 ms]
            (582) QDP
            (583) TransformationProof [EQUIVALENT, 0 ms]
            (584) QDP
            (585) TransformationProof [EQUIVALENT, 0 ms]
            (586) QDP
            (587) TransformationProof [EQUIVALENT, 0 ms]
            (588) QDP
            (589) DependencyGraphProof [EQUIVALENT, 0 ms]
            (590) QDP
            (591) TransformationProof [EQUIVALENT, 0 ms]
            (592) QDP
            (593) TransformationProof [EQUIVALENT, 0 ms]
            (594) QDP
            (595) DependencyGraphProof [EQUIVALENT, 0 ms]
            (596) QDP
            (597) TransformationProof [EQUIVALENT, 0 ms]
            (598) QDP
            (599) TransformationProof [EQUIVALENT, 0 ms]
            (600) QDP
            (601) TransformationProof [EQUIVALENT, 0 ms]
            (602) QDP
            (603) TransformationProof [EQUIVALENT, 0 ms]
            (604) QDP
            (605) TransformationProof [EQUIVALENT, 0 ms]
            (606) QDP
            (607) TransformationProof [EQUIVALENT, 0 ms]
            (608) QDP
            (609) TransformationProof [EQUIVALENT, 0 ms]
            (610) QDP
            (611) TransformationProof [EQUIVALENT, 0 ms]
            (612) QDP
            (613) TransformationProof [EQUIVALENT, 0 ms]
            (614) QDP
            (615) TransformationProof [EQUIVALENT, 0 ms]
            (616) QDP
            (617) DependencyGraphProof [EQUIVALENT, 0 ms]
            (618) QDP
            (619) TransformationProof [EQUIVALENT, 0 ms]
            (620) QDP
            (621) TransformationProof [EQUIVALENT, 0 ms]
            (622) QDP
            (623) TransformationProof [EQUIVALENT, 0 ms]
            (624) QDP
            (625) DependencyGraphProof [EQUIVALENT, 0 ms]
            (626) QDP
            (627) TransformationProof [EQUIVALENT, 0 ms]
            (628) QDP
            (629) TransformationProof [EQUIVALENT, 0 ms]
            (630) QDP
            (631) TransformationProof [EQUIVALENT, 0 ms]
            (632) QDP
            (633) TransformationProof [EQUIVALENT, 0 ms]
            (634) QDP
            (635) TransformationProof [EQUIVALENT, 0 ms]
            (636) QDP
            (637) TransformationProof [EQUIVALENT, 0 ms]
            (638) QDP
            (639) TransformationProof [EQUIVALENT, 0 ms]
            (640) QDP
            (641) TransformationProof [EQUIVALENT, 0 ms]
            (642) QDP
            (643) TransformationProof [EQUIVALENT, 0 ms]
            (644) QDP
            (645) TransformationProof [EQUIVALENT, 0 ms]
            (646) QDP
            (647) DependencyGraphProof [EQUIVALENT, 0 ms]
            (648) QDP
            (649) TransformationProof [EQUIVALENT, 0 ms]
            (650) QDP
            (651) TransformationProof [EQUIVALENT, 0 ms]
            (652) QDP
            (653) TransformationProof [EQUIVALENT, 0 ms]
            (654) QDP
            (655) DependencyGraphProof [EQUIVALENT, 0 ms]
            (656) QDP
            (657) TransformationProof [EQUIVALENT, 0 ms]
            (658) QDP
            (659) TransformationProof [EQUIVALENT, 0 ms]
            (660) QDP
            (661) TransformationProof [EQUIVALENT, 0 ms]
            (662) QDP
            (663) TransformationProof [EQUIVALENT, 0 ms]
            (664) QDP
            (665) TransformationProof [EQUIVALENT, 0 ms]
            (666) QDP
            (667) TransformationProof [EQUIVALENT, 0 ms]
            (668) QDP
            (669) TransformationProof [EQUIVALENT, 0 ms]
            (670) QDP
            (671) TransformationProof [EQUIVALENT, 0 ms]
            (672) QDP
            (673) TransformationProof [EQUIVALENT, 0 ms]
            (674) QDP
            (675) TransformationProof [EQUIVALENT, 0 ms]
            (676) QDP
            (677) DependencyGraphProof [EQUIVALENT, 0 ms]
            (678) QDP
            (679) TransformationProof [EQUIVALENT, 0 ms]
            (680) QDP
            (681) TransformationProof [EQUIVALENT, 0 ms]
            (682) QDP
            (683) TransformationProof [EQUIVALENT, 0 ms]
            (684) QDP
            (685) DependencyGraphProof [EQUIVALENT, 0 ms]
            (686) QDP
            (687) TransformationProof [EQUIVALENT, 0 ms]
            (688) QDP
            (689) TransformationProof [EQUIVALENT, 0 ms]
            (690) QDP
            (691) TransformationProof [EQUIVALENT, 0 ms]
            (692) QDP
            (693) TransformationProof [EQUIVALENT, 0 ms]
            (694) QDP
            (695) TransformationProof [EQUIVALENT, 0 ms]
            (696) QDP
            (697) TransformationProof [EQUIVALENT, 0 ms]
            (698) QDP
            (699) TransformationProof [EQUIVALENT, 0 ms]
            (700) QDP
            (701) TransformationProof [EQUIVALENT, 0 ms]
            (702) QDP
            (703) TransformationProof [EQUIVALENT, 0 ms]
            (704) QDP
            (705) TransformationProof [EQUIVALENT, 0 ms]
            (706) QDP
            (707) DependencyGraphProof [EQUIVALENT, 0 ms]
            (708) QDP
            (709) TransformationProof [EQUIVALENT, 0 ms]
            (710) QDP
            (711) TransformationProof [EQUIVALENT, 0 ms]
            (712) QDP
            (713) TransformationProof [EQUIVALENT, 0 ms]
            (714) QDP
            (715) DependencyGraphProof [EQUIVALENT, 0 ms]
            (716) QDP
            (717) TransformationProof [EQUIVALENT, 0 ms]
            (718) QDP
            (719) QDPOrderProof [EQUIVALENT, 0 ms]
            (720) QDP
            (721) QDPOrderProof [EQUIVALENT, 0 ms]
            (722) QDP
            (723) MNOCProof [EQUIVALENT, 0 ms]
            (724) QDP
            (725) QDPOrderProof [EQUIVALENT, 795 ms]
            (726) QDP
                (727) MNOCProof [EQUIVALENT, 0 ms]
                (728) QDP
                (729) InductionCalculusProof [EQUIVALENT, 0 ms]
                (730) QDP
(731) Narrow [COMPLETE, 0 ms]
(732) QDP
    (733) PisEmptyProof [EQUIVALENT, 0 ms]
    (734) YES


----------------------------------------

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(1) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero"
is transformed to
"primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y));
primDivNatS0 x y False = Zero;
"
The following If expression
"if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x"
is transformed to
"primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y);
primModNatS0 x y False = Succ x;
"

----------------------------------------

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(3) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(5) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"absReal x|x >= 0x|otherwise`negate` x;
"
is transformed to
"absReal x = absReal2 x;
"
"absReal1 x True = x;
absReal1 x False = absReal0 x otherwise;
"
"absReal0 x True = `negate` x;
"
"absReal2 x = absReal1 x (x >= 0);
"
The following Function with conditions
"gcd' x 0 = x;
gcd' x y = gcd' y (x `rem` y);
"
is transformed to
"gcd' x xz = gcd'2 x xz;
gcd' x y = gcd'0 x y;
"
"gcd'0 x y = gcd' y (x `rem` y);
"
"gcd'1 True x xz = x;
gcd'1 yu yv yw = gcd'0 yv yw;
"
"gcd'2 x xz = gcd'1 (xz == 0) x xz;
gcd'2 yx yy = gcd'0 yx yy;
"
The following Function with conditions
"gcd 0 0 = error [];
gcd x y = gcd' (abs x) (abs y) where {
gcd' x 0 = x;
gcd' x y = gcd' y (x `rem` y);
} 
;
"
is transformed to
"gcd yz zu = gcd3 yz zu;
gcd x y = gcd0 x y;
"
"gcd0 x y = gcd' (abs x) (abs y) where {
gcd' x xz = gcd'2 x xz;
gcd' x y = gcd'0 x y;
;
gcd'0 x y = gcd' y (x `rem` y);
;
gcd'1 True x xz = x;
gcd'1 yu yv yw = gcd'0 yv yw;
;
gcd'2 x xz = gcd'1 (xz == 0) x xz;
gcd'2 yx yy = gcd'0 yx yy;
} 
;
"
"gcd1 True yz zu = error [];
gcd1 zv zw zx = gcd0 zw zx;
"
"gcd2 True yz zu = gcd1 (zu == 0) yz zu;
gcd2 zy zz vuu = gcd0 zz vuu;
"
"gcd3 yz zu = gcd2 (yz == 0) yz zu;
gcd3 vuv vuw = gcd0 vuv vuw;
"
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"
The following Function with conditions
"reduce x y|y == 0error []|otherwisex `quot` d :% (y `quot` d) where {
d  = gcd x y;
} 
;
"
is transformed to
"reduce x y = reduce2 x y;
"
"reduce2 x y = reduce1 x y (y == 0) where {
d  = gcd x y;
;
reduce0 x y True = x `quot` d :% (y `quot` d);
;
reduce1 x y True = error [];
reduce1 x y False = reduce0 x y otherwise;
} 
;
"

----------------------------------------

(6)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(7) LetRed (EQUIVALENT)
Let/Where Reductions:
The bindings of the following Let/Where expression
"gcd' (abs x) (abs y) where {
gcd' x xz = gcd'2 x xz;
gcd' x y = gcd'0 x y;
;
gcd'0 x y = gcd' y (x `rem` y);
;
gcd'1 True x xz = x;
gcd'1 yu yv yw = gcd'0 yv yw;
;
gcd'2 x xz = gcd'1 (xz == 0) x xz;
gcd'2 yx yy = gcd'0 yx yy;
} 
"
are unpacked to the following functions on top level
"gcd0Gcd'2 x xz = gcd0Gcd'1 (xz == 0) x xz;
gcd0Gcd'2 yx yy = gcd0Gcd'0 yx yy;
"
"gcd0Gcd'0 x y = gcd0Gcd' y (x `rem` y);
"
"gcd0Gcd'1 True x xz = x;
gcd0Gcd'1 yu yv yw = gcd0Gcd'0 yv yw;
"
"gcd0Gcd' x xz = gcd0Gcd'2 x xz;
gcd0Gcd' x y = gcd0Gcd'0 x y;
"
The bindings of the following Let/Where expression
"reduce1 x y (y == 0) where {
d  = gcd x y;
;
reduce0 x y True = x `quot` d :% (y `quot` d);
;
reduce1 x y True = error [];
reduce1 x y False = reduce0 x y otherwise;
} 
"
are unpacked to the following functions on top level
"reduce2D vux vuy = gcd vux vuy;
"
"reduce2Reduce0 vux vuy x y True = x `quot` reduce2D vux vuy :% (y `quot` reduce2D vux vuy);
"
"reduce2Reduce1 vux vuy x y True = error [];
reduce2Reduce1 vux vuy x y False = reduce2Reduce0 vux vuy x y otherwise;
"

----------------------------------------

(8)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(9) NumRed (SOUND)
Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
----------------------------------------

(10)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(11) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="(*)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="(*) vuz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="(*) vuz3 vuz4",fontsize=16,color="blue",shape="box"];1023[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4 -> 1023[label="",style="solid", color="blue", weight=9];
1023 -> 5[label="",style="solid", color="blue", weight=3];
1024[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4 -> 1024[label="",style="solid", color="blue", weight=9];
1024 -> 6[label="",style="solid", color="blue", weight=3];
1025[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4 -> 1025[label="",style="solid", color="blue", weight=9];
1025 -> 7[label="",style="solid", color="blue", weight=3];
1026[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4 -> 1026[label="",style="solid", color="blue", weight=9];
1026 -> 8[label="",style="solid", color="blue", weight=3];
1027[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4 -> 1027[label="",style="solid", color="blue", weight=9];
1027 -> 9[label="",style="solid", color="blue", weight=3];
5[label="(*) vuz3 vuz4",fontsize=16,color="burlywood",shape="box"];1028[label="vuz3/vuz30 :% vuz31",fontsize=10,color="white",style="solid",shape="box"];5 -> 1028[label="",style="solid", color="burlywood", weight=9];
1028 -> 10[label="",style="solid", color="burlywood", weight=3];
6[label="(*) vuz3 vuz4",fontsize=16,color="black",shape="box"];6 -> 11[label="",style="solid", color="black", weight=3];
7[label="(*) vuz3 vuz4",fontsize=16,color="black",shape="triangle"];7 -> 12[label="",style="solid", color="black", weight=3];
8[label="(*) vuz3 vuz4",fontsize=16,color="burlywood",shape="triangle"];1029[label="vuz3/Integer vuz30",fontsize=10,color="white",style="solid",shape="box"];8 -> 1029[label="",style="solid", color="burlywood", weight=9];
1029 -> 13[label="",style="solid", color="burlywood", weight=3];
9[label="(*) vuz3 vuz4",fontsize=16,color="black",shape="box"];9 -> 14[label="",style="solid", color="black", weight=3];
10[label="(*) (vuz30 :% vuz31) vuz4",fontsize=16,color="burlywood",shape="box"];1030[label="vuz4/vuz40 :% vuz41",fontsize=10,color="white",style="solid",shape="box"];10 -> 1030[label="",style="solid", color="burlywood", weight=9];
1030 -> 15[label="",style="solid", color="burlywood", weight=3];
11[label="primMulDouble vuz3 vuz4",fontsize=16,color="burlywood",shape="box"];1031[label="vuz3/Double vuz30 vuz31",fontsize=10,color="white",style="solid",shape="box"];11 -> 1031[label="",style="solid", color="burlywood", weight=9];
1031 -> 16[label="",style="solid", color="burlywood", weight=3];
12[label="primMulInt vuz3 vuz4",fontsize=16,color="burlywood",shape="triangle"];1032[label="vuz3/Pos vuz30",fontsize=10,color="white",style="solid",shape="box"];12 -> 1032[label="",style="solid", color="burlywood", weight=9];
1032 -> 17[label="",style="solid", color="burlywood", weight=3];
1033[label="vuz3/Neg vuz30",fontsize=10,color="white",style="solid",shape="box"];12 -> 1033[label="",style="solid", color="burlywood", weight=9];
1033 -> 18[label="",style="solid", color="burlywood", weight=3];
13[label="(*) Integer vuz30 vuz4",fontsize=16,color="burlywood",shape="box"];1034[label="vuz4/Integer vuz40",fontsize=10,color="white",style="solid",shape="box"];13 -> 1034[label="",style="solid", color="burlywood", weight=9];
1034 -> 19[label="",style="solid", color="burlywood", weight=3];
14[label="primMulFloat vuz3 vuz4",fontsize=16,color="burlywood",shape="box"];1035[label="vuz3/Float vuz30 vuz31",fontsize=10,color="white",style="solid",shape="box"];14 -> 1035[label="",style="solid", color="burlywood", weight=9];
1035 -> 20[label="",style="solid", color="burlywood", weight=3];
15[label="(*) (vuz30 :% vuz31) (vuz40 :% vuz41)",fontsize=16,color="black",shape="box"];15 -> 21[label="",style="solid", color="black", weight=3];
16[label="primMulDouble (Double vuz30 vuz31) vuz4",fontsize=16,color="burlywood",shape="box"];1036[label="vuz4/Double vuz40 vuz41",fontsize=10,color="white",style="solid",shape="box"];16 -> 1036[label="",style="solid", color="burlywood", weight=9];
1036 -> 22[label="",style="solid", color="burlywood", weight=3];
17[label="primMulInt (Pos vuz30) vuz4",fontsize=16,color="burlywood",shape="box"];1037[label="vuz4/Pos vuz40",fontsize=10,color="white",style="solid",shape="box"];17 -> 1037[label="",style="solid", color="burlywood", weight=9];
1037 -> 23[label="",style="solid", color="burlywood", weight=3];
1038[label="vuz4/Neg vuz40",fontsize=10,color="white",style="solid",shape="box"];17 -> 1038[label="",style="solid", color="burlywood", weight=9];
1038 -> 24[label="",style="solid", color="burlywood", weight=3];
18[label="primMulInt (Neg vuz30) vuz4",fontsize=16,color="burlywood",shape="box"];1039[label="vuz4/Pos vuz40",fontsize=10,color="white",style="solid",shape="box"];18 -> 1039[label="",style="solid", color="burlywood", weight=9];
1039 -> 25[label="",style="solid", color="burlywood", weight=3];
1040[label="vuz4/Neg vuz40",fontsize=10,color="white",style="solid",shape="box"];18 -> 1040[label="",style="solid", color="burlywood", weight=9];
1040 -> 26[label="",style="solid", color="burlywood", weight=3];
19[label="(*) Integer vuz30 Integer vuz40",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
20[label="primMulFloat (Float vuz30 vuz31) vuz4",fontsize=16,color="burlywood",shape="box"];1041[label="vuz4/Float vuz40 vuz41",fontsize=10,color="white",style="solid",shape="box"];20 -> 1041[label="",style="solid", color="burlywood", weight=9];
1041 -> 28[label="",style="solid", color="burlywood", weight=3];
21[label="reduce (vuz30 * vuz40) (vuz31 * vuz41)",fontsize=16,color="black",shape="box"];21 -> 29[label="",style="solid", color="black", weight=3];
22[label="primMulDouble (Double vuz30 vuz31) (Double vuz40 vuz41)",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3];
23[label="primMulInt (Pos vuz30) (Pos vuz40)",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
24[label="primMulInt (Pos vuz30) (Neg vuz40)",fontsize=16,color="black",shape="box"];24 -> 32[label="",style="solid", color="black", weight=3];
25[label="primMulInt (Neg vuz30) (Pos vuz40)",fontsize=16,color="black",shape="box"];25 -> 33[label="",style="solid", color="black", weight=3];
26[label="primMulInt (Neg vuz30) (Neg vuz40)",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
27[label="Integer (primMulInt vuz30 vuz40)",fontsize=16,color="green",shape="box"];27 -> 35[label="",style="dashed", color="green", weight=3];
28[label="primMulFloat (Float vuz30 vuz31) (Float vuz40 vuz41)",fontsize=16,color="black",shape="box"];28 -> 36[label="",style="solid", color="black", weight=3];
29[label="reduce2 (vuz30 * vuz40) (vuz31 * vuz41)",fontsize=16,color="black",shape="box"];29 -> 37[label="",style="solid", color="black", weight=3];
30[label="Double (vuz30 * vuz40) (vuz31 * vuz41)",fontsize=16,color="green",shape="box"];30 -> 38[label="",style="dashed", color="green", weight=3];
30 -> 39[label="",style="dashed", color="green", weight=3];
31[label="Pos (primMulNat vuz30 vuz40)",fontsize=16,color="green",shape="box"];31 -> 40[label="",style="dashed", color="green", weight=3];
32[label="Neg (primMulNat vuz30 vuz40)",fontsize=16,color="green",shape="box"];32 -> 41[label="",style="dashed", color="green", weight=3];
33[label="Neg (primMulNat vuz30 vuz40)",fontsize=16,color="green",shape="box"];33 -> 42[label="",style="dashed", color="green", weight=3];
34[label="Pos (primMulNat vuz30 vuz40)",fontsize=16,color="green",shape="box"];34 -> 43[label="",style="dashed", color="green", weight=3];
35 -> 12[label="",style="dashed", color="red", weight=0];
35[label="primMulInt vuz30 vuz40",fontsize=16,color="magenta"];35 -> 44[label="",style="dashed", color="magenta", weight=3];
35 -> 45[label="",style="dashed", color="magenta", weight=3];
36[label="Float (vuz30 * vuz40) (vuz31 * vuz41)",fontsize=16,color="green",shape="box"];36 -> 46[label="",style="dashed", color="green", weight=3];
36 -> 47[label="",style="dashed", color="green", weight=3];
37 -> 48[label="",style="dashed", color="red", weight=0];
37[label="reduce2Reduce1 (vuz30 * vuz40) (vuz31 * vuz41) (vuz30 * vuz40) (vuz31 * vuz41) (vuz31 * vuz41 == fromInt (Pos Zero))",fontsize=16,color="magenta"];37 -> 49[label="",style="dashed", color="magenta", weight=3];
37 -> 50[label="",style="dashed", color="magenta", weight=3];
37 -> 51[label="",style="dashed", color="magenta", weight=3];
37 -> 52[label="",style="dashed", color="magenta", weight=3];
37 -> 53[label="",style="dashed", color="magenta", weight=3];
38 -> 7[label="",style="dashed", color="red", weight=0];
38[label="vuz30 * vuz40",fontsize=16,color="magenta"];38 -> 54[label="",style="dashed", color="magenta", weight=3];
38 -> 55[label="",style="dashed", color="magenta", weight=3];
39 -> 7[label="",style="dashed", color="red", weight=0];
39[label="vuz31 * vuz41",fontsize=16,color="magenta"];39 -> 56[label="",style="dashed", color="magenta", weight=3];
39 -> 57[label="",style="dashed", color="magenta", weight=3];
40[label="primMulNat vuz30 vuz40",fontsize=16,color="burlywood",shape="triangle"];1042[label="vuz30/Succ vuz300",fontsize=10,color="white",style="solid",shape="box"];40 -> 1042[label="",style="solid", color="burlywood", weight=9];
1042 -> 58[label="",style="solid", color="burlywood", weight=3];
1043[label="vuz30/Zero",fontsize=10,color="white",style="solid",shape="box"];40 -> 1043[label="",style="solid", color="burlywood", weight=9];
1043 -> 59[label="",style="solid", color="burlywood", weight=3];
41 -> 40[label="",style="dashed", color="red", weight=0];
41[label="primMulNat vuz30 vuz40",fontsize=16,color="magenta"];41 -> 60[label="",style="dashed", color="magenta", weight=3];
42 -> 40[label="",style="dashed", color="red", weight=0];
42[label="primMulNat vuz30 vuz40",fontsize=16,color="magenta"];42 -> 61[label="",style="dashed", color="magenta", weight=3];
43 -> 40[label="",style="dashed", color="red", weight=0];
43[label="primMulNat vuz30 vuz40",fontsize=16,color="magenta"];43 -> 62[label="",style="dashed", color="magenta", weight=3];
43 -> 63[label="",style="dashed", color="magenta", weight=3];
44[label="vuz40",fontsize=16,color="green",shape="box"];45[label="vuz30",fontsize=16,color="green",shape="box"];46 -> 7[label="",style="dashed", color="red", weight=0];
46[label="vuz30 * vuz40",fontsize=16,color="magenta"];46 -> 64[label="",style="dashed", color="magenta", weight=3];
46 -> 65[label="",style="dashed", color="magenta", weight=3];
47 -> 7[label="",style="dashed", color="red", weight=0];
47[label="vuz31 * vuz41",fontsize=16,color="magenta"];47 -> 66[label="",style="dashed", color="magenta", weight=3];
47 -> 67[label="",style="dashed", color="magenta", weight=3];
49[label="vuz41",fontsize=16,color="green",shape="box"];50[label="vuz31 * vuz41 == fromInt (Pos Zero)",fontsize=16,color="blue",shape="box"];1044[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];50 -> 1044[label="",style="solid", color="blue", weight=9];
1044 -> 68[label="",style="solid", color="blue", weight=3];
1045[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];50 -> 1045[label="",style="solid", color="blue", weight=9];
1045 -> 69[label="",style="solid", color="blue", weight=3];
51[label="vuz40",fontsize=16,color="green",shape="box"];52[label="vuz30",fontsize=16,color="green",shape="box"];53[label="vuz31",fontsize=16,color="green",shape="box"];48[label="reduce2Reduce1 (vuz11 * vuz12) (vuz13 * vuz14) (vuz11 * vuz12) (vuz13 * vuz14) vuz15",fontsize=16,color="burlywood",shape="triangle"];1046[label="vuz15/False",fontsize=10,color="white",style="solid",shape="box"];48 -> 1046[label="",style="solid", color="burlywood", weight=9];
1046 -> 70[label="",style="solid", color="burlywood", weight=3];
1047[label="vuz15/True",fontsize=10,color="white",style="solid",shape="box"];48 -> 1047[label="",style="solid", color="burlywood", weight=9];
1047 -> 71[label="",style="solid", color="burlywood", weight=3];
54[label="vuz40",fontsize=16,color="green",shape="box"];55[label="vuz30",fontsize=16,color="green",shape="box"];56[label="vuz41",fontsize=16,color="green",shape="box"];57[label="vuz31",fontsize=16,color="green",shape="box"];58[label="primMulNat (Succ vuz300) vuz40",fontsize=16,color="burlywood",shape="box"];1048[label="vuz40/Succ vuz400",fontsize=10,color="white",style="solid",shape="box"];58 -> 1048[label="",style="solid", color="burlywood", weight=9];
1048 -> 72[label="",style="solid", color="burlywood", weight=3];
1049[label="vuz40/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 1049[label="",style="solid", color="burlywood", weight=9];
1049 -> 73[label="",style="solid", color="burlywood", weight=3];
59[label="primMulNat Zero vuz40",fontsize=16,color="burlywood",shape="box"];1050[label="vuz40/Succ vuz400",fontsize=10,color="white",style="solid",shape="box"];59 -> 1050[label="",style="solid", color="burlywood", weight=9];
1050 -> 74[label="",style="solid", color="burlywood", weight=3];
1051[label="vuz40/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 1051[label="",style="solid", color="burlywood", weight=9];
1051 -> 75[label="",style="solid", color="burlywood", weight=3];
60[label="vuz40",fontsize=16,color="green",shape="box"];61[label="vuz30",fontsize=16,color="green",shape="box"];62[label="vuz30",fontsize=16,color="green",shape="box"];63[label="vuz40",fontsize=16,color="green",shape="box"];64[label="vuz40",fontsize=16,color="green",shape="box"];65[label="vuz30",fontsize=16,color="green",shape="box"];66[label="vuz41",fontsize=16,color="green",shape="box"];67[label="vuz31",fontsize=16,color="green",shape="box"];68 -> 76[label="",style="dashed", color="red", weight=0];
68[label="vuz31 * vuz41 == fromInt (Pos Zero)",fontsize=16,color="magenta"];68 -> 77[label="",style="dashed", color="magenta", weight=3];
69 -> 78[label="",style="dashed", color="red", weight=0];
69[label="vuz31 * vuz41 == fromInt (Pos Zero)",fontsize=16,color="magenta"];69 -> 79[label="",style="dashed", color="magenta", weight=3];
70[label="reduce2Reduce1 (vuz11 * vuz12) (vuz13 * vuz14) (vuz11 * vuz12) (vuz13 * vuz14) False",fontsize=16,color="black",shape="box"];70 -> 80[label="",style="solid", color="black", weight=3];
71[label="reduce2Reduce1 (vuz11 * vuz12) (vuz13 * vuz14) (vuz11 * vuz12) (vuz13 * vuz14) True",fontsize=16,color="black",shape="box"];71 -> 81[label="",style="solid", color="black", weight=3];
72[label="primMulNat (Succ vuz300) (Succ vuz400)",fontsize=16,color="black",shape="box"];72 -> 82[label="",style="solid", color="black", weight=3];
73[label="primMulNat (Succ vuz300) Zero",fontsize=16,color="black",shape="box"];73 -> 83[label="",style="solid", color="black", weight=3];
74[label="primMulNat Zero (Succ vuz400)",fontsize=16,color="black",shape="box"];74 -> 84[label="",style="solid", color="black", weight=3];
75[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];75 -> 85[label="",style="solid", color="black", weight=3];
77 -> 8[label="",style="dashed", color="red", weight=0];
77[label="vuz31 * vuz41",fontsize=16,color="magenta"];77 -> 86[label="",style="dashed", color="magenta", weight=3];
77 -> 87[label="",style="dashed", color="magenta", weight=3];
76[label="vuz16 == fromInt (Pos Zero)",fontsize=16,color="burlywood",shape="triangle"];1052[label="vuz16/Integer vuz160",fontsize=10,color="white",style="solid",shape="box"];76 -> 1052[label="",style="solid", color="burlywood", weight=9];
1052 -> 88[label="",style="solid", color="burlywood", weight=3];
79 -> 7[label="",style="dashed", color="red", weight=0];
79[label="vuz31 * vuz41",fontsize=16,color="magenta"];79 -> 89[label="",style="dashed", color="magenta", weight=3];
79 -> 90[label="",style="dashed", color="magenta", weight=3];
78[label="vuz17 == fromInt (Pos Zero)",fontsize=16,color="black",shape="triangle"];78 -> 91[label="",style="solid", color="black", weight=3];
80[label="reduce2Reduce0 (vuz11 * vuz12) (vuz13 * vuz14) (vuz11 * vuz12) (vuz13 * vuz14) otherwise",fontsize=16,color="black",shape="box"];80 -> 92[label="",style="solid", color="black", weight=3];
81[label="error []",fontsize=16,color="black",shape="box"];81 -> 93[label="",style="solid", color="black", weight=3];
82 -> 94[label="",style="dashed", color="red", weight=0];
82[label="primPlusNat (primMulNat vuz300 (Succ vuz400)) (Succ vuz400)",fontsize=16,color="magenta"];82 -> 95[label="",style="dashed", color="magenta", weight=3];
83[label="Zero",fontsize=16,color="green",shape="box"];84[label="Zero",fontsize=16,color="green",shape="box"];85[label="Zero",fontsize=16,color="green",shape="box"];86[label="vuz41",fontsize=16,color="green",shape="box"];87[label="vuz31",fontsize=16,color="green",shape="box"];88[label="Integer vuz160 == fromInt (Pos Zero)",fontsize=16,color="black",shape="box"];88 -> 96[label="",style="solid", color="black", weight=3];
89[label="vuz41",fontsize=16,color="green",shape="box"];90[label="vuz31",fontsize=16,color="green",shape="box"];91[label="primEqInt vuz17 (fromInt (Pos Zero))",fontsize=16,color="burlywood",shape="box"];1053[label="vuz17/Pos vuz170",fontsize=10,color="white",style="solid",shape="box"];91 -> 1053[label="",style="solid", color="burlywood", weight=9];
1053 -> 97[label="",style="solid", color="burlywood", weight=3];
1054[label="vuz17/Neg vuz170",fontsize=10,color="white",style="solid",shape="box"];91 -> 1054[label="",style="solid", color="burlywood", weight=9];
1054 -> 98[label="",style="solid", color="burlywood", weight=3];
92[label="reduce2Reduce0 (vuz11 * vuz12) (vuz13 * vuz14) (vuz11 * vuz12) (vuz13 * vuz14) True",fontsize=16,color="black",shape="box"];92 -> 99[label="",style="solid", color="black", weight=3];
93[label="error []",fontsize=16,color="red",shape="box"];95 -> 40[label="",style="dashed", color="red", weight=0];
95[label="primMulNat vuz300 (Succ vuz400)",fontsize=16,color="magenta"];95 -> 100[label="",style="dashed", color="magenta", weight=3];
95 -> 101[label="",style="dashed", color="magenta", weight=3];
94[label="primPlusNat vuz18 (Succ vuz400)",fontsize=16,color="burlywood",shape="triangle"];1055[label="vuz18/Succ vuz180",fontsize=10,color="white",style="solid",shape="box"];94 -> 1055[label="",style="solid", color="burlywood", weight=9];
1055 -> 102[label="",style="solid", color="burlywood", weight=3];
1056[label="vuz18/Zero",fontsize=10,color="white",style="solid",shape="box"];94 -> 1056[label="",style="solid", color="burlywood", weight=9];
1056 -> 103[label="",style="solid", color="burlywood", weight=3];
96[label="Integer vuz160 == Integer (Pos Zero)",fontsize=16,color="black",shape="box"];96 -> 104[label="",style="solid", color="black", weight=3];
97[label="primEqInt (Pos vuz170) (fromInt (Pos Zero))",fontsize=16,color="burlywood",shape="box"];1057[label="vuz170/Succ vuz1700",fontsize=10,color="white",style="solid",shape="box"];97 -> 1057[label="",style="solid", color="burlywood", weight=9];
1057 -> 105[label="",style="solid", color="burlywood", weight=3];
1058[label="vuz170/Zero",fontsize=10,color="white",style="solid",shape="box"];97 -> 1058[label="",style="solid", color="burlywood", weight=9];
1058 -> 106[label="",style="solid", color="burlywood", weight=3];
98[label="primEqInt (Neg vuz170) (fromInt (Pos Zero))",fontsize=16,color="burlywood",shape="box"];1059[label="vuz170/Succ vuz1700",fontsize=10,color="white",style="solid",shape="box"];98 -> 1059[label="",style="solid", color="burlywood", weight=9];
1059 -> 107[label="",style="solid", color="burlywood", weight=3];
1060[label="vuz170/Zero",fontsize=10,color="white",style="solid",shape="box"];98 -> 1060[label="",style="solid", color="burlywood", weight=9];
1060 -> 108[label="",style="solid", color="burlywood", weight=3];
99[label="vuz11 * vuz12 `quot` reduce2D (vuz11 * vuz12) (vuz13 * vuz14) :% (vuz13 * vuz14 `quot` reduce2D (vuz11 * vuz12) (vuz13 * vuz14))",fontsize=16,color="green",shape="box"];99 -> 109[label="",style="dashed", color="green", weight=3];
99 -> 110[label="",style="dashed", color="green", weight=3];
100[label="vuz300",fontsize=16,color="green",shape="box"];101[label="Succ vuz400",fontsize=16,color="green",shape="box"];102[label="primPlusNat (Succ vuz180) (Succ vuz400)",fontsize=16,color="black",shape="box"];102 -> 111[label="",style="solid", color="black", weight=3];
103[label="primPlusNat Zero (Succ vuz400)",fontsize=16,color="black",shape="box"];103 -> 112[label="",style="solid", color="black", weight=3];
104[label="primEqInt vuz160 (Pos Zero)",fontsize=16,color="burlywood",shape="triangle"];1061[label="vuz160/Pos vuz1600",fontsize=10,color="white",style="solid",shape="box"];104 -> 1061[label="",style="solid", color="burlywood", weight=9];
1061 -> 113[label="",style="solid", color="burlywood", weight=3];
1062[label="vuz160/Neg vuz1600",fontsize=10,color="white",style="solid",shape="box"];104 -> 1062[label="",style="solid", color="burlywood", weight=9];
1062 -> 114[label="",style="solid", color="burlywood", weight=3];
105[label="primEqInt (Pos (Succ vuz1700)) (fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];105 -> 115[label="",style="solid", color="black", weight=3];
106[label="primEqInt (Pos Zero) (fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];106 -> 116[label="",style="solid", color="black", weight=3];
107[label="primEqInt (Neg (Succ vuz1700)) (fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];107 -> 117[label="",style="solid", color="black", weight=3];
108[label="primEqInt (Neg Zero) (fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];108 -> 118[label="",style="solid", color="black", weight=3];
109[label="vuz11 * vuz12 `quot` reduce2D (vuz11 * vuz12) (vuz13 * vuz14)",fontsize=16,color="blue",shape="box"];1063[label="`quot` :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];109 -> 1063[label="",style="solid", color="blue", weight=9];
1063 -> 119[label="",style="solid", color="blue", weight=3];
1064[label="`quot` :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];109 -> 1064[label="",style="solid", color="blue", weight=9];
1064 -> 120[label="",style="solid", color="blue", weight=3];
110[label="vuz13 * vuz14 `quot` reduce2D (vuz11 * vuz12) (vuz13 * vuz14)",fontsize=16,color="blue",shape="box"];1065[label="`quot` :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];110 -> 1065[label="",style="solid", color="blue", weight=9];
1065 -> 121[label="",style="solid", color="blue", weight=3];
1066[label="`quot` :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];110 -> 1066[label="",style="solid", color="blue", weight=9];
1066 -> 122[label="",style="solid", color="blue", weight=3];
111[label="Succ (Succ (primPlusNat vuz180 vuz400))",fontsize=16,color="green",shape="box"];111 -> 123[label="",style="dashed", color="green", weight=3];
112[label="Succ vuz400",fontsize=16,color="green",shape="box"];113[label="primEqInt (Pos vuz1600) (Pos Zero)",fontsize=16,color="burlywood",shape="box"];1067[label="vuz1600/Succ vuz16000",fontsize=10,color="white",style="solid",shape="box"];113 -> 1067[label="",style="solid", color="burlywood", weight=9];
1067 -> 124[label="",style="solid", color="burlywood", weight=3];
1068[label="vuz1600/Zero",fontsize=10,color="white",style="solid",shape="box"];113 -> 1068[label="",style="solid", color="burlywood", weight=9];
1068 -> 125[label="",style="solid", color="burlywood", weight=3];
114[label="primEqInt (Neg vuz1600) (Pos Zero)",fontsize=16,color="burlywood",shape="box"];1069[label="vuz1600/Succ vuz16000",fontsize=10,color="white",style="solid",shape="box"];114 -> 1069[label="",style="solid", color="burlywood", weight=9];
1069 -> 126[label="",style="solid", color="burlywood", weight=3];
1070[label="vuz1600/Zero",fontsize=10,color="white",style="solid",shape="box"];114 -> 1070[label="",style="solid", color="burlywood", weight=9];
1070 -> 127[label="",style="solid", color="burlywood", weight=3];
115 -> 104[label="",style="dashed", color="red", weight=0];
115[label="primEqInt (Pos (Succ vuz1700)) (Pos Zero)",fontsize=16,color="magenta"];115 -> 128[label="",style="dashed", color="magenta", weight=3];
116 -> 104[label="",style="dashed", color="red", weight=0];
116[label="primEqInt (Pos Zero) (Pos Zero)",fontsize=16,color="magenta"];116 -> 129[label="",style="dashed", color="magenta", weight=3];
117 -> 104[label="",style="dashed", color="red", weight=0];
117[label="primEqInt (Neg (Succ vuz1700)) (Pos Zero)",fontsize=16,color="magenta"];117 -> 130[label="",style="dashed", color="magenta", weight=3];
118 -> 104[label="",style="dashed", color="red", weight=0];
118[label="primEqInt (Neg Zero) (Pos Zero)",fontsize=16,color="magenta"];118 -> 131[label="",style="dashed", color="magenta", weight=3];
119 -> 132[label="",style="dashed", color="red", weight=0];
119[label="vuz11 * vuz12 `quot` reduce2D (vuz11 * vuz12) (vuz13 * vuz14)",fontsize=16,color="magenta"];119 -> 133[label="",style="dashed", color="magenta", weight=3];
119 -> 134[label="",style="dashed", color="magenta", weight=3];
119 -> 135[label="",style="dashed", color="magenta", weight=3];
120 -> 139[label="",style="dashed", color="red", weight=0];
120[label="vuz11 * vuz12 `quot` reduce2D (vuz11 * vuz12) (vuz13 * vuz14)",fontsize=16,color="magenta"];120 -> 140[label="",style="dashed", color="magenta", weight=3];
120 -> 141[label="",style="dashed", color="magenta", weight=3];
120 -> 142[label="",style="dashed", color="magenta", weight=3];
121 -> 132[label="",style="dashed", color="red", weight=0];
121[label="vuz13 * vuz14 `quot` reduce2D (vuz11 * vuz12) (vuz13 * vuz14)",fontsize=16,color="magenta"];121 -> 136[label="",style="dashed", color="magenta", weight=3];
121 -> 137[label="",style="dashed", color="magenta", weight=3];
121 -> 138[label="",style="dashed", color="magenta", weight=3];
122 -> 139[label="",style="dashed", color="red", weight=0];
122[label="vuz13 * vuz14 `quot` reduce2D (vuz11 * vuz12) (vuz13 * vuz14)",fontsize=16,color="magenta"];122 -> 143[label="",style="dashed", color="magenta", weight=3];
122 -> 144[label="",style="dashed", color="magenta", weight=3];
122 -> 145[label="",style="dashed", color="magenta", weight=3];
123[label="primPlusNat vuz180 vuz400",fontsize=16,color="burlywood",shape="triangle"];1071[label="vuz180/Succ vuz1800",fontsize=10,color="white",style="solid",shape="box"];123 -> 1071[label="",style="solid", color="burlywood", weight=9];
1071 -> 146[label="",style="solid", color="burlywood", weight=3];
1072[label="vuz180/Zero",fontsize=10,color="white",style="solid",shape="box"];123 -> 1072[label="",style="solid", color="burlywood", weight=9];
1072 -> 147[label="",style="solid", color="burlywood", weight=3];
124[label="primEqInt (Pos (Succ vuz16000)) (Pos Zero)",fontsize=16,color="black",shape="box"];124 -> 148[label="",style="solid", color="black", weight=3];
125[label="primEqInt (Pos Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];125 -> 149[label="",style="solid", color="black", weight=3];
126[label="primEqInt (Neg (Succ vuz16000)) (Pos Zero)",fontsize=16,color="black",shape="box"];126 -> 150[label="",style="solid", color="black", weight=3];
127[label="primEqInt (Neg Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];127 -> 151[label="",style="solid", color="black", weight=3];
128[label="Pos (Succ vuz1700)",fontsize=16,color="green",shape="box"];129[label="Pos Zero",fontsize=16,color="green",shape="box"];130[label="Neg (Succ vuz1700)",fontsize=16,color="green",shape="box"];131[label="Neg Zero",fontsize=16,color="green",shape="box"];133 -> 8[label="",style="dashed", color="red", weight=0];
133[label="vuz11 * vuz12",fontsize=16,color="magenta"];133 -> 152[label="",style="dashed", color="magenta", weight=3];
133 -> 153[label="",style="dashed", color="magenta", weight=3];
134 -> 8[label="",style="dashed", color="red", weight=0];
134[label="vuz11 * vuz12",fontsize=16,color="magenta"];134 -> 154[label="",style="dashed", color="magenta", weight=3];
134 -> 155[label="",style="dashed", color="magenta", weight=3];
135 -> 8[label="",style="dashed", color="red", weight=0];
135[label="vuz13 * vuz14",fontsize=16,color="magenta"];135 -> 156[label="",style="dashed", color="magenta", weight=3];
135 -> 157[label="",style="dashed", color="magenta", weight=3];
132[label="vuz19 `quot` reduce2D vuz21 vuz20",fontsize=16,color="burlywood",shape="triangle"];1073[label="vuz19/Integer vuz190",fontsize=10,color="white",style="solid",shape="box"];132 -> 1073[label="",style="solid", color="burlywood", weight=9];
1073 -> 158[label="",style="solid", color="burlywood", weight=3];
140 -> 7[label="",style="dashed", color="red", weight=0];
140[label="vuz11 * vuz12",fontsize=16,color="magenta"];140 -> 159[label="",style="dashed", color="magenta", weight=3];
140 -> 160[label="",style="dashed", color="magenta", weight=3];
141 -> 7[label="",style="dashed", color="red", weight=0];
141[label="vuz11 * vuz12",fontsize=16,color="magenta"];141 -> 161[label="",style="dashed", color="magenta", weight=3];
141 -> 162[label="",style="dashed", color="magenta", weight=3];
142 -> 7[label="",style="dashed", color="red", weight=0];
142[label="vuz13 * vuz14",fontsize=16,color="magenta"];142 -> 163[label="",style="dashed", color="magenta", weight=3];
142 -> 164[label="",style="dashed", color="magenta", weight=3];
139[label="vuz22 `quot` reduce2D vuz24 vuz23",fontsize=16,color="black",shape="triangle"];139 -> 165[label="",style="solid", color="black", weight=3];
136 -> 8[label="",style="dashed", color="red", weight=0];
136[label="vuz13 * vuz14",fontsize=16,color="magenta"];136 -> 166[label="",style="dashed", color="magenta", weight=3];
136 -> 167[label="",style="dashed", color="magenta", weight=3];
137 -> 8[label="",style="dashed", color="red", weight=0];
137[label="vuz11 * vuz12",fontsize=16,color="magenta"];137 -> 168[label="",style="dashed", color="magenta", weight=3];
137 -> 169[label="",style="dashed", color="magenta", weight=3];
138 -> 8[label="",style="dashed", color="red", weight=0];
138[label="vuz13 * vuz14",fontsize=16,color="magenta"];138 -> 170[label="",style="dashed", color="magenta", weight=3];
138 -> 171[label="",style="dashed", color="magenta", weight=3];
143 -> 7[label="",style="dashed", color="red", weight=0];
143[label="vuz13 * vuz14",fontsize=16,color="magenta"];143 -> 172[label="",style="dashed", color="magenta", weight=3];
143 -> 173[label="",style="dashed", color="magenta", weight=3];
144 -> 7[label="",style="dashed", color="red", weight=0];
144[label="vuz11 * vuz12",fontsize=16,color="magenta"];144 -> 174[label="",style="dashed", color="magenta", weight=3];
144 -> 175[label="",style="dashed", color="magenta", weight=3];
145 -> 7[label="",style="dashed", color="red", weight=0];
145[label="vuz13 * vuz14",fontsize=16,color="magenta"];145 -> 176[label="",style="dashed", color="magenta", weight=3];
145 -> 177[label="",style="dashed", color="magenta", weight=3];
146[label="primPlusNat (Succ vuz1800) vuz400",fontsize=16,color="burlywood",shape="box"];1074[label="vuz400/Succ vuz4000",fontsize=10,color="white",style="solid",shape="box"];146 -> 1074[label="",style="solid", color="burlywood", weight=9];
1074 -> 178[label="",style="solid", color="burlywood", weight=3];
1075[label="vuz400/Zero",fontsize=10,color="white",style="solid",shape="box"];146 -> 1075[label="",style="solid", color="burlywood", weight=9];
1075 -> 179[label="",style="solid", color="burlywood", weight=3];
147[label="primPlusNat Zero vuz400",fontsize=16,color="burlywood",shape="box"];1076[label="vuz400/Succ vuz4000",fontsize=10,color="white",style="solid",shape="box"];147 -> 1076[label="",style="solid", color="burlywood", weight=9];
1076 -> 180[label="",style="solid", color="burlywood", weight=3];
1077[label="vuz400/Zero",fontsize=10,color="white",style="solid",shape="box"];147 -> 1077[label="",style="solid", color="burlywood", weight=9];
1077 -> 181[label="",style="solid", color="burlywood", weight=3];
148[label="False",fontsize=16,color="green",shape="box"];149[label="True",fontsize=16,color="green",shape="box"];150[label="False",fontsize=16,color="green",shape="box"];151[label="True",fontsize=16,color="green",shape="box"];152[label="vuz12",fontsize=16,color="green",shape="box"];153[label="vuz11",fontsize=16,color="green",shape="box"];154[label="vuz12",fontsize=16,color="green",shape="box"];155[label="vuz11",fontsize=16,color="green",shape="box"];156[label="vuz14",fontsize=16,color="green",shape="box"];157[label="vuz13",fontsize=16,color="green",shape="box"];158[label="Integer vuz190 `quot` reduce2D vuz21 vuz20",fontsize=16,color="black",shape="box"];158 -> 182[label="",style="solid", color="black", weight=3];
159[label="vuz12",fontsize=16,color="green",shape="box"];160[label="vuz11",fontsize=16,color="green",shape="box"];161[label="vuz12",fontsize=16,color="green",shape="box"];162[label="vuz11",fontsize=16,color="green",shape="box"];163[label="vuz14",fontsize=16,color="green",shape="box"];164[label="vuz13",fontsize=16,color="green",shape="box"];165[label="primQuotInt vuz22 (reduce2D vuz24 vuz23)",fontsize=16,color="burlywood",shape="box"];1078[label="vuz22/Pos vuz220",fontsize=10,color="white",style="solid",shape="box"];165 -> 1078[label="",style="solid", color="burlywood", weight=9];
1078 -> 183[label="",style="solid", color="burlywood", weight=3];
1079[label="vuz22/Neg vuz220",fontsize=10,color="white",style="solid",shape="box"];165 -> 1079[label="",style="solid", color="burlywood", weight=9];
1079 -> 184[label="",style="solid", color="burlywood", weight=3];
166[label="vuz14",fontsize=16,color="green",shape="box"];167[label="vuz13",fontsize=16,color="green",shape="box"];168[label="vuz12",fontsize=16,color="green",shape="box"];169[label="vuz11",fontsize=16,color="green",shape="box"];170[label="vuz14",fontsize=16,color="green",shape="box"];171[label="vuz13",fontsize=16,color="green",shape="box"];172[label="vuz14",fontsize=16,color="green",shape="box"];173[label="vuz13",fontsize=16,color="green",shape="box"];174[label="vuz12",fontsize=16,color="green",shape="box"];175[label="vuz11",fontsize=16,color="green",shape="box"];176[label="vuz14",fontsize=16,color="green",shape="box"];177[label="vuz13",fontsize=16,color="green",shape="box"];178[label="primPlusNat (Succ vuz1800) (Succ vuz4000)",fontsize=16,color="black",shape="box"];178 -> 185[label="",style="solid", color="black", weight=3];
179[label="primPlusNat (Succ vuz1800) Zero",fontsize=16,color="black",shape="box"];179 -> 186[label="",style="solid", color="black", weight=3];
180[label="primPlusNat Zero (Succ vuz4000)",fontsize=16,color="black",shape="box"];180 -> 187[label="",style="solid", color="black", weight=3];
181[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];181 -> 188[label="",style="solid", color="black", weight=3];
182 -> 276[label="",style="dashed", color="red", weight=0];
182[label="Integer vuz190 `quot` gcd vuz21 vuz20",fontsize=16,color="magenta"];182 -> 277[label="",style="dashed", color="magenta", weight=3];
183[label="primQuotInt (Pos vuz220) (reduce2D vuz24 vuz23)",fontsize=16,color="black",shape="box"];183 -> 190[label="",style="solid", color="black", weight=3];
184[label="primQuotInt (Neg vuz220) (reduce2D vuz24 vuz23)",fontsize=16,color="black",shape="box"];184 -> 191[label="",style="solid", color="black", weight=3];
185[label="Succ (Succ (primPlusNat vuz1800 vuz4000))",fontsize=16,color="green",shape="box"];185 -> 192[label="",style="dashed", color="green", weight=3];
186[label="Succ vuz1800",fontsize=16,color="green",shape="box"];187[label="Succ vuz4000",fontsize=16,color="green",shape="box"];188[label="Zero",fontsize=16,color="green",shape="box"];277[label="gcd vuz21 vuz20",fontsize=16,color="black",shape="box"];277 -> 293[label="",style="solid", color="black", weight=3];
276[label="Integer vuz190 `quot` vuz36",fontsize=16,color="burlywood",shape="triangle"];1080[label="vuz36/Integer vuz360",fontsize=10,color="white",style="solid",shape="box"];276 -> 1080[label="",style="solid", color="burlywood", weight=9];
1080 -> 294[label="",style="solid", color="burlywood", weight=3];
190 -> 312[label="",style="dashed", color="red", weight=0];
190[label="primQuotInt (Pos vuz220) (gcd vuz24 vuz23)",fontsize=16,color="magenta"];190 -> 313[label="",style="dashed", color="magenta", weight=3];
191[label="primQuotInt (Neg vuz220) (gcd vuz24 vuz23)",fontsize=16,color="black",shape="box"];191 -> 195[label="",style="solid", color="black", weight=3];
192 -> 123[label="",style="dashed", color="red", weight=0];
192[label="primPlusNat vuz1800 vuz4000",fontsize=16,color="magenta"];192 -> 196[label="",style="dashed", color="magenta", weight=3];
192 -> 197[label="",style="dashed", color="magenta", weight=3];
293[label="gcd3 vuz21 vuz20",fontsize=16,color="black",shape="box"];293 -> 300[label="",style="solid", color="black", weight=3];
294[label="Integer vuz190 `quot` Integer vuz360",fontsize=16,color="black",shape="box"];294 -> 301[label="",style="solid", color="black", weight=3];
313[label="gcd vuz24 vuz23",fontsize=16,color="black",shape="box"];313 -> 329[label="",style="solid", color="black", weight=3];
312[label="primQuotInt (Pos vuz220) vuz40",fontsize=16,color="burlywood",shape="triangle"];1081[label="vuz40/Pos vuz400",fontsize=10,color="white",style="solid",shape="box"];312 -> 1081[label="",style="solid", color="burlywood", weight=9];
1081 -> 330[label="",style="solid", color="burlywood", weight=3];
1082[label="vuz40/Neg vuz400",fontsize=10,color="white",style="solid",shape="box"];312 -> 1082[label="",style="solid", color="burlywood", weight=9];
1082 -> 331[label="",style="solid", color="burlywood", weight=3];
195[label="primQuotInt (Neg vuz220) (gcd3 vuz24 vuz23)",fontsize=16,color="black",shape="box"];195 -> 201[label="",style="solid", color="black", weight=3];
196[label="vuz4000",fontsize=16,color="green",shape="box"];197[label="vuz1800",fontsize=16,color="green",shape="box"];300 -> 306[label="",style="dashed", color="red", weight=0];
300[label="gcd2 (vuz21 == fromInt (Pos Zero)) vuz21 vuz20",fontsize=16,color="magenta"];300 -> 307[label="",style="dashed", color="magenta", weight=3];
301[label="Integer (primQuotInt vuz190 vuz360)",fontsize=16,color="green",shape="box"];301 -> 308[label="",style="dashed", color="green", weight=3];
329[label="gcd3 vuz24 vuz23",fontsize=16,color="black",shape="box"];329 -> 341[label="",style="solid", color="black", weight=3];
330[label="primQuotInt (Pos vuz220) (Pos vuz400)",fontsize=16,color="burlywood",shape="box"];1083[label="vuz400/Succ vuz4000",fontsize=10,color="white",style="solid",shape="box"];330 -> 1083[label="",style="solid", color="burlywood", weight=9];
1083 -> 342[label="",style="solid", color="burlywood", weight=3];
1084[label="vuz400/Zero",fontsize=10,color="white",style="solid",shape="box"];330 -> 1084[label="",style="solid", color="burlywood", weight=9];
1084 -> 343[label="",style="solid", color="burlywood", weight=3];
331[label="primQuotInt (Pos vuz220) (Neg vuz400)",fontsize=16,color="burlywood",shape="box"];1085[label="vuz400/Succ vuz4000",fontsize=10,color="white",style="solid",shape="box"];331 -> 1085[label="",style="solid", color="burlywood", weight=9];
1085 -> 344[label="",style="solid", color="burlywood", weight=3];
1086[label="vuz400/Zero",fontsize=10,color="white",style="solid",shape="box"];331 -> 1086[label="",style="solid", color="burlywood", weight=9];
1086 -> 345[label="",style="solid", color="burlywood", weight=3];
201 -> 207[label="",style="dashed", color="red", weight=0];
201[label="primQuotInt (Neg vuz220) (gcd2 (vuz24 == fromInt (Pos Zero)) vuz24 vuz23)",fontsize=16,color="magenta"];201 -> 208[label="",style="dashed", color="magenta", weight=3];
307 -> 76[label="",style="dashed", color="red", weight=0];
307[label="vuz21 == fromInt (Pos Zero)",fontsize=16,color="magenta"];307 -> 309[label="",style="dashed", color="magenta", weight=3];
306[label="gcd2 vuz39 vuz21 vuz20",fontsize=16,color="burlywood",shape="triangle"];1087[label="vuz39/False",fontsize=10,color="white",style="solid",shape="box"];306 -> 1087[label="",style="solid", color="burlywood", weight=9];
1087 -> 310[label="",style="solid", color="burlywood", weight=3];
1088[label="vuz39/True",fontsize=10,color="white",style="solid",shape="box"];306 -> 1088[label="",style="solid", color="burlywood", weight=9];
1088 -> 311[label="",style="solid", color="burlywood", weight=3];
308[label="primQuotInt vuz190 vuz360",fontsize=16,color="burlywood",shape="triangle"];1089[label="vuz190/Pos vuz1900",fontsize=10,color="white",style="solid",shape="box"];308 -> 1089[label="",style="solid", color="burlywood", weight=9];
1089 -> 332[label="",style="solid", color="burlywood", weight=3];
1090[label="vuz190/Neg vuz1900",fontsize=10,color="white",style="solid",shape="box"];308 -> 1090[label="",style="solid", color="burlywood", weight=9];
1090 -> 333[label="",style="solid", color="burlywood", weight=3];
341 -> 353[label="",style="dashed", color="red", weight=0];
341[label="gcd2 (vuz24 == fromInt (Pos Zero)) vuz24 vuz23",fontsize=16,color="magenta"];341 -> 354[label="",style="dashed", color="magenta", weight=3];
342[label="primQuotInt (Pos vuz220) (Pos (Succ vuz4000))",fontsize=16,color="black",shape="box"];342 -> 355[label="",style="solid", color="black", weight=3];
343[label="primQuotInt (Pos vuz220) (Pos Zero)",fontsize=16,color="black",shape="box"];343 -> 356[label="",style="solid", color="black", weight=3];
344[label="primQuotInt (Pos vuz220) (Neg (Succ vuz4000))",fontsize=16,color="black",shape="box"];344 -> 357[label="",style="solid", color="black", weight=3];
345[label="primQuotInt (Pos vuz220) (Neg Zero)",fontsize=16,color="black",shape="box"];345 -> 358[label="",style="solid", color="black", weight=3];
208 -> 78[label="",style="dashed", color="red", weight=0];
208[label="vuz24 == fromInt (Pos Zero)",fontsize=16,color="magenta"];208 -> 214[label="",style="dashed", color="magenta", weight=3];
207[label="primQuotInt (Neg vuz220) (gcd2 vuz27 vuz24 vuz23)",fontsize=16,color="burlywood",shape="triangle"];1091[label="vuz27/False",fontsize=10,color="white",style="solid",shape="box"];207 -> 1091[label="",style="solid", color="burlywood", weight=9];
1091 -> 215[label="",style="solid", color="burlywood", weight=3];
1092[label="vuz27/True",fontsize=10,color="white",style="solid",shape="box"];207 -> 1092[label="",style="solid", color="burlywood", weight=9];
1092 -> 216[label="",style="solid", color="burlywood", weight=3];
309[label="vuz21",fontsize=16,color="green",shape="box"];310[label="gcd2 False vuz21 vuz20",fontsize=16,color="black",shape="box"];310 -> 334[label="",style="solid", color="black", weight=3];
311[label="gcd2 True vuz21 vuz20",fontsize=16,color="black",shape="box"];311 -> 335[label="",style="solid", color="black", weight=3];
332[label="primQuotInt (Pos vuz1900) vuz360",fontsize=16,color="burlywood",shape="box"];1093[label="vuz360/Pos vuz3600",fontsize=10,color="white",style="solid",shape="box"];332 -> 1093[label="",style="solid", color="burlywood", weight=9];
1093 -> 346[label="",style="solid", color="burlywood", weight=3];
1094[label="vuz360/Neg vuz3600",fontsize=10,color="white",style="solid",shape="box"];332 -> 1094[label="",style="solid", color="burlywood", weight=9];
1094 -> 347[label="",style="solid", color="burlywood", weight=3];
333[label="primQuotInt (Neg vuz1900) vuz360",fontsize=16,color="burlywood",shape="box"];1095[label="vuz360/Pos vuz3600",fontsize=10,color="white",style="solid",shape="box"];333 -> 1095[label="",style="solid", color="burlywood", weight=9];
1095 -> 348[label="",style="solid", color="burlywood", weight=3];
1096[label="vuz360/Neg vuz3600",fontsize=10,color="white",style="solid",shape="box"];333 -> 1096[label="",style="solid", color="burlywood", weight=9];
1096 -> 349[label="",style="solid", color="burlywood", weight=3];
354 -> 78[label="",style="dashed", color="red", weight=0];
354[label="vuz24 == fromInt (Pos Zero)",fontsize=16,color="magenta"];354 -> 359[label="",style="dashed", color="magenta", weight=3];
353[label="gcd2 vuz42 vuz24 vuz23",fontsize=16,color="burlywood",shape="triangle"];1097[label="vuz42/False",fontsize=10,color="white",style="solid",shape="box"];353 -> 1097[label="",style="solid", color="burlywood", weight=9];
1097 -> 360[label="",style="solid", color="burlywood", weight=3];
1098[label="vuz42/True",fontsize=10,color="white",style="solid",shape="box"];353 -> 1098[label="",style="solid", color="burlywood", weight=9];
1098 -> 361[label="",style="solid", color="burlywood", weight=3];
355[label="Pos (primDivNatS vuz220 (Succ vuz4000))",fontsize=16,color="green",shape="box"];355 -> 377[label="",style="dashed", color="green", weight=3];
356[label="error []",fontsize=16,color="black",shape="triangle"];356 -> 378[label="",style="solid", color="black", weight=3];
357[label="Neg (primDivNatS vuz220 (Succ vuz4000))",fontsize=16,color="green",shape="box"];357 -> 379[label="",style="dashed", color="green", weight=3];
358 -> 356[label="",style="dashed", color="red", weight=0];
358[label="error []",fontsize=16,color="magenta"];214[label="vuz24",fontsize=16,color="green",shape="box"];215[label="primQuotInt (Neg vuz220) (gcd2 False vuz24 vuz23)",fontsize=16,color="black",shape="box"];215 -> 222[label="",style="solid", color="black", weight=3];
216[label="primQuotInt (Neg vuz220) (gcd2 True vuz24 vuz23)",fontsize=16,color="black",shape="box"];216 -> 223[label="",style="solid", color="black", weight=3];
334[label="gcd0 vuz21 vuz20",fontsize=16,color="black",shape="triangle"];334 -> 350[label="",style="solid", color="black", weight=3];
335 -> 351[label="",style="dashed", color="red", weight=0];
335[label="gcd1 (vuz20 == fromInt (Pos Zero)) vuz21 vuz20",fontsize=16,color="magenta"];335 -> 352[label="",style="dashed", color="magenta", weight=3];
346[label="primQuotInt (Pos vuz1900) (Pos vuz3600)",fontsize=16,color="burlywood",shape="box"];1099[label="vuz3600/Succ vuz36000",fontsize=10,color="white",style="solid",shape="box"];346 -> 1099[label="",style="solid", color="burlywood", weight=9];
1099 -> 362[label="",style="solid", color="burlywood", weight=3];
1100[label="vuz3600/Zero",fontsize=10,color="white",style="solid",shape="box"];346 -> 1100[label="",style="solid", color="burlywood", weight=9];
1100 -> 363[label="",style="solid", color="burlywood", weight=3];
347[label="primQuotInt (Pos vuz1900) (Neg vuz3600)",fontsize=16,color="burlywood",shape="box"];1101[label="vuz3600/Succ vuz36000",fontsize=10,color="white",style="solid",shape="box"];347 -> 1101[label="",style="solid", color="burlywood", weight=9];
1101 -> 364[label="",style="solid", color="burlywood", weight=3];
1102[label="vuz3600/Zero",fontsize=10,color="white",style="solid",shape="box"];347 -> 1102[label="",style="solid", color="burlywood", weight=9];
1102 -> 365[label="",style="solid", color="burlywood", weight=3];
348[label="primQuotInt (Neg vuz1900) (Pos vuz3600)",fontsize=16,color="burlywood",shape="box"];1103[label="vuz3600/Succ vuz36000",fontsize=10,color="white",style="solid",shape="box"];348 -> 1103[label="",style="solid", color="burlywood", weight=9];
1103 -> 366[label="",style="solid", color="burlywood", weight=3];
1104[label="vuz3600/Zero",fontsize=10,color="white",style="solid",shape="box"];348 -> 1104[label="",style="solid", color="burlywood", weight=9];
1104 -> 367[label="",style="solid", color="burlywood", weight=3];
349[label="primQuotInt (Neg vuz1900) (Neg vuz3600)",fontsize=16,color="burlywood",shape="box"];1105[label="vuz3600/Succ vuz36000",fontsize=10,color="white",style="solid",shape="box"];349 -> 1105[label="",style="solid", color="burlywood", weight=9];
1105 -> 368[label="",style="solid", color="burlywood", weight=3];
1106[label="vuz3600/Zero",fontsize=10,color="white",style="solid",shape="box"];349 -> 1106[label="",style="solid", color="burlywood", weight=9];
1106 -> 369[label="",style="solid", color="burlywood", weight=3];
359[label="vuz24",fontsize=16,color="green",shape="box"];360[label="gcd2 False vuz24 vuz23",fontsize=16,color="black",shape="box"];360 -> 380[label="",style="solid", color="black", weight=3];
361[label="gcd2 True vuz24 vuz23",fontsize=16,color="black",shape="box"];361 -> 381[label="",style="solid", color="black", weight=3];
377[label="primDivNatS vuz220 (Succ vuz4000)",fontsize=16,color="burlywood",shape="triangle"];1107[label="vuz220/Succ vuz2200",fontsize=10,color="white",style="solid",shape="box"];377 -> 1107[label="",style="solid", color="burlywood", weight=9];
1107 -> 397[label="",style="solid", color="burlywood", weight=3];
1108[label="vuz220/Zero",fontsize=10,color="white",style="solid",shape="box"];377 -> 1108[label="",style="solid", color="burlywood", weight=9];
1108 -> 398[label="",style="solid", color="burlywood", weight=3];
378[label="error []",fontsize=16,color="red",shape="box"];379 -> 377[label="",style="dashed", color="red", weight=0];
379[label="primDivNatS vuz220 (Succ vuz4000)",fontsize=16,color="magenta"];379 -> 399[label="",style="dashed", color="magenta", weight=3];
222[label="primQuotInt (Neg vuz220) (gcd0 vuz24 vuz23)",fontsize=16,color="black",shape="triangle"];222 -> 231[label="",style="solid", color="black", weight=3];
223 -> 232[label="",style="dashed", color="red", weight=0];
223[label="primQuotInt (Neg vuz220) (gcd1 (vuz23 == fromInt (Pos Zero)) vuz24 vuz23)",fontsize=16,color="magenta"];223 -> 233[label="",style="dashed", color="magenta", weight=3];
350[label="gcd0Gcd' (abs vuz21) (abs vuz20)",fontsize=16,color="black",shape="box"];350 -> 370[label="",style="solid", color="black", weight=3];
352 -> 76[label="",style="dashed", color="red", weight=0];
352[label="vuz20 == fromInt (Pos Zero)",fontsize=16,color="magenta"];352 -> 371[label="",style="dashed", color="magenta", weight=3];
351[label="gcd1 vuz41 vuz21 vuz20",fontsize=16,color="burlywood",shape="triangle"];1109[label="vuz41/False",fontsize=10,color="white",style="solid",shape="box"];351 -> 1109[label="",style="solid", color="burlywood", weight=9];
1109 -> 372[label="",style="solid", color="burlywood", weight=3];
1110[label="vuz41/True",fontsize=10,color="white",style="solid",shape="box"];351 -> 1110[label="",style="solid", color="burlywood", weight=9];
1110 -> 373[label="",style="solid", color="burlywood", weight=3];
362[label="primQuotInt (Pos vuz1900) (Pos (Succ vuz36000))",fontsize=16,color="black",shape="box"];362 -> 382[label="",style="solid", color="black", weight=3];
363[label="primQuotInt (Pos vuz1900) (Pos Zero)",fontsize=16,color="black",shape="box"];363 -> 383[label="",style="solid", color="black", weight=3];
364[label="primQuotInt (Pos vuz1900) (Neg (Succ vuz36000))",fontsize=16,color="black",shape="box"];364 -> 384[label="",style="solid", color="black", weight=3];
365[label="primQuotInt (Pos vuz1900) (Neg Zero)",fontsize=16,color="black",shape="box"];365 -> 385[label="",style="solid", color="black", weight=3];
366[label="primQuotInt (Neg vuz1900) (Pos (Succ vuz36000))",fontsize=16,color="black",shape="box"];366 -> 386[label="",style="solid", color="black", weight=3];
367[label="primQuotInt (Neg vuz1900) (Pos Zero)",fontsize=16,color="black",shape="box"];367 -> 387[label="",style="solid", color="black", weight=3];
368[label="primQuotInt (Neg vuz1900) (Neg (Succ vuz36000))",fontsize=16,color="black",shape="box"];368 -> 388[label="",style="solid", color="black", weight=3];
369[label="primQuotInt (Neg vuz1900) (Neg Zero)",fontsize=16,color="black",shape="box"];369 -> 389[label="",style="solid", color="black", weight=3];
380[label="gcd0 vuz24 vuz23",fontsize=16,color="black",shape="triangle"];380 -> 400[label="",style="solid", color="black", weight=3];
381 -> 401[label="",style="dashed", color="red", weight=0];
381[label="gcd1 (vuz23 == fromInt (Pos Zero)) vuz24 vuz23",fontsize=16,color="magenta"];381 -> 402[label="",style="dashed", color="magenta", weight=3];
397[label="primDivNatS (Succ vuz2200) (Succ vuz4000)",fontsize=16,color="black",shape="box"];397 -> 403[label="",style="solid", color="black", weight=3];
398[label="primDivNatS Zero (Succ vuz4000)",fontsize=16,color="black",shape="box"];398 -> 404[label="",style="solid", color="black", weight=3];
399[label="vuz4000",fontsize=16,color="green",shape="box"];231[label="primQuotInt (Neg vuz220) (gcd0Gcd' (abs vuz24) (abs vuz23))",fontsize=16,color="black",shape="box"];231 -> 241[label="",style="solid", color="black", weight=3];
233 -> 78[label="",style="dashed", color="red", weight=0];
233[label="vuz23 == fromInt (Pos Zero)",fontsize=16,color="magenta"];233 -> 242[label="",style="dashed", color="magenta", weight=3];
232[label="primQuotInt (Neg vuz220) (gcd1 vuz30 vuz24 vuz23)",fontsize=16,color="burlywood",shape="triangle"];1111[label="vuz30/False",fontsize=10,color="white",style="solid",shape="box"];232 -> 1111[label="",style="solid", color="burlywood", weight=9];
1111 -> 243[label="",style="solid", color="burlywood", weight=3];
1112[label="vuz30/True",fontsize=10,color="white",style="solid",shape="box"];232 -> 1112[label="",style="solid", color="burlywood", weight=9];
1112 -> 244[label="",style="solid", color="burlywood", weight=3];
370[label="gcd0Gcd'2 (abs vuz21) (abs vuz20)",fontsize=16,color="black",shape="box"];370 -> 390[label="",style="solid", color="black", weight=3];
371[label="vuz20",fontsize=16,color="green",shape="box"];372[label="gcd1 False vuz21 vuz20",fontsize=16,color="black",shape="box"];372 -> 391[label="",style="solid", color="black", weight=3];
373[label="gcd1 True vuz21 vuz20",fontsize=16,color="black",shape="box"];373 -> 392[label="",style="solid", color="black", weight=3];
382[label="Pos (primDivNatS vuz1900 (Succ vuz36000))",fontsize=16,color="green",shape="box"];382 -> 405[label="",style="dashed", color="green", weight=3];
383 -> 356[label="",style="dashed", color="red", weight=0];
383[label="error []",fontsize=16,color="magenta"];384[label="Neg (primDivNatS vuz1900 (Succ vuz36000))",fontsize=16,color="green",shape="box"];384 -> 406[label="",style="dashed", color="green", weight=3];
385 -> 356[label="",style="dashed", color="red", weight=0];
385[label="error []",fontsize=16,color="magenta"];386[label="Neg (primDivNatS vuz1900 (Succ vuz36000))",fontsize=16,color="green",shape="box"];386 -> 407[label="",style="dashed", color="green", weight=3];
387 -> 356[label="",style="dashed", color="red", weight=0];
387[label="error []",fontsize=16,color="magenta"];388[label="Pos (primDivNatS vuz1900 (Succ vuz36000))",fontsize=16,color="green",shape="box"];388 -> 408[label="",style="dashed", color="green", weight=3];
389 -> 356[label="",style="dashed", color="red", weight=0];
389[label="error []",fontsize=16,color="magenta"];400 -> 409[label="",style="dashed", color="red", weight=0];
400[label="gcd0Gcd' (abs vuz24) (abs vuz23)",fontsize=16,color="magenta"];400 -> 410[label="",style="dashed", color="magenta", weight=3];
400 -> 411[label="",style="dashed", color="magenta", weight=3];
402 -> 78[label="",style="dashed", color="red", weight=0];
402[label="vuz23 == fromInt (Pos Zero)",fontsize=16,color="magenta"];402 -> 414[label="",style="dashed", color="magenta", weight=3];
401[label="gcd1 vuz45 vuz24 vuz23",fontsize=16,color="burlywood",shape="triangle"];1113[label="vuz45/False",fontsize=10,color="white",style="solid",shape="box"];401 -> 1113[label="",style="solid", color="burlywood", weight=9];
1113 -> 415[label="",style="solid", color="burlywood", weight=3];
1114[label="vuz45/True",fontsize=10,color="white",style="solid",shape="box"];401 -> 1114[label="",style="solid", color="burlywood", weight=9];
1114 -> 416[label="",style="solid", color="burlywood", weight=3];
403[label="primDivNatS0 vuz2200 vuz4000 (primGEqNatS vuz2200 vuz4000)",fontsize=16,color="burlywood",shape="box"];1115[label="vuz2200/Succ vuz22000",fontsize=10,color="white",style="solid",shape="box"];403 -> 1115[label="",style="solid", color="burlywood", weight=9];
1115 -> 417[label="",style="solid", color="burlywood", weight=3];
1116[label="vuz2200/Zero",fontsize=10,color="white",style="solid",shape="box"];403 -> 1116[label="",style="solid", color="burlywood", weight=9];
1116 -> 418[label="",style="solid", color="burlywood", weight=3];
404[label="Zero",fontsize=16,color="green",shape="box"];241[label="primQuotInt (Neg vuz220) (gcd0Gcd'2 (abs vuz24) (abs vuz23))",fontsize=16,color="black",shape="box"];241 -> 251[label="",style="solid", color="black", weight=3];
242[label="vuz23",fontsize=16,color="green",shape="box"];243[label="primQuotInt (Neg vuz220) (gcd1 False vuz24 vuz23)",fontsize=16,color="black",shape="box"];243 -> 252[label="",style="solid", color="black", weight=3];
244[label="primQuotInt (Neg vuz220) (gcd1 True vuz24 vuz23)",fontsize=16,color="black",shape="box"];244 -> 253[label="",style="solid", color="black", weight=3];
390 -> 419[label="",style="dashed", color="red", weight=0];
390[label="gcd0Gcd'1 (abs vuz20 == fromInt (Pos Zero)) (abs vuz21) (abs vuz20)",fontsize=16,color="magenta"];390 -> 420[label="",style="dashed", color="magenta", weight=3];
391 -> 334[label="",style="dashed", color="red", weight=0];
391[label="gcd0 vuz21 vuz20",fontsize=16,color="magenta"];392[label="error []",fontsize=16,color="black",shape="box"];392 -> 421[label="",style="solid", color="black", weight=3];
405 -> 377[label="",style="dashed", color="red", weight=0];
405[label="primDivNatS vuz1900 (Succ vuz36000)",fontsize=16,color="magenta"];405 -> 422[label="",style="dashed", color="magenta", weight=3];
405 -> 423[label="",style="dashed", color="magenta", weight=3];
406 -> 377[label="",style="dashed", color="red", weight=0];
406[label="primDivNatS vuz1900 (Succ vuz36000)",fontsize=16,color="magenta"];406 -> 424[label="",style="dashed", color="magenta", weight=3];
406 -> 425[label="",style="dashed", color="magenta", weight=3];
407 -> 377[label="",style="dashed", color="red", weight=0];
407[label="primDivNatS vuz1900 (Succ vuz36000)",fontsize=16,color="magenta"];407 -> 426[label="",style="dashed", color="magenta", weight=3];
407 -> 427[label="",style="dashed", color="magenta", weight=3];
408 -> 377[label="",style="dashed", color="red", weight=0];
408[label="primDivNatS vuz1900 (Succ vuz36000)",fontsize=16,color="magenta"];408 -> 428[label="",style="dashed", color="magenta", weight=3];
408 -> 429[label="",style="dashed", color="magenta", weight=3];
410 -> 266[label="",style="dashed", color="red", weight=0];
410[label="abs vuz23",fontsize=16,color="magenta"];411 -> 266[label="",style="dashed", color="red", weight=0];
411[label="abs vuz24",fontsize=16,color="magenta"];411 -> 430[label="",style="dashed", color="magenta", weight=3];
409[label="gcd0Gcd' vuz47 vuz46",fontsize=16,color="black",shape="triangle"];409 -> 431[label="",style="solid", color="black", weight=3];
414[label="vuz23",fontsize=16,color="green",shape="box"];415[label="gcd1 False vuz24 vuz23",fontsize=16,color="black",shape="box"];415 -> 432[label="",style="solid", color="black", weight=3];
416[label="gcd1 True vuz24 vuz23",fontsize=16,color="black",shape="box"];416 -> 433[label="",style="solid", color="black", weight=3];
417[label="primDivNatS0 (Succ vuz22000) vuz4000 (primGEqNatS (Succ vuz22000) vuz4000)",fontsize=16,color="burlywood",shape="box"];1117[label="vuz4000/Succ vuz40000",fontsize=10,color="white",style="solid",shape="box"];417 -> 1117[label="",style="solid", color="burlywood", weight=9];
1117 -> 434[label="",style="solid", color="burlywood", weight=3];
1118[label="vuz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];417 -> 1118[label="",style="solid", color="burlywood", weight=9];
1118 -> 435[label="",style="solid", color="burlywood", weight=3];
418[label="primDivNatS0 Zero vuz4000 (primGEqNatS Zero vuz4000)",fontsize=16,color="burlywood",shape="box"];1119[label="vuz4000/Succ vuz40000",fontsize=10,color="white",style="solid",shape="box"];418 -> 1119[label="",style="solid", color="burlywood", weight=9];
1119 -> 436[label="",style="solid", color="burlywood", weight=3];
1120[label="vuz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];418 -> 1120[label="",style="solid", color="burlywood", weight=9];
1120 -> 437[label="",style="solid", color="burlywood", weight=3];
251 -> 260[label="",style="dashed", color="red", weight=0];
251[label="primQuotInt (Neg vuz220) (gcd0Gcd'1 (abs vuz23 == fromInt (Pos Zero)) (abs vuz24) (abs vuz23))",fontsize=16,color="magenta"];251 -> 261[label="",style="dashed", color="magenta", weight=3];
252 -> 222[label="",style="dashed", color="red", weight=0];
252[label="primQuotInt (Neg vuz220) (gcd0 vuz24 vuz23)",fontsize=16,color="magenta"];253[label="primQuotInt (Neg vuz220) (error [])",fontsize=16,color="black",shape="box"];253 -> 262[label="",style="solid", color="black", weight=3];
420 -> 76[label="",style="dashed", color="red", weight=0];
420[label="abs vuz20 == fromInt (Pos Zero)",fontsize=16,color="magenta"];420 -> 438[label="",style="dashed", color="magenta", weight=3];
419[label="gcd0Gcd'1 vuz48 (abs vuz21) (abs vuz20)",fontsize=16,color="burlywood",shape="triangle"];1121[label="vuz48/False",fontsize=10,color="white",style="solid",shape="box"];419 -> 1121[label="",style="solid", color="burlywood", weight=9];
1121 -> 439[label="",style="solid", color="burlywood", weight=3];
1122[label="vuz48/True",fontsize=10,color="white",style="solid",shape="box"];419 -> 1122[label="",style="solid", color="burlywood", weight=9];
1122 -> 440[label="",style="solid", color="burlywood", weight=3];
421[label="error []",fontsize=16,color="red",shape="box"];422[label="vuz1900",fontsize=16,color="green",shape="box"];423[label="vuz36000",fontsize=16,color="green",shape="box"];424[label="vuz1900",fontsize=16,color="green",shape="box"];425[label="vuz36000",fontsize=16,color="green",shape="box"];426[label="vuz1900",fontsize=16,color="green",shape="box"];427[label="vuz36000",fontsize=16,color="green",shape="box"];428[label="vuz1900",fontsize=16,color="green",shape="box"];429[label="vuz36000",fontsize=16,color="green",shape="box"];266[label="abs vuz23",fontsize=16,color="black",shape="triangle"];266 -> 295[label="",style="solid", color="black", weight=3];
430[label="vuz24",fontsize=16,color="green",shape="box"];431[label="gcd0Gcd'2 vuz47 vuz46",fontsize=16,color="black",shape="box"];431 -> 443[label="",style="solid", color="black", weight=3];
432 -> 380[label="",style="dashed", color="red", weight=0];
432[label="gcd0 vuz24 vuz23",fontsize=16,color="magenta"];433 -> 356[label="",style="dashed", color="red", weight=0];
433[label="error []",fontsize=16,color="magenta"];434[label="primDivNatS0 (Succ vuz22000) (Succ vuz40000) (primGEqNatS (Succ vuz22000) (Succ vuz40000))",fontsize=16,color="black",shape="box"];434 -> 444[label="",style="solid", color="black", weight=3];
435[label="primDivNatS0 (Succ vuz22000) Zero (primGEqNatS (Succ vuz22000) Zero)",fontsize=16,color="black",shape="box"];435 -> 445[label="",style="solid", color="black", weight=3];
436[label="primDivNatS0 Zero (Succ vuz40000) (primGEqNatS Zero (Succ vuz40000))",fontsize=16,color="black",shape="box"];436 -> 446[label="",style="solid", color="black", weight=3];
437[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];437 -> 447[label="",style="solid", color="black", weight=3];
261 -> 78[label="",style="dashed", color="red", weight=0];
261[label="abs vuz23 == fromInt (Pos Zero)",fontsize=16,color="magenta"];261 -> 269[label="",style="dashed", color="magenta", weight=3];
260[label="primQuotInt (Neg vuz220) (gcd0Gcd'1 vuz33 (abs vuz24) (abs vuz23))",fontsize=16,color="burlywood",shape="triangle"];1123[label="vuz33/False",fontsize=10,color="white",style="solid",shape="box"];260 -> 1123[label="",style="solid", color="burlywood", weight=9];
1123 -> 270[label="",style="solid", color="burlywood", weight=3];
1124[label="vuz33/True",fontsize=10,color="white",style="solid",shape="box"];260 -> 1124[label="",style="solid", color="burlywood", weight=9];
1124 -> 271[label="",style="solid", color="burlywood", weight=3];
262[label="error []",fontsize=16,color="red",shape="box"];438[label="abs vuz20",fontsize=16,color="black",shape="triangle"];438 -> 448[label="",style="solid", color="black", weight=3];
439[label="gcd0Gcd'1 False (abs vuz21) (abs vuz20)",fontsize=16,color="black",shape="box"];439 -> 449[label="",style="solid", color="black", weight=3];
440[label="gcd0Gcd'1 True (abs vuz21) (abs vuz20)",fontsize=16,color="black",shape="box"];440 -> 450[label="",style="solid", color="black", weight=3];
295[label="absReal vuz23",fontsize=16,color="black",shape="box"];295 -> 302[label="",style="solid", color="black", weight=3];
443 -> 454[label="",style="dashed", color="red", weight=0];
443[label="gcd0Gcd'1 (vuz46 == fromInt (Pos Zero)) vuz47 vuz46",fontsize=16,color="magenta"];443 -> 455[label="",style="dashed", color="magenta", weight=3];
444 -> 791[label="",style="dashed", color="red", weight=0];
444[label="primDivNatS0 (Succ vuz22000) (Succ vuz40000) (primGEqNatS vuz22000 vuz40000)",fontsize=16,color="magenta"];444 -> 792[label="",style="dashed", color="magenta", weight=3];
444 -> 793[label="",style="dashed", color="magenta", weight=3];
444 -> 794[label="",style="dashed", color="magenta", weight=3];
444 -> 795[label="",style="dashed", color="magenta", weight=3];
445[label="primDivNatS0 (Succ vuz22000) Zero True",fontsize=16,color="black",shape="box"];445 -> 458[label="",style="solid", color="black", weight=3];
446[label="primDivNatS0 Zero (Succ vuz40000) False",fontsize=16,color="black",shape="box"];446 -> 459[label="",style="solid", color="black", weight=3];
447[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];447 -> 460[label="",style="solid", color="black", weight=3];
269 -> 266[label="",style="dashed", color="red", weight=0];
269[label="abs vuz23",fontsize=16,color="magenta"];270[label="primQuotInt (Neg vuz220) (gcd0Gcd'1 False (abs vuz24) (abs vuz23))",fontsize=16,color="black",shape="box"];270 -> 298[label="",style="solid", color="black", weight=3];
271[label="primQuotInt (Neg vuz220) (gcd0Gcd'1 True (abs vuz24) (abs vuz23))",fontsize=16,color="black",shape="box"];271 -> 299[label="",style="solid", color="black", weight=3];
448[label="absReal vuz20",fontsize=16,color="black",shape="box"];448 -> 461[label="",style="solid", color="black", weight=3];
449 -> 462[label="",style="dashed", color="red", weight=0];
449[label="gcd0Gcd'0 (abs vuz21) (abs vuz20)",fontsize=16,color="magenta"];449 -> 463[label="",style="dashed", color="magenta", weight=3];
449 -> 464[label="",style="dashed", color="magenta", weight=3];
450 -> 438[label="",style="dashed", color="red", weight=0];
450[label="abs vuz21",fontsize=16,color="magenta"];450 -> 465[label="",style="dashed", color="magenta", weight=3];
302[label="absReal2 vuz23",fontsize=16,color="black",shape="box"];302 -> 340[label="",style="solid", color="black", weight=3];
455 -> 78[label="",style="dashed", color="red", weight=0];
455[label="vuz46 == fromInt (Pos Zero)",fontsize=16,color="magenta"];455 -> 466[label="",style="dashed", color="magenta", weight=3];
454[label="gcd0Gcd'1 vuz49 vuz47 vuz46",fontsize=16,color="burlywood",shape="triangle"];1125[label="vuz49/False",fontsize=10,color="white",style="solid",shape="box"];454 -> 1125[label="",style="solid", color="burlywood", weight=9];
1125 -> 467[label="",style="solid", color="burlywood", weight=3];
1126[label="vuz49/True",fontsize=10,color="white",style="solid",shape="box"];454 -> 1126[label="",style="solid", color="burlywood", weight=9];
1126 -> 468[label="",style="solid", color="burlywood", weight=3];
792[label="vuz40000",fontsize=16,color="green",shape="box"];793[label="vuz22000",fontsize=16,color="green",shape="box"];794[label="vuz22000",fontsize=16,color="green",shape="box"];795[label="vuz40000",fontsize=16,color="green",shape="box"];791[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS vuz71 vuz72)",fontsize=16,color="burlywood",shape="triangle"];1127[label="vuz71/Succ vuz710",fontsize=10,color="white",style="solid",shape="box"];791 -> 1127[label="",style="solid", color="burlywood", weight=9];
1127 -> 824[label="",style="solid", color="burlywood", weight=3];
1128[label="vuz71/Zero",fontsize=10,color="white",style="solid",shape="box"];791 -> 1128[label="",style="solid", color="burlywood", weight=9];
1128 -> 825[label="",style="solid", color="burlywood", weight=3];
458[label="Succ (primDivNatS (primMinusNatS (Succ vuz22000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];458 -> 473[label="",style="dashed", color="green", weight=3];
459[label="Zero",fontsize=16,color="green",shape="box"];460[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];460 -> 474[label="",style="dashed", color="green", weight=3];
298 -> 308[label="",style="dashed", color="red", weight=0];
298[label="primQuotInt (Neg vuz220) (gcd0Gcd'0 (abs vuz24) (abs vuz23))",fontsize=16,color="magenta"];298 -> 336[label="",style="dashed", color="magenta", weight=3];
298 -> 337[label="",style="dashed", color="magenta", weight=3];
299 -> 308[label="",style="dashed", color="red", weight=0];
299[label="primQuotInt (Neg vuz220) (abs vuz24)",fontsize=16,color="magenta"];299 -> 338[label="",style="dashed", color="magenta", weight=3];
299 -> 339[label="",style="dashed", color="magenta", weight=3];
461[label="absReal2 vuz20",fontsize=16,color="black",shape="box"];461 -> 475[label="",style="solid", color="black", weight=3];
463 -> 438[label="",style="dashed", color="red", weight=0];
463[label="abs vuz20",fontsize=16,color="magenta"];464 -> 438[label="",style="dashed", color="red", weight=0];
464[label="abs vuz21",fontsize=16,color="magenta"];464 -> 476[label="",style="dashed", color="magenta", weight=3];
462[label="gcd0Gcd'0 vuz51 vuz50",fontsize=16,color="black",shape="triangle"];462 -> 477[label="",style="solid", color="black", weight=3];
465[label="vuz21",fontsize=16,color="green",shape="box"];340[label="absReal1 vuz23 (vuz23 >= fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];340 -> 396[label="",style="solid", color="black", weight=3];
466[label="vuz46",fontsize=16,color="green",shape="box"];467[label="gcd0Gcd'1 False vuz47 vuz46",fontsize=16,color="black",shape="box"];467 -> 484[label="",style="solid", color="black", weight=3];
468[label="gcd0Gcd'1 True vuz47 vuz46",fontsize=16,color="black",shape="box"];468 -> 485[label="",style="solid", color="black", weight=3];
824[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS (Succ vuz710) vuz72)",fontsize=16,color="burlywood",shape="box"];1129[label="vuz72/Succ vuz720",fontsize=10,color="white",style="solid",shape="box"];824 -> 1129[label="",style="solid", color="burlywood", weight=9];
1129 -> 832[label="",style="solid", color="burlywood", weight=3];
1130[label="vuz72/Zero",fontsize=10,color="white",style="solid",shape="box"];824 -> 1130[label="",style="solid", color="burlywood", weight=9];
1130 -> 833[label="",style="solid", color="burlywood", weight=3];
825[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS Zero vuz72)",fontsize=16,color="burlywood",shape="box"];1131[label="vuz72/Succ vuz720",fontsize=10,color="white",style="solid",shape="box"];825 -> 1131[label="",style="solid", color="burlywood", weight=9];
1131 -> 834[label="",style="solid", color="burlywood", weight=3];
1132[label="vuz72/Zero",fontsize=10,color="white",style="solid",shape="box"];825 -> 1132[label="",style="solid", color="burlywood", weight=9];
1132 -> 835[label="",style="solid", color="burlywood", weight=3];
473 -> 377[label="",style="dashed", color="red", weight=0];
473[label="primDivNatS (primMinusNatS (Succ vuz22000) Zero) (Succ Zero)",fontsize=16,color="magenta"];473 -> 490[label="",style="dashed", color="magenta", weight=3];
473 -> 491[label="",style="dashed", color="magenta", weight=3];
474 -> 377[label="",style="dashed", color="red", weight=0];
474[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];474 -> 492[label="",style="dashed", color="magenta", weight=3];
474 -> 493[label="",style="dashed", color="magenta", weight=3];
336 -> 374[label="",style="dashed", color="red", weight=0];
336[label="gcd0Gcd'0 (abs vuz24) (abs vuz23)",fontsize=16,color="magenta"];336 -> 375[label="",style="dashed", color="magenta", weight=3];
336 -> 376[label="",style="dashed", color="magenta", weight=3];
337[label="Neg vuz220",fontsize=16,color="green",shape="box"];338 -> 266[label="",style="dashed", color="red", weight=0];
338[label="abs vuz24",fontsize=16,color="magenta"];338 -> 393[label="",style="dashed", color="magenta", weight=3];
339[label="Neg vuz220",fontsize=16,color="green",shape="box"];475[label="absReal1 vuz20 (vuz20 >= fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];475 -> 494[label="",style="solid", color="black", weight=3];
476[label="vuz21",fontsize=16,color="green",shape="box"];477[label="gcd0Gcd' vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="black",shape="box"];477 -> 495[label="",style="solid", color="black", weight=3];
396[label="absReal1 vuz23 (compare vuz23 (fromInt (Pos Zero)) /= LT)",fontsize=16,color="black",shape="box"];396 -> 441[label="",style="solid", color="black", weight=3];
484 -> 374[label="",style="dashed", color="red", weight=0];
484[label="gcd0Gcd'0 vuz47 vuz46",fontsize=16,color="magenta"];484 -> 508[label="",style="dashed", color="magenta", weight=3];
484 -> 509[label="",style="dashed", color="magenta", weight=3];
485[label="vuz47",fontsize=16,color="green",shape="box"];832[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS (Succ vuz710) (Succ vuz720))",fontsize=16,color="black",shape="box"];832 -> 844[label="",style="solid", color="black", weight=3];
833[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS (Succ vuz710) Zero)",fontsize=16,color="black",shape="box"];833 -> 845[label="",style="solid", color="black", weight=3];
834[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS Zero (Succ vuz720))",fontsize=16,color="black",shape="box"];834 -> 846[label="",style="solid", color="black", weight=3];
835[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];835 -> 847[label="",style="solid", color="black", weight=3];
490[label="primMinusNatS (Succ vuz22000) Zero",fontsize=16,color="black",shape="triangle"];490 -> 515[label="",style="solid", color="black", weight=3];
491[label="Zero",fontsize=16,color="green",shape="box"];492[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];492 -> 516[label="",style="solid", color="black", weight=3];
493[label="Zero",fontsize=16,color="green",shape="box"];375 -> 266[label="",style="dashed", color="red", weight=0];
375[label="abs vuz24",fontsize=16,color="magenta"];375 -> 394[label="",style="dashed", color="magenta", weight=3];
376 -> 266[label="",style="dashed", color="red", weight=0];
376[label="abs vuz23",fontsize=16,color="magenta"];374[label="gcd0Gcd'0 vuz44 vuz43",fontsize=16,color="black",shape="triangle"];374 -> 395[label="",style="solid", color="black", weight=3];
393[label="vuz24",fontsize=16,color="green",shape="box"];494[label="absReal1 vuz20 (compare vuz20 (fromInt (Pos Zero)) /= LT)",fontsize=16,color="black",shape="box"];494 -> 517[label="",style="solid", color="black", weight=3];
495[label="gcd0Gcd'2 vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="black",shape="box"];495 -> 518[label="",style="solid", color="black", weight=3];
441[label="absReal1 vuz23 (not (compare vuz23 (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];441 -> 451[label="",style="solid", color="black", weight=3];
508[label="vuz47",fontsize=16,color="green",shape="box"];509[label="vuz46",fontsize=16,color="green",shape="box"];844 -> 791[label="",style="dashed", color="red", weight=0];
844[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS vuz710 vuz720)",fontsize=16,color="magenta"];844 -> 854[label="",style="dashed", color="magenta", weight=3];
844 -> 855[label="",style="dashed", color="magenta", weight=3];
845[label="primDivNatS0 (Succ vuz69) (Succ vuz70) True",fontsize=16,color="black",shape="triangle"];845 -> 856[label="",style="solid", color="black", weight=3];
846[label="primDivNatS0 (Succ vuz69) (Succ vuz70) False",fontsize=16,color="black",shape="box"];846 -> 857[label="",style="solid", color="black", weight=3];
847 -> 845[label="",style="dashed", color="red", weight=0];
847[label="primDivNatS0 (Succ vuz69) (Succ vuz70) True",fontsize=16,color="magenta"];515[label="Succ vuz22000",fontsize=16,color="green",shape="box"];516[label="Zero",fontsize=16,color="green",shape="box"];394[label="vuz24",fontsize=16,color="green",shape="box"];395 -> 409[label="",style="dashed", color="red", weight=0];
395[label="gcd0Gcd' vuz43 (vuz44 `rem` vuz43)",fontsize=16,color="magenta"];395 -> 412[label="",style="dashed", color="magenta", weight=3];
395 -> 413[label="",style="dashed", color="magenta", weight=3];
517[label="absReal1 vuz20 (not (compare vuz20 (fromInt (Pos Zero)) == LT))",fontsize=16,color="burlywood",shape="box"];1133[label="vuz20/Integer vuz200",fontsize=10,color="white",style="solid",shape="box"];517 -> 1133[label="",style="solid", color="burlywood", weight=9];
1133 -> 537[label="",style="solid", color="burlywood", weight=3];
518 -> 538[label="",style="dashed", color="red", weight=0];
518[label="gcd0Gcd'1 (vuz51 `rem` vuz50 == fromInt (Pos Zero)) vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="magenta"];518 -> 539[label="",style="dashed", color="magenta", weight=3];
451[label="absReal1 vuz23 (not (primCmpInt vuz23 (fromInt (Pos Zero)) == LT))",fontsize=16,color="burlywood",shape="box"];1134[label="vuz23/Pos vuz230",fontsize=10,color="white",style="solid",shape="box"];451 -> 1134[label="",style="solid", color="burlywood", weight=9];
1134 -> 478[label="",style="solid", color="burlywood", weight=3];
1135[label="vuz23/Neg vuz230",fontsize=10,color="white",style="solid",shape="box"];451 -> 1135[label="",style="solid", color="burlywood", weight=9];
1135 -> 479[label="",style="solid", color="burlywood", weight=3];
854[label="vuz710",fontsize=16,color="green",shape="box"];855[label="vuz720",fontsize=16,color="green",shape="box"];856[label="Succ (primDivNatS (primMinusNatS (Succ vuz69) (Succ vuz70)) (Succ (Succ vuz70)))",fontsize=16,color="green",shape="box"];856 -> 864[label="",style="dashed", color="green", weight=3];
857[label="Zero",fontsize=16,color="green",shape="box"];412[label="vuz44 `rem` vuz43",fontsize=16,color="black",shape="box"];412 -> 442[label="",style="solid", color="black", weight=3];
413[label="vuz43",fontsize=16,color="green",shape="box"];537[label="absReal1 (Integer vuz200) (not (compare (Integer vuz200) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];537 -> 548[label="",style="solid", color="black", weight=3];
539 -> 76[label="",style="dashed", color="red", weight=0];
539[label="vuz51 `rem` vuz50 == fromInt (Pos Zero)",fontsize=16,color="magenta"];539 -> 549[label="",style="dashed", color="magenta", weight=3];
538[label="gcd0Gcd'1 vuz52 vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="burlywood",shape="triangle"];1136[label="vuz52/False",fontsize=10,color="white",style="solid",shape="box"];538 -> 1136[label="",style="solid", color="burlywood", weight=9];
1136 -> 550[label="",style="solid", color="burlywood", weight=3];
1137[label="vuz52/True",fontsize=10,color="white",style="solid",shape="box"];538 -> 1137[label="",style="solid", color="burlywood", weight=9];
1137 -> 551[label="",style="solid", color="burlywood", weight=3];
478[label="absReal1 (Pos vuz230) (not (primCmpInt (Pos vuz230) (fromInt (Pos Zero)) == LT))",fontsize=16,color="burlywood",shape="box"];1138[label="vuz230/Succ vuz2300",fontsize=10,color="white",style="solid",shape="box"];478 -> 1138[label="",style="solid", color="burlywood", weight=9];
1138 -> 496[label="",style="solid", color="burlywood", weight=3];
1139[label="vuz230/Zero",fontsize=10,color="white",style="solid",shape="box"];478 -> 1139[label="",style="solid", color="burlywood", weight=9];
1139 -> 497[label="",style="solid", color="burlywood", weight=3];
479[label="absReal1 (Neg vuz230) (not (primCmpInt (Neg vuz230) (fromInt (Pos Zero)) == LT))",fontsize=16,color="burlywood",shape="box"];1140[label="vuz230/Succ vuz2300",fontsize=10,color="white",style="solid",shape="box"];479 -> 1140[label="",style="solid", color="burlywood", weight=9];
1140 -> 498[label="",style="solid", color="burlywood", weight=3];
1141[label="vuz230/Zero",fontsize=10,color="white",style="solid",shape="box"];479 -> 1141[label="",style="solid", color="burlywood", weight=9];
1141 -> 499[label="",style="solid", color="burlywood", weight=3];
864 -> 377[label="",style="dashed", color="red", weight=0];
864[label="primDivNatS (primMinusNatS (Succ vuz69) (Succ vuz70)) (Succ (Succ vuz70))",fontsize=16,color="magenta"];864 -> 873[label="",style="dashed", color="magenta", weight=3];
864 -> 874[label="",style="dashed", color="magenta", weight=3];
442[label="primRemInt vuz44 vuz43",fontsize=16,color="burlywood",shape="triangle"];1142[label="vuz44/Pos vuz440",fontsize=10,color="white",style="solid",shape="box"];442 -> 1142[label="",style="solid", color="burlywood", weight=9];
1142 -> 452[label="",style="solid", color="burlywood", weight=3];
1143[label="vuz44/Neg vuz440",fontsize=10,color="white",style="solid",shape="box"];442 -> 1143[label="",style="solid", color="burlywood", weight=9];
1143 -> 453[label="",style="solid", color="burlywood", weight=3];
548[label="absReal1 (Integer vuz200) (not (compare (Integer vuz200) (Integer (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];548 -> 567[label="",style="solid", color="black", weight=3];
549[label="vuz51 `rem` vuz50",fontsize=16,color="burlywood",shape="triangle"];1144[label="vuz51/Integer vuz510",fontsize=10,color="white",style="solid",shape="box"];549 -> 1144[label="",style="solid", color="burlywood", weight=9];
1144 -> 568[label="",style="solid", color="burlywood", weight=3];
550[label="gcd0Gcd'1 False vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="black",shape="box"];550 -> 569[label="",style="solid", color="black", weight=3];
551[label="gcd0Gcd'1 True vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="black",shape="box"];551 -> 570[label="",style="solid", color="black", weight=3];
496[label="absReal1 (Pos (Succ vuz2300)) (not (primCmpInt (Pos (Succ vuz2300)) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];496 -> 519[label="",style="solid", color="black", weight=3];
497[label="absReal1 (Pos Zero) (not (primCmpInt (Pos Zero) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];497 -> 520[label="",style="solid", color="black", weight=3];
498[label="absReal1 (Neg (Succ vuz2300)) (not (primCmpInt (Neg (Succ vuz2300)) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];498 -> 521[label="",style="solid", color="black", weight=3];
499[label="absReal1 (Neg Zero) (not (primCmpInt (Neg Zero) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];499 -> 522[label="",style="solid", color="black", weight=3];
873[label="primMinusNatS (Succ vuz69) (Succ vuz70)",fontsize=16,color="black",shape="triangle"];873 -> 881[label="",style="solid", color="black", weight=3];
874[label="Succ vuz70",fontsize=16,color="green",shape="box"];452[label="primRemInt (Pos vuz440) vuz43",fontsize=16,color="burlywood",shape="box"];1145[label="vuz43/Pos vuz430",fontsize=10,color="white",style="solid",shape="box"];452 -> 1145[label="",style="solid", color="burlywood", weight=9];
1145 -> 480[label="",style="solid", color="burlywood", weight=3];
1146[label="vuz43/Neg vuz430",fontsize=10,color="white",style="solid",shape="box"];452 -> 1146[label="",style="solid", color="burlywood", weight=9];
1146 -> 481[label="",style="solid", color="burlywood", weight=3];
453[label="primRemInt (Neg vuz440) vuz43",fontsize=16,color="burlywood",shape="box"];1147[label="vuz43/Pos vuz430",fontsize=10,color="white",style="solid",shape="box"];453 -> 1147[label="",style="solid", color="burlywood", weight=9];
1147 -> 482[label="",style="solid", color="burlywood", weight=3];
1148[label="vuz43/Neg vuz430",fontsize=10,color="white",style="solid",shape="box"];453 -> 1148[label="",style="solid", color="burlywood", weight=9];
1148 -> 483[label="",style="solid", color="burlywood", weight=3];
567[label="absReal1 (Integer vuz200) (not (primCmpInt vuz200 (Pos Zero) == LT))",fontsize=16,color="burlywood",shape="box"];1149[label="vuz200/Pos vuz2000",fontsize=10,color="white",style="solid",shape="box"];567 -> 1149[label="",style="solid", color="burlywood", weight=9];
1149 -> 588[label="",style="solid", color="burlywood", weight=3];
1150[label="vuz200/Neg vuz2000",fontsize=10,color="white",style="solid",shape="box"];567 -> 1150[label="",style="solid", color="burlywood", weight=9];
1150 -> 589[label="",style="solid", color="burlywood", weight=3];
568[label="Integer vuz510 `rem` vuz50",fontsize=16,color="burlywood",shape="box"];1151[label="vuz50/Integer vuz500",fontsize=10,color="white",style="solid",shape="box"];568 -> 1151[label="",style="solid", color="burlywood", weight=9];
1151 -> 590[label="",style="solid", color="burlywood", weight=3];
569 -> 462[label="",style="dashed", color="red", weight=0];
569[label="gcd0Gcd'0 vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="magenta"];569 -> 591[label="",style="dashed", color="magenta", weight=3];
569 -> 592[label="",style="dashed", color="magenta", weight=3];
570[label="vuz50",fontsize=16,color="green",shape="box"];519[label="absReal1 (Pos (Succ vuz2300)) (not (primCmpInt (Pos (Succ vuz2300)) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];519 -> 552[label="",style="solid", color="black", weight=3];
520[label="absReal1 (Pos Zero) (not (primCmpInt (Pos Zero) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];520 -> 553[label="",style="solid", color="black", weight=3];
521[label="absReal1 (Neg (Succ vuz2300)) (not (primCmpInt (Neg (Succ vuz2300)) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];521 -> 554[label="",style="solid", color="black", weight=3];
522[label="absReal1 (Neg Zero) (not (primCmpInt (Neg Zero) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];522 -> 555[label="",style="solid", color="black", weight=3];
881[label="primMinusNatS vuz69 vuz70",fontsize=16,color="burlywood",shape="triangle"];1152[label="vuz69/Succ vuz690",fontsize=10,color="white",style="solid",shape="box"];881 -> 1152[label="",style="solid", color="burlywood", weight=9];
1152 -> 891[label="",style="solid", color="burlywood", weight=3];
1153[label="vuz69/Zero",fontsize=10,color="white",style="solid",shape="box"];881 -> 1153[label="",style="solid", color="burlywood", weight=9];
1153 -> 892[label="",style="solid", color="burlywood", weight=3];
480[label="primRemInt (Pos vuz440) (Pos vuz430)",fontsize=16,color="burlywood",shape="box"];1154[label="vuz430/Succ vuz4300",fontsize=10,color="white",style="solid",shape="box"];480 -> 1154[label="",style="solid", color="burlywood", weight=9];
1154 -> 500[label="",style="solid", color="burlywood", weight=3];
1155[label="vuz430/Zero",fontsize=10,color="white",style="solid",shape="box"];480 -> 1155[label="",style="solid", color="burlywood", weight=9];
1155 -> 501[label="",style="solid", color="burlywood", weight=3];
481[label="primRemInt (Pos vuz440) (Neg vuz430)",fontsize=16,color="burlywood",shape="box"];1156[label="vuz430/Succ vuz4300",fontsize=10,color="white",style="solid",shape="box"];481 -> 1156[label="",style="solid", color="burlywood", weight=9];
1156 -> 502[label="",style="solid", color="burlywood", weight=3];
1157[label="vuz430/Zero",fontsize=10,color="white",style="solid",shape="box"];481 -> 1157[label="",style="solid", color="burlywood", weight=9];
1157 -> 503[label="",style="solid", color="burlywood", weight=3];
482[label="primRemInt (Neg vuz440) (Pos vuz430)",fontsize=16,color="burlywood",shape="box"];1158[label="vuz430/Succ vuz4300",fontsize=10,color="white",style="solid",shape="box"];482 -> 1158[label="",style="solid", color="burlywood", weight=9];
1158 -> 504[label="",style="solid", color="burlywood", weight=3];
1159[label="vuz430/Zero",fontsize=10,color="white",style="solid",shape="box"];482 -> 1159[label="",style="solid", color="burlywood", weight=9];
1159 -> 505[label="",style="solid", color="burlywood", weight=3];
483[label="primRemInt (Neg vuz440) (Neg vuz430)",fontsize=16,color="burlywood",shape="box"];1160[label="vuz430/Succ vuz4300",fontsize=10,color="white",style="solid",shape="box"];483 -> 1160[label="",style="solid", color="burlywood", weight=9];
1160 -> 506[label="",style="solid", color="burlywood", weight=3];
1161[label="vuz430/Zero",fontsize=10,color="white",style="solid",shape="box"];483 -> 1161[label="",style="solid", color="burlywood", weight=9];
1161 -> 507[label="",style="solid", color="burlywood", weight=3];
588[label="absReal1 (Integer (Pos vuz2000)) (not (primCmpInt (Pos vuz2000) (Pos Zero) == LT))",fontsize=16,color="burlywood",shape="box"];1162[label="vuz2000/Succ vuz20000",fontsize=10,color="white",style="solid",shape="box"];588 -> 1162[label="",style="solid", color="burlywood", weight=9];
1162 -> 607[label="",style="solid", color="burlywood", weight=3];
1163[label="vuz2000/Zero",fontsize=10,color="white",style="solid",shape="box"];588 -> 1163[label="",style="solid", color="burlywood", weight=9];
1163 -> 608[label="",style="solid", color="burlywood", weight=3];
589[label="absReal1 (Integer (Neg vuz2000)) (not (primCmpInt (Neg vuz2000) (Pos Zero) == LT))",fontsize=16,color="burlywood",shape="box"];1164[label="vuz2000/Succ vuz20000",fontsize=10,color="white",style="solid",shape="box"];589 -> 1164[label="",style="solid", color="burlywood", weight=9];
1164 -> 609[label="",style="solid", color="burlywood", weight=3];
1165[label="vuz2000/Zero",fontsize=10,color="white",style="solid",shape="box"];589 -> 1165[label="",style="solid", color="burlywood", weight=9];
1165 -> 610[label="",style="solid", color="burlywood", weight=3];
590[label="Integer vuz510 `rem` Integer vuz500",fontsize=16,color="black",shape="box"];590 -> 611[label="",style="solid", color="black", weight=3];
591 -> 549[label="",style="dashed", color="red", weight=0];
591[label="vuz51 `rem` vuz50",fontsize=16,color="magenta"];592[label="vuz50",fontsize=16,color="green",shape="box"];552[label="absReal1 (Pos (Succ vuz2300)) (not (primCmpNat (Succ vuz2300) Zero == LT))",fontsize=16,color="black",shape="box"];552 -> 571[label="",style="solid", color="black", weight=3];
553[label="absReal1 (Pos Zero) (not (EQ == LT))",fontsize=16,color="black",shape="box"];553 -> 572[label="",style="solid", color="black", weight=3];
554[label="absReal1 (Neg (Succ vuz2300)) (not (LT == LT))",fontsize=16,color="black",shape="box"];554 -> 573[label="",style="solid", color="black", weight=3];
555[label="absReal1 (Neg Zero) (not (EQ == LT))",fontsize=16,color="black",shape="box"];555 -> 574[label="",style="solid", color="black", weight=3];
891[label="primMinusNatS (Succ vuz690) vuz70",fontsize=16,color="burlywood",shape="box"];1166[label="vuz70/Succ vuz700",fontsize=10,color="white",style="solid",shape="box"];891 -> 1166[label="",style="solid", color="burlywood", weight=9];
1166 -> 901[label="",style="solid", color="burlywood", weight=3];
1167[label="vuz70/Zero",fontsize=10,color="white",style="solid",shape="box"];891 -> 1167[label="",style="solid", color="burlywood", weight=9];
1167 -> 902[label="",style="solid", color="burlywood", weight=3];
892[label="primMinusNatS Zero vuz70",fontsize=16,color="burlywood",shape="box"];1168[label="vuz70/Succ vuz700",fontsize=10,color="white",style="solid",shape="box"];892 -> 1168[label="",style="solid", color="burlywood", weight=9];
1168 -> 903[label="",style="solid", color="burlywood", weight=3];
1169[label="vuz70/Zero",fontsize=10,color="white",style="solid",shape="box"];892 -> 1169[label="",style="solid", color="burlywood", weight=9];
1169 -> 904[label="",style="solid", color="burlywood", weight=3];
500[label="primRemInt (Pos vuz440) (Pos (Succ vuz4300))",fontsize=16,color="black",shape="box"];500 -> 523[label="",style="solid", color="black", weight=3];
501[label="primRemInt (Pos vuz440) (Pos Zero)",fontsize=16,color="black",shape="box"];501 -> 524[label="",style="solid", color="black", weight=3];
502[label="primRemInt (Pos vuz440) (Neg (Succ vuz4300))",fontsize=16,color="black",shape="box"];502 -> 525[label="",style="solid", color="black", weight=3];
503[label="primRemInt (Pos vuz440) (Neg Zero)",fontsize=16,color="black",shape="box"];503 -> 526[label="",style="solid", color="black", weight=3];
504[label="primRemInt (Neg vuz440) (Pos (Succ vuz4300))",fontsize=16,color="black",shape="box"];504 -> 527[label="",style="solid", color="black", weight=3];
505[label="primRemInt (Neg vuz440) (Pos Zero)",fontsize=16,color="black",shape="box"];505 -> 528[label="",style="solid", color="black", weight=3];
506[label="primRemInt (Neg vuz440) (Neg (Succ vuz4300))",fontsize=16,color="black",shape="box"];506 -> 529[label="",style="solid", color="black", weight=3];
507[label="primRemInt (Neg vuz440) (Neg Zero)",fontsize=16,color="black",shape="box"];507 -> 530[label="",style="solid", color="black", weight=3];
607[label="absReal1 (Integer (Pos (Succ vuz20000))) (not (primCmpInt (Pos (Succ vuz20000)) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];607 -> 625[label="",style="solid", color="black", weight=3];
608[label="absReal1 (Integer (Pos Zero)) (not (primCmpInt (Pos Zero) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];608 -> 626[label="",style="solid", color="black", weight=3];
609[label="absReal1 (Integer (Neg (Succ vuz20000))) (not (primCmpInt (Neg (Succ vuz20000)) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];609 -> 627[label="",style="solid", color="black", weight=3];
610[label="absReal1 (Integer (Neg Zero)) (not (primCmpInt (Neg Zero) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];610 -> 628[label="",style="solid", color="black", weight=3];
611[label="Integer (primRemInt vuz510 vuz500)",fontsize=16,color="green",shape="box"];611 -> 629[label="",style="dashed", color="green", weight=3];
571[label="absReal1 (Pos (Succ vuz2300)) (not (GT == LT))",fontsize=16,color="black",shape="box"];571 -> 593[label="",style="solid", color="black", weight=3];
572[label="absReal1 (Pos Zero) (not False)",fontsize=16,color="black",shape="box"];572 -> 594[label="",style="solid", color="black", weight=3];
573[label="absReal1 (Neg (Succ vuz2300)) (not True)",fontsize=16,color="black",shape="box"];573 -> 595[label="",style="solid", color="black", weight=3];
574[label="absReal1 (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];574 -> 596[label="",style="solid", color="black", weight=3];
901[label="primMinusNatS (Succ vuz690) (Succ vuz700)",fontsize=16,color="black",shape="box"];901 -> 913[label="",style="solid", color="black", weight=3];
902[label="primMinusNatS (Succ vuz690) Zero",fontsize=16,color="black",shape="box"];902 -> 914[label="",style="solid", color="black", weight=3];
903[label="primMinusNatS Zero (Succ vuz700)",fontsize=16,color="black",shape="box"];903 -> 915[label="",style="solid", color="black", weight=3];
904[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];904 -> 916[label="",style="solid", color="black", weight=3];
523[label="Pos (primModNatS vuz440 (Succ vuz4300))",fontsize=16,color="green",shape="box"];523 -> 556[label="",style="dashed", color="green", weight=3];
524 -> 356[label="",style="dashed", color="red", weight=0];
524[label="error []",fontsize=16,color="magenta"];525[label="Pos (primModNatS vuz440 (Succ vuz4300))",fontsize=16,color="green",shape="box"];525 -> 557[label="",style="dashed", color="green", weight=3];
526 -> 356[label="",style="dashed", color="red", weight=0];
526[label="error []",fontsize=16,color="magenta"];527[label="Neg (primModNatS vuz440 (Succ vuz4300))",fontsize=16,color="green",shape="box"];527 -> 558[label="",style="dashed", color="green", weight=3];
528 -> 356[label="",style="dashed", color="red", weight=0];
528[label="error []",fontsize=16,color="magenta"];529[label="Neg (primModNatS vuz440 (Succ vuz4300))",fontsize=16,color="green",shape="box"];529 -> 559[label="",style="dashed", color="green", weight=3];
530 -> 356[label="",style="dashed", color="red", weight=0];
530[label="error []",fontsize=16,color="magenta"];625[label="absReal1 (Integer (Pos (Succ vuz20000))) (not (primCmpNat (Succ vuz20000) Zero == LT))",fontsize=16,color="black",shape="box"];625 -> 643[label="",style="solid", color="black", weight=3];
626[label="absReal1 (Integer (Pos Zero)) (not (EQ == LT))",fontsize=16,color="black",shape="box"];626 -> 644[label="",style="solid", color="black", weight=3];
627[label="absReal1 (Integer (Neg (Succ vuz20000))) (not (LT == LT))",fontsize=16,color="black",shape="box"];627 -> 645[label="",style="solid", color="black", weight=3];
628[label="absReal1 (Integer (Neg Zero)) (not (EQ == LT))",fontsize=16,color="black",shape="box"];628 -> 646[label="",style="solid", color="black", weight=3];
629 -> 442[label="",style="dashed", color="red", weight=0];
629[label="primRemInt vuz510 vuz500",fontsize=16,color="magenta"];629 -> 647[label="",style="dashed", color="magenta", weight=3];
629 -> 648[label="",style="dashed", color="magenta", weight=3];
593[label="absReal1 (Pos (Succ vuz2300)) (not False)",fontsize=16,color="black",shape="box"];593 -> 612[label="",style="solid", color="black", weight=3];
594[label="absReal1 (Pos Zero) True",fontsize=16,color="black",shape="box"];594 -> 613[label="",style="solid", color="black", weight=3];
595[label="absReal1 (Neg (Succ vuz2300)) False",fontsize=16,color="black",shape="box"];595 -> 614[label="",style="solid", color="black", weight=3];
596[label="absReal1 (Neg Zero) True",fontsize=16,color="black",shape="box"];596 -> 615[label="",style="solid", color="black", weight=3];
913 -> 881[label="",style="dashed", color="red", weight=0];
913[label="primMinusNatS vuz690 vuz700",fontsize=16,color="magenta"];913 -> 921[label="",style="dashed", color="magenta", weight=3];
913 -> 922[label="",style="dashed", color="magenta", weight=3];
914[label="Succ vuz690",fontsize=16,color="green",shape="box"];915[label="Zero",fontsize=16,color="green",shape="box"];916[label="Zero",fontsize=16,color="green",shape="box"];556[label="primModNatS vuz440 (Succ vuz4300)",fontsize=16,color="burlywood",shape="triangle"];1170[label="vuz440/Succ vuz4400",fontsize=10,color="white",style="solid",shape="box"];556 -> 1170[label="",style="solid", color="burlywood", weight=9];
1170 -> 575[label="",style="solid", color="burlywood", weight=3];
1171[label="vuz440/Zero",fontsize=10,color="white",style="solid",shape="box"];556 -> 1171[label="",style="solid", color="burlywood", weight=9];
1171 -> 576[label="",style="solid", color="burlywood", weight=3];
557 -> 556[label="",style="dashed", color="red", weight=0];
557[label="primModNatS vuz440 (Succ vuz4300)",fontsize=16,color="magenta"];557 -> 577[label="",style="dashed", color="magenta", weight=3];
558 -> 556[label="",style="dashed", color="red", weight=0];
558[label="primModNatS vuz440 (Succ vuz4300)",fontsize=16,color="magenta"];558 -> 578[label="",style="dashed", color="magenta", weight=3];
559 -> 556[label="",style="dashed", color="red", weight=0];
559[label="primModNatS vuz440 (Succ vuz4300)",fontsize=16,color="magenta"];559 -> 579[label="",style="dashed", color="magenta", weight=3];
559 -> 580[label="",style="dashed", color="magenta", weight=3];
643[label="absReal1 (Integer (Pos (Succ vuz20000))) (not (GT == LT))",fontsize=16,color="black",shape="box"];643 -> 662[label="",style="solid", color="black", weight=3];
644[label="absReal1 (Integer (Pos Zero)) (not False)",fontsize=16,color="black",shape="box"];644 -> 663[label="",style="solid", color="black", weight=3];
645[label="absReal1 (Integer (Neg (Succ vuz20000))) (not True)",fontsize=16,color="black",shape="box"];645 -> 664[label="",style="solid", color="black", weight=3];
646[label="absReal1 (Integer (Neg Zero)) (not False)",fontsize=16,color="black",shape="box"];646 -> 665[label="",style="solid", color="black", weight=3];
647[label="vuz510",fontsize=16,color="green",shape="box"];648[label="vuz500",fontsize=16,color="green",shape="box"];612[label="absReal1 (Pos (Succ vuz2300)) True",fontsize=16,color="black",shape="box"];612 -> 630[label="",style="solid", color="black", weight=3];
613[label="Pos Zero",fontsize=16,color="green",shape="box"];614[label="absReal0 (Neg (Succ vuz2300)) otherwise",fontsize=16,color="black",shape="box"];614 -> 631[label="",style="solid", color="black", weight=3];
615[label="Neg Zero",fontsize=16,color="green",shape="box"];921[label="vuz700",fontsize=16,color="green",shape="box"];922[label="vuz690",fontsize=16,color="green",shape="box"];575[label="primModNatS (Succ vuz4400) (Succ vuz4300)",fontsize=16,color="black",shape="box"];575 -> 597[label="",style="solid", color="black", weight=3];
576[label="primModNatS Zero (Succ vuz4300)",fontsize=16,color="black",shape="box"];576 -> 598[label="",style="solid", color="black", weight=3];
577[label="vuz4300",fontsize=16,color="green",shape="box"];578[label="vuz440",fontsize=16,color="green",shape="box"];579[label="vuz4300",fontsize=16,color="green",shape="box"];580[label="vuz440",fontsize=16,color="green",shape="box"];662[label="absReal1 (Integer (Pos (Succ vuz20000))) (not False)",fontsize=16,color="black",shape="box"];662 -> 679[label="",style="solid", color="black", weight=3];
663[label="absReal1 (Integer (Pos Zero)) True",fontsize=16,color="black",shape="box"];663 -> 680[label="",style="solid", color="black", weight=3];
664[label="absReal1 (Integer (Neg (Succ vuz20000))) False",fontsize=16,color="black",shape="box"];664 -> 681[label="",style="solid", color="black", weight=3];
665[label="absReal1 (Integer (Neg Zero)) True",fontsize=16,color="black",shape="box"];665 -> 682[label="",style="solid", color="black", weight=3];
630[label="Pos (Succ vuz2300)",fontsize=16,color="green",shape="box"];631[label="absReal0 (Neg (Succ vuz2300)) True",fontsize=16,color="black",shape="box"];631 -> 649[label="",style="solid", color="black", weight=3];
597[label="primModNatS0 vuz4400 vuz4300 (primGEqNatS vuz4400 vuz4300)",fontsize=16,color="burlywood",shape="box"];1172[label="vuz4400/Succ vuz44000",fontsize=10,color="white",style="solid",shape="box"];597 -> 1172[label="",style="solid", color="burlywood", weight=9];
1172 -> 616[label="",style="solid", color="burlywood", weight=3];
1173[label="vuz4400/Zero",fontsize=10,color="white",style="solid",shape="box"];597 -> 1173[label="",style="solid", color="burlywood", weight=9];
1173 -> 617[label="",style="solid", color="burlywood", weight=3];
598[label="Zero",fontsize=16,color="green",shape="box"];679[label="absReal1 (Integer (Pos (Succ vuz20000))) True",fontsize=16,color="black",shape="box"];679 -> 699[label="",style="solid", color="black", weight=3];
680[label="Integer (Pos Zero)",fontsize=16,color="green",shape="box"];681[label="absReal0 (Integer (Neg (Succ vuz20000))) otherwise",fontsize=16,color="black",shape="box"];681 -> 700[label="",style="solid", color="black", weight=3];
682[label="Integer (Neg Zero)",fontsize=16,color="green",shape="box"];649[label="`negate` Neg (Succ vuz2300)",fontsize=16,color="black",shape="box"];649 -> 666[label="",style="solid", color="black", weight=3];
616[label="primModNatS0 (Succ vuz44000) vuz4300 (primGEqNatS (Succ vuz44000) vuz4300)",fontsize=16,color="burlywood",shape="box"];1174[label="vuz4300/Succ vuz43000",fontsize=10,color="white",style="solid",shape="box"];616 -> 1174[label="",style="solid", color="burlywood", weight=9];
1174 -> 632[label="",style="solid", color="burlywood", weight=3];
1175[label="vuz4300/Zero",fontsize=10,color="white",style="solid",shape="box"];616 -> 1175[label="",style="solid", color="burlywood", weight=9];
1175 -> 633[label="",style="solid", color="burlywood", weight=3];
617[label="primModNatS0 Zero vuz4300 (primGEqNatS Zero vuz4300)",fontsize=16,color="burlywood",shape="box"];1176[label="vuz4300/Succ vuz43000",fontsize=10,color="white",style="solid",shape="box"];617 -> 1176[label="",style="solid", color="burlywood", weight=9];
1176 -> 634[label="",style="solid", color="burlywood", weight=3];
1177[label="vuz4300/Zero",fontsize=10,color="white",style="solid",shape="box"];617 -> 1177[label="",style="solid", color="burlywood", weight=9];
1177 -> 635[label="",style="solid", color="burlywood", weight=3];
699[label="Integer (Pos (Succ vuz20000))",fontsize=16,color="green",shape="box"];700[label="absReal0 (Integer (Neg (Succ vuz20000))) True",fontsize=16,color="black",shape="box"];700 -> 714[label="",style="solid", color="black", weight=3];
666[label="primNegInt (Neg (Succ vuz2300))",fontsize=16,color="black",shape="triangle"];666 -> 683[label="",style="solid", color="black", weight=3];
632[label="primModNatS0 (Succ vuz44000) (Succ vuz43000) (primGEqNatS (Succ vuz44000) (Succ vuz43000))",fontsize=16,color="black",shape="box"];632 -> 650[label="",style="solid", color="black", weight=3];
633[label="primModNatS0 (Succ vuz44000) Zero (primGEqNatS (Succ vuz44000) Zero)",fontsize=16,color="black",shape="box"];633 -> 651[label="",style="solid", color="black", weight=3];
634[label="primModNatS0 Zero (Succ vuz43000) (primGEqNatS Zero (Succ vuz43000))",fontsize=16,color="black",shape="box"];634 -> 652[label="",style="solid", color="black", weight=3];
635[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];635 -> 653[label="",style="solid", color="black", weight=3];
714[label="`negate` Integer (Neg (Succ vuz20000))",fontsize=16,color="black",shape="box"];714 -> 734[label="",style="solid", color="black", weight=3];
683[label="Pos (Succ vuz2300)",fontsize=16,color="green",shape="box"];650 -> 972[label="",style="dashed", color="red", weight=0];
650[label="primModNatS0 (Succ vuz44000) (Succ vuz43000) (primGEqNatS vuz44000 vuz43000)",fontsize=16,color="magenta"];650 -> 973[label="",style="dashed", color="magenta", weight=3];
650 -> 974[label="",style="dashed", color="magenta", weight=3];
650 -> 975[label="",style="dashed", color="magenta", weight=3];
650 -> 976[label="",style="dashed", color="magenta", weight=3];
651[label="primModNatS0 (Succ vuz44000) Zero True",fontsize=16,color="black",shape="box"];651 -> 669[label="",style="solid", color="black", weight=3];
652[label="primModNatS0 Zero (Succ vuz43000) False",fontsize=16,color="black",shape="box"];652 -> 670[label="",style="solid", color="black", weight=3];
653[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];653 -> 671[label="",style="solid", color="black", weight=3];
734[label="Integer (primNegInt (Neg (Succ vuz20000)))",fontsize=16,color="green",shape="box"];734 -> 764[label="",style="dashed", color="green", weight=3];
973[label="vuz44000",fontsize=16,color="green",shape="box"];974[label="vuz43000",fontsize=16,color="green",shape="box"];975[label="vuz44000",fontsize=16,color="green",shape="box"];976[label="vuz43000",fontsize=16,color="green",shape="box"];972[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS vuz91 vuz92)",fontsize=16,color="burlywood",shape="triangle"];1178[label="vuz91/Succ vuz910",fontsize=10,color="white",style="solid",shape="box"];972 -> 1178[label="",style="solid", color="burlywood", weight=9];
1178 -> 1005[label="",style="solid", color="burlywood", weight=3];
1179[label="vuz91/Zero",fontsize=10,color="white",style="solid",shape="box"];972 -> 1179[label="",style="solid", color="burlywood", weight=9];
1179 -> 1006[label="",style="solid", color="burlywood", weight=3];
669 -> 556[label="",style="dashed", color="red", weight=0];
669[label="primModNatS (primMinusNatS (Succ vuz44000) Zero) (Succ Zero)",fontsize=16,color="magenta"];669 -> 688[label="",style="dashed", color="magenta", weight=3];
669 -> 689[label="",style="dashed", color="magenta", weight=3];
670[label="Succ Zero",fontsize=16,color="green",shape="box"];671 -> 556[label="",style="dashed", color="red", weight=0];
671[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];671 -> 690[label="",style="dashed", color="magenta", weight=3];
671 -> 691[label="",style="dashed", color="magenta", weight=3];
764 -> 666[label="",style="dashed", color="red", weight=0];
764[label="primNegInt (Neg (Succ vuz20000))",fontsize=16,color="magenta"];764 -> 778[label="",style="dashed", color="magenta", weight=3];
1005[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS (Succ vuz910) vuz92)",fontsize=16,color="burlywood",shape="box"];1180[label="vuz92/Succ vuz920",fontsize=10,color="white",style="solid",shape="box"];1005 -> 1180[label="",style="solid", color="burlywood", weight=9];
1180 -> 1007[label="",style="solid", color="burlywood", weight=3];
1181[label="vuz92/Zero",fontsize=10,color="white",style="solid",shape="box"];1005 -> 1181[label="",style="solid", color="burlywood", weight=9];
1181 -> 1008[label="",style="solid", color="burlywood", weight=3];
1006[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS Zero vuz92)",fontsize=16,color="burlywood",shape="box"];1182[label="vuz92/Succ vuz920",fontsize=10,color="white",style="solid",shape="box"];1006 -> 1182[label="",style="solid", color="burlywood", weight=9];
1182 -> 1009[label="",style="solid", color="burlywood", weight=3];
1183[label="vuz92/Zero",fontsize=10,color="white",style="solid",shape="box"];1006 -> 1183[label="",style="solid", color="burlywood", weight=9];
1183 -> 1010[label="",style="solid", color="burlywood", weight=3];
688[label="Zero",fontsize=16,color="green",shape="box"];689 -> 490[label="",style="dashed", color="red", weight=0];
689[label="primMinusNatS (Succ vuz44000) Zero",fontsize=16,color="magenta"];689 -> 705[label="",style="dashed", color="magenta", weight=3];
690[label="Zero",fontsize=16,color="green",shape="box"];691 -> 492[label="",style="dashed", color="red", weight=0];
691[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];778[label="vuz20000",fontsize=16,color="green",shape="box"];1007[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS (Succ vuz910) (Succ vuz920))",fontsize=16,color="black",shape="box"];1007 -> 1011[label="",style="solid", color="black", weight=3];
1008[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS (Succ vuz910) Zero)",fontsize=16,color="black",shape="box"];1008 -> 1012[label="",style="solid", color="black", weight=3];
1009[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS Zero (Succ vuz920))",fontsize=16,color="black",shape="box"];1009 -> 1013[label="",style="solid", color="black", weight=3];
1010[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];1010 -> 1014[label="",style="solid", color="black", weight=3];
705[label="vuz44000",fontsize=16,color="green",shape="box"];1011 -> 972[label="",style="dashed", color="red", weight=0];
1011[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS vuz910 vuz920)",fontsize=16,color="magenta"];1011 -> 1015[label="",style="dashed", color="magenta", weight=3];
1011 -> 1016[label="",style="dashed", color="magenta", weight=3];
1012[label="primModNatS0 (Succ vuz89) (Succ vuz90) True",fontsize=16,color="black",shape="triangle"];1012 -> 1017[label="",style="solid", color="black", weight=3];
1013[label="primModNatS0 (Succ vuz89) (Succ vuz90) False",fontsize=16,color="black",shape="box"];1013 -> 1018[label="",style="solid", color="black", weight=3];
1014 -> 1012[label="",style="dashed", color="red", weight=0];
1014[label="primModNatS0 (Succ vuz89) (Succ vuz90) True",fontsize=16,color="magenta"];1015[label="vuz910",fontsize=16,color="green",shape="box"];1016[label="vuz920",fontsize=16,color="green",shape="box"];1017 -> 556[label="",style="dashed", color="red", weight=0];
1017[label="primModNatS (primMinusNatS (Succ vuz89) (Succ vuz90)) (Succ (Succ vuz90))",fontsize=16,color="magenta"];1017 -> 1019[label="",style="dashed", color="magenta", weight=3];
1017 -> 1020[label="",style="dashed", color="magenta", weight=3];
1018[label="Succ (Succ vuz89)",fontsize=16,color="green",shape="box"];1019[label="Succ vuz90",fontsize=16,color="green",shape="box"];1020 -> 881[label="",style="dashed", color="red", weight=0];
1020[label="primMinusNatS (Succ vuz89) (Succ vuz90)",fontsize=16,color="magenta"];1020 -> 1021[label="",style="dashed", color="magenta", weight=3];
1020 -> 1022[label="",style="dashed", color="magenta", weight=3];
1021[label="Succ vuz90",fontsize=16,color="green",shape="box"];1022[label="Succ vuz89",fontsize=16,color="green",shape="box"];}

----------------------------------------

(12)
Complex Obligation (AND)

----------------------------------------

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS(Succ(Succ(vuz44000)), Zero) -> new_primModNatS(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS0(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS(Succ(Zero), Zero) -> new_primModNatS(new_primMinusNatS1, Zero)
   new_primModNatS0(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS0(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS0(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS0(vuz89, vuz90, Zero, Zero) -> new_primModNatS00(vuz89, vuz90)
   new_primModNatS00(vuz89, vuz90) -> new_primModNatS(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))

The TRS R consists of the following rules:

   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primMinusNatS2(Zero, Zero)
   new_primMinusNatS1
   new_primMinusNatS0(x0)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(14) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
----------------------------------------

(15)
Complex Obligation (AND)

----------------------------------------

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS0(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS0(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS0(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS0(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS0(vuz89, vuz90, Zero, Zero) -> new_primModNatS00(vuz89, vuz90)
   new_primModNatS00(vuz89, vuz90) -> new_primModNatS(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))

The TRS R consists of the following rules:

   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primMinusNatS2(Zero, Zero)
   new_primMinusNatS1
   new_primMinusNatS0(x0)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(17) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   new_primModNatS(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS0(vuz44000, vuz43000, vuz44000, vuz43000)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(Succ(x_1)) = 1 + x_1
   POL(Zero) = 1
   POL(new_primMinusNatS2(x_1, x_2)) = x_1
   POL(new_primModNatS(x_1, x_2)) = x_1
   POL(new_primModNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1
   POL(new_primModNatS00(x_1, x_2)) = 1 + x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero


----------------------------------------

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS0(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS0(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS0(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS0(vuz89, vuz90, Zero, Zero) -> new_primModNatS00(vuz89, vuz90)
   new_primModNatS00(vuz89, vuz90) -> new_primModNatS(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))

The TRS R consists of the following rules:

   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primMinusNatS2(Zero, Zero)
   new_primMinusNatS1
   new_primMinusNatS0(x0)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(19) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
----------------------------------------

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS0(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS0(vuz89, vuz90, vuz910, vuz920)

The TRS R consists of the following rules:

   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primMinusNatS2(Zero, Zero)
   new_primMinusNatS1
   new_primMinusNatS0(x0)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(21) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_primModNatS0(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS0(vuz89, vuz90, vuz910, vuz920)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


----------------------------------------

(22)
YES

----------------------------------------

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS(Succ(Succ(vuz44000)), Zero) -> new_primModNatS(new_primMinusNatS0(vuz44000), Zero)

The TRS R consists of the following rules:

   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primMinusNatS2(Zero, Zero)
   new_primMinusNatS1
   new_primMinusNatS0(x0)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(24) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented dependency pairs:

   new_primModNatS(Succ(Succ(vuz44000)), Zero) -> new_primModNatS(new_primMinusNatS0(vuz44000), Zero)

Strictly oriented rules of the TRS R:

   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero

Used ordering: Polynomial interpretation [POLO]:

   POL(Succ(x_1)) = 1 + x_1
   POL(Zero) = 2
   POL(new_primMinusNatS0(x_1)) = 1 + x_1
   POL(new_primMinusNatS1) = 2
   POL(new_primMinusNatS2(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(new_primModNatS(x_1, x_2)) = x_1 + x_2


----------------------------------------

(25)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primMinusNatS2(Zero, Zero)
   new_primMinusNatS1
   new_primMinusNatS0(x0)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(26) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(27)
YES

----------------------------------------

(28)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(vuz51, vuz50) -> new_gcd0Gcd'10(new_esEs0(new_rem(vuz51, vuz50)), vuz50, vuz51)
   new_gcd0Gcd'10(False, vuz50, vuz51) -> new_gcd0Gcd'00(vuz50, new_rem(vuz51, vuz50))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_rem(Integer(vuz510), Integer(vuz500)) -> Integer(new_primRemInt(vuz510, vuz500))
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_error -> error([])
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_esEs0(Integer(vuz160)) -> new_primEqInt(vuz160)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_rem(Integer(x0), Integer(x1))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_esEs0(Integer(x0))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(29) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
----------------------------------------

(30)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(vuz51, vuz50) -> new_gcd0Gcd'10(new_esEs0(new_rem(vuz51, vuz50)), vuz50, vuz51)
   new_gcd0Gcd'10(False, vuz50, vuz51) -> new_gcd0Gcd'00(vuz50, new_rem(vuz51, vuz50))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_rem(Integer(vuz510), Integer(vuz500)) -> Integer(new_primRemInt(vuz510, vuz500))
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_error -> error([])
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_esEs0(Integer(vuz160)) -> new_primEqInt(vuz160)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)

Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(31) InductionCalculusProof (EQUIVALENT)
Note that final constraints are written in bold face.



For Pair new_gcd0Gcd'00(vuz51, vuz50) -> new_gcd0Gcd'10(new_esEs0(new_rem(vuz51, vuz50)), vuz50, vuz51) the following chains were created:
*We consider the chain new_gcd0Gcd'00(x2, x3) -> new_gcd0Gcd'10(new_esEs0(new_rem(x2, x3)), x3, x2), new_gcd0Gcd'10(False, x4, x5) -> new_gcd0Gcd'00(x4, new_rem(x5, x4)) which results in the following constraint:

(1)    (new_gcd0Gcd'10(new_esEs0(new_rem(x2, x3)), x3, x2)=new_gcd0Gcd'10(False, x4, x5)  ==>  new_gcd0Gcd'00(x2, x3)_>=_new_gcd0Gcd'10(new_esEs0(new_rem(x2, x3)), x3, x2))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (new_rem(x2, x3)=x12 & new_esEs0(x12)=False  ==>  new_gcd0Gcd'00(x2, x3)_>=_new_gcd0Gcd'10(new_esEs0(new_rem(x2, x3)), x3, x2))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_esEs0(x12)=False which results in the following new constraint:

(3)    (new_primEqInt(x13)=False & new_rem(x2, x3)=Integer(x13)  ==>  new_gcd0Gcd'00(x2, x3)_>=_new_gcd0Gcd'10(new_esEs0(new_rem(x2, x3)), x3, x2))



We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primEqInt(x13)=False which results in the following new constraints:

(4)    (False=False & new_rem(x2, x3)=Integer(Pos(Succ(x14)))  ==>  new_gcd0Gcd'00(x2, x3)_>=_new_gcd0Gcd'10(new_esEs0(new_rem(x2, x3)), x3, x2))

(5)    (False=False & new_rem(x2, x3)=Integer(Neg(Succ(x15)))  ==>  new_gcd0Gcd'00(x2, x3)_>=_new_gcd0Gcd'10(new_esEs0(new_rem(x2, x3)), x3, x2))



We simplified constraint (4) using rules (I), (II) which results in the following new constraint:

(6)    (new_rem(x2, x3)=Integer(Pos(Succ(x14)))  ==>  new_gcd0Gcd'00(x2, x3)_>=_new_gcd0Gcd'10(new_esEs0(new_rem(x2, x3)), x3, x2))



We simplified constraint (5) using rules (I), (II) which results in the following new constraint:

(7)    (new_rem(x2, x3)=Integer(Neg(Succ(x15)))  ==>  new_gcd0Gcd'00(x2, x3)_>=_new_gcd0Gcd'10(new_esEs0(new_rem(x2, x3)), x3, x2))



We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_rem(x2, x3)=Integer(Pos(Succ(x14))) which results in the following new constraint:

(8)    (Integer(new_primRemInt(x17, x16))=Integer(Pos(Succ(x14)))  ==>  new_gcd0Gcd'00(Integer(x17), Integer(x16))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(x17), Integer(x16))), Integer(x16), Integer(x17)))



We simplified constraint (8) using rules (I), (II) which results in the following new constraint:

(9)    (new_primRemInt(x17, x16)=Pos(Succ(x14))  ==>  new_gcd0Gcd'00(Integer(x17), Integer(x16))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(x17), Integer(x16))), Integer(x16), Integer(x17)))



We simplified constraint (9) using rule (V) (with possible (I) afterwards) using induction on new_primRemInt(x17, x16)=Pos(Succ(x14)) which results in the following new constraints:

(10)    (Pos(new_primModNatS1(x19, x18))=Pos(Succ(x14))  ==>  new_gcd0Gcd'00(Integer(Pos(x19)), Integer(Neg(Succ(x18))))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x19)), Integer(Neg(Succ(x18))))), Integer(Neg(Succ(x18))), Integer(Pos(x19))))

(11)    (Pos(new_primModNatS1(x21, x20))=Pos(Succ(x14))  ==>  new_gcd0Gcd'00(Integer(Pos(x21)), Integer(Pos(Succ(x20))))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x21)), Integer(Pos(Succ(x20))))), Integer(Pos(Succ(x20))), Integer(Pos(x21))))

(12)    (new_error=Pos(Succ(x14))  ==>  new_gcd0Gcd'00(Integer(Neg(x22)), Integer(Neg(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x22)), Integer(Neg(Zero)))), Integer(Neg(Zero)), Integer(Neg(x22))))

(13)    (new_error=Pos(Succ(x14))  ==>  new_gcd0Gcd'00(Integer(Pos(x25)), Integer(Pos(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x25)), Integer(Pos(Zero)))), Integer(Pos(Zero)), Integer(Pos(x25))))

(14)    (new_error=Pos(Succ(x14))  ==>  new_gcd0Gcd'00(Integer(Pos(x28)), Integer(Neg(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x28)), Integer(Neg(Zero)))), Integer(Neg(Zero)), Integer(Pos(x28))))

(15)    (new_error=Pos(Succ(x14))  ==>  new_gcd0Gcd'00(Integer(Neg(x29)), Integer(Pos(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x29)), Integer(Pos(Zero)))), Integer(Pos(Zero)), Integer(Neg(x29))))



We simplified constraint (10) using rules (I), (II), (IV) which results in the following new constraint:

(16)    (new_gcd0Gcd'00(Integer(Pos(x19)), Integer(Neg(Succ(x18))))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x19)), Integer(Neg(Succ(x18))))), Integer(Neg(Succ(x18))), Integer(Pos(x19))))



We simplified constraint (11) using rules (I), (II), (IV) which results in the following new constraint:

(17)    (new_gcd0Gcd'00(Integer(Pos(x21)), Integer(Pos(Succ(x20))))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x21)), Integer(Pos(Succ(x20))))), Integer(Pos(Succ(x20))), Integer(Pos(x21))))



We simplified constraint (12) using rule (IV) which results in the following new constraint:

(18)    (new_gcd0Gcd'00(Integer(Neg(x22)), Integer(Neg(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x22)), Integer(Neg(Zero)))), Integer(Neg(Zero)), Integer(Neg(x22))))



We simplified constraint (13) using rule (IV) which results in the following new constraint:

(19)    (new_gcd0Gcd'00(Integer(Pos(x25)), Integer(Pos(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x25)), Integer(Pos(Zero)))), Integer(Pos(Zero)), Integer(Pos(x25))))



We simplified constraint (14) using rule (IV) which results in the following new constraint:

(20)    (new_gcd0Gcd'00(Integer(Pos(x28)), Integer(Neg(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x28)), Integer(Neg(Zero)))), Integer(Neg(Zero)), Integer(Pos(x28))))



We simplified constraint (15) using rule (IV) which results in the following new constraint:

(21)    (new_gcd0Gcd'00(Integer(Neg(x29)), Integer(Pos(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x29)), Integer(Pos(Zero)))), Integer(Pos(Zero)), Integer(Neg(x29))))



We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_rem(x2, x3)=Integer(Neg(Succ(x15))) which results in the following new constraint:

(22)    (Integer(new_primRemInt(x31, x30))=Integer(Neg(Succ(x15)))  ==>  new_gcd0Gcd'00(Integer(x31), Integer(x30))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(x31), Integer(x30))), Integer(x30), Integer(x31)))



We simplified constraint (22) using rules (I), (II) which results in the following new constraint:

(23)    (new_primRemInt(x31, x30)=Neg(Succ(x15))  ==>  new_gcd0Gcd'00(Integer(x31), Integer(x30))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(x31), Integer(x30))), Integer(x30), Integer(x31)))



We simplified constraint (23) using rule (V) (with possible (I) afterwards) using induction on new_primRemInt(x31, x30)=Neg(Succ(x15)) which results in the following new constraints:

(24)    (new_error=Neg(Succ(x15))  ==>  new_gcd0Gcd'00(Integer(Neg(x36)), Integer(Neg(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x36)), Integer(Neg(Zero)))), Integer(Neg(Zero)), Integer(Neg(x36))))

(25)    (Neg(new_primModNatS1(x38, x37))=Neg(Succ(x15))  ==>  new_gcd0Gcd'00(Integer(Neg(x38)), Integer(Pos(Succ(x37))))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x38)), Integer(Pos(Succ(x37))))), Integer(Pos(Succ(x37))), Integer(Neg(x38))))

(26)    (new_error=Neg(Succ(x15))  ==>  new_gcd0Gcd'00(Integer(Pos(x39)), Integer(Pos(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x39)), Integer(Pos(Zero)))), Integer(Pos(Zero)), Integer(Pos(x39))))

(27)    (Neg(new_primModNatS1(x41, x40))=Neg(Succ(x15))  ==>  new_gcd0Gcd'00(Integer(Neg(x41)), Integer(Neg(Succ(x40))))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x41)), Integer(Neg(Succ(x40))))), Integer(Neg(Succ(x40))), Integer(Neg(x41))))

(28)    (new_error=Neg(Succ(x15))  ==>  new_gcd0Gcd'00(Integer(Pos(x42)), Integer(Neg(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x42)), Integer(Neg(Zero)))), Integer(Neg(Zero)), Integer(Pos(x42))))

(29)    (new_error=Neg(Succ(x15))  ==>  new_gcd0Gcd'00(Integer(Neg(x43)), Integer(Pos(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x43)), Integer(Pos(Zero)))), Integer(Pos(Zero)), Integer(Neg(x43))))



We simplified constraint (24) using rule (IV) which results in the following new constraint:

(30)    (new_gcd0Gcd'00(Integer(Neg(x36)), Integer(Neg(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x36)), Integer(Neg(Zero)))), Integer(Neg(Zero)), Integer(Neg(x36))))



We simplified constraint (25) using rules (I), (II), (IV) which results in the following new constraint:

(31)    (new_gcd0Gcd'00(Integer(Neg(x38)), Integer(Pos(Succ(x37))))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x38)), Integer(Pos(Succ(x37))))), Integer(Pos(Succ(x37))), Integer(Neg(x38))))



We simplified constraint (26) using rule (IV) which results in the following new constraint:

(32)    (new_gcd0Gcd'00(Integer(Pos(x39)), Integer(Pos(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x39)), Integer(Pos(Zero)))), Integer(Pos(Zero)), Integer(Pos(x39))))



We simplified constraint (27) using rules (I), (II), (IV) which results in the following new constraint:

(33)    (new_gcd0Gcd'00(Integer(Neg(x41)), Integer(Neg(Succ(x40))))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x41)), Integer(Neg(Succ(x40))))), Integer(Neg(Succ(x40))), Integer(Neg(x41))))



We simplified constraint (28) using rule (IV) which results in the following new constraint:

(34)    (new_gcd0Gcd'00(Integer(Pos(x42)), Integer(Neg(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x42)), Integer(Neg(Zero)))), Integer(Neg(Zero)), Integer(Pos(x42))))



We solved constraint (29) using rule (IV).




For Pair new_gcd0Gcd'10(False, vuz50, vuz51) -> new_gcd0Gcd'00(vuz50, new_rem(vuz51, vuz50)) the following chains were created:
*We consider the chain new_gcd0Gcd'10(False, x6, x7) -> new_gcd0Gcd'00(x6, new_rem(x7, x6)), new_gcd0Gcd'00(x8, x9) -> new_gcd0Gcd'10(new_esEs0(new_rem(x8, x9)), x9, x8) which results in the following constraint:

(1)    (new_gcd0Gcd'00(x6, new_rem(x7, x6))=new_gcd0Gcd'00(x8, x9)  ==>  new_gcd0Gcd'10(False, x6, x7)_>=_new_gcd0Gcd'00(x6, new_rem(x7, x6)))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'10(False, x6, x7)_>=_new_gcd0Gcd'00(x6, new_rem(x7, x6)))








To summarize, we get the following constraints P__>=_ for the following pairs.

*new_gcd0Gcd'00(vuz51, vuz50) -> new_gcd0Gcd'10(new_esEs0(new_rem(vuz51, vuz50)), vuz50, vuz51)

*(new_gcd0Gcd'00(Integer(Pos(x19)), Integer(Neg(Succ(x18))))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x19)), Integer(Neg(Succ(x18))))), Integer(Neg(Succ(x18))), Integer(Pos(x19))))


*(new_gcd0Gcd'00(Integer(Pos(x21)), Integer(Pos(Succ(x20))))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x21)), Integer(Pos(Succ(x20))))), Integer(Pos(Succ(x20))), Integer(Pos(x21))))


*(new_gcd0Gcd'00(Integer(Neg(x22)), Integer(Neg(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x22)), Integer(Neg(Zero)))), Integer(Neg(Zero)), Integer(Neg(x22))))


*(new_gcd0Gcd'00(Integer(Pos(x25)), Integer(Pos(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x25)), Integer(Pos(Zero)))), Integer(Pos(Zero)), Integer(Pos(x25))))


*(new_gcd0Gcd'00(Integer(Pos(x28)), Integer(Neg(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x28)), Integer(Neg(Zero)))), Integer(Neg(Zero)), Integer(Pos(x28))))


*(new_gcd0Gcd'00(Integer(Neg(x29)), Integer(Pos(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x29)), Integer(Pos(Zero)))), Integer(Pos(Zero)), Integer(Neg(x29))))


*(new_gcd0Gcd'00(Integer(Neg(x36)), Integer(Neg(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x36)), Integer(Neg(Zero)))), Integer(Neg(Zero)), Integer(Neg(x36))))


*(new_gcd0Gcd'00(Integer(Neg(x38)), Integer(Pos(Succ(x37))))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x38)), Integer(Pos(Succ(x37))))), Integer(Pos(Succ(x37))), Integer(Neg(x38))))


*(new_gcd0Gcd'00(Integer(Pos(x39)), Integer(Pos(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x39)), Integer(Pos(Zero)))), Integer(Pos(Zero)), Integer(Pos(x39))))


*(new_gcd0Gcd'00(Integer(Neg(x41)), Integer(Neg(Succ(x40))))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Neg(x41)), Integer(Neg(Succ(x40))))), Integer(Neg(Succ(x40))), Integer(Neg(x41))))


*(new_gcd0Gcd'00(Integer(Pos(x42)), Integer(Neg(Zero)))_>=_new_gcd0Gcd'10(new_esEs0(new_rem(Integer(Pos(x42)), Integer(Neg(Zero)))), Integer(Neg(Zero)), Integer(Pos(x42))))




*new_gcd0Gcd'10(False, vuz50, vuz51) -> new_gcd0Gcd'00(vuz50, new_rem(vuz51, vuz50))

*(new_gcd0Gcd'10(False, x6, x7)_>=_new_gcd0Gcd'00(x6, new_rem(x7, x6)))








The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
----------------------------------------

(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(vuz51, vuz50) -> new_gcd0Gcd'10(new_esEs0(new_rem(vuz51, vuz50)), vuz50, vuz51)
   new_gcd0Gcd'10(False, vuz50, vuz51) -> new_gcd0Gcd'00(vuz50, new_rem(vuz51, vuz50))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_rem(Integer(vuz510), Integer(vuz500)) -> Integer(new_primRemInt(vuz510, vuz500))
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_error -> error([])
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_esEs0(Integer(vuz160)) -> new_primEqInt(vuz160)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_rem(Integer(x0), Integer(x1))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_esEs0(Integer(x0))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(33) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(vuz51, vuz50) -> new_gcd0Gcd'10(new_esEs0(new_rem(vuz51, vuz50)), vuz50, vuz51) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(x0), Integer(x1)) -> new_gcd0Gcd'10(new_esEs0(Integer(new_primRemInt(x0, x1))), Integer(x1), Integer(x0)),new_gcd0Gcd'00(Integer(x0), Integer(x1)) -> new_gcd0Gcd'10(new_esEs0(Integer(new_primRemInt(x0, x1))), Integer(x1), Integer(x0)))


----------------------------------------

(34)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, vuz50, vuz51) -> new_gcd0Gcd'00(vuz50, new_rem(vuz51, vuz50))
   new_gcd0Gcd'00(Integer(x0), Integer(x1)) -> new_gcd0Gcd'10(new_esEs0(Integer(new_primRemInt(x0, x1))), Integer(x1), Integer(x0))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_rem(Integer(vuz510), Integer(vuz500)) -> Integer(new_primRemInt(vuz510, vuz500))
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_error -> error([])
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_esEs0(Integer(vuz160)) -> new_primEqInt(vuz160)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_rem(Integer(x0), Integer(x1))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_esEs0(Integer(x0))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(35) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(x0), Integer(x1)) -> new_gcd0Gcd'10(new_esEs0(Integer(new_primRemInt(x0, x1))), Integer(x1), Integer(x0)) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(x0), Integer(x1)) -> new_gcd0Gcd'10(new_primEqInt(new_primRemInt(x0, x1)), Integer(x1), Integer(x0)),new_gcd0Gcd'00(Integer(x0), Integer(x1)) -> new_gcd0Gcd'10(new_primEqInt(new_primRemInt(x0, x1)), Integer(x1), Integer(x0)))


----------------------------------------

(36)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, vuz50, vuz51) -> new_gcd0Gcd'00(vuz50, new_rem(vuz51, vuz50))
   new_gcd0Gcd'00(Integer(x0), Integer(x1)) -> new_gcd0Gcd'10(new_primEqInt(new_primRemInt(x0, x1)), Integer(x1), Integer(x0))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_rem(Integer(vuz510), Integer(vuz500)) -> Integer(new_primRemInt(vuz510, vuz500))
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_error -> error([])
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_esEs0(Integer(vuz160)) -> new_primEqInt(vuz160)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_rem(Integer(x0), Integer(x1))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_esEs0(Integer(x0))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(37) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(38)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, vuz50, vuz51) -> new_gcd0Gcd'00(vuz50, new_rem(vuz51, vuz50))
   new_gcd0Gcd'00(Integer(x0), Integer(x1)) -> new_gcd0Gcd'10(new_primEqInt(new_primRemInt(x0, x1)), Integer(x1), Integer(x0))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_rem(Integer(vuz510), Integer(vuz500)) -> Integer(new_primRemInt(vuz510, vuz500))

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_rem(Integer(x0), Integer(x1))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_esEs0(Integer(x0))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(39) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_esEs0(Integer(x0))


----------------------------------------

(40)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, vuz50, vuz51) -> new_gcd0Gcd'00(vuz50, new_rem(vuz51, vuz50))
   new_gcd0Gcd'00(Integer(x0), Integer(x1)) -> new_gcd0Gcd'10(new_primEqInt(new_primRemInt(x0, x1)), Integer(x1), Integer(x0))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_rem(Integer(vuz510), Integer(vuz500)) -> Integer(new_primRemInt(vuz510, vuz500))

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_rem(Integer(x0), Integer(x1))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(41) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, vuz50, vuz51) -> new_gcd0Gcd'00(vuz50, new_rem(vuz51, vuz50)) at position [1] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1))),new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1))))


----------------------------------------

(42)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(x0), Integer(x1)) -> new_gcd0Gcd'10(new_primEqInt(new_primRemInt(x0, x1)), Integer(x1), Integer(x0))
   new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_rem(Integer(vuz510), Integer(vuz500)) -> Integer(new_primRemInt(vuz510, vuz500))

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_rem(Integer(x0), Integer(x1))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(43) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(44)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(x0), Integer(x1)) -> new_gcd0Gcd'10(new_primEqInt(new_primRemInt(x0, x1)), Integer(x1), Integer(x0))
   new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_rem(Integer(x0), Integer(x1))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(45) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_rem(Integer(x0), Integer(x1))


----------------------------------------

(46)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(x0), Integer(x1)) -> new_gcd0Gcd'10(new_primEqInt(new_primRemInt(x0, x1)), Integer(x1), Integer(x0))
   new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(47) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(Integer(x0), Integer(x1)) -> new_gcd0Gcd'10(new_primEqInt(new_primRemInt(x0, x1)), Integer(x1), Integer(x0)) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0))),new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0))))
   (new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0))),new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0))))
   (new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Neg(Zero)), Integer(Neg(x0))),new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Neg(Zero)), Integer(Neg(x0))))
   (new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0))),new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0))))
   (new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Pos(x0))),new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Pos(x0))))
   (new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0))),new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0))))
   (new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Neg(Zero)), Integer(Pos(x0))),new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Neg(Zero)), Integer(Pos(x0))))
   (new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Neg(x0))),new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Neg(x0))))


----------------------------------------

(48)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Neg(Zero)), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Neg(Zero)), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Neg(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(49) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Neg(Zero)), Integer(Neg(x0))) at position [0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(error([])), Integer(Neg(Zero)), Integer(Neg(x0))),new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(error([])), Integer(Neg(Zero)), Integer(Neg(x0))))


----------------------------------------

(50)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Neg(Zero)), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(error([])), Integer(Neg(Zero)), Integer(Neg(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(51) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(52)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Neg(Zero)), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Neg(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(53) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Pos(x0))) at position [0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(error([])), Integer(Pos(Zero)), Integer(Pos(x0))),new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(error([])), Integer(Pos(Zero)), Integer(Pos(x0))))


----------------------------------------

(54)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Neg(Zero)), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(error([])), Integer(Pos(Zero)), Integer(Pos(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(55) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(56)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Neg(Zero)), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Neg(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(57) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Neg(Zero)), Integer(Pos(x0))) at position [0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(error([])), Integer(Neg(Zero)), Integer(Pos(x0))),new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(error([])), Integer(Neg(Zero)), Integer(Pos(x0))))


----------------------------------------

(58)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Zero))) -> new_gcd0Gcd'10(new_primEqInt(error([])), Integer(Neg(Zero)), Integer(Pos(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(59) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(60)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Neg(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(61) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(new_error), Integer(Pos(Zero)), Integer(Neg(x0))) at position [0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(error([])), Integer(Pos(Zero)), Integer(Neg(x0))),new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(error([])), Integer(Pos(Zero)), Integer(Neg(x0))))


----------------------------------------

(62)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Zero))) -> new_gcd0Gcd'10(new_primEqInt(error([])), Integer(Pos(Zero)), Integer(Neg(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(63) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(64)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(65) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, Integer(x1), Integer(x0)) -> new_gcd0Gcd'00(Integer(x1), Integer(new_primRemInt(x0, x1))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Zero)), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Neg(Zero)), Integer(new_error)),new_gcd0Gcd'10(False, Integer(Neg(Zero)), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Neg(Zero)), Integer(new_error)))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Zero)), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Pos(Zero)), Integer(new_error)),new_gcd0Gcd'10(False, Integer(Pos(Zero)), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Pos(Zero)), Integer(new_error)))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Zero)), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Neg(Zero)), Integer(new_error)),new_gcd0Gcd'10(False, Integer(Neg(Zero)), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Neg(Zero)), Integer(new_error)))
   (new_gcd0Gcd'10(False, Integer(Pos(Zero)), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Zero)), Integer(new_error)),new_gcd0Gcd'10(False, Integer(Pos(Zero)), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Zero)), Integer(new_error)))


----------------------------------------

(66)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Zero)), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Neg(Zero)), Integer(new_error))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'10(False, Integer(Pos(Zero)), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Pos(Zero)), Integer(new_error))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Zero)), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Neg(Zero)), Integer(new_error))
   new_gcd0Gcd'10(False, Integer(Pos(Zero)), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Zero)), Integer(new_error))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(67) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 4 less nodes.
----------------------------------------

(68)
Complex Obligation (AND)

----------------------------------------

(69)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(70) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(71)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(72) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primRemInt(Neg(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_error
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))


----------------------------------------

(73)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(74) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(x0))), Integer(Neg(Zero))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x0))), Integer(Neg(Zero))),new_gcd0Gcd'10(False, Integer(Neg(Succ(x0))), Integer(Neg(Zero))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x0))), Integer(Neg(Zero))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1)))))


----------------------------------------

(75)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(x0))), Integer(Neg(Zero))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x0))), Integer(Neg(Zero)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(76) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(77)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(78) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Zero, Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Zero, Zero)))))


----------------------------------------

(79)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Zero, Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(80) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(81)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(82) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero)))))


----------------------------------------

(83)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(84) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Neg(x0))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))),new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))))
   (new_gcd0Gcd'00(Integer(Neg(Zero)), Integer(Neg(Succ(x0)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Neg(Succ(x0))), Integer(Neg(Zero))),new_gcd0Gcd'00(Integer(Neg(Zero)), Integer(Neg(Succ(x0)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Neg(Succ(x0))), Integer(Neg(Zero))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero)))),new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero)))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))))


----------------------------------------

(85)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Zero)), Integer(Neg(Succ(x0)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Neg(Succ(x0))), Integer(Neg(Zero)))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(86) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
----------------------------------------

(87)
Complex Obligation (AND)

----------------------------------------

(88)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(89) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(90)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(91) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS02(x0, x1, Zero, Zero)


----------------------------------------

(92)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(93) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))))


----------------------------------------

(94)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(95) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))),new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))))


----------------------------------------

(96)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(97) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero)))))


----------------------------------------

(98)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(99) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero)))))


----------------------------------------

(100)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(101) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Zero, Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Zero, Zero)))))


----------------------------------------

(102)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Zero, Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(103) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(104)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(105) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(106)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(107) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(108)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(109) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))))


----------------------------------------

(110)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0))))))

The TRS R consists of the following rules:

   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(111) QDPSizeChangeProof (EQUIVALENT)
We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 

Order:Polynomial interpretation [POLO]:

   POL(False) = 1
   POL(Integer(x_1)) = x_1
   POL(Neg(x_1)) = x_1
   POL(Succ(x_1)) = 1 + x_1
   POL(Zero) = 1
   POL(new_primMinusNatS0(x_1)) = 1 + x_1
   POL(new_primMinusNatS1) = 1
   POL(new_primModNatS1(x_1, x_2)) = x_1




From the DPs we obtained the following set of size-change graphs:
*new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 >= 1, 3 >= 2


*new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero)))) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 >= 1, 3 > 2


*new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) (allowed arguments on rhs = {2, 3})
The graph contains the following edges 2 >= 2, 1 >= 3


*new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) (allowed arguments on rhs = {2, 3})
The graph contains the following edges 2 >= 2, 1 >= 3



We oriented the following set of usable rules [AAECC05,FROCOS05].

   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

----------------------------------------

(112)
YES

----------------------------------------

(113)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(114) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(115)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(116) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primMinusNatS0(x0)
   new_primMinusNatS1


----------------------------------------

(117)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(118) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Succ(x2), Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Succ(x2), Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Zero, Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Zero, Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(119)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Succ(x2), Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Zero, Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(120) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Succ(x2), Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) at position [0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))))


----------------------------------------

(121)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Zero, Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(122) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Zero, Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) at position [0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(123)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(124) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(125)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(126) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))))


----------------------------------------

(127)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(128) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(129)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(130) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))))


----------------------------------------

(131)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(132) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(133)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(134) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) at position [0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(135)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(136) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(137)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(138) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Succ(x2), Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Succ(x2), Zero)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Zero, Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Zero, Zero)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(139)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Succ(x2), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Zero, Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(140) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
----------------------------------------

(141)
Complex Obligation (AND)

----------------------------------------

(142)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Succ(x2), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(143) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Succ(x2), Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))))


----------------------------------------

(144)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(145) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))))


----------------------------------------

(146)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(147) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero))))))


----------------------------------------

(148)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(149) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(False) = 0
   POL(Integer(x_1)) = 2*x_1
   POL(Neg(x_1)) = 2 + 2*x_1
   POL(Succ(x_1)) = 1 + x_1
   POL(True) = 3
   POL(Zero) = 0
   POL(new_gcd0Gcd'00(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(new_gcd0Gcd'10(x_1, x_2, x_3)) = 2*x_2 + 2*x_3
   POL(new_primEqInt(x_1)) = 0
   POL(new_primMinusNatS2(x_1, x_2)) = x_1
   POL(new_primModNatS01(x_1, x_2)) = 3 + x_1
   POL(new_primModNatS02(x_1, x_2, x_3, x_4)) = 3 + x_1
   POL(new_primModNatS1(x_1, x_2)) = 1 + x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)


----------------------------------------

(150)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(151) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
----------------------------------------

(152)
TRUE

----------------------------------------

(153)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(154) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].

   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))


----------------------------------------

(155)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(156) InductionCalculusProof (EQUIVALENT)
Note that final constraints are written in bold face.



For Pair new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))) the following chains were created:
*We consider the chain new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(new_primModNatS02(Succ(x3), Succ(x2), x3, x2)))), new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x4))))), Integer(Neg(Succ(Succ(Succ(x5)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x4), Succ(x5), x4, x5))), Integer(Neg(Succ(Succ(Succ(x5))))), Integer(Neg(Succ(Succ(Succ(x4)))))) which results in the following constraint:

(1)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(new_primModNatS02(Succ(x3), Succ(x2), x3, x2))))=new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x4))))), Integer(Neg(Succ(Succ(Succ(x5))))))  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(new_primModNatS02(Succ(x3), Succ(x2), x3, x2)))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Succ(x3)=x12 & Succ(x2)=x13 & new_primModNatS02(x12, x13, x3, x2)=Succ(Succ(Succ(x5)))  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(new_primModNatS02(Succ(x3), Succ(x2), x3, x2)))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x12, x13, x3, x2)=Succ(Succ(Succ(x5))) which results in the following new constraints:

(3)    (new_primModNatS01(x16, x15)=Succ(Succ(Succ(x5))) & Succ(Succ(x14))=x16 & Succ(Zero)=x15  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x14)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Succ(x14)), Succ(Zero), Succ(x14), Zero)))))

(4)    (new_primModNatS02(x20, x19, x18, x17)=Succ(Succ(Succ(x5))) & Succ(Succ(x18))=x20 & Succ(Succ(x17))=x19 & (\/x21:new_primModNatS02(x20, x19, x18, x17)=Succ(Succ(Succ(x21))) & Succ(x18)=x20 & Succ(x17)=x19  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x17))))), Integer(Neg(Succ(Succ(Succ(x18))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x17))))), Integer(Neg(new_primModNatS02(Succ(x18), Succ(x17), x18, x17)))))  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(x17)))))), Integer(Neg(Succ(Succ(Succ(Succ(x18)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x17)))))), Integer(Neg(new_primModNatS02(Succ(Succ(x18)), Succ(Succ(x17)), Succ(x18), Succ(x17))))))

(5)    (new_primModNatS01(x23, x22)=Succ(Succ(Succ(x5))) & Succ(Zero)=x23 & Succ(Zero)=x22  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))

(6)    (Succ(Succ(x26))=Succ(Succ(Succ(x5))) & Succ(Zero)=x26 & Succ(Succ(x24))=x25  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(x24)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x24)))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Succ(x24)), Zero, Succ(x24))))))



We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x16, x15)=Succ(Succ(Succ(x5))) which results in the following new constraint:

(7)    (new_primModNatS1(new_primMinusNatS2(Succ(x28), Succ(x27)), Succ(x27))=Succ(Succ(Succ(x5))) & Succ(Succ(x14))=x28 & Succ(Zero)=x27  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x14)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Succ(x14)), Succ(Zero), Succ(x14), Zero)))))



We simplified constraint (4) using rule (IV) which results in the following new constraint:

(8)    (new_primModNatS02(x20, x19, x18, x17)=Succ(Succ(Succ(x5))) & Succ(Succ(x18))=x20 & Succ(Succ(x17))=x19  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(x17)))))), Integer(Neg(Succ(Succ(Succ(Succ(x18)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x17)))))), Integer(Neg(new_primModNatS02(Succ(Succ(x18)), Succ(Succ(x17)), Succ(x18), Succ(x17))))))



We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x23, x22)=Succ(Succ(Succ(x5))) which results in the following new constraint:

(9)    (new_primModNatS1(new_primMinusNatS2(Succ(x47), Succ(x46)), Succ(x46))=Succ(Succ(Succ(x5))) & Succ(Zero)=x47 & Succ(Zero)=x46  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))



We simplified constraint (6) using rules (I), (II), (III), (IV) which results in the following new constraint:

(10)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(x24)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x24)))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Succ(x24)), Zero, Succ(x24))))))



We simplified constraint (7) using rules (III), (IV), (VII) which results in the following new constraint:

(11)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x14)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Succ(x14)), Succ(Zero), Succ(x14), Zero)))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x20, x19, x18, x17)=Succ(Succ(Succ(x5))) which results in the following new constraints:

(12)    (new_primModNatS01(x35, x34)=Succ(Succ(Succ(x5))) & Succ(Succ(Succ(x33)))=x35 & Succ(Succ(Zero))=x34  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x33))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(new_primModNatS02(Succ(Succ(Succ(x33))), Succ(Succ(Zero)), Succ(Succ(x33)), Succ(Zero))))))

(13)    (new_primModNatS02(x39, x38, x37, x36)=Succ(Succ(Succ(x5))) & Succ(Succ(Succ(x37)))=x39 & Succ(Succ(Succ(x36)))=x38 & (\/x40:new_primModNatS02(x39, x38, x37, x36)=Succ(Succ(Succ(x40))) & Succ(Succ(x37))=x39 & Succ(Succ(x36))=x38  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(x36)))))), Integer(Neg(Succ(Succ(Succ(Succ(x37)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x36)))))), Integer(Neg(new_primModNatS02(Succ(Succ(x37)), Succ(Succ(x36)), Succ(x37), Succ(x36))))))  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Succ(x36))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x37))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x36))))))), Integer(Neg(new_primModNatS02(Succ(Succ(Succ(x37))), Succ(Succ(Succ(x36))), Succ(Succ(x37)), Succ(Succ(x36)))))))

(14)    (new_primModNatS01(x42, x41)=Succ(Succ(Succ(x5))) & Succ(Succ(Zero))=x42 & Succ(Succ(Zero))=x41  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero))))))

(15)    (Succ(Succ(x45))=Succ(Succ(Succ(x5))) & Succ(Succ(Zero))=x45 & Succ(Succ(Succ(x43)))=x44  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Succ(x43))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x43))))))), Integer(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x43))), Succ(Zero), Succ(Succ(x43)))))))



We simplified constraint (12) using rules (III), (IV) which results in the following new constraint:

(16)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x33))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(new_primModNatS02(Succ(Succ(Succ(x33))), Succ(Succ(Zero)), Succ(Succ(x33)), Succ(Zero))))))



We simplified constraint (13) using rules (III), (IV) which results in the following new constraint:

(17)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Succ(x36))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x37))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x36))))))), Integer(Neg(new_primModNatS02(Succ(Succ(Succ(x37))), Succ(Succ(Succ(x36))), Succ(Succ(x37)), Succ(Succ(x36)))))))



We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:

(18)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero))))))



We simplified constraint (15) using rules (I), (II), (III), (IV) which results in the following new constraint:

(19)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Succ(x43))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x43))))))), Integer(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x43))), Succ(Zero), Succ(Succ(x43)))))))



We simplified constraint (9) using rules (III), (IV), (VII) which results in the following new constraint:

(20)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))








For Pair new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) the following chains were created:
*We consider the chain new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x6))))), Integer(Neg(Succ(Succ(Succ(x7)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))), Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Neg(Succ(Succ(Succ(x6)))))), new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x8))))), Integer(Neg(Succ(Succ(Succ(x9)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x8))))), Integer(Neg(new_primModNatS02(Succ(x9), Succ(x8), x9, x8)))) which results in the following constraint:

(1)    (new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))), Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Neg(Succ(Succ(Succ(x6))))))=new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x8))))), Integer(Neg(Succ(Succ(Succ(x9))))))  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x6))))), Integer(Neg(Succ(Succ(Succ(x7))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))), Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Neg(Succ(Succ(Succ(x6)))))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Neg(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))=x52 & new_primEqInt(x52)=False  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x6))))), Integer(Neg(Succ(Succ(Succ(x7))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))), Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Neg(Succ(Succ(Succ(x6)))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primEqInt(x52)=False which results in the following new constraint:

(3)    (False=False & Neg(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))=Neg(Succ(x53))  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x6))))), Integer(Neg(Succ(Succ(Succ(x7))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))), Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Neg(Succ(Succ(Succ(x6)))))))



We simplified constraint (3) using rules (I), (II), (VII) which results in the following new constraint:

(4)    (Succ(x6)=x54 & Succ(x7)=x55 & new_primModNatS02(x54, x55, x6, x7)=Succ(x53)  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x6))))), Integer(Neg(Succ(Succ(Succ(x7))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))), Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Neg(Succ(Succ(Succ(x6)))))))



We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x54, x55, x6, x7)=Succ(x53) which results in the following new constraints:

(5)    (new_primModNatS01(x58, x57)=Succ(x53) & Succ(Succ(x56))=x58 & Succ(Zero)=x57  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x56)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x56)), Succ(Zero), Succ(x56), Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x56))))))))

(6)    (new_primModNatS02(x62, x61, x60, x59)=Succ(x53) & Succ(Succ(x60))=x62 & Succ(Succ(x59))=x61 & (\/x63:new_primModNatS02(x62, x61, x60, x59)=Succ(x63) & Succ(x60)=x62 & Succ(x59)=x61  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x60))))), Integer(Neg(Succ(Succ(Succ(x59))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x60), Succ(x59), x60, x59))), Integer(Neg(Succ(Succ(Succ(x59))))), Integer(Neg(Succ(Succ(Succ(x60)))))))  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x60)))))), Integer(Neg(Succ(Succ(Succ(Succ(x59)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x60)), Succ(Succ(x59)), Succ(x60), Succ(x59)))), Integer(Neg(Succ(Succ(Succ(Succ(x59)))))), Integer(Neg(Succ(Succ(Succ(Succ(x60))))))))

(7)    (new_primModNatS01(x65, x64)=Succ(x53) & Succ(Zero)=x65 & Succ(Zero)=x64  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))

(8)    (Succ(Succ(x68))=Succ(x53) & Succ(Zero)=x68 & Succ(Succ(x66))=x67  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x66)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Succ(x66)), Zero, Succ(x66)))), Integer(Neg(Succ(Succ(Succ(Succ(x66)))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))



We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x58, x57)=Succ(x53) which results in the following new constraint:

(9)    (new_primModNatS1(new_primMinusNatS2(Succ(x70), Succ(x69)), Succ(x69))=Succ(x53) & Succ(Succ(x56))=x70 & Succ(Zero)=x69  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x56)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x56)), Succ(Zero), Succ(x56), Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x56))))))))



We simplified constraint (6) using rule (IV) which results in the following new constraint:

(10)    (new_primModNatS02(x62, x61, x60, x59)=Succ(x53) & Succ(Succ(x60))=x62 & Succ(Succ(x59))=x61  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x60)))))), Integer(Neg(Succ(Succ(Succ(Succ(x59)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x60)), Succ(Succ(x59)), Succ(x60), Succ(x59)))), Integer(Neg(Succ(Succ(Succ(Succ(x59)))))), Integer(Neg(Succ(Succ(Succ(Succ(x60))))))))



We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x65, x64)=Succ(x53) which results in the following new constraint:

(11)    (new_primModNatS1(new_primMinusNatS2(Succ(x89), Succ(x88)), Succ(x88))=Succ(x53) & Succ(Zero)=x89 & Succ(Zero)=x88  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))



We simplified constraint (8) using rules (I), (II), (IV) which results in the following new constraint:

(12)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x66)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Succ(x66)), Zero, Succ(x66)))), Integer(Neg(Succ(Succ(Succ(Succ(x66)))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))



We simplified constraint (9) using rules (III), (IV), (VII) which results in the following new constraint:

(13)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x56)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x56)), Succ(Zero), Succ(x56), Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x56))))))))



We simplified constraint (10) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x62, x61, x60, x59)=Succ(x53) which results in the following new constraints:

(14)    (new_primModNatS01(x77, x76)=Succ(x53) & Succ(Succ(Succ(x75)))=x77 & Succ(Succ(Zero))=x76  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x75))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Succ(x75))), Succ(Succ(Zero)), Succ(Succ(x75)), Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x75)))))))))

(15)    (new_primModNatS02(x81, x80, x79, x78)=Succ(x53) & Succ(Succ(Succ(x79)))=x81 & Succ(Succ(Succ(x78)))=x80 & (\/x82:new_primModNatS02(x81, x80, x79, x78)=Succ(x82) & Succ(Succ(x79))=x81 & Succ(Succ(x78))=x80  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x79)))))), Integer(Neg(Succ(Succ(Succ(Succ(x78)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x79)), Succ(Succ(x78)), Succ(x79), Succ(x78)))), Integer(Neg(Succ(Succ(Succ(Succ(x78)))))), Integer(Neg(Succ(Succ(Succ(Succ(x79))))))))  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x79))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x78))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Succ(x79))), Succ(Succ(Succ(x78))), Succ(Succ(x79)), Succ(Succ(x78))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x78))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x79)))))))))

(16)    (new_primModNatS01(x84, x83)=Succ(x53) & Succ(Succ(Zero))=x84 & Succ(Succ(Zero))=x83  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))

(17)    (Succ(Succ(x87))=Succ(x53) & Succ(Succ(Zero))=x87 & Succ(Succ(Succ(x85)))=x86  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x85))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x85))), Succ(Zero), Succ(Succ(x85))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x85))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))



We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:

(18)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x75))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Succ(x75))), Succ(Succ(Zero)), Succ(Succ(x75)), Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x75)))))))))



We simplified constraint (15) using rules (III), (IV) which results in the following new constraint:

(19)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x79))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x78))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Succ(x79))), Succ(Succ(Succ(x78))), Succ(Succ(x79)), Succ(Succ(x78))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x78))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x79)))))))))



We simplified constraint (16) using rules (III), (IV) which results in the following new constraint:

(20)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))



We simplified constraint (17) using rules (I), (II), (IV) which results in the following new constraint:

(21)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x85))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x85))), Succ(Zero), Succ(Succ(x85))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x85))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))



We simplified constraint (11) using rules (III), (IV), (VII) which results in the following new constraint:

(22)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))








To summarize, we get the following constraints P__>=_ for the following pairs.

*new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))

*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Succ(x43))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x43))))))), Integer(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x43))), Succ(Zero), Succ(Succ(x43)))))))


*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(x24)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x24)))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Succ(x24)), Zero, Succ(x24))))))


*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x14)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Succ(x14)), Succ(Zero), Succ(x14), Zero)))))


*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x33))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(new_primModNatS02(Succ(Succ(Succ(x33))), Succ(Succ(Zero)), Succ(Succ(x33)), Succ(Zero))))))


*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Succ(x36))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x37))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x36))))))), Integer(Neg(new_primModNatS02(Succ(Succ(Succ(x37))), Succ(Succ(Succ(x36))), Succ(Succ(x37)), Succ(Succ(x36)))))))


*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero))))))


*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))




*new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))

*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x85))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x85))), Succ(Zero), Succ(Succ(x85))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x85))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x66)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Succ(x66)), Zero, Succ(x66)))), Integer(Neg(Succ(Succ(Succ(Succ(x66)))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x56)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x56)), Succ(Zero), Succ(x56), Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x56))))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x75))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Succ(x75))), Succ(Succ(Zero)), Succ(Succ(x75)), Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x75)))))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x79))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x78))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Succ(x79))), Succ(Succ(Succ(x78))), Succ(Succ(x79)), Succ(Succ(x78))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x78))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x79)))))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))








The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
----------------------------------------

(157)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(158)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(159) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(160)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(161) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primRemInt(Neg(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_error
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))


----------------------------------------

(162)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(163) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(x0))), Integer(Pos(Zero))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x0))), Integer(Pos(Zero))),new_gcd0Gcd'10(False, Integer(Pos(Succ(x0))), Integer(Pos(Zero))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x0))), Integer(Pos(Zero))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1)))))


----------------------------------------

(164)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x0))), Integer(Pos(Zero))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x0))), Integer(Pos(Zero)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(165) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(166)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(167) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Zero, Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Zero, Zero)))))


----------------------------------------

(168)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Zero, Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(169) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(170)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(171) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero)))))


----------------------------------------

(172)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(173) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Pos(x0))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))),new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))))
   (new_gcd0Gcd'00(Integer(Pos(Zero)), Integer(Pos(Succ(x0)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Zero)), Integer(Pos(Succ(x0))), Integer(Pos(Zero))),new_gcd0Gcd'00(Integer(Pos(Zero)), Integer(Pos(Succ(x0)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Zero)), Integer(Pos(Succ(x0))), Integer(Pos(Zero))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero)))),new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero)))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))))


----------------------------------------

(174)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Zero)), Integer(Pos(Succ(x0)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Zero)), Integer(Pos(Succ(x0))), Integer(Pos(Zero)))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(175) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
----------------------------------------

(176)
Complex Obligation (AND)

----------------------------------------

(177)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(178) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(179)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(180) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS02(x0, x1, Zero, Zero)


----------------------------------------

(181)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(182) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))))


----------------------------------------

(183)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(184) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))),new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))))


----------------------------------------

(185)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(186) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero)))))


----------------------------------------

(187)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(188) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero)))))


----------------------------------------

(189)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(190) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Zero, Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Zero, Zero)))))


----------------------------------------

(191)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Zero, Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(192) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(193)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(194) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))))


----------------------------------------

(195)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(196) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(197)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0))))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(198) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))))


----------------------------------------

(199)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0))))))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(200) QDPSizeChangeProof (EQUIVALENT)
We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 

Order:Polynomial interpretation [POLO]:

   POL(False) = 1
   POL(Integer(x_1)) = x_1
   POL(Pos(x_1)) = x_1
   POL(Succ(x_1)) = 1 + x_1
   POL(Zero) = 1
   POL(new_primMinusNatS0(x_1)) = 1 + x_1
   POL(new_primMinusNatS1) = 1
   POL(new_primModNatS1(x_1, x_2)) = x_1




From the DPs we obtained the following set of size-change graphs:
*new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 >= 1, 3 >= 2


*new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero)))) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 >= 1, 3 > 2


*new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) (allowed arguments on rhs = {2, 3})
The graph contains the following edges 2 >= 2, 1 >= 3


*new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) (allowed arguments on rhs = {2, 3})
The graph contains the following edges 2 >= 2, 1 >= 3



We oriented the following set of usable rules [AAECC05,FROCOS05].

   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

----------------------------------------

(201)
YES

----------------------------------------

(202)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(203) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(204)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(205) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primMinusNatS0(x0)
   new_primMinusNatS1


----------------------------------------

(206)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(207) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Succ(x2), Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Succ(x2), Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Zero, Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Zero, Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))))


----------------------------------------

(208)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Succ(x2), Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Zero, Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(209) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Succ(x2), Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) at position [0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))))


----------------------------------------

(210)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Zero, Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(211) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Zero, Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) at position [0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))))


----------------------------------------

(212)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(213) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))))


----------------------------------------

(214)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(215) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))))


----------------------------------------

(216)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(217) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))))


----------------------------------------

(218)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(219) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))))


----------------------------------------

(220)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(221) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Zero, Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Zero, Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))))


----------------------------------------

(222)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Zero, Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(223) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Zero, Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) at position [0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Zero)), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Zero)), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))))


----------------------------------------

(224)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Zero)), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(225) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(226)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(227) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Succ(x2), Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Succ(x2), Zero)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Zero, Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Zero, Zero)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))))


----------------------------------------

(228)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Succ(x2), Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Zero, Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(229) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
----------------------------------------

(230)
Complex Obligation (AND)

----------------------------------------

(231)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Succ(x2), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(232) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Succ(x2), Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))))


----------------------------------------

(233)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(234) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))))


----------------------------------------

(235)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(236) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero))))))


----------------------------------------

(237)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(238) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].

   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Neg(Zero))


----------------------------------------

(239)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(240) InductionCalculusProof (EQUIVALENT)
Note that final constraints are written in bold face.



For Pair new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Pos(Succ(Succ(Zero))))), new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) which results in the following constraint:

(1)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Pos(Succ(Succ(Zero)))))=new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Pos(Succ(Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Pos(Succ(Succ(Zero))))))








For Pair new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) the following chains were created:
*We consider the chain new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x8), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x8)))))), new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x9)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x9), Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x8), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x8))))))=new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x9))))))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x8), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x8)))))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Pos(new_primModNatS1(Succ(x8), Succ(Zero)))=x20 & new_primEqInt(x20)=False  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x8), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x8)))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primEqInt(x20)=False which results in the following new constraint:

(3)    (False=False & Pos(new_primModNatS1(Succ(x8), Succ(Zero)))=Pos(Succ(x21))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x8), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x8)))))))



We simplified constraint (3) using rules (I), (II), (VII) which results in the following new constraint:

(4)    (Succ(x8)=x22 & Succ(Zero)=x23 & new_primModNatS1(x22, x23)=Succ(x21)  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x8), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x8)))))))



We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS1(x22, x23)=Succ(x21) which results in the following new constraints:

(5)    (new_primModNatS02(x25, x24, x25, x24)=Succ(x21) & Succ(x8)=Succ(Succ(x25)) & Succ(Zero)=Succ(x24)  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x8), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x8)))))))

(6)    (Succ(Zero)=Succ(x21) & Succ(x8)=Succ(Zero) & Succ(Zero)=Succ(x26)  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x8), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x8)))))))



We simplified constraint (5) using rules (I), (II), (III), (VII) which results in the following new constraint:

(7)    (x25=x28 & x24=x29 & new_primModNatS02(x25, x24, x28, x29)=Succ(x21) & Zero=x24  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x25)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(x25)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x25))))))))



We simplified constraint (6) using rules (I), (II), (III), (IV) which results in the following new constraint:

(8)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Zero)))))))



We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x25, x24, x28, x29)=Succ(x21) which results in the following new constraints:

(9)    (new_primModNatS01(x32, x31)=Succ(x21) & x32=Succ(x30) & x31=Zero & Zero=x31  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x32)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(x32)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x32))))))))

(10)    (new_primModNatS02(x36, x35, x34, x33)=Succ(x21) & x36=Succ(x34) & x35=Succ(x33) & Zero=x35 & (\/x37:new_primModNatS02(x36, x35, x34, x33)=Succ(x37) & x36=x34 & x35=x33 & Zero=x35  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x36)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(x36)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x36))))))))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x36)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(x36)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x36))))))))

(11)    (new_primModNatS01(x39, x38)=Succ(x21) & x39=Zero & x38=Zero & Zero=x38  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x39)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(x39)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x39))))))))

(12)    (Succ(Succ(x42))=Succ(x21) & x42=Zero & x41=Succ(x40) & Zero=x41  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x42)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(x42)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x42))))))))



We simplified constraint (9) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(13)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x30))))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(Succ(x30))), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x30)))))))))



We solved constraint (10) using rules (I), (II), (III).We simplified constraint (11) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(14)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))



We solved constraint (12) using rules (I), (II), (III), (IV).




For Pair new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x10)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x10))))), Integer(Pos(Succ(Succ(Zero))))), new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x11))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x11))))), Integer(Pos(Succ(Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x10))))), Integer(Pos(Succ(Succ(Zero)))))=new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x11))))), Integer(Pos(Succ(Succ(Zero)))))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x10))))))_>=_new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x10))))), Integer(Pos(Succ(Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x10))))))_>=_new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x10))))), Integer(Pos(Succ(Succ(Zero))))))








For Pair new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x17)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x17), Succ(Zero))))), new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x18)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x18))))), Integer(Pos(Succ(Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x17), Succ(Zero)))))=new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x18))))))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x17))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x17), Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (VII) which results in the following new constraint:

(2)    (Succ(x17)=x47 & Succ(Zero)=x48 & new_primModNatS1(x47, x48)=Succ(Succ(Succ(x18)))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x17))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x17), Succ(Zero))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS1(x47, x48)=Succ(Succ(Succ(x18))) which results in the following new constraints:

(3)    (new_primModNatS02(x50, x49, x50, x49)=Succ(Succ(Succ(x18))) & Succ(x17)=Succ(Succ(x50)) & Succ(Zero)=Succ(x49)  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x17))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x17), Succ(Zero))))))

(4)    (Succ(Zero)=Succ(Succ(Succ(x18))) & Succ(x17)=Succ(Zero) & Succ(Zero)=Succ(x51)  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x17))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x17), Succ(Zero))))))



We simplified constraint (3) using rules (I), (II), (III), (VII) which results in the following new constraint:

(5)    (x50=x53 & x49=x54 & new_primModNatS02(x50, x49, x53, x54)=Succ(Succ(Succ(x18))) & Zero=x49  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x50)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(x50)), Succ(Zero))))))



We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x50, x49, x53, x54)=Succ(Succ(Succ(x18))) which results in the following new constraints:

(6)    (new_primModNatS01(x57, x56)=Succ(Succ(Succ(x18))) & x57=Succ(x55) & x56=Zero & Zero=x56  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x57)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(x57)), Succ(Zero))))))

(7)    (new_primModNatS02(x61, x60, x59, x58)=Succ(Succ(Succ(x18))) & x61=Succ(x59) & x60=Succ(x58) & Zero=x60 & (\/x62:new_primModNatS02(x61, x60, x59, x58)=Succ(Succ(Succ(x62))) & x61=x59 & x60=x58 & Zero=x60  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x61)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(x61)), Succ(Zero))))))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x61)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(x61)), Succ(Zero))))))

(8)    (new_primModNatS01(x64, x63)=Succ(Succ(Succ(x18))) & x64=Zero & x63=Zero & Zero=x63  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x64)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(x64)), Succ(Zero))))))

(9)    (Succ(Succ(x67))=Succ(Succ(Succ(x18))) & x67=Zero & x66=Succ(x65) & Zero=x66  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x67)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(x67)), Succ(Zero))))))



We simplified constraint (6) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(10)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x55))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(Succ(x55))), Succ(Zero))))))



We solved constraint (7) using rules (I), (II), (III).We simplified constraint (8) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(11)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(Zero)), Succ(Zero))))))



We solved constraint (9) using rules (I), (II), (III).




To summarize, we get the following constraints P__>=_ for the following pairs.

*new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))

*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Pos(Succ(Succ(Zero))))))




*new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))

*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Zero)))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x30))))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(Succ(x30))), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x30)))))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))




*new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))

*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x10))))))_>=_new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x10))))), Integer(Pos(Succ(Succ(Zero))))))




*new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))

*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x55))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(Succ(x55))), Succ(Zero))))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(Zero)), Succ(Zero))))))








The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
----------------------------------------

(241)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(242)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(243) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].

   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Neg(Zero))


----------------------------------------

(244)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(245) InductionCalculusProof (EQUIVALENT)
Note that final constraints are written in bold face.



For Pair new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))) the following chains were created:
*We consider the chain new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(new_primModNatS02(Succ(x3), Succ(x2), x3, x2)))), new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x4))))), Integer(Pos(Succ(Succ(Succ(x5)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x4), Succ(x5), x4, x5))), Integer(Pos(Succ(Succ(Succ(x5))))), Integer(Pos(Succ(Succ(Succ(x4)))))) which results in the following constraint:

(1)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(new_primModNatS02(Succ(x3), Succ(x2), x3, x2))))=new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x4))))), Integer(Pos(Succ(Succ(Succ(x5))))))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(new_primModNatS02(Succ(x3), Succ(x2), x3, x2)))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Succ(x3)=x12 & Succ(x2)=x13 & new_primModNatS02(x12, x13, x3, x2)=Succ(Succ(Succ(x5)))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(new_primModNatS02(Succ(x3), Succ(x2), x3, x2)))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x12, x13, x3, x2)=Succ(Succ(Succ(x5))) which results in the following new constraints:

(3)    (new_primModNatS01(x16, x15)=Succ(Succ(Succ(x5))) & Succ(Succ(x14))=x16 & Succ(Zero)=x15  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x14)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Succ(x14)), Succ(Zero), Succ(x14), Zero)))))

(4)    (new_primModNatS02(x20, x19, x18, x17)=Succ(Succ(Succ(x5))) & Succ(Succ(x18))=x20 & Succ(Succ(x17))=x19 & (\/x21:new_primModNatS02(x20, x19, x18, x17)=Succ(Succ(Succ(x21))) & Succ(x18)=x20 & Succ(x17)=x19  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x17))))), Integer(Pos(Succ(Succ(Succ(x18))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x17))))), Integer(Pos(new_primModNatS02(Succ(x18), Succ(x17), x18, x17)))))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(x17)))))), Integer(Pos(Succ(Succ(Succ(Succ(x18)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x17)))))), Integer(Pos(new_primModNatS02(Succ(Succ(x18)), Succ(Succ(x17)), Succ(x18), Succ(x17))))))

(5)    (new_primModNatS01(x23, x22)=Succ(Succ(Succ(x5))) & Succ(Zero)=x23 & Succ(Zero)=x22  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))

(6)    (Succ(Succ(x26))=Succ(Succ(Succ(x5))) & Succ(Zero)=x26 & Succ(Succ(x24))=x25  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(x24)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x24)))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Succ(x24)), Zero, Succ(x24))))))



We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x16, x15)=Succ(Succ(Succ(x5))) which results in the following new constraint:

(7)    (new_primModNatS1(new_primMinusNatS2(Succ(x28), Succ(x27)), Succ(x27))=Succ(Succ(Succ(x5))) & Succ(Succ(x14))=x28 & Succ(Zero)=x27  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x14)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Succ(x14)), Succ(Zero), Succ(x14), Zero)))))



We simplified constraint (4) using rule (IV) which results in the following new constraint:

(8)    (new_primModNatS02(x20, x19, x18, x17)=Succ(Succ(Succ(x5))) & Succ(Succ(x18))=x20 & Succ(Succ(x17))=x19  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(x17)))))), Integer(Pos(Succ(Succ(Succ(Succ(x18)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x17)))))), Integer(Pos(new_primModNatS02(Succ(Succ(x18)), Succ(Succ(x17)), Succ(x18), Succ(x17))))))



We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x23, x22)=Succ(Succ(Succ(x5))) which results in the following new constraint:

(9)    (new_primModNatS1(new_primMinusNatS2(Succ(x47), Succ(x46)), Succ(x46))=Succ(Succ(Succ(x5))) & Succ(Zero)=x47 & Succ(Zero)=x46  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))



We simplified constraint (6) using rules (I), (II), (III), (IV) which results in the following new constraint:

(10)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(x24)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x24)))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Succ(x24)), Zero, Succ(x24))))))



We simplified constraint (7) using rules (III), (IV), (VII) which results in the following new constraint:

(11)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x14)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Succ(x14)), Succ(Zero), Succ(x14), Zero)))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x20, x19, x18, x17)=Succ(Succ(Succ(x5))) which results in the following new constraints:

(12)    (new_primModNatS01(x35, x34)=Succ(Succ(Succ(x5))) & Succ(Succ(Succ(x33)))=x35 & Succ(Succ(Zero))=x34  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x33))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(new_primModNatS02(Succ(Succ(Succ(x33))), Succ(Succ(Zero)), Succ(Succ(x33)), Succ(Zero))))))

(13)    (new_primModNatS02(x39, x38, x37, x36)=Succ(Succ(Succ(x5))) & Succ(Succ(Succ(x37)))=x39 & Succ(Succ(Succ(x36)))=x38 & (\/x40:new_primModNatS02(x39, x38, x37, x36)=Succ(Succ(Succ(x40))) & Succ(Succ(x37))=x39 & Succ(Succ(x36))=x38  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(x36)))))), Integer(Pos(Succ(Succ(Succ(Succ(x37)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x36)))))), Integer(Pos(new_primModNatS02(Succ(Succ(x37)), Succ(Succ(x36)), Succ(x37), Succ(x36))))))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Succ(x36))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x37))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x36))))))), Integer(Pos(new_primModNatS02(Succ(Succ(Succ(x37))), Succ(Succ(Succ(x36))), Succ(Succ(x37)), Succ(Succ(x36)))))))

(14)    (new_primModNatS01(x42, x41)=Succ(Succ(Succ(x5))) & Succ(Succ(Zero))=x42 & Succ(Succ(Zero))=x41  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero))))))

(15)    (Succ(Succ(x45))=Succ(Succ(Succ(x5))) & Succ(Succ(Zero))=x45 & Succ(Succ(Succ(x43)))=x44  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Succ(x43))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x43))))))), Integer(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x43))), Succ(Zero), Succ(Succ(x43)))))))



We simplified constraint (12) using rules (III), (IV) which results in the following new constraint:

(16)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x33))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(new_primModNatS02(Succ(Succ(Succ(x33))), Succ(Succ(Zero)), Succ(Succ(x33)), Succ(Zero))))))



We simplified constraint (13) using rules (III), (IV) which results in the following new constraint:

(17)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Succ(x36))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x37))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x36))))))), Integer(Pos(new_primModNatS02(Succ(Succ(Succ(x37))), Succ(Succ(Succ(x36))), Succ(Succ(x37)), Succ(Succ(x36)))))))



We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:

(18)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero))))))



We simplified constraint (15) using rules (I), (II), (III), (IV) which results in the following new constraint:

(19)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Succ(x43))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x43))))))), Integer(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x43))), Succ(Zero), Succ(Succ(x43)))))))



We simplified constraint (9) using rules (III), (IV), (VII) which results in the following new constraint:

(20)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))








For Pair new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) the following chains were created:
*We consider the chain new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x6))))), Integer(Pos(Succ(Succ(Succ(x7)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))), Integer(Pos(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Succ(x6)))))), new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Pos(Succ(Succ(Succ(x9)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Pos(new_primModNatS02(Succ(x9), Succ(x8), x9, x8)))) which results in the following constraint:

(1)    (new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))), Integer(Pos(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Succ(x6))))))=new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Pos(Succ(Succ(Succ(x9))))))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x6))))), Integer(Pos(Succ(Succ(Succ(x7))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))), Integer(Pos(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Succ(x6)))))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Pos(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))=x52 & new_primEqInt(x52)=False  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x6))))), Integer(Pos(Succ(Succ(Succ(x7))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))), Integer(Pos(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Succ(x6)))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primEqInt(x52)=False which results in the following new constraint:

(3)    (False=False & Pos(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))=Pos(Succ(x53))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x6))))), Integer(Pos(Succ(Succ(Succ(x7))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))), Integer(Pos(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Succ(x6)))))))



We simplified constraint (3) using rules (I), (II), (VII) which results in the following new constraint:

(4)    (Succ(x6)=x54 & Succ(x7)=x55 & new_primModNatS02(x54, x55, x6, x7)=Succ(x53)  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x6))))), Integer(Pos(Succ(Succ(Succ(x7))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x6), Succ(x7), x6, x7))), Integer(Pos(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Succ(x6)))))))



We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x54, x55, x6, x7)=Succ(x53) which results in the following new constraints:

(5)    (new_primModNatS01(x58, x57)=Succ(x53) & Succ(Succ(x56))=x58 & Succ(Zero)=x57  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x56)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x56)), Succ(Zero), Succ(x56), Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x56))))))))

(6)    (new_primModNatS02(x62, x61, x60, x59)=Succ(x53) & Succ(Succ(x60))=x62 & Succ(Succ(x59))=x61 & (\/x63:new_primModNatS02(x62, x61, x60, x59)=Succ(x63) & Succ(x60)=x62 & Succ(x59)=x61  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x60))))), Integer(Pos(Succ(Succ(Succ(x59))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x60), Succ(x59), x60, x59))), Integer(Pos(Succ(Succ(Succ(x59))))), Integer(Pos(Succ(Succ(Succ(x60)))))))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x60)))))), Integer(Pos(Succ(Succ(Succ(Succ(x59)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x60)), Succ(Succ(x59)), Succ(x60), Succ(x59)))), Integer(Pos(Succ(Succ(Succ(Succ(x59)))))), Integer(Pos(Succ(Succ(Succ(Succ(x60))))))))

(7)    (new_primModNatS01(x65, x64)=Succ(x53) & Succ(Zero)=x65 & Succ(Zero)=x64  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))

(8)    (Succ(Succ(x68))=Succ(x53) & Succ(Zero)=x68 & Succ(Succ(x66))=x67  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x66)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Succ(x66)), Zero, Succ(x66)))), Integer(Pos(Succ(Succ(Succ(Succ(x66)))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))



We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x58, x57)=Succ(x53) which results in the following new constraint:

(9)    (new_primModNatS1(new_primMinusNatS2(Succ(x70), Succ(x69)), Succ(x69))=Succ(x53) & Succ(Succ(x56))=x70 & Succ(Zero)=x69  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x56)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x56)), Succ(Zero), Succ(x56), Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x56))))))))



We simplified constraint (6) using rule (IV) which results in the following new constraint:

(10)    (new_primModNatS02(x62, x61, x60, x59)=Succ(x53) & Succ(Succ(x60))=x62 & Succ(Succ(x59))=x61  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x60)))))), Integer(Pos(Succ(Succ(Succ(Succ(x59)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x60)), Succ(Succ(x59)), Succ(x60), Succ(x59)))), Integer(Pos(Succ(Succ(Succ(Succ(x59)))))), Integer(Pos(Succ(Succ(Succ(Succ(x60))))))))



We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x65, x64)=Succ(x53) which results in the following new constraint:

(11)    (new_primModNatS1(new_primMinusNatS2(Succ(x89), Succ(x88)), Succ(x88))=Succ(x53) & Succ(Zero)=x89 & Succ(Zero)=x88  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))



We simplified constraint (8) using rules (I), (II), (IV) which results in the following new constraint:

(12)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x66)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Succ(x66)), Zero, Succ(x66)))), Integer(Pos(Succ(Succ(Succ(Succ(x66)))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))



We simplified constraint (9) using rules (III), (IV), (VII) which results in the following new constraint:

(13)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x56)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x56)), Succ(Zero), Succ(x56), Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x56))))))))



We simplified constraint (10) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x62, x61, x60, x59)=Succ(x53) which results in the following new constraints:

(14)    (new_primModNatS01(x77, x76)=Succ(x53) & Succ(Succ(Succ(x75)))=x77 & Succ(Succ(Zero))=x76  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x75))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Succ(x75))), Succ(Succ(Zero)), Succ(Succ(x75)), Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x75)))))))))

(15)    (new_primModNatS02(x81, x80, x79, x78)=Succ(x53) & Succ(Succ(Succ(x79)))=x81 & Succ(Succ(Succ(x78)))=x80 & (\/x82:new_primModNatS02(x81, x80, x79, x78)=Succ(x82) & Succ(Succ(x79))=x81 & Succ(Succ(x78))=x80  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x79)))))), Integer(Pos(Succ(Succ(Succ(Succ(x78)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x79)), Succ(Succ(x78)), Succ(x79), Succ(x78)))), Integer(Pos(Succ(Succ(Succ(Succ(x78)))))), Integer(Pos(Succ(Succ(Succ(Succ(x79))))))))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x79))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x78))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Succ(x79))), Succ(Succ(Succ(x78))), Succ(Succ(x79)), Succ(Succ(x78))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x78))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x79)))))))))

(16)    (new_primModNatS01(x84, x83)=Succ(x53) & Succ(Succ(Zero))=x84 & Succ(Succ(Zero))=x83  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))

(17)    (Succ(Succ(x87))=Succ(x53) & Succ(Succ(Zero))=x87 & Succ(Succ(Succ(x85)))=x86  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x85))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x85))), Succ(Zero), Succ(Succ(x85))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x85))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))



We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:

(18)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x75))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Succ(x75))), Succ(Succ(Zero)), Succ(Succ(x75)), Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x75)))))))))



We simplified constraint (15) using rules (III), (IV) which results in the following new constraint:

(19)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x79))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x78))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Succ(x79))), Succ(Succ(Succ(x78))), Succ(Succ(x79)), Succ(Succ(x78))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x78))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x79)))))))))



We simplified constraint (16) using rules (III), (IV) which results in the following new constraint:

(20)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))



We simplified constraint (17) using rules (I), (II), (IV) which results in the following new constraint:

(21)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x85))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x85))), Succ(Zero), Succ(Succ(x85))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x85))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))



We simplified constraint (11) using rules (III), (IV), (VII) which results in the following new constraint:

(22)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))








To summarize, we get the following constraints P__>=_ for the following pairs.

*new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))

*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Succ(x43))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x43))))))), Integer(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x43))), Succ(Zero), Succ(Succ(x43)))))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(x24)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x24)))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Succ(x24)), Zero, Succ(x24))))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x14)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Succ(x14)), Succ(Zero), Succ(x14), Zero)))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x33))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(new_primModNatS02(Succ(Succ(Succ(x33))), Succ(Succ(Zero)), Succ(Succ(x33)), Succ(Zero))))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Succ(x36))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x37))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x36))))))), Integer(Pos(new_primModNatS02(Succ(Succ(Succ(x37))), Succ(Succ(Succ(x36))), Succ(Succ(x37)), Succ(Succ(x36)))))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero))))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))




*new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))

*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x85))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x85))), Succ(Zero), Succ(Succ(x85))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x85))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x66)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Succ(x66)), Zero, Succ(x66)))), Integer(Pos(Succ(Succ(Succ(Succ(x66)))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x56)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x56)), Succ(Zero), Succ(x56), Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x56))))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x75))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Succ(x75))), Succ(Succ(Zero)), Succ(Succ(x75)), Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x75)))))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x79))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x78))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Succ(x79))), Succ(Succ(Succ(x78))), Succ(Succ(x79)), Succ(Succ(x78))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x78))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x79)))))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))








The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
----------------------------------------

(246)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(247)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(248) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(249)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(250) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primRemInt(Neg(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_error
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))


----------------------------------------

(251)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(252) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(x1))), Integer(Pos(x0))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x1))), Integer(Pos(new_primModNatS1(x0, x1)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(x0))), Integer(Pos(Zero))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x0))), Integer(Pos(Zero))),new_gcd0Gcd'10(False, Integer(Neg(Succ(x0))), Integer(Pos(Zero))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x0))), Integer(Pos(Zero))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1)))))


----------------------------------------

(253)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(x0))), Integer(Pos(Zero))) -> new_gcd0Gcd'00(Integer(Neg(Succ(x0))), Integer(Pos(Zero)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(254) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(255)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(256) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Zero, Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Zero, Zero)))))


----------------------------------------

(257)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Zero, Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(258) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(259)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(260) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero)))))


----------------------------------------

(261)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0)))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(262) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(x0)), Integer(Neg(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(x0, x1))), Integer(Neg(Succ(x1))), Integer(Pos(x0))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))),new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))))
   (new_gcd0Gcd'00(Integer(Pos(Zero)), Integer(Neg(Succ(x0)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Zero)), Integer(Neg(Succ(x0))), Integer(Pos(Zero))),new_gcd0Gcd'00(Integer(Pos(Zero)), Integer(Neg(Succ(x0)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Zero)), Integer(Neg(Succ(x0))), Integer(Pos(Zero))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))),new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))))


----------------------------------------

(263)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Zero)), Integer(Neg(Succ(x0)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Zero)), Integer(Neg(Succ(x0))), Integer(Pos(Zero)))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(264) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(265)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(266) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))),new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))))


----------------------------------------

(267)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(268) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))))


----------------------------------------

(269)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(270) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(x0)), Integer(Pos(Succ(x1)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(x0, x1))), Integer(Pos(Succ(x1))), Integer(Neg(x0))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))),new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))))
   (new_gcd0Gcd'00(Integer(Neg(Zero)), Integer(Pos(Succ(x0)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Pos(Succ(x0))), Integer(Neg(Zero))),new_gcd0Gcd'00(Integer(Neg(Zero)), Integer(Pos(Succ(x0)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Pos(Succ(x0))), Integer(Neg(Zero))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))),new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))))


----------------------------------------

(271)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Zero)), Integer(Pos(Succ(x0)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Pos(Succ(x0))), Integer(Neg(Zero)))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(272) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(273)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(274) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))))


----------------------------------------

(275)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(276) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))),new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))))


----------------------------------------

(277)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(278) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))),new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))))


----------------------------------------

(279)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(280) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))) at position [0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))),new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))))


----------------------------------------

(281)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(282) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(283)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(284) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x1)))), Integer(Pos(new_primModNatS02(x0, x1, x0, x1)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Succ(x2), Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Succ(x2), Zero)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Zero, Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Zero, Zero)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))))


----------------------------------------

(285)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Succ(x2), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Zero, Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(286) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Succ(x2), Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))))


----------------------------------------

(287)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Zero, Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(288) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS01(Zero, Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero))))))


----------------------------------------

(289)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(290) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))))


----------------------------------------

(291)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(292) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero))))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero))))))


----------------------------------------

(293)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(294) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero))))))


----------------------------------------

(295)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(296) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero))))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Zero, Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Zero, Succ(Zero))))))


----------------------------------------

(297)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Zero, Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(298) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(299)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(300) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(x1))), Integer(Neg(x0))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x1))), Integer(Neg(new_primModNatS1(x0, x1)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(x0))), Integer(Neg(Zero))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x0))), Integer(Neg(Zero))),new_gcd0Gcd'10(False, Integer(Pos(Succ(x0))), Integer(Neg(Zero))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x0))), Integer(Neg(Zero))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1)))))


----------------------------------------

(301)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(x0))), Integer(Neg(Zero))) -> new_gcd0Gcd'00(Integer(Pos(Succ(x0))), Integer(Neg(Zero)))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(302) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 2 less nodes.
----------------------------------------

(303)
Complex Obligation (AND)

----------------------------------------

(304)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(305) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(306)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(307) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS02(x0, x1, Zero, Zero)


----------------------------------------

(308)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(309) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))))
   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero)))))


----------------------------------------

(310)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(311) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero)))))


----------------------------------------

(312)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(313) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(new_primMinusNatS1, Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Zero, Zero)))),new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Zero, Zero)))))


----------------------------------------

(314)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Zero, Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(315) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(316)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(317) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Zero))))))


----------------------------------------

(318)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(319) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(320)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(321) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))))


----------------------------------------

(322)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(323) QDPSizeChangeProof (EQUIVALENT)
We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 

Order:Polynomial interpretation [POLO]:

   POL(False) = 1
   POL(Integer(x_1)) = x_1
   POL(Neg(x_1)) = 1
   POL(Pos(x_1)) = x_1
   POL(Succ(x_1)) = 1 + x_1
   POL(Zero) = 1
   POL(new_primMinusNatS0(x_1)) = 1 + x_1
   POL(new_primMinusNatS1) = 1
   POL(new_primModNatS1(x_1, x_2)) = x_1




From the DPs we obtained the following set of size-change graphs:
*new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) (allowed arguments on rhs = {2, 3})
The graph contains the following edges 2 >= 2, 1 >= 3


*new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x0))))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x0), Zero))), Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) (allowed arguments on rhs = {2, 3})
The graph contains the following edges 2 >= 2, 1 >= 3


*new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Zero)))) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 >= 1, 3 >= 2


*new_gcd0Gcd'10(False, Integer(Neg(Succ(Zero))), Integer(Pos(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Zero))), Integer(Pos(new_primModNatS1(Succ(x0), Zero)))) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 >= 1, 3 > 2



We oriented the following set of usable rules [AAECC05,FROCOS05].

   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

----------------------------------------

(324)
YES

----------------------------------------

(325)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(326) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(327)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(328) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS02(x0, x1, Zero, Zero)


----------------------------------------

(329)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(330) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero)))))


----------------------------------------

(331)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(332) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(333)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(334) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))))


----------------------------------------

(335)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(336) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS1, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(337)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(338) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))) at position [0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(339)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(340) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(341)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0))))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(342) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero)))))


----------------------------------------

(343)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS1, Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(344) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(345)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0))))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(346) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero)))))


----------------------------------------

(347)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primModNatS1(Zero, x0)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(348) QDPSizeChangeProof (EQUIVALENT)
We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 

Order:Polynomial interpretation [POLO]:

   POL(False) = 1
   POL(Integer(x_1)) = x_1
   POL(Neg(x_1)) = x_1
   POL(Pos(x_1)) = 1
   POL(Succ(x_1)) = 1 + x_1
   POL(Zero) = 1
   POL(new_primMinusNatS0(x_1)) = 1 + x_1
   POL(new_primMinusNatS1) = 1
   POL(new_primModNatS1(x_1, x_2)) = x_1




From the DPs we obtained the following set of size-change graphs:
*new_gcd0Gcd'10(False, Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(new_primModNatS1(Succ(x0), Zero)))) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 >= 1, 3 > 2


*new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 >= 1, 3 >= 2


*new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x0))))), Integer(Pos(Succ(Zero)))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x0), Zero))), Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(Succ(x0)))))) (allowed arguments on rhs = {2, 3})
The graph contains the following edges 2 >= 2, 1 >= 3


*new_gcd0Gcd'00(Integer(Pos(Succ(Zero))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Zero)))) (allowed arguments on rhs = {2, 3})
The graph contains the following edges 2 >= 2, 1 >= 3



We oriented the following set of usable rules [AAECC05,FROCOS05].

   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

----------------------------------------

(349)
YES

----------------------------------------

(350)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(351) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(352)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(353) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primMinusNatS0(x0)
   new_primMinusNatS1


----------------------------------------

(354)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(355) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(x0)))), Integer(Pos(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(x0, x1, x0, x1))), Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Succ(x2), Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Succ(x2), Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Zero, Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Zero, Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))))
   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(356)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Succ(x2), Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Zero, Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(357) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Succ(x2), Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) at position [0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))))


----------------------------------------

(358)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Zero, Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(359) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS01(Zero, Zero))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) at position [0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(360)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(361) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(362)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(363) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))))


----------------------------------------

(364)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(365) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(366)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(367) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))))


----------------------------------------

(368)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(369) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(370)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(371) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Zero, Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) at position [0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(372)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(Zero)), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(373) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(374)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(375) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(x1)))), Integer(Neg(Succ(Succ(x0))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x1)))), Integer(Neg(new_primModNatS02(x0, x1, x0, x1)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Succ(x2), Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Succ(x2), Zero)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Zero, Zero)))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Zero, Zero)))))
   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))))


----------------------------------------

(376)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Succ(x2), Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Zero, Zero))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(377) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(378)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Succ(x2), Zero))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(379) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS01(Succ(x2), Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))))


----------------------------------------

(380)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(381) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))))


----------------------------------------

(382)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(383) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero))))),new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero))))))


----------------------------------------

(384)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(385) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(x0)))), Integer(Neg(Succ(Succ(x1))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(x0, x1, x0, x1))), Integer(Neg(Succ(Succ(x1)))), Integer(Pos(Succ(Succ(x0))))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Succ(x2), Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Succ(x2), Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Zero, Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Zero, Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero))))))
   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))))


----------------------------------------

(386)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Succ(x2), Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Zero, Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(387) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 1 less node.
----------------------------------------

(388)
Complex Obligation (AND)

----------------------------------------

(389)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Succ(x2), Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(390) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(391)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Succ(x2), Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(392) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS01(Succ(x2), Zero))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) at position [0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))))


----------------------------------------

(393)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(394) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))))


----------------------------------------

(395)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(396) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))))


----------------------------------------

(397)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(398) InductionCalculusProof (EQUIVALENT)
Note that final constraints are written in bold face.



For Pair new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x1)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Neg(Succ(Succ(Zero))))), new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Neg(Succ(Succ(Zero)))))=new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x1))))))_>=_new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Neg(Succ(Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x1))))))_>=_new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Neg(Succ(Succ(Zero))))))








For Pair new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Neg(Succ(Succ(Zero))))), new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x9))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x9), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x9)))))) which results in the following constraint:

(1)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Neg(Succ(Succ(Zero)))))=new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x9))))), Integer(Neg(Succ(Succ(Zero)))))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Neg(Succ(Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Neg(Succ(Succ(Zero))))))








For Pair new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x10)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x10), Succ(Zero))))), new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x11)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x11))))), Integer(Neg(Succ(Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x10), Succ(Zero)))))=new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x11))))))  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x10))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x10), Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (VII) which results in the following new constraint:

(2)    (Succ(x10)=x20 & Succ(Zero)=x21 & new_primModNatS1(x20, x21)=Succ(Succ(Succ(x11)))  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x10))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x10), Succ(Zero))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS1(x20, x21)=Succ(Succ(Succ(x11))) which results in the following new constraints:

(3)    (Succ(Zero)=Succ(Succ(Succ(x11))) & Succ(x10)=Succ(Zero) & Succ(Zero)=Succ(x22)  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x10))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x10), Succ(Zero))))))

(4)    (new_primModNatS02(x24, x23, x24, x23)=Succ(Succ(Succ(x11))) & Succ(x10)=Succ(Succ(x24)) & Succ(Zero)=Succ(x23)  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x10))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x10), Succ(Zero))))))



We solved constraint (3) using rules (I), (II).We simplified constraint (4) using rules (I), (II), (III), (VII) which results in the following new constraint:

(5)    (x24=x26 & x23=x27 & new_primModNatS02(x24, x23, x26, x27)=Succ(Succ(Succ(x11))) & Zero=x23  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x24)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(x24)), Succ(Zero))))))



We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x24, x23, x26, x27)=Succ(Succ(Succ(x11))) which results in the following new constraints:

(6)    (new_primModNatS01(x30, x29)=Succ(Succ(Succ(x11))) & x30=Succ(x28) & x29=Zero & Zero=x29  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x30)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(x30)), Succ(Zero))))))

(7)    (new_primModNatS02(x34, x33, x32, x31)=Succ(Succ(Succ(x11))) & x34=Succ(x32) & x33=Succ(x31) & Zero=x33 & (\/x35:new_primModNatS02(x34, x33, x32, x31)=Succ(Succ(Succ(x35))) & x34=x32 & x33=x31 & Zero=x33  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x34)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(x34)), Succ(Zero))))))  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x34)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(x34)), Succ(Zero))))))

(8)    (new_primModNatS01(x37, x36)=Succ(Succ(Succ(x11))) & x37=Zero & x36=Zero & Zero=x36  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x37)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(x37)), Succ(Zero))))))

(9)    (Succ(Succ(x40))=Succ(Succ(Succ(x11))) & x40=Zero & x39=Succ(x38) & Zero=x39  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x40)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(x40)), Succ(Zero))))))



We simplified constraint (6) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(10)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x28))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(Succ(x28))), Succ(Zero))))))



We solved constraint (7) using rules (I), (II), (III).We simplified constraint (8) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(11)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(Zero)), Succ(Zero))))))



We solved constraint (9) using rules (I), (II), (III).




For Pair new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) the following chains were created:
*We consider the chain new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x17))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x17), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x17)))))), new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x18)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x18), Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x17), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x17))))))=new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x18))))))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x17))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x17), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x17)))))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Pos(new_primModNatS1(Succ(x17), Succ(Zero)))=x45 & new_primEqInt(x45)=False  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x17))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x17), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x17)))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primEqInt(x45)=False which results in the following new constraint:

(3)    (False=False & Pos(new_primModNatS1(Succ(x17), Succ(Zero)))=Pos(Succ(x46))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x17))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x17), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x17)))))))



We simplified constraint (3) using rules (I), (II), (VII) which results in the following new constraint:

(4)    (Succ(x17)=x47 & Succ(Zero)=x48 & new_primModNatS1(x47, x48)=Succ(x46)  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x17))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x17), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x17)))))))



We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS1(x47, x48)=Succ(x46) which results in the following new constraints:

(5)    (Succ(Zero)=Succ(x46) & Succ(x17)=Succ(Zero) & Succ(Zero)=Succ(x49)  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x17))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x17), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x17)))))))

(6)    (new_primModNatS02(x51, x50, x51, x50)=Succ(x46) & Succ(x17)=Succ(Succ(x51)) & Succ(Zero)=Succ(x50)  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x17))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x17), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x17)))))))



We simplified constraint (5) using rules (I), (II), (III), (IV) which results in the following new constraint:

(7)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Zero)))))))



We simplified constraint (6) using rules (I), (II), (III), (VII) which results in the following new constraint:

(8)    (x51=x53 & x50=x54 & new_primModNatS02(x51, x50, x53, x54)=Succ(x46) & Zero=x50  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x51)))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(x51)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x51))))))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x51, x50, x53, x54)=Succ(x46) which results in the following new constraints:

(9)    (new_primModNatS01(x57, x56)=Succ(x46) & x57=Succ(x55) & x56=Zero & Zero=x56  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x57)))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(x57)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x57))))))))

(10)    (new_primModNatS02(x61, x60, x59, x58)=Succ(x46) & x61=Succ(x59) & x60=Succ(x58) & Zero=x60 & (\/x62:new_primModNatS02(x61, x60, x59, x58)=Succ(x62) & x61=x59 & x60=x58 & Zero=x60  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x61)))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(x61)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x61))))))))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x61)))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(x61)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x61))))))))

(11)    (new_primModNatS01(x64, x63)=Succ(x46) & x64=Zero & x63=Zero & Zero=x63  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x64)))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(x64)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x64))))))))

(12)    (Succ(Succ(x67))=Succ(x46) & x67=Zero & x66=Succ(x65) & Zero=x66  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x67)))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(x67)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(x67))))))))



We simplified constraint (9) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(13)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x55))))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(Succ(x55))), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x55)))))))))



We solved constraint (10) using rules (I), (II), (III).We simplified constraint (11) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(14)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))



We solved constraint (12) using rules (I), (II), (III), (IV).




To summarize, we get the following constraints P__>=_ for the following pairs.

*new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))

*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x1))))))_>=_new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x1))))), Integer(Neg(Succ(Succ(Zero))))))




*new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))

*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x8))))), Integer(Neg(Succ(Succ(Zero))))))




*new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))

*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x28))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(Succ(x28))), Succ(Zero))))))


*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(Succ(Zero)), Succ(Zero))))))




*new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))

*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Zero)))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x55))))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(Succ(x55))), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x55)))))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(Succ(Zero)), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))








The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
----------------------------------------

(399)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Neg(Succ(Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(400)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(401) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(402)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(403) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))),new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))))


----------------------------------------

(404)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(405) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(406)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(407) InductionCalculusProof (EQUIVALENT)
Note that final constraints are written in bold face.



For Pair new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x3), Succ(Zero))))), new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x4)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x4))))), Integer(Pos(Succ(Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x3), Succ(Zero)))))=new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x4))))))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x3))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x3), Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (VII) which results in the following new constraint:

(2)    (Succ(x3)=x20 & Succ(Zero)=x21 & new_primModNatS1(x20, x21)=Succ(Succ(Succ(x4)))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x3))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x3), Succ(Zero))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS1(x20, x21)=Succ(Succ(Succ(x4))) which results in the following new constraints:

(3)    (Succ(Zero)=Succ(Succ(Succ(x4))) & Succ(x3)=Succ(Zero) & Succ(Zero)=Succ(x22)  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x3))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x3), Succ(Zero))))))

(4)    (new_primModNatS02(x24, x23, x24, x23)=Succ(Succ(Succ(x4))) & Succ(x3)=Succ(Succ(x24)) & Succ(Zero)=Succ(x23)  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x3))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x3), Succ(Zero))))))



We solved constraint (3) using rules (I), (II).We simplified constraint (4) using rules (I), (II), (III), (VII) which results in the following new constraint:

(5)    (x24=x26 & x23=x27 & new_primModNatS02(x24, x23, x26, x27)=Succ(Succ(Succ(x4))) & Zero=x23  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(x24)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(Succ(x24)), Succ(Zero))))))



We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x24, x23, x26, x27)=Succ(Succ(Succ(x4))) which results in the following new constraints:

(6)    (new_primModNatS01(x30, x29)=Succ(Succ(Succ(x4))) & x30=Succ(x28) & x29=Zero & Zero=x29  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(x30)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(Succ(x30)), Succ(Zero))))))

(7)    (new_primModNatS02(x34, x33, x32, x31)=Succ(Succ(Succ(x4))) & x34=Succ(x32) & x33=Succ(x31) & Zero=x33 & (\/x35:new_primModNatS02(x34, x33, x32, x31)=Succ(Succ(Succ(x35))) & x34=x32 & x33=x31 & Zero=x33  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(x34)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(Succ(x34)), Succ(Zero))))))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(x34)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(Succ(x34)), Succ(Zero))))))

(8)    (new_primModNatS01(x37, x36)=Succ(Succ(Succ(x4))) & x37=Zero & x36=Zero & Zero=x36  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(x37)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(Succ(x37)), Succ(Zero))))))

(9)    (Succ(Succ(x40))=Succ(Succ(Succ(x4))) & x40=Zero & x39=Succ(x38) & Zero=x39  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(x40)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(Succ(x40)), Succ(Zero))))))



We simplified constraint (6) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(10)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x28))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(Succ(Succ(x28))), Succ(Zero))))))



We solved constraint (7) using rules (I), (II), (III).We simplified constraint (8) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(11)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(Succ(Zero)), Succ(Zero))))))



We solved constraint (9) using rules (I), (II), (III).




For Pair new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Zero))))), new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x8))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x8), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x8)))))) which results in the following constraint:

(1)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Zero)))))=new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x8))))), Integer(Pos(Succ(Succ(Zero)))))  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Zero))))))








For Pair new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) the following chains were created:
*We consider the chain new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x10))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x10), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x10)))))), new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x11)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x11), Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x10), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x10))))))=new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x11))))))  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x10))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x10), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x10)))))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Neg(new_primModNatS1(Succ(x10), Succ(Zero)))=x45 & new_primEqInt(x45)=False  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x10))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x10), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x10)))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primEqInt(x45)=False which results in the following new constraint:

(3)    (False=False & Neg(new_primModNatS1(Succ(x10), Succ(Zero)))=Neg(Succ(x46))  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x10))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x10), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x10)))))))



We simplified constraint (3) using rules (I), (II), (VII) which results in the following new constraint:

(4)    (Succ(x10)=x47 & Succ(Zero)=x48 & new_primModNatS1(x47, x48)=Succ(x46)  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x10))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x10), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x10)))))))



We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS1(x47, x48)=Succ(x46) which results in the following new constraints:

(5)    (Succ(Zero)=Succ(x46) & Succ(x10)=Succ(Zero) & Succ(Zero)=Succ(x49)  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x10))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x10), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x10)))))))

(6)    (new_primModNatS02(x51, x50, x51, x50)=Succ(x46) & Succ(x10)=Succ(Succ(x51)) & Succ(Zero)=Succ(x50)  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x10))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x10), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x10)))))))



We simplified constraint (5) using rules (I), (II), (III), (IV) which results in the following new constraint:

(7)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Zero)))))))



We simplified constraint (6) using rules (I), (II), (III), (VII) which results in the following new constraint:

(8)    (x51=x53 & x50=x54 & new_primModNatS02(x51, x50, x53, x54)=Succ(x46) & Zero=x50  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x51)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(Succ(x51)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(x51))))))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x51, x50, x53, x54)=Succ(x46) which results in the following new constraints:

(9)    (new_primModNatS01(x57, x56)=Succ(x46) & x57=Succ(x55) & x56=Zero & Zero=x56  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x57)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(Succ(x57)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(x57))))))))

(10)    (new_primModNatS02(x61, x60, x59, x58)=Succ(x46) & x61=Succ(x59) & x60=Succ(x58) & Zero=x60 & (\/x62:new_primModNatS02(x61, x60, x59, x58)=Succ(x62) & x61=x59 & x60=x58 & Zero=x60  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x61)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(Succ(x61)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(x61))))))))  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x61)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(Succ(x61)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(x61))))))))

(11)    (new_primModNatS01(x64, x63)=Succ(x46) & x64=Zero & x63=Zero & Zero=x63  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x64)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(Succ(x64)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(x64))))))))

(12)    (Succ(Succ(x67))=Succ(x46) & x67=Zero & x66=Succ(x65) & Zero=x66  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x67)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(Succ(x67)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(x67))))))))



We simplified constraint (9) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(13)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x55))))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(Succ(Succ(x55))), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x55)))))))))



We solved constraint (10) using rules (I), (II), (III).We simplified constraint (11) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(14)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))



We solved constraint (12) using rules (I), (II), (III), (IV).




For Pair new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x16)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x16))))), Integer(Pos(Succ(Succ(Zero))))), new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x17))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x17))))), Integer(Pos(Succ(Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x16))))), Integer(Pos(Succ(Succ(Zero)))))=new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x17))))), Integer(Pos(Succ(Succ(Zero)))))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x16))))))_>=_new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x16))))), Integer(Pos(Succ(Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x16))))))_>=_new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x16))))), Integer(Pos(Succ(Succ(Zero))))))








To summarize, we get the following constraints P__>=_ for the following pairs.

*new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))))

*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x28))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(Succ(Succ(x28))), Succ(Zero))))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(Succ(Zero)), Succ(Zero))))))




*new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))

*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x7))))), Integer(Pos(Succ(Succ(Zero))))))




*new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))

*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Zero)))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x55))))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(Succ(Succ(x55))), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x55)))))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(Succ(Zero)), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))




*new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))

*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x16))))))_>=_new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x16))))), Integer(Pos(Succ(Succ(Zero))))))








The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
----------------------------------------

(408)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS1(Succ(x2), Succ(Zero)))), Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(409)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(410) InductionCalculusProof (EQUIVALENT)
Note that final constraints are written in bold face.



For Pair new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) the following chains were created:
*We consider the chain new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))), new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x4))))), Integer(Pos(Succ(Succ(Succ(x5)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x4))))), Integer(Pos(new_primModNatS02(Succ(x5), Succ(x4), x5, x4)))) which results in the following constraint:

(1)    (new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))=new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x4))))), Integer(Pos(Succ(Succ(Succ(x5))))))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))=x40 & new_primEqInt(x40)=False  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primEqInt(x40)=False which results in the following new constraints:

(3)    (False=False & Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))=Pos(Succ(x41))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))))

(4)    (False=False & Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))=Neg(Succ(x42))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))))



We simplified constraint (3) using rules (I), (II), (VII) which results in the following new constraint:

(5)    (Succ(x2)=x43 & Succ(x3)=x44 & new_primModNatS02(x43, x44, x2, x3)=Succ(x41)  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))))



We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x43, x44, x2, x3)=Succ(x41) which results in the following new constraints:

(6)    (new_primModNatS01(x47, x46)=Succ(x41) & Succ(Succ(x45))=x47 & Succ(Zero)=x46  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x45)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x45)), Succ(Zero), Succ(x45), Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x45))))))))

(7)    (new_primModNatS02(x51, x50, x49, x48)=Succ(x41) & Succ(Succ(x49))=x51 & Succ(Succ(x48))=x50 & (\/x52:new_primModNatS02(x51, x50, x49, x48)=Succ(x52) & Succ(x49)=x51 & Succ(x48)=x50  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x49))))), Integer(Neg(Succ(Succ(Succ(x48))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x49), Succ(x48), x49, x48))), Integer(Neg(Succ(Succ(Succ(x48))))), Integer(Pos(Succ(Succ(Succ(x49)))))))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x49)))))), Integer(Neg(Succ(Succ(Succ(Succ(x48)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x49)), Succ(Succ(x48)), Succ(x49), Succ(x48)))), Integer(Neg(Succ(Succ(Succ(Succ(x48)))))), Integer(Pos(Succ(Succ(Succ(Succ(x49))))))))

(8)    (new_primModNatS01(x54, x53)=Succ(x41) & Succ(Zero)=x54 & Succ(Zero)=x53  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))

(9)    (Succ(Succ(x57))=Succ(x41) & Succ(Zero)=x57 & Succ(Succ(x55))=x56  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x55)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Succ(x55)), Zero, Succ(x55)))), Integer(Neg(Succ(Succ(Succ(Succ(x55)))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))



We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x47, x46)=Succ(x41) which results in the following new constraint:

(10)    (new_primModNatS1(new_primMinusNatS2(Succ(x59), Succ(x58)), Succ(x58))=Succ(x41) & Succ(Succ(x45))=x59 & Succ(Zero)=x58  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x45)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x45)), Succ(Zero), Succ(x45), Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x45))))))))



We simplified constraint (7) using rule (IV) which results in the following new constraint:

(11)    (new_primModNatS02(x51, x50, x49, x48)=Succ(x41) & Succ(Succ(x49))=x51 & Succ(Succ(x48))=x50  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x49)))))), Integer(Neg(Succ(Succ(Succ(Succ(x48)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x49)), Succ(Succ(x48)), Succ(x49), Succ(x48)))), Integer(Neg(Succ(Succ(Succ(Succ(x48)))))), Integer(Pos(Succ(Succ(Succ(Succ(x49))))))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x54, x53)=Succ(x41) which results in the following new constraint:

(12)    (new_primModNatS1(new_primMinusNatS2(Succ(x78), Succ(x77)), Succ(x77))=Succ(x41) & Succ(Zero)=x78 & Succ(Zero)=x77  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))



We simplified constraint (9) using rules (I), (II), (IV) which results in the following new constraint:

(13)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x55)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Succ(x55)), Zero, Succ(x55)))), Integer(Neg(Succ(Succ(Succ(Succ(x55)))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))



We simplified constraint (10) using rules (III), (IV), (VII) which results in the following new constraint:

(14)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x45)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x45)), Succ(Zero), Succ(x45), Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x45))))))))



We simplified constraint (11) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x51, x50, x49, x48)=Succ(x41) which results in the following new constraints:

(15)    (new_primModNatS01(x66, x65)=Succ(x41) & Succ(Succ(Succ(x64)))=x66 & Succ(Succ(Zero))=x65  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x64))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Succ(x64))), Succ(Succ(Zero)), Succ(Succ(x64)), Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x64)))))))))

(16)    (new_primModNatS02(x70, x69, x68, x67)=Succ(x41) & Succ(Succ(Succ(x68)))=x70 & Succ(Succ(Succ(x67)))=x69 & (\/x71:new_primModNatS02(x70, x69, x68, x67)=Succ(x71) & Succ(Succ(x68))=x70 & Succ(Succ(x67))=x69  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x68)))))), Integer(Neg(Succ(Succ(Succ(Succ(x67)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x68)), Succ(Succ(x67)), Succ(x68), Succ(x67)))), Integer(Neg(Succ(Succ(Succ(Succ(x67)))))), Integer(Pos(Succ(Succ(Succ(Succ(x68))))))))  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x68))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x67))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Succ(x68))), Succ(Succ(Succ(x67))), Succ(Succ(x68)), Succ(Succ(x67))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x67))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x68)))))))))

(17)    (new_primModNatS01(x73, x72)=Succ(x41) & Succ(Succ(Zero))=x73 & Succ(Succ(Zero))=x72  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))

(18)    (Succ(Succ(x76))=Succ(x41) & Succ(Succ(Zero))=x76 & Succ(Succ(Succ(x74)))=x75  ==>  new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x74))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x74))), Succ(Zero), Succ(Succ(x74))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x74))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))



We simplified constraint (15) using rules (III), (IV) which results in the following new constraint:

(19)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x64))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Succ(x64))), Succ(Succ(Zero)), Succ(Succ(x64)), Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x64)))))))))



We simplified constraint (16) using rules (III), (IV) which results in the following new constraint:

(20)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x68))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x67))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Succ(x68))), Succ(Succ(Succ(x67))), Succ(Succ(x68)), Succ(Succ(x67))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x67))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x68)))))))))



We simplified constraint (17) using rules (III), (IV) which results in the following new constraint:

(21)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))



We simplified constraint (18) using rules (I), (II), (IV) which results in the following new constraint:

(22)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x74))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x74))), Succ(Zero), Succ(Succ(x74))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x74))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))



We simplified constraint (12) using rules (III), (IV), (VII) which results in the following new constraint:

(23)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))








For Pair new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))) the following chains were created:
*We consider the chain new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x14))))), Integer(Pos(Succ(Succ(Succ(x15)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x14))))), Integer(Pos(new_primModNatS02(Succ(x15), Succ(x14), x15, x14)))), new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x16))))), Integer(Pos(Succ(Succ(Succ(x17)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x16), Succ(x17), x16, x17))), Integer(Pos(Succ(Succ(Succ(x17))))), Integer(Neg(Succ(Succ(Succ(x16)))))) which results in the following constraint:

(1)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x14))))), Integer(Pos(new_primModNatS02(Succ(x15), Succ(x14), x15, x14))))=new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x16))))), Integer(Pos(Succ(Succ(Succ(x17))))))  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x14))))), Integer(Pos(Succ(Succ(Succ(x15))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x14))))), Integer(Pos(new_primModNatS02(Succ(x15), Succ(x14), x15, x14)))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Succ(x15)=x83 & Succ(x14)=x84 & new_primModNatS02(x83, x84, x15, x14)=Succ(Succ(Succ(x17)))  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x14))))), Integer(Pos(Succ(Succ(Succ(x15))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x14))))), Integer(Pos(new_primModNatS02(Succ(x15), Succ(x14), x15, x14)))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x83, x84, x15, x14)=Succ(Succ(Succ(x17))) which results in the following new constraints:

(3)    (new_primModNatS01(x87, x86)=Succ(Succ(Succ(x17))) & Succ(Succ(x85))=x87 & Succ(Zero)=x86  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x85)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Succ(x85)), Succ(Zero), Succ(x85), Zero)))))

(4)    (new_primModNatS02(x91, x90, x89, x88)=Succ(Succ(Succ(x17))) & Succ(Succ(x89))=x91 & Succ(Succ(x88))=x90 & (\/x92:new_primModNatS02(x91, x90, x89, x88)=Succ(Succ(Succ(x92))) & Succ(x89)=x91 & Succ(x88)=x90  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x88))))), Integer(Pos(Succ(Succ(Succ(x89))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x88))))), Integer(Pos(new_primModNatS02(Succ(x89), Succ(x88), x89, x88)))))  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(x88)))))), Integer(Pos(Succ(Succ(Succ(Succ(x89)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x88)))))), Integer(Pos(new_primModNatS02(Succ(Succ(x89)), Succ(Succ(x88)), Succ(x89), Succ(x88))))))

(5)    (new_primModNatS01(x94, x93)=Succ(Succ(Succ(x17))) & Succ(Zero)=x94 & Succ(Zero)=x93  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))

(6)    (Succ(Succ(x97))=Succ(Succ(Succ(x17))) & Succ(Zero)=x97 & Succ(Succ(x95))=x96  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(x95)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x95)))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Succ(x95)), Zero, Succ(x95))))))



We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x87, x86)=Succ(Succ(Succ(x17))) which results in the following new constraint:

(7)    (new_primModNatS1(new_primMinusNatS2(Succ(x99), Succ(x98)), Succ(x98))=Succ(Succ(Succ(x17))) & Succ(Succ(x85))=x99 & Succ(Zero)=x98  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x85)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Succ(x85)), Succ(Zero), Succ(x85), Zero)))))



We simplified constraint (4) using rule (IV) which results in the following new constraint:

(8)    (new_primModNatS02(x91, x90, x89, x88)=Succ(Succ(Succ(x17))) & Succ(Succ(x89))=x91 & Succ(Succ(x88))=x90  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(x88)))))), Integer(Pos(Succ(Succ(Succ(Succ(x89)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x88)))))), Integer(Pos(new_primModNatS02(Succ(Succ(x89)), Succ(Succ(x88)), Succ(x89), Succ(x88))))))



We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x94, x93)=Succ(Succ(Succ(x17))) which results in the following new constraint:

(9)    (new_primModNatS1(new_primMinusNatS2(Succ(x118), Succ(x117)), Succ(x117))=Succ(Succ(Succ(x17))) & Succ(Zero)=x118 & Succ(Zero)=x117  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))



We simplified constraint (6) using rules (I), (II), (III), (IV) which results in the following new constraint:

(10)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(x95)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x95)))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Succ(x95)), Zero, Succ(x95))))))



We simplified constraint (7) using rules (III), (IV), (VII) which results in the following new constraint:

(11)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x85)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Succ(x85)), Succ(Zero), Succ(x85), Zero)))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x91, x90, x89, x88)=Succ(Succ(Succ(x17))) which results in the following new constraints:

(12)    (new_primModNatS01(x106, x105)=Succ(Succ(Succ(x17))) & Succ(Succ(Succ(x104)))=x106 & Succ(Succ(Zero))=x105  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x104))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(new_primModNatS02(Succ(Succ(Succ(x104))), Succ(Succ(Zero)), Succ(Succ(x104)), Succ(Zero))))))

(13)    (new_primModNatS02(x110, x109, x108, x107)=Succ(Succ(Succ(x17))) & Succ(Succ(Succ(x108)))=x110 & Succ(Succ(Succ(x107)))=x109 & (\/x111:new_primModNatS02(x110, x109, x108, x107)=Succ(Succ(Succ(x111))) & Succ(Succ(x108))=x110 & Succ(Succ(x107))=x109  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(x107)))))), Integer(Pos(Succ(Succ(Succ(Succ(x108)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x107)))))), Integer(Pos(new_primModNatS02(Succ(Succ(x108)), Succ(Succ(x107)), Succ(x108), Succ(x107))))))  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Succ(x107))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x108))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x107))))))), Integer(Pos(new_primModNatS02(Succ(Succ(Succ(x108))), Succ(Succ(Succ(x107))), Succ(Succ(x108)), Succ(Succ(x107)))))))

(14)    (new_primModNatS01(x113, x112)=Succ(Succ(Succ(x17))) & Succ(Succ(Zero))=x113 & Succ(Succ(Zero))=x112  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero))))))

(15)    (Succ(Succ(x116))=Succ(Succ(Succ(x17))) & Succ(Succ(Zero))=x116 & Succ(Succ(Succ(x114)))=x115  ==>  new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Succ(x114))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x114))))))), Integer(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x114))), Succ(Zero), Succ(Succ(x114)))))))



We simplified constraint (12) using rules (III), (IV) which results in the following new constraint:

(16)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x104))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(new_primModNatS02(Succ(Succ(Succ(x104))), Succ(Succ(Zero)), Succ(Succ(x104)), Succ(Zero))))))



We simplified constraint (13) using rules (III), (IV) which results in the following new constraint:

(17)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Succ(x107))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x108))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x107))))))), Integer(Pos(new_primModNatS02(Succ(Succ(Succ(x108))), Succ(Succ(Succ(x107))), Succ(Succ(x108)), Succ(Succ(x107)))))))



We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:

(18)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero))))))



We simplified constraint (15) using rules (I), (II), (III), (IV) which results in the following new constraint:

(19)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Succ(x114))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x114))))))), Integer(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x114))), Succ(Zero), Succ(Succ(x114)))))))



We simplified constraint (9) using rules (III), (IV), (VII) which results in the following new constraint:

(20)    (new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))








For Pair new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) the following chains were created:
*We consider the chain new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x26))))), Integer(Pos(Succ(Succ(Succ(x27)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x26), Succ(x27), x26, x27))), Integer(Pos(Succ(Succ(Succ(x27))))), Integer(Neg(Succ(Succ(Succ(x26)))))), new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x28))))), Integer(Neg(Succ(Succ(Succ(x29)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x28))))), Integer(Neg(new_primModNatS02(Succ(x29), Succ(x28), x29, x28)))) which results in the following constraint:

(1)    (new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x26), Succ(x27), x26, x27))), Integer(Pos(Succ(Succ(Succ(x27))))), Integer(Neg(Succ(Succ(Succ(x26))))))=new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x28))))), Integer(Neg(Succ(Succ(Succ(x29))))))  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x26))))), Integer(Pos(Succ(Succ(Succ(x27))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x26), Succ(x27), x26, x27))), Integer(Pos(Succ(Succ(Succ(x27))))), Integer(Neg(Succ(Succ(Succ(x26)))))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Neg(new_primModNatS02(Succ(x26), Succ(x27), x26, x27))=x123 & new_primEqInt(x123)=False  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x26))))), Integer(Pos(Succ(Succ(Succ(x27))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x26), Succ(x27), x26, x27))), Integer(Pos(Succ(Succ(Succ(x27))))), Integer(Neg(Succ(Succ(Succ(x26)))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primEqInt(x123)=False which results in the following new constraints:

(3)    (False=False & Neg(new_primModNatS02(Succ(x26), Succ(x27), x26, x27))=Pos(Succ(x124))  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x26))))), Integer(Pos(Succ(Succ(Succ(x27))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x26), Succ(x27), x26, x27))), Integer(Pos(Succ(Succ(Succ(x27))))), Integer(Neg(Succ(Succ(Succ(x26)))))))

(4)    (False=False & Neg(new_primModNatS02(Succ(x26), Succ(x27), x26, x27))=Neg(Succ(x125))  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x26))))), Integer(Pos(Succ(Succ(Succ(x27))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x26), Succ(x27), x26, x27))), Integer(Pos(Succ(Succ(Succ(x27))))), Integer(Neg(Succ(Succ(Succ(x26)))))))



We solved constraint (3) using rules (I), (II).We simplified constraint (4) using rules (I), (II), (VII) which results in the following new constraint:

(5)    (Succ(x26)=x126 & Succ(x27)=x127 & new_primModNatS02(x126, x127, x26, x27)=Succ(x125)  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x26))))), Integer(Pos(Succ(Succ(Succ(x27))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x26), Succ(x27), x26, x27))), Integer(Pos(Succ(Succ(Succ(x27))))), Integer(Neg(Succ(Succ(Succ(x26)))))))



We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x126, x127, x26, x27)=Succ(x125) which results in the following new constraints:

(6)    (new_primModNatS01(x130, x129)=Succ(x125) & Succ(Succ(x128))=x130 & Succ(Zero)=x129  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x128)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x128)), Succ(Zero), Succ(x128), Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x128))))))))

(7)    (new_primModNatS02(x134, x133, x132, x131)=Succ(x125) & Succ(Succ(x132))=x134 & Succ(Succ(x131))=x133 & (\/x135:new_primModNatS02(x134, x133, x132, x131)=Succ(x135) & Succ(x132)=x134 & Succ(x131)=x133  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x132))))), Integer(Pos(Succ(Succ(Succ(x131))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x132), Succ(x131), x132, x131))), Integer(Pos(Succ(Succ(Succ(x131))))), Integer(Neg(Succ(Succ(Succ(x132)))))))  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x132)))))), Integer(Pos(Succ(Succ(Succ(Succ(x131)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x132)), Succ(Succ(x131)), Succ(x132), Succ(x131)))), Integer(Pos(Succ(Succ(Succ(Succ(x131)))))), Integer(Neg(Succ(Succ(Succ(Succ(x132))))))))

(8)    (new_primModNatS01(x137, x136)=Succ(x125) & Succ(Zero)=x137 & Succ(Zero)=x136  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))

(9)    (Succ(Succ(x140))=Succ(x125) & Succ(Zero)=x140 & Succ(Succ(x138))=x139  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x138)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Succ(x138)), Zero, Succ(x138)))), Integer(Pos(Succ(Succ(Succ(Succ(x138)))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))



We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x130, x129)=Succ(x125) which results in the following new constraint:

(10)    (new_primModNatS1(new_primMinusNatS2(Succ(x142), Succ(x141)), Succ(x141))=Succ(x125) & Succ(Succ(x128))=x142 & Succ(Zero)=x141  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x128)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x128)), Succ(Zero), Succ(x128), Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x128))))))))



We simplified constraint (7) using rule (IV) which results in the following new constraint:

(11)    (new_primModNatS02(x134, x133, x132, x131)=Succ(x125) & Succ(Succ(x132))=x134 & Succ(Succ(x131))=x133  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x132)))))), Integer(Pos(Succ(Succ(Succ(Succ(x131)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x132)), Succ(Succ(x131)), Succ(x132), Succ(x131)))), Integer(Pos(Succ(Succ(Succ(Succ(x131)))))), Integer(Neg(Succ(Succ(Succ(Succ(x132))))))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x137, x136)=Succ(x125) which results in the following new constraint:

(12)    (new_primModNatS1(new_primMinusNatS2(Succ(x161), Succ(x160)), Succ(x160))=Succ(x125) & Succ(Zero)=x161 & Succ(Zero)=x160  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))



We simplified constraint (9) using rules (I), (II), (IV) which results in the following new constraint:

(13)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x138)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Succ(x138)), Zero, Succ(x138)))), Integer(Pos(Succ(Succ(Succ(Succ(x138)))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))



We simplified constraint (10) using rules (III), (IV), (VII) which results in the following new constraint:

(14)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x128)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x128)), Succ(Zero), Succ(x128), Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x128))))))))



We simplified constraint (11) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x134, x133, x132, x131)=Succ(x125) which results in the following new constraints:

(15)    (new_primModNatS01(x149, x148)=Succ(x125) & Succ(Succ(Succ(x147)))=x149 & Succ(Succ(Zero))=x148  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x147))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Succ(x147))), Succ(Succ(Zero)), Succ(Succ(x147)), Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x147)))))))))

(16)    (new_primModNatS02(x153, x152, x151, x150)=Succ(x125) & Succ(Succ(Succ(x151)))=x153 & Succ(Succ(Succ(x150)))=x152 & (\/x154:new_primModNatS02(x153, x152, x151, x150)=Succ(x154) & Succ(Succ(x151))=x153 & Succ(Succ(x150))=x152  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x151)))))), Integer(Pos(Succ(Succ(Succ(Succ(x150)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x151)), Succ(Succ(x150)), Succ(x151), Succ(x150)))), Integer(Pos(Succ(Succ(Succ(Succ(x150)))))), Integer(Neg(Succ(Succ(Succ(Succ(x151))))))))  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x151))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x150))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Succ(x151))), Succ(Succ(Succ(x150))), Succ(Succ(x151)), Succ(Succ(x150))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x150))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x151)))))))))

(17)    (new_primModNatS01(x156, x155)=Succ(x125) & Succ(Succ(Zero))=x156 & Succ(Succ(Zero))=x155  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))

(18)    (Succ(Succ(x159))=Succ(x125) & Succ(Succ(Zero))=x159 & Succ(Succ(Succ(x157)))=x158  ==>  new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x157))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x157))), Succ(Zero), Succ(Succ(x157))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x157))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))



We simplified constraint (15) using rules (III), (IV) which results in the following new constraint:

(19)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x147))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Succ(x147))), Succ(Succ(Zero)), Succ(Succ(x147)), Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x147)))))))))



We simplified constraint (16) using rules (III), (IV) which results in the following new constraint:

(20)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x151))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x150))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Succ(x151))), Succ(Succ(Succ(x150))), Succ(Succ(x151)), Succ(Succ(x150))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x150))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x151)))))))))



We simplified constraint (17) using rules (III), (IV) which results in the following new constraint:

(21)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))



We simplified constraint (18) using rules (I), (II), (IV) which results in the following new constraint:

(22)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x157))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x157))), Succ(Zero), Succ(Succ(x157))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x157))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))



We simplified constraint (12) using rules (III), (IV), (VII) which results in the following new constraint:

(23)    (new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))








For Pair new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))) the following chains were created:
*We consider the chain new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x30))))), Integer(Neg(Succ(Succ(Succ(x31)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x30))))), Integer(Neg(new_primModNatS02(Succ(x31), Succ(x30), x31, x30)))), new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x32))))), Integer(Neg(Succ(Succ(Succ(x33)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x32), Succ(x33), x32, x33))), Integer(Neg(Succ(Succ(Succ(x33))))), Integer(Pos(Succ(Succ(Succ(x32)))))) which results in the following constraint:

(1)    (new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x30))))), Integer(Neg(new_primModNatS02(Succ(x31), Succ(x30), x31, x30))))=new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x32))))), Integer(Neg(Succ(Succ(Succ(x33))))))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x30))))), Integer(Neg(Succ(Succ(Succ(x31))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x30))))), Integer(Neg(new_primModNatS02(Succ(x31), Succ(x30), x31, x30)))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Succ(x31)=x166 & Succ(x30)=x167 & new_primModNatS02(x166, x167, x31, x30)=Succ(Succ(Succ(x33)))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x30))))), Integer(Neg(Succ(Succ(Succ(x31))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x30))))), Integer(Neg(new_primModNatS02(Succ(x31), Succ(x30), x31, x30)))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x166, x167, x31, x30)=Succ(Succ(Succ(x33))) which results in the following new constraints:

(3)    (new_primModNatS01(x170, x169)=Succ(Succ(Succ(x33))) & Succ(Succ(x168))=x170 & Succ(Zero)=x169  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x168)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Succ(x168)), Succ(Zero), Succ(x168), Zero)))))

(4)    (new_primModNatS02(x174, x173, x172, x171)=Succ(Succ(Succ(x33))) & Succ(Succ(x172))=x174 & Succ(Succ(x171))=x173 & (\/x175:new_primModNatS02(x174, x173, x172, x171)=Succ(Succ(Succ(x175))) & Succ(x172)=x174 & Succ(x171)=x173  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x171))))), Integer(Neg(Succ(Succ(Succ(x172))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x171))))), Integer(Neg(new_primModNatS02(Succ(x172), Succ(x171), x172, x171)))))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(x171)))))), Integer(Neg(Succ(Succ(Succ(Succ(x172)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x171)))))), Integer(Neg(new_primModNatS02(Succ(Succ(x172)), Succ(Succ(x171)), Succ(x172), Succ(x171))))))

(5)    (new_primModNatS01(x177, x176)=Succ(Succ(Succ(x33))) & Succ(Zero)=x177 & Succ(Zero)=x176  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))

(6)    (Succ(Succ(x180))=Succ(Succ(Succ(x33))) & Succ(Zero)=x180 & Succ(Succ(x178))=x179  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(x178)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x178)))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Succ(x178)), Zero, Succ(x178))))))



We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x170, x169)=Succ(Succ(Succ(x33))) which results in the following new constraint:

(7)    (new_primModNatS1(new_primMinusNatS2(Succ(x182), Succ(x181)), Succ(x181))=Succ(Succ(Succ(x33))) & Succ(Succ(x168))=x182 & Succ(Zero)=x181  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x168)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Succ(x168)), Succ(Zero), Succ(x168), Zero)))))



We simplified constraint (4) using rule (IV) which results in the following new constraint:

(8)    (new_primModNatS02(x174, x173, x172, x171)=Succ(Succ(Succ(x33))) & Succ(Succ(x172))=x174 & Succ(Succ(x171))=x173  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(x171)))))), Integer(Neg(Succ(Succ(Succ(Succ(x172)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x171)))))), Integer(Neg(new_primModNatS02(Succ(Succ(x172)), Succ(Succ(x171)), Succ(x172), Succ(x171))))))



We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x177, x176)=Succ(Succ(Succ(x33))) which results in the following new constraint:

(9)    (new_primModNatS1(new_primMinusNatS2(Succ(x201), Succ(x200)), Succ(x200))=Succ(Succ(Succ(x33))) & Succ(Zero)=x201 & Succ(Zero)=x200  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))



We simplified constraint (6) using rules (I), (II), (III), (IV) which results in the following new constraint:

(10)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(x178)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x178)))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Succ(x178)), Zero, Succ(x178))))))



We simplified constraint (7) using rules (III), (IV), (VII) which results in the following new constraint:

(11)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x168)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Succ(x168)), Succ(Zero), Succ(x168), Zero)))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x174, x173, x172, x171)=Succ(Succ(Succ(x33))) which results in the following new constraints:

(12)    (new_primModNatS01(x189, x188)=Succ(Succ(Succ(x33))) & Succ(Succ(Succ(x187)))=x189 & Succ(Succ(Zero))=x188  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x187))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(new_primModNatS02(Succ(Succ(Succ(x187))), Succ(Succ(Zero)), Succ(Succ(x187)), Succ(Zero))))))

(13)    (new_primModNatS02(x193, x192, x191, x190)=Succ(Succ(Succ(x33))) & Succ(Succ(Succ(x191)))=x193 & Succ(Succ(Succ(x190)))=x192 & (\/x194:new_primModNatS02(x193, x192, x191, x190)=Succ(Succ(Succ(x194))) & Succ(Succ(x191))=x193 & Succ(Succ(x190))=x192  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(x190)))))), Integer(Neg(Succ(Succ(Succ(Succ(x191)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x190)))))), Integer(Neg(new_primModNatS02(Succ(Succ(x191)), Succ(Succ(x190)), Succ(x191), Succ(x190))))))  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Succ(x190))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x191))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x190))))))), Integer(Neg(new_primModNatS02(Succ(Succ(Succ(x191))), Succ(Succ(Succ(x190))), Succ(Succ(x191)), Succ(Succ(x190)))))))

(14)    (new_primModNatS01(x196, x195)=Succ(Succ(Succ(x33))) & Succ(Succ(Zero))=x196 & Succ(Succ(Zero))=x195  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero))))))

(15)    (Succ(Succ(x199))=Succ(Succ(Succ(x33))) & Succ(Succ(Zero))=x199 & Succ(Succ(Succ(x197)))=x198  ==>  new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Succ(x197))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x197))))))), Integer(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x197))), Succ(Zero), Succ(Succ(x197)))))))



We simplified constraint (12) using rules (III), (IV) which results in the following new constraint:

(16)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x187))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(new_primModNatS02(Succ(Succ(Succ(x187))), Succ(Succ(Zero)), Succ(Succ(x187)), Succ(Zero))))))



We simplified constraint (13) using rules (III), (IV) which results in the following new constraint:

(17)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Succ(x190))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x191))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x190))))))), Integer(Neg(new_primModNatS02(Succ(Succ(Succ(x191))), Succ(Succ(Succ(x190))), Succ(Succ(x191)), Succ(Succ(x190)))))))



We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:

(18)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero))))))



We simplified constraint (15) using rules (I), (II), (III), (IV) which results in the following new constraint:

(19)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Succ(x197))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x197))))))), Integer(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x197))), Succ(Zero), Succ(Succ(x197)))))))



We simplified constraint (9) using rules (III), (IV), (VII) which results in the following new constraint:

(20)    (new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))








To summarize, we get the following constraints P__>=_ for the following pairs.

*new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))

*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x74))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x74))), Succ(Zero), Succ(Succ(x74))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x74))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x55)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Succ(x55)), Zero, Succ(x55)))), Integer(Neg(Succ(Succ(Succ(Succ(x55)))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x45)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(x45)), Succ(Zero), Succ(x45), Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x45))))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x64))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Succ(x64))), Succ(Succ(Zero)), Succ(Succ(x64)), Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x64)))))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x68))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x67))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Succ(x68))), Succ(Succ(Succ(x67))), Succ(Succ(x68)), Succ(Succ(x67))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x67))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x68)))))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero)))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero))))))))


*(new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero)))))))




*new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))

*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Succ(x114))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x114))))))), Integer(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x114))), Succ(Zero), Succ(Succ(x114)))))))


*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(x95)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x95)))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Succ(x95)), Zero, Succ(x95))))))


*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x85)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Succ(x85)), Succ(Zero), Succ(x85), Zero)))))


*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x104))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(new_primModNatS02(Succ(Succ(Succ(x104))), Succ(Succ(Zero)), Succ(Succ(x104)), Succ(Zero))))))


*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Succ(x107))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x108))))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x107))))))), Integer(Pos(new_primModNatS02(Succ(Succ(Succ(x108))), Succ(Succ(Succ(x107))), Succ(Succ(x108)), Succ(Succ(x107)))))))


*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero))))))


*(new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))




*new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))

*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x157))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x157))), Succ(Zero), Succ(Succ(x157))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x157))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Succ(x138)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Succ(x138)), Zero, Succ(x138)))), Integer(Pos(Succ(Succ(Succ(Succ(x138)))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(x128)))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(x128)), Succ(Zero), Succ(x128), Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x128))))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x147))))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Succ(x147))), Succ(Succ(Zero)), Succ(Succ(x147)), Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x147)))))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Succ(x151))))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x150))))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Succ(x151))), Succ(Succ(Succ(x150))), Succ(Succ(x151)), Succ(Succ(x150))))), Integer(Pos(Succ(Succ(Succ(Succ(Succ(x150))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x151)))))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero)))), Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero))))))))


*(new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(Zero))))), Integer(Pos(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero))), Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero)))))))




*new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))

*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Succ(x197))))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x197))))))), Integer(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x197))), Succ(Zero), Succ(Succ(x197)))))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(x178)))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(x178)))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Succ(x178)), Zero, Succ(x178))))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Succ(x168)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Succ(x168)), Succ(Zero), Succ(x168), Zero)))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x187))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(new_primModNatS02(Succ(Succ(Succ(x187))), Succ(Succ(Zero)), Succ(Succ(x187)), Succ(Zero))))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Succ(x190))))))), Integer(Neg(Succ(Succ(Succ(Succ(Succ(x191))))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Succ(x190))))))), Integer(Neg(new_primModNatS02(Succ(Succ(Succ(x191))), Succ(Succ(Succ(x190))), Succ(Succ(x191)), Succ(Succ(x190)))))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Succ(Zero)))))), Integer(Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Succ(Zero), Succ(Zero))))))


*(new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(Zero))))), Integer(Neg(new_primModNatS02(Succ(Zero), Succ(Zero), Zero, Zero)))))








The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
----------------------------------------

(411)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x2))))), Integer(Neg(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x3))))), Integer(Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   new_gcd0Gcd'00(Integer(Neg(Succ(Succ(Succ(x2))))), Integer(Pos(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'10(new_primEqInt(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))), Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2))))))
   new_gcd0Gcd'10(False, Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'00(Integer(Pos(Succ(Succ(Succ(x3))))), Integer(Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_primEqInt(Neg(Zero)) -> True
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(412)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS(Succ(Succ(vuz22000)), Succ(vuz40000)) -> new_primDivNatS0(vuz22000, vuz40000, vuz22000, vuz40000)
   new_primDivNatS0(vuz69, vuz70, Zero, Zero) -> new_primDivNatS00(vuz69, vuz70)
   new_primDivNatS(Succ(Succ(vuz22000)), Zero) -> new_primDivNatS(new_primMinusNatS0(vuz22000), Zero)
   new_primDivNatS0(vuz69, vuz70, Succ(vuz710), Zero) -> new_primDivNatS(new_primMinusNatS3(vuz69, vuz70), Succ(vuz70))
   new_primDivNatS0(vuz69, vuz70, Succ(vuz710), Succ(vuz720)) -> new_primDivNatS0(vuz69, vuz70, vuz710, vuz720)
   new_primDivNatS00(vuz69, vuz70) -> new_primDivNatS(new_primMinusNatS3(vuz69, vuz70), Succ(vuz70))
   new_primDivNatS(Succ(Zero), Zero) -> new_primDivNatS(new_primMinusNatS1, Zero)

The TRS R consists of the following rules:

   new_primMinusNatS3(vuz69, vuz70) -> new_primMinusNatS2(vuz69, vuz70)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primMinusNatS2(Zero, Zero)
   new_primMinusNatS1
   new_primMinusNatS0(x0)
   new_primMinusNatS3(x0, x1)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(413) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
----------------------------------------

(414)
Complex Obligation (AND)

----------------------------------------

(415)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS(Succ(Succ(vuz22000)), Zero) -> new_primDivNatS(new_primMinusNatS0(vuz22000), Zero)

The TRS R consists of the following rules:

   new_primMinusNatS3(vuz69, vuz70) -> new_primMinusNatS2(vuz69, vuz70)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primMinusNatS2(Zero, Zero)
   new_primMinusNatS1
   new_primMinusNatS0(x0)
   new_primMinusNatS3(x0, x1)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(416) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented dependency pairs:

   new_primDivNatS(Succ(Succ(vuz22000)), Zero) -> new_primDivNatS(new_primMinusNatS0(vuz22000), Zero)

Strictly oriented rules of the TRS R:

   new_primMinusNatS3(vuz69, vuz70) -> new_primMinusNatS2(vuz69, vuz70)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero

Used ordering: Polynomial interpretation [POLO]:

   POL(Succ(x_1)) = 1 + x_1
   POL(Zero) = 1
   POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2
   POL(new_primMinusNatS0(x_1)) = 1 + x_1
   POL(new_primMinusNatS1) = 2
   POL(new_primMinusNatS2(x_1, x_2)) = 1 + 2*x_1 + x_2
   POL(new_primMinusNatS3(x_1, x_2)) = 2 + 2*x_1 + 2*x_2


----------------------------------------

(417)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primMinusNatS2(Zero, Zero)
   new_primMinusNatS1
   new_primMinusNatS0(x0)
   new_primMinusNatS3(x0, x1)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(418) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(419)
YES

----------------------------------------

(420)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS0(vuz69, vuz70, Zero, Zero) -> new_primDivNatS00(vuz69, vuz70)
   new_primDivNatS00(vuz69, vuz70) -> new_primDivNatS(new_primMinusNatS3(vuz69, vuz70), Succ(vuz70))
   new_primDivNatS(Succ(Succ(vuz22000)), Succ(vuz40000)) -> new_primDivNatS0(vuz22000, vuz40000, vuz22000, vuz40000)
   new_primDivNatS0(vuz69, vuz70, Succ(vuz710), Zero) -> new_primDivNatS(new_primMinusNatS3(vuz69, vuz70), Succ(vuz70))
   new_primDivNatS0(vuz69, vuz70, Succ(vuz710), Succ(vuz720)) -> new_primDivNatS0(vuz69, vuz70, vuz710, vuz720)

The TRS R consists of the following rules:

   new_primMinusNatS3(vuz69, vuz70) -> new_primMinusNatS2(vuz69, vuz70)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primMinusNatS2(Zero, Zero)
   new_primMinusNatS1
   new_primMinusNatS0(x0)
   new_primMinusNatS3(x0, x1)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(421) QDPSizeChangeProof (EQUIVALENT)
We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 

Order:Polynomial interpretation [POLO]:

   POL(Succ(x_1)) = 1 + x_1
   POL(Zero) = 1
   POL(new_primMinusNatS2(x_1, x_2)) = x_1
   POL(new_primMinusNatS3(x_1, x_2)) = x_1




From the DPs we obtained the following set of size-change graphs:
*new_primDivNatS00(vuz69, vuz70) -> new_primDivNatS(new_primMinusNatS3(vuz69, vuz70), Succ(vuz70)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1


*new_primDivNatS(Succ(Succ(vuz22000)), Succ(vuz40000)) -> new_primDivNatS0(vuz22000, vuz40000, vuz22000, vuz40000) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4


*new_primDivNatS0(vuz69, vuz70, Succ(vuz710), Succ(vuz720)) -> new_primDivNatS0(vuz69, vuz70, vuz710, vuz720) (allowed arguments on rhs = {1, 2, 3, 4})
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


*new_primDivNatS0(vuz69, vuz70, Zero, Zero) -> new_primDivNatS00(vuz69, vuz70) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1, 2 >= 2


*new_primDivNatS0(vuz69, vuz70, Succ(vuz710), Zero) -> new_primDivNatS(new_primMinusNatS3(vuz69, vuz70), Succ(vuz70)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1



We oriented the following set of usable rules [AAECC05,FROCOS05].

   new_primMinusNatS3(vuz69, vuz70) -> new_primMinusNatS2(vuz69, vuz70)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)

----------------------------------------

(422)
YES

----------------------------------------

(423)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primMulNat(Succ(vuz300), Succ(vuz400)) -> new_primMulNat(vuz300, Succ(vuz400))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(424) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_primMulNat(Succ(vuz300), Succ(vuz400)) -> new_primMulNat(vuz300, Succ(vuz400))
The graph contains the following edges 1 > 1, 2 >= 2


----------------------------------------

(425)
YES

----------------------------------------

(426)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primMinusNatS(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS(vuz690, vuz700)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(427) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_primMinusNatS(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS(vuz690, vuz700)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(428)
YES

----------------------------------------

(429)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primPlusNat(Succ(vuz1800), Succ(vuz4000)) -> new_primPlusNat(vuz1800, vuz4000)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(430) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_primPlusNat(Succ(vuz1800), Succ(vuz4000)) -> new_primPlusNat(vuz1800, vuz4000)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(431)
YES

----------------------------------------

(432)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'(vuz47, vuz46) -> new_gcd0Gcd'1(new_esEs(vuz46), vuz47, vuz46)

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_esEs(Neg(Succ(vuz1700))) -> new_primEqInt(Neg(Succ(vuz1700)))
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_esEs(Neg(Zero)) -> new_primEqInt(Neg(Zero))
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_error -> error([])
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_esEs(Pos(Succ(vuz1700))) -> new_primEqInt(Pos(Succ(vuz1700)))
   new_esEs(Pos(Zero)) -> new_primEqInt(Pos(Zero))
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_esEs(Pos(Zero))
   new_esEs(Pos(Succ(x0)))
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_esEs(Neg(Succ(x0)))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))
   new_esEs(Neg(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(433) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'(vuz47, vuz46) -> new_gcd0Gcd'1(new_esEs(vuz46), vuz47, vuz46) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Neg(Succ(x0))), y0, Neg(Succ(x0))),new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Neg(Succ(x0))), y0, Neg(Succ(x0))))
   (new_gcd0Gcd'(y0, Neg(Zero)) -> new_gcd0Gcd'1(new_primEqInt(Neg(Zero)), y0, Neg(Zero)),new_gcd0Gcd'(y0, Neg(Zero)) -> new_gcd0Gcd'1(new_primEqInt(Neg(Zero)), y0, Neg(Zero)))
   (new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Pos(Succ(x0))), y0, Pos(Succ(x0))),new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Pos(Succ(x0))), y0, Pos(Succ(x0))))
   (new_gcd0Gcd'(y0, Pos(Zero)) -> new_gcd0Gcd'1(new_primEqInt(Pos(Zero)), y0, Pos(Zero)),new_gcd0Gcd'(y0, Pos(Zero)) -> new_gcd0Gcd'1(new_primEqInt(Pos(Zero)), y0, Pos(Zero)))


----------------------------------------

(434)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Neg(Succ(x0))), y0, Neg(Succ(x0)))
   new_gcd0Gcd'(y0, Neg(Zero)) -> new_gcd0Gcd'1(new_primEqInt(Neg(Zero)), y0, Neg(Zero))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Pos(Succ(x0))), y0, Pos(Succ(x0)))
   new_gcd0Gcd'(y0, Pos(Zero)) -> new_gcd0Gcd'1(new_primEqInt(Pos(Zero)), y0, Pos(Zero))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_esEs(Neg(Succ(vuz1700))) -> new_primEqInt(Neg(Succ(vuz1700)))
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_esEs(Neg(Zero)) -> new_primEqInt(Neg(Zero))
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_error -> error([])
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_esEs(Pos(Succ(vuz1700))) -> new_primEqInt(Pos(Succ(vuz1700)))
   new_esEs(Pos(Zero)) -> new_primEqInt(Pos(Zero))
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_esEs(Pos(Zero))
   new_esEs(Pos(Succ(x0)))
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_esEs(Neg(Succ(x0)))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))
   new_esEs(Neg(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(435) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(436)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Neg(Succ(x0))), y0, Neg(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Pos(Succ(x0))), y0, Pos(Succ(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primEqInt(Neg(Zero)) -> True
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_esEs(Neg(Succ(vuz1700))) -> new_primEqInt(Neg(Succ(vuz1700)))
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_esEs(Neg(Zero)) -> new_primEqInt(Neg(Zero))
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primEqInt(Pos(Zero)) -> True
   new_error -> error([])
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primEqInt(Neg(Succ(vuz16000))) -> False
   new_esEs(Pos(Succ(vuz1700))) -> new_primEqInt(Pos(Succ(vuz1700)))
   new_esEs(Pos(Zero)) -> new_primEqInt(Pos(Zero))
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primMinusNatS1 -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_esEs(Pos(Zero))
   new_esEs(Pos(Succ(x0)))
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_esEs(Neg(Succ(x0)))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))
   new_esEs(Neg(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(437) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(438)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Neg(Succ(x0))), y0, Neg(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Pos(Succ(x0))), y0, Pos(Succ(x0)))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_esEs(Pos(Zero))
   new_esEs(Pos(Succ(x0)))
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_esEs(Neg(Succ(x0)))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))
   new_esEs(Neg(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(439) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_esEs(Pos(Zero))
   new_esEs(Pos(Succ(x0)))
   new_esEs(Neg(Succ(x0)))
   new_esEs(Neg(Zero))


----------------------------------------

(440)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Neg(Succ(x0))), y0, Neg(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Pos(Succ(x0))), y0, Pos(Succ(x0)))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(441) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Neg(Succ(x0))), y0, Neg(Succ(x0))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0))),new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0))))


----------------------------------------

(442)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Pos(Succ(x0))), y0, Pos(Succ(x0)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primEqInt(Neg(Succ(vuz16000))) -> False

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(443) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(444)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Pos(Succ(x0))), y0, Pos(Succ(x0)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(445) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(new_primEqInt(Pos(Succ(x0))), y0, Pos(Succ(x0))) at position [0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0))),new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0))))


----------------------------------------

(446)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))

The TRS R consists of the following rules:

   new_primEqInt(Pos(Succ(vuz16000))) -> False
   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(447) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(448)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primEqInt(Neg(Succ(x0)))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primEqInt(Pos(Zero))
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primEqInt(Pos(Succ(x0)))
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primEqInt(Neg(Zero))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(449) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primEqInt(Neg(Succ(x0)))
   new_primEqInt(Pos(Zero))
   new_primEqInt(Pos(Succ(x0)))
   new_primEqInt(Neg(Zero))


----------------------------------------

(450)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(451) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
----------------------------------------

(452)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero

Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(453) InductionCalculusProof (EQUIVALENT)
Note that final constraints are written in bold face.



For Pair new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46) the following chains were created:
*We consider the chain new_gcd0Gcd'1(False, x2, x3) -> new_gcd0Gcd'0(x2, x3), new_gcd0Gcd'0(x4, x5) -> new_gcd0Gcd'(x5, new_primRemInt(x4, x5)) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x2, x3)=new_gcd0Gcd'0(x4, x5)  ==>  new_gcd0Gcd'1(False, x2, x3)_>=_new_gcd0Gcd'0(x2, x3))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, x2, x3)_>=_new_gcd0Gcd'0(x2, x3))








For Pair new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43)) the following chains were created:
*We consider the chain new_gcd0Gcd'0(x14, x15) -> new_gcd0Gcd'(x15, new_primRemInt(x14, x15)), new_gcd0Gcd'(x16, Neg(Succ(x17))) -> new_gcd0Gcd'1(False, x16, Neg(Succ(x17))) which results in the following constraint:

(1)    (new_gcd0Gcd'(x15, new_primRemInt(x14, x15))=new_gcd0Gcd'(x16, Neg(Succ(x17)))  ==>  new_gcd0Gcd'0(x14, x15)_>=_new_gcd0Gcd'(x15, new_primRemInt(x14, x15)))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_primRemInt(x14, x15)=Neg(Succ(x17))  ==>  new_gcd0Gcd'0(x14, x15)_>=_new_gcd0Gcd'(x15, new_primRemInt(x14, x15)))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primRemInt(x14, x15)=Neg(Succ(x17)) which results in the following new constraints:

(3)    (new_error=Neg(Succ(x17))  ==>  new_gcd0Gcd'0(Neg(x46), Neg(Zero))_>=_new_gcd0Gcd'(Neg(Zero), new_primRemInt(Neg(x46), Neg(Zero))))

(4)    (Neg(new_primModNatS1(x48, x47))=Neg(Succ(x17))  ==>  new_gcd0Gcd'0(Neg(x48), Pos(Succ(x47)))_>=_new_gcd0Gcd'(Pos(Succ(x47)), new_primRemInt(Neg(x48), Pos(Succ(x47)))))

(5)    (new_error=Neg(Succ(x17))  ==>  new_gcd0Gcd'0(Pos(x49), Pos(Zero))_>=_new_gcd0Gcd'(Pos(Zero), new_primRemInt(Pos(x49), Pos(Zero))))

(6)    (Neg(new_primModNatS1(x51, x50))=Neg(Succ(x17))  ==>  new_gcd0Gcd'0(Neg(x51), Neg(Succ(x50)))_>=_new_gcd0Gcd'(Neg(Succ(x50)), new_primRemInt(Neg(x51), Neg(Succ(x50)))))

(7)    (new_error=Neg(Succ(x17))  ==>  new_gcd0Gcd'0(Pos(x52), Neg(Zero))_>=_new_gcd0Gcd'(Neg(Zero), new_primRemInt(Pos(x52), Neg(Zero))))

(8)    (new_error=Neg(Succ(x17))  ==>  new_gcd0Gcd'0(Neg(x53), Pos(Zero))_>=_new_gcd0Gcd'(Pos(Zero), new_primRemInt(Neg(x53), Pos(Zero))))



We solved constraint (3) using rule (V) (with possible (I) afterwards).We simplified constraint (4) using rules (I), (II) which results in the following new constraint:

(9)    (new_primModNatS1(x48, x47)=Succ(x17)  ==>  new_gcd0Gcd'0(Neg(x48), Pos(Succ(x47)))_>=_new_gcd0Gcd'(Pos(Succ(x47)), new_primRemInt(Neg(x48), Pos(Succ(x47)))))



We solved constraint (5) using rule (V) (with possible (I) afterwards).We simplified constraint (6) using rules (I), (II) which results in the following new constraint:

(10)    (new_primModNatS1(x51, x50)=Succ(x17)  ==>  new_gcd0Gcd'0(Neg(x51), Neg(Succ(x50)))_>=_new_gcd0Gcd'(Neg(Succ(x50)), new_primRemInt(Neg(x51), Neg(Succ(x50)))))



We solved constraint (7) using rule (V) (with possible (I) afterwards).We solved constraint (8) using rule (V) (with possible (I) afterwards).We simplified constraint (9) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS1(x48, x47)=Succ(x17) which results in the following new constraints:

(11)    (Succ(Zero)=Succ(x17)  ==>  new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x54))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x54))), new_primRemInt(Neg(Succ(Zero)), Pos(Succ(Succ(x54))))))

(12)    (new_primModNatS1(new_primMinusNatS0(x56), Zero)=Succ(x17)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(x56))), Pos(Succ(Zero)))_>=_new_gcd0Gcd'(Pos(Succ(Zero)), new_primRemInt(Neg(Succ(Succ(x56))), Pos(Succ(Zero)))))

(13)    (new_primModNatS1(new_primMinusNatS1, Zero)=Succ(x17)  ==>  new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Zero)))_>=_new_gcd0Gcd'(Pos(Succ(Zero)), new_primRemInt(Neg(Succ(Zero)), Pos(Succ(Zero)))))

(14)    (new_primModNatS02(x58, x57, x58, x57)=Succ(x17)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(x58))), Pos(Succ(Succ(x57))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x57))), new_primRemInt(Neg(Succ(Succ(x58))), Pos(Succ(Succ(x57))))))



We simplified constraint (11) using rules (I), (II), (IV) which results in the following new constraint:

(15)    (new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x54))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x54))), new_primRemInt(Neg(Succ(Zero)), Pos(Succ(Succ(x54))))))



We simplified constraint (12) using rules (III), (IV), (VII) which results in the following new constraint:

(16)    (new_gcd0Gcd'0(Neg(Succ(Succ(x56))), Pos(Succ(Zero)))_>=_new_gcd0Gcd'(Pos(Succ(Zero)), new_primRemInt(Neg(Succ(Succ(x56))), Pos(Succ(Zero)))))



We simplified constraint (13) using rules (III), (IV), (VII) which results in the following new constraint:

(17)    (new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Zero)))_>=_new_gcd0Gcd'(Pos(Succ(Zero)), new_primRemInt(Neg(Succ(Zero)), Pos(Succ(Zero)))))



We simplified constraint (14) using rules (III), (IV), (VII) which results in the following new constraint:

(18)    (new_gcd0Gcd'0(Neg(Succ(Succ(x63))), Pos(Succ(Succ(x64))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x64))), new_primRemInt(Neg(Succ(Succ(x63))), Pos(Succ(Succ(x64))))))



We simplified constraint (10) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS1(x51, x50)=Succ(x17) which results in the following new constraints:

(19)    (Succ(Zero)=Succ(x17)  ==>  new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x65))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x65))), new_primRemInt(Neg(Succ(Zero)), Neg(Succ(Succ(x65))))))

(20)    (new_primModNatS1(new_primMinusNatS0(x67), Zero)=Succ(x17)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(x67))), Neg(Succ(Zero)))_>=_new_gcd0Gcd'(Neg(Succ(Zero)), new_primRemInt(Neg(Succ(Succ(x67))), Neg(Succ(Zero)))))

(21)    (new_primModNatS1(new_primMinusNatS1, Zero)=Succ(x17)  ==>  new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Zero)))_>=_new_gcd0Gcd'(Neg(Succ(Zero)), new_primRemInt(Neg(Succ(Zero)), Neg(Succ(Zero)))))

(22)    (new_primModNatS02(x69, x68, x69, x68)=Succ(x17)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(x69))), Neg(Succ(Succ(x68))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x68))), new_primRemInt(Neg(Succ(Succ(x69))), Neg(Succ(Succ(x68))))))



We simplified constraint (19) using rules (I), (II), (IV) which results in the following new constraint:

(23)    (new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x65))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x65))), new_primRemInt(Neg(Succ(Zero)), Neg(Succ(Succ(x65))))))



We simplified constraint (20) using rules (III), (IV), (VII) which results in the following new constraint:

(24)    (new_gcd0Gcd'0(Neg(Succ(Succ(x67))), Neg(Succ(Zero)))_>=_new_gcd0Gcd'(Neg(Succ(Zero)), new_primRemInt(Neg(Succ(Succ(x67))), Neg(Succ(Zero)))))



We simplified constraint (21) using rules (III), (IV), (VII) which results in the following new constraint:

(25)    (new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Zero)))_>=_new_gcd0Gcd'(Neg(Succ(Zero)), new_primRemInt(Neg(Succ(Zero)), Neg(Succ(Zero)))))



We simplified constraint (22) using rules (III), (IV), (VII) which results in the following new constraint:

(26)    (new_gcd0Gcd'0(Neg(Succ(Succ(x74))), Neg(Succ(Succ(x75))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x75))), new_primRemInt(Neg(Succ(Succ(x74))), Neg(Succ(Succ(x75))))))




*We consider the chain new_gcd0Gcd'0(x18, x19) -> new_gcd0Gcd'(x19, new_primRemInt(x18, x19)), new_gcd0Gcd'(x20, Pos(Succ(x21))) -> new_gcd0Gcd'1(False, x20, Pos(Succ(x21))) which results in the following constraint:

(1)    (new_gcd0Gcd'(x19, new_primRemInt(x18, x19))=new_gcd0Gcd'(x20, Pos(Succ(x21)))  ==>  new_gcd0Gcd'0(x18, x19)_>=_new_gcd0Gcd'(x19, new_primRemInt(x18, x19)))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_primRemInt(x18, x19)=Pos(Succ(x21))  ==>  new_gcd0Gcd'0(x18, x19)_>=_new_gcd0Gcd'(x19, new_primRemInt(x18, x19)))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primRemInt(x18, x19)=Pos(Succ(x21)) which results in the following new constraints:

(3)    (Pos(new_primModNatS1(x77, x76))=Pos(Succ(x21))  ==>  new_gcd0Gcd'0(Pos(x77), Neg(Succ(x76)))_>=_new_gcd0Gcd'(Neg(Succ(x76)), new_primRemInt(Pos(x77), Neg(Succ(x76)))))

(4)    (Pos(new_primModNatS1(x79, x78))=Pos(Succ(x21))  ==>  new_gcd0Gcd'0(Pos(x79), Pos(Succ(x78)))_>=_new_gcd0Gcd'(Pos(Succ(x78)), new_primRemInt(Pos(x79), Pos(Succ(x78)))))

(5)    (new_error=Pos(Succ(x21))  ==>  new_gcd0Gcd'0(Neg(x80), Neg(Zero))_>=_new_gcd0Gcd'(Neg(Zero), new_primRemInt(Neg(x80), Neg(Zero))))

(6)    (new_error=Pos(Succ(x21))  ==>  new_gcd0Gcd'0(Pos(x83), Pos(Zero))_>=_new_gcd0Gcd'(Pos(Zero), new_primRemInt(Pos(x83), Pos(Zero))))

(7)    (new_error=Pos(Succ(x21))  ==>  new_gcd0Gcd'0(Pos(x86), Neg(Zero))_>=_new_gcd0Gcd'(Neg(Zero), new_primRemInt(Pos(x86), Neg(Zero))))

(8)    (new_error=Pos(Succ(x21))  ==>  new_gcd0Gcd'0(Neg(x87), Pos(Zero))_>=_new_gcd0Gcd'(Pos(Zero), new_primRemInt(Neg(x87), Pos(Zero))))



We simplified constraint (3) using rules (I), (II) which results in the following new constraint:

(9)    (new_primModNatS1(x77, x76)=Succ(x21)  ==>  new_gcd0Gcd'0(Pos(x77), Neg(Succ(x76)))_>=_new_gcd0Gcd'(Neg(Succ(x76)), new_primRemInt(Pos(x77), Neg(Succ(x76)))))



We simplified constraint (4) using rules (I), (II) which results in the following new constraint:

(10)    (new_primModNatS1(x79, x78)=Succ(x21)  ==>  new_gcd0Gcd'0(Pos(x79), Pos(Succ(x78)))_>=_new_gcd0Gcd'(Pos(Succ(x78)), new_primRemInt(Pos(x79), Pos(Succ(x78)))))



We solved constraint (5) using rule (V) (with possible (I) afterwards).We solved constraint (6) using rule (V) (with possible (I) afterwards).We solved constraint (7) using rule (V) (with possible (I) afterwards).We solved constraint (8) using rule (V) (with possible (I) afterwards).We simplified constraint (9) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS1(x77, x76)=Succ(x21) which results in the following new constraints:

(11)    (Succ(Zero)=Succ(x21)  ==>  new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x88))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x88))), new_primRemInt(Pos(Succ(Zero)), Neg(Succ(Succ(x88))))))

(12)    (new_primModNatS1(new_primMinusNatS0(x90), Zero)=Succ(x21)  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(x90))), Neg(Succ(Zero)))_>=_new_gcd0Gcd'(Neg(Succ(Zero)), new_primRemInt(Pos(Succ(Succ(x90))), Neg(Succ(Zero)))))

(13)    (new_primModNatS1(new_primMinusNatS1, Zero)=Succ(x21)  ==>  new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Zero)))_>=_new_gcd0Gcd'(Neg(Succ(Zero)), new_primRemInt(Pos(Succ(Zero)), Neg(Succ(Zero)))))

(14)    (new_primModNatS02(x92, x91, x92, x91)=Succ(x21)  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(x92))), Neg(Succ(Succ(x91))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x91))), new_primRemInt(Pos(Succ(Succ(x92))), Neg(Succ(Succ(x91))))))



We simplified constraint (11) using rules (I), (II), (IV) which results in the following new constraint:

(15)    (new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x88))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x88))), new_primRemInt(Pos(Succ(Zero)), Neg(Succ(Succ(x88))))))



We simplified constraint (12) using rules (III), (IV), (VII) which results in the following new constraint:

(16)    (new_gcd0Gcd'0(Pos(Succ(Succ(x90))), Neg(Succ(Zero)))_>=_new_gcd0Gcd'(Neg(Succ(Zero)), new_primRemInt(Pos(Succ(Succ(x90))), Neg(Succ(Zero)))))



We simplified constraint (13) using rules (III), (IV), (VII) which results in the following new constraint:

(17)    (new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Zero)))_>=_new_gcd0Gcd'(Neg(Succ(Zero)), new_primRemInt(Pos(Succ(Zero)), Neg(Succ(Zero)))))



We simplified constraint (14) using rules (III), (IV), (VII) which results in the following new constraint:

(18)    (new_gcd0Gcd'0(Pos(Succ(Succ(x97))), Neg(Succ(Succ(x98))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x98))), new_primRemInt(Pos(Succ(Succ(x97))), Neg(Succ(Succ(x98))))))



We simplified constraint (10) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS1(x79, x78)=Succ(x21) which results in the following new constraints:

(19)    (Succ(Zero)=Succ(x21)  ==>  new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x99))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x99))), new_primRemInt(Pos(Succ(Zero)), Pos(Succ(Succ(x99))))))

(20)    (new_primModNatS1(new_primMinusNatS0(x101), Zero)=Succ(x21)  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(x101))), Pos(Succ(Zero)))_>=_new_gcd0Gcd'(Pos(Succ(Zero)), new_primRemInt(Pos(Succ(Succ(x101))), Pos(Succ(Zero)))))

(21)    (new_primModNatS1(new_primMinusNatS1, Zero)=Succ(x21)  ==>  new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Zero)))_>=_new_gcd0Gcd'(Pos(Succ(Zero)), new_primRemInt(Pos(Succ(Zero)), Pos(Succ(Zero)))))

(22)    (new_primModNatS02(x103, x102, x103, x102)=Succ(x21)  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(x103))), Pos(Succ(Succ(x102))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x102))), new_primRemInt(Pos(Succ(Succ(x103))), Pos(Succ(Succ(x102))))))



We simplified constraint (19) using rules (I), (II), (IV) which results in the following new constraint:

(23)    (new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x99))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x99))), new_primRemInt(Pos(Succ(Zero)), Pos(Succ(Succ(x99))))))



We simplified constraint (20) using rules (III), (IV), (VII) which results in the following new constraint:

(24)    (new_gcd0Gcd'0(Pos(Succ(Succ(x101))), Pos(Succ(Zero)))_>=_new_gcd0Gcd'(Pos(Succ(Zero)), new_primRemInt(Pos(Succ(Succ(x101))), Pos(Succ(Zero)))))



We simplified constraint (21) using rules (III), (IV), (VII) which results in the following new constraint:

(25)    (new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Zero)))_>=_new_gcd0Gcd'(Pos(Succ(Zero)), new_primRemInt(Pos(Succ(Zero)), Pos(Succ(Zero)))))



We simplified constraint (22) using rules (III), (IV), (VII) which results in the following new constraint:

(26)    (new_gcd0Gcd'0(Pos(Succ(Succ(x108))), Pos(Succ(Succ(x109))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x109))), new_primRemInt(Pos(Succ(Succ(x108))), Pos(Succ(Succ(x109))))))








For Pair new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0))) the following chains were created:
*We consider the chain new_gcd0Gcd'(x22, Neg(Succ(x23))) -> new_gcd0Gcd'1(False, x22, Neg(Succ(x23))), new_gcd0Gcd'1(False, x24, x25) -> new_gcd0Gcd'0(x24, x25) which results in the following constraint:

(1)    (new_gcd0Gcd'1(False, x22, Neg(Succ(x23)))=new_gcd0Gcd'1(False, x24, x25)  ==>  new_gcd0Gcd'(x22, Neg(Succ(x23)))_>=_new_gcd0Gcd'1(False, x22, Neg(Succ(x23))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'(x22, Neg(Succ(x23)))_>=_new_gcd0Gcd'1(False, x22, Neg(Succ(x23))))








For Pair new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0))) the following chains were created:
*We consider the chain new_gcd0Gcd'(x32, Pos(Succ(x33))) -> new_gcd0Gcd'1(False, x32, Pos(Succ(x33))), new_gcd0Gcd'1(False, x34, x35) -> new_gcd0Gcd'0(x34, x35) which results in the following constraint:

(1)    (new_gcd0Gcd'1(False, x32, Pos(Succ(x33)))=new_gcd0Gcd'1(False, x34, x35)  ==>  new_gcd0Gcd'(x32, Pos(Succ(x33)))_>=_new_gcd0Gcd'1(False, x32, Pos(Succ(x33))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'(x32, Pos(Succ(x33)))_>=_new_gcd0Gcd'1(False, x32, Pos(Succ(x33))))








To summarize, we get the following constraints P__>=_ for the following pairs.

*new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)

*(new_gcd0Gcd'1(False, x2, x3)_>=_new_gcd0Gcd'0(x2, x3))




*new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43))

*(new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x54))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x54))), new_primRemInt(Neg(Succ(Zero)), Pos(Succ(Succ(x54))))))


*(new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x65))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x65))), new_primRemInt(Neg(Succ(Zero)), Neg(Succ(Succ(x65))))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(x56))), Pos(Succ(Zero)))_>=_new_gcd0Gcd'(Pos(Succ(Zero)), new_primRemInt(Neg(Succ(Succ(x56))), Pos(Succ(Zero)))))


*(new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Zero)))_>=_new_gcd0Gcd'(Pos(Succ(Zero)), new_primRemInt(Neg(Succ(Zero)), Pos(Succ(Zero)))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(x63))), Pos(Succ(Succ(x64))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x64))), new_primRemInt(Neg(Succ(Succ(x63))), Pos(Succ(Succ(x64))))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(x67))), Neg(Succ(Zero)))_>=_new_gcd0Gcd'(Neg(Succ(Zero)), new_primRemInt(Neg(Succ(Succ(x67))), Neg(Succ(Zero)))))


*(new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Zero)))_>=_new_gcd0Gcd'(Neg(Succ(Zero)), new_primRemInt(Neg(Succ(Zero)), Neg(Succ(Zero)))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(x74))), Neg(Succ(Succ(x75))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x75))), new_primRemInt(Neg(Succ(Succ(x74))), Neg(Succ(Succ(x75))))))


*(new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x88))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x88))), new_primRemInt(Pos(Succ(Zero)), Neg(Succ(Succ(x88))))))


*(new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x99))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x99))), new_primRemInt(Pos(Succ(Zero)), Pos(Succ(Succ(x99))))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(x90))), Neg(Succ(Zero)))_>=_new_gcd0Gcd'(Neg(Succ(Zero)), new_primRemInt(Pos(Succ(Succ(x90))), Neg(Succ(Zero)))))


*(new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Zero)))_>=_new_gcd0Gcd'(Neg(Succ(Zero)), new_primRemInt(Pos(Succ(Zero)), Neg(Succ(Zero)))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(x97))), Neg(Succ(Succ(x98))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x98))), new_primRemInt(Pos(Succ(Succ(x97))), Neg(Succ(Succ(x98))))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(x101))), Pos(Succ(Zero)))_>=_new_gcd0Gcd'(Pos(Succ(Zero)), new_primRemInt(Pos(Succ(Succ(x101))), Pos(Succ(Zero)))))


*(new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Zero)))_>=_new_gcd0Gcd'(Pos(Succ(Zero)), new_primRemInt(Pos(Succ(Zero)), Pos(Succ(Zero)))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(x108))), Pos(Succ(Succ(x109))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x109))), new_primRemInt(Pos(Succ(Succ(x108))), Pos(Succ(Succ(x109))))))




*new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))

*(new_gcd0Gcd'(x22, Neg(Succ(x23)))_>=_new_gcd0Gcd'1(False, x22, Neg(Succ(x23))))




*new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))

*(new_gcd0Gcd'(x32, Pos(Succ(x33)))_>=_new_gcd0Gcd'1(False, x32, Pos(Succ(x33))))








The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
----------------------------------------

(454)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(455) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(vuz44, vuz43) -> new_gcd0Gcd'(vuz43, new_primRemInt(vuz44, vuz43)) at position [1] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1))),new_gcd0Gcd'0(Pos(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1))))
   (new_gcd0Gcd'0(Pos(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1))),new_gcd0Gcd'0(Pos(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1))))
   (new_gcd0Gcd'0(Neg(x0), Neg(Zero)) -> new_gcd0Gcd'(Neg(Zero), new_error),new_gcd0Gcd'0(Neg(x0), Neg(Zero)) -> new_gcd0Gcd'(Neg(Zero), new_error))
   (new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1))),new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1))))
   (new_gcd0Gcd'0(Pos(x0), Pos(Zero)) -> new_gcd0Gcd'(Pos(Zero), new_error),new_gcd0Gcd'0(Pos(x0), Pos(Zero)) -> new_gcd0Gcd'(Pos(Zero), new_error))
   (new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1))),new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1))))
   (new_gcd0Gcd'0(Pos(x0), Neg(Zero)) -> new_gcd0Gcd'(Neg(Zero), new_error),new_gcd0Gcd'0(Pos(x0), Neg(Zero)) -> new_gcd0Gcd'(Neg(Zero), new_error))
   (new_gcd0Gcd'0(Neg(x0), Pos(Zero)) -> new_gcd0Gcd'(Pos(Zero), new_error),new_gcd0Gcd'0(Neg(x0), Pos(Zero)) -> new_gcd0Gcd'(Pos(Zero), new_error))


----------------------------------------

(456)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Neg(x0), Neg(Zero)) -> new_gcd0Gcd'(Neg(Zero), new_error)
   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(x0), Pos(Zero)) -> new_gcd0Gcd'(Pos(Zero), new_error)
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(x0), Neg(Zero)) -> new_gcd0Gcd'(Neg(Zero), new_error)
   new_gcd0Gcd'0(Neg(x0), Pos(Zero)) -> new_gcd0Gcd'(Pos(Zero), new_error)

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(457) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
----------------------------------------

(458)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))

The TRS R consists of the following rules:

   new_primRemInt(Pos(vuz440), Neg(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Succ(vuz4300))) -> Pos(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Neg(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Pos(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Neg(Succ(vuz4300))) -> Neg(new_primModNatS1(vuz440, vuz4300))
   new_primRemInt(Pos(vuz440), Neg(Zero)) -> new_error
   new_primRemInt(Neg(vuz440), Pos(Zero)) -> new_error
   new_error -> error([])
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)
   new_primMinusNatS1 -> Zero

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(459) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(460)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primRemInt(Neg(x0), Neg(Zero))
   new_primMinusNatS0(x0)
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_error
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primModNatS02(x0, x1, Zero, Zero)
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(461) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primRemInt(Neg(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Neg(Succ(x1)))
   new_primRemInt(Pos(x0), Neg(Succ(x1)))
   new_primRemInt(Neg(x0), Pos(Succ(x1)))
   new_error
   new_primRemInt(Pos(x0), Pos(Zero))
   new_primRemInt(Pos(x0), Pos(Succ(x1)))
   new_primRemInt(Pos(x0), Neg(Zero))
   new_primRemInt(Neg(x0), Pos(Zero))


----------------------------------------

(462)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(463) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Pos(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero))),new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero))))
   (new_gcd0Gcd'0(Pos(Zero), Neg(Succ(x0))) -> new_gcd0Gcd'(Neg(Succ(x0)), Pos(Zero)),new_gcd0Gcd'0(Pos(Zero), Neg(Succ(x0))) -> new_gcd0Gcd'(Neg(Succ(x0)), Pos(Zero)))
   (new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))),new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1))),new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1))))


----------------------------------------

(464)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Zero), Neg(Succ(x0))) -> new_gcd0Gcd'(Neg(Succ(x0)), Pos(Zero))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(465) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(466)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(467) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero))),new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero))))


----------------------------------------

(468)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(469) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(470)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(471) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))))


----------------------------------------

(472)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(473) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Pos(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero))),new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero))))
   (new_gcd0Gcd'0(Pos(Zero), Pos(Succ(x0))) -> new_gcd0Gcd'(Pos(Succ(x0)), Pos(Zero)),new_gcd0Gcd'0(Pos(Zero), Pos(Succ(x0))) -> new_gcd0Gcd'(Pos(Succ(x0)), Pos(Zero)))
   (new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))),new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1))),new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1))))


----------------------------------------

(474)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Zero), Pos(Succ(x0))) -> new_gcd0Gcd'(Pos(Succ(x0)), Pos(Zero))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(475) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(476)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(477) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero))),new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero))))


----------------------------------------

(478)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(479) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(480)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(481) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))))


----------------------------------------

(482)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(483) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Neg(x0), Pos(Succ(x1))) -> new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero))),new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero))))
   (new_gcd0Gcd'0(Neg(Zero), Pos(Succ(x0))) -> new_gcd0Gcd'(Pos(Succ(x0)), Neg(Zero)),new_gcd0Gcd'0(Neg(Zero), Pos(Succ(x0))) -> new_gcd0Gcd'(Pos(Succ(x0)), Neg(Zero)))
   (new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))),new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1))),new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1))))


----------------------------------------

(484)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Zero), Pos(Succ(x0))) -> new_gcd0Gcd'(Pos(Succ(x0)), Neg(Zero))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(485) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(486)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(487) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero))),new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero))))


----------------------------------------

(488)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(489) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(490)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(491) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))))


----------------------------------------

(492)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(493) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Neg(x0), Neg(Succ(x1))) -> new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero))),new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero))))
   (new_gcd0Gcd'0(Neg(Zero), Neg(Succ(x0))) -> new_gcd0Gcd'(Neg(Succ(x0)), Neg(Zero)),new_gcd0Gcd'0(Neg(Zero), Neg(Succ(x0))) -> new_gcd0Gcd'(Neg(Succ(x0)), Neg(Zero)))
   (new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))),new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1))),new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1))))


----------------------------------------

(494)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Zero), Neg(Succ(x0))) -> new_gcd0Gcd'(Neg(Succ(x0)), Neg(Zero))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(495) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(496)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(497) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero))),new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero))))


----------------------------------------

(498)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(499) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(500)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(501) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))))


----------------------------------------

(502)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(503) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS01(Succ(x2), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS01(Succ(x2), Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS01(Zero, Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS01(Zero, Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))))


----------------------------------------

(504)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS01(Succ(x2), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS01(Zero, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(505) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS01(Succ(x2), Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))))


----------------------------------------

(506)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS01(Zero, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(507) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS01(Zero, Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))))


----------------------------------------

(508)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(509) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))))


----------------------------------------

(510)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(511) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))))


----------------------------------------

(512)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(513) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))


----------------------------------------

(514)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(515) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Zero, Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Zero, Succ(Zero)))))


----------------------------------------

(516)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Zero, Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(517) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(518)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(519) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))


----------------------------------------

(520)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(521) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero))))


----------------------------------------

(522)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(523) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(524)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(525) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))))


----------------------------------------

(526)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(527) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Pos(new_primModNatS02(x0, x1, x0, x1))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS01(Succ(x2), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS01(Succ(x2), Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS01(Zero, Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS01(Zero, Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))))


----------------------------------------

(528)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS01(Succ(x2), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS01(Zero, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(529) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS01(Succ(x2), Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))))


----------------------------------------

(530)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS01(Zero, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(531) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS01(Zero, Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))))


----------------------------------------

(532)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(533) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))))


----------------------------------------

(534)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(535) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))))


----------------------------------------

(536)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(537) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero)))))


----------------------------------------

(538)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(539) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Zero, Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Zero, Succ(Zero)))))


----------------------------------------

(540)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Zero, Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(541) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(542)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(543) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))


----------------------------------------

(544)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(545) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero))))


----------------------------------------

(546)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(547) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(548)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(549) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))))


----------------------------------------

(550)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(551) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Succ(x1)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS01(Succ(x2), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS01(Succ(x2), Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS01(Zero, Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS01(Zero, Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))))


----------------------------------------

(552)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS01(Succ(x2), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS01(Zero, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(553) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS01(Succ(x2), Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))))


----------------------------------------

(554)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS01(Zero, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(555) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS01(Zero, Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))))


----------------------------------------

(556)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(557) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))))


----------------------------------------

(558)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(559) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))))


----------------------------------------

(560)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(561) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero)))))


----------------------------------------

(562)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(563) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Zero, Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Zero, Succ(Zero)))))


----------------------------------------

(564)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Zero, Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(565) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(566)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(567) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))


----------------------------------------

(568)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(569) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero))))


----------------------------------------

(570)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(571) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(572)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(573) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))))


----------------------------------------

(574)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(575) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Succ(x1)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x1))), Neg(new_primModNatS02(x0, x1, x0, x1))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS01(Succ(x2), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS01(Succ(x2), Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS01(Zero, Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS01(Zero, Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))))


----------------------------------------

(576)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS01(Succ(x2), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS01(Zero, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(577) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS01(Succ(x2), Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))))


----------------------------------------

(578)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS01(Zero, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(579) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS01(Zero, Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))))


----------------------------------------

(580)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(581) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))))


----------------------------------------

(582)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(583) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))))


----------------------------------------

(584)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(585) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero)))))


----------------------------------------

(586)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(587) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Zero)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Zero, Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Zero, Succ(Zero)))))


----------------------------------------

(588)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Zero, Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(589) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(590)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(591) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))


----------------------------------------

(592)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(593) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero))))


----------------------------------------

(594)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(595) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(596)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(597) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))))


----------------------------------------

(598)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(599) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Succ(x2)), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Succ(x2)), Succ(Zero)))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Zero), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Zero), Succ(Zero)))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))))


----------------------------------------

(600)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Succ(x2)), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Zero), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(601) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Succ(x2)), Succ(Zero)))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero))))))


----------------------------------------

(602)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Zero), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(603) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Zero), Succ(Zero)))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero))))))


----------------------------------------

(604)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(605) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero))))))


----------------------------------------

(606)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(607) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero))))))


----------------------------------------

(608)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(609) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero))))))


----------------------------------------

(610)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(611) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero))))))


----------------------------------------

(612)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(613) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero))))))


----------------------------------------

(614)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(615) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Zero, Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Zero, Succ(Succ(Zero))))))


----------------------------------------

(616)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Zero, Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(617) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(618)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(619) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero)))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero))))


----------------------------------------

(620)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(621) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))


----------------------------------------

(622)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(623) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero))))


----------------------------------------

(624)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(625) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(626)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(627) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))))


----------------------------------------

(628)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(629) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Pos(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Succ(x2)), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Succ(x2)), Succ(Zero)))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Zero), Succ(Zero)))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Zero), Succ(Zero)))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))))


----------------------------------------

(630)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Succ(x2)), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Zero), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(631) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Succ(x2)), Succ(Zero)))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero))))))


----------------------------------------

(632)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Zero), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(633) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS01(Succ(Zero), Succ(Zero)))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero))))))


----------------------------------------

(634)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(635) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero))))))


----------------------------------------

(636)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(637) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero))))))


----------------------------------------

(638)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(639) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero))))))


----------------------------------------

(640)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(641) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero))))))


----------------------------------------

(642)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(643) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero))))))


----------------------------------------

(644)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(645) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Zero, Succ(Succ(Zero))))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Zero, Succ(Succ(Zero))))))


----------------------------------------

(646)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Zero, Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(647) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(648)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(649) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS1(Succ(x2), Succ(Zero)))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero))))


----------------------------------------

(650)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(651) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))))
   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))))


----------------------------------------

(652)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(653) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS1, Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero))))


----------------------------------------

(654)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Zero, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(655) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(656)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(657) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(new_primMinusNatS0(x0), Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))),new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero))))


----------------------------------------

(658)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(659) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Succ(x2)), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Succ(x2)), Succ(Zero)))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Zero), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Zero), Succ(Zero)))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))))


----------------------------------------

(660)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Succ(x2)), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Zero), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(661) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Succ(x2)), Succ(Zero)))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero))))))


----------------------------------------

(662)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Zero), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(663) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Zero), Succ(Zero)))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero))))))


----------------------------------------

(664)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(665) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero))))))


----------------------------------------

(666)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(667) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero))))))


----------------------------------------

(668)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(669) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero))))))


----------------------------------------

(670)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(671) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero))))))


----------------------------------------

(672)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(673) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero))))))


----------------------------------------

(674)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(675) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Zero, Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Zero, Succ(Succ(Zero))))))


----------------------------------------

(676)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Zero, Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(677) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(678)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(679) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero)))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero))))


----------------------------------------

(680)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(681) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))


----------------------------------------

(682)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(683) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero))))


----------------------------------------

(684)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(685) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(686)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(687) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))))


----------------------------------------

(688)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(689) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Succ(x3))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x3)))), Neg(new_primModNatS02(Succ(x2), Succ(x3), x2, x3))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Succ(x2)), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Succ(x2)), Succ(Zero)))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Zero), Succ(Zero)))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Zero), Succ(Zero)))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))))


----------------------------------------

(690)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Succ(x2)), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Zero), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(691) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Succ(x2)), Succ(Zero)))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero))))))


----------------------------------------

(692)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Zero), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(693) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS01(Succ(Zero), Succ(Zero)))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero))))))


----------------------------------------

(694)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(695) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Succ(x2))), Succ(Succ(Zero))), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero))))))


----------------------------------------

(696)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(697) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(Zero)), Succ(Succ(Zero))), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero))))))


----------------------------------------

(698)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(699) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Succ(x2)), Succ(Zero)), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero))))))


----------------------------------------

(700)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(701) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(Zero), Succ(Zero)), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero))))))


----------------------------------------

(702)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(703) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Succ(x2), Zero), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero))))))


----------------------------------------

(704)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(705) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(new_primMinusNatS2(Zero, Zero), Succ(Succ(Zero))))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Zero, Succ(Succ(Zero))))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Zero, Succ(Succ(Zero))))))


----------------------------------------

(706)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Zero, Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(707) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(708)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(709) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS1(Succ(x2), Succ(Zero)))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero))))


----------------------------------------

(710)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(711) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x0)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))) at position [1,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))))
   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))))


----------------------------------------

(712)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(713) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS1, Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero))))


----------------------------------------

(714)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Zero, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(715) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(716)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(717) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(new_primMinusNatS0(x0), Zero))) at position [1,0,0] we obtained the following new rules [LPAR04]:

   (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))),new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero))))


----------------------------------------

(718)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(719) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Pos(new_primModNatS1(Succ(x0), Zero)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(False) = 1
   POL(Neg(x_1)) = 1
   POL(Pos(x_1)) = x_1
   POL(Succ(x_1)) = 1
   POL(Zero) = 0
   POL(new_gcd0Gcd'(x_1, x_2)) = x_2
   POL(new_gcd0Gcd'0(x_1, x_2)) = 1
   POL(new_gcd0Gcd'1(x_1, x_2, x_3)) = x_1
   POL(new_primMinusNatS0(x_1)) = 1 + x_1
   POL(new_primMinusNatS1) = 1
   POL(new_primMinusNatS2(x_1, x_2)) = 0
   POL(new_primModNatS01(x_1, x_2)) = 1
   POL(new_primModNatS02(x_1, x_2, x_3, x_4)) = 1
   POL(new_primModNatS1(x_1, x_2)) = x_2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero


----------------------------------------

(720)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(721) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Zero))) -> new_gcd0Gcd'(Pos(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Zero))) -> new_gcd0Gcd'(Neg(Succ(Zero)), Neg(new_primModNatS1(Succ(x0), Zero)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(False) = 1
   POL(Neg(x_1)) = 1 + x_1
   POL(Pos(x_1)) = 1 + x_1
   POL(Succ(x_1)) = 1
   POL(Zero) = 0
   POL(new_gcd0Gcd'(x_1, x_2)) = x_2
   POL(new_gcd0Gcd'0(x_1, x_2)) = x_2
   POL(new_gcd0Gcd'1(x_1, x_2, x_3)) = x_3
   POL(new_primMinusNatS0(x_1)) = 1 + x_1
   POL(new_primMinusNatS1) = 1
   POL(new_primMinusNatS2(x_1, x_2)) = 0
   POL(new_primModNatS01(x_1, x_2)) = 1
   POL(new_primModNatS02(x_1, x_2, x_3, x_4)) = 1
   POL(new_primModNatS1(x_1, x_2)) = x_2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero


----------------------------------------

(722)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(723) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
----------------------------------------

(724)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(725) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Pos(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Pos(new_primModNatS02(x0, Zero, x0, Zero)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(False) = 2
   POL(Neg(x_1)) = 0
   POL(Pos(x_1)) = 2*x_1
   POL(Succ(x_1)) = 1 + x_1
   POL(Zero) = 0
   POL(new_gcd0Gcd'(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(new_gcd0Gcd'0(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(new_gcd0Gcd'1(x_1, x_2, x_3)) = 2*x_2 + 2*x_3
   POL(new_primMinusNatS2(x_1, x_2)) = x_1
   POL(new_primModNatS01(x_1, x_2)) = 1 + x_1
   POL(new_primModNatS02(x_1, x_2, x_3, x_4)) = 2 + x_1
   POL(new_primModNatS1(x_1, x_2)) = x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)


----------------------------------------

(726)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(727) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
----------------------------------------

(728)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(729) InductionCalculusProof (EQUIVALENT)
Note that final constraints are written in bold face.



For Pair new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0))) the following chains were created:
*We consider the chain new_gcd0Gcd'(x2, Pos(Succ(x3))) -> new_gcd0Gcd'1(False, x2, Pos(Succ(x3))), new_gcd0Gcd'1(False, x4, x5) -> new_gcd0Gcd'0(x4, x5) which results in the following constraint:

(1)    (new_gcd0Gcd'1(False, x2, Pos(Succ(x3)))=new_gcd0Gcd'1(False, x4, x5)  ==>  new_gcd0Gcd'(x2, Pos(Succ(x3)))_>=_new_gcd0Gcd'1(False, x2, Pos(Succ(x3))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'(x2, Pos(Succ(x3)))_>=_new_gcd0Gcd'1(False, x2, Pos(Succ(x3))))








For Pair new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46) the following chains were created:
*We consider the chain new_gcd0Gcd'1(False, x56, x57) -> new_gcd0Gcd'0(x56, x57), new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x58)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x58))), Pos(Succ(Zero))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x56, x57)=new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x58))))  ==>  new_gcd0Gcd'1(False, x56, x57)_>=_new_gcd0Gcd'0(x56, x57))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Pos(Succ(Zero)), Neg(Succ(Succ(x58))))_>=_new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x58)))))




*We consider the chain new_gcd0Gcd'1(False, x59, x60) -> new_gcd0Gcd'0(x59, x60), new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x61)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x61))), Pos(Succ(Zero))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x59, x60)=new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x61))))  ==>  new_gcd0Gcd'1(False, x59, x60)_>=_new_gcd0Gcd'0(x59, x60))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Pos(Succ(Zero)), Pos(Succ(Succ(x61))))_>=_new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x61)))))




*We consider the chain new_gcd0Gcd'1(False, x62, x63) -> new_gcd0Gcd'0(x62, x63), new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x64)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x64))), Neg(Succ(Zero))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x62, x63)=new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x64))))  ==>  new_gcd0Gcd'1(False, x62, x63)_>=_new_gcd0Gcd'0(x62, x63))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Zero)), Pos(Succ(Succ(x64))))_>=_new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x64)))))




*We consider the chain new_gcd0Gcd'1(False, x67, x68) -> new_gcd0Gcd'0(x67, x68), new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x69)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x69))), Neg(Succ(Zero))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x67, x68)=new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x69))))  ==>  new_gcd0Gcd'1(False, x67, x68)_>=_new_gcd0Gcd'0(x67, x68))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Zero)), Neg(Succ(Succ(x69))))_>=_new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x69)))))




*We consider the chain new_gcd0Gcd'1(False, x70, x71) -> new_gcd0Gcd'0(x70, x71), new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x72))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x72)))), Pos(Succ(Succ(Zero)))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x70, x71)=new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x72)))))  ==>  new_gcd0Gcd'1(False, x70, x71)_>=_new_gcd0Gcd'0(x70, x71))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x72)))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x72))))))




*We consider the chain new_gcd0Gcd'1(False, x73, x74) -> new_gcd0Gcd'0(x73, x74), new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x75))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x75)))), Pos(Succ(Succ(Zero)))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x73, x74)=new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x75)))))  ==>  new_gcd0Gcd'1(False, x73, x74)_>=_new_gcd0Gcd'0(x73, x74))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x75)))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x75))))))




*We consider the chain new_gcd0Gcd'1(False, x76, x77) -> new_gcd0Gcd'0(x76, x77), new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x78))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x78)))), Neg(Succ(Succ(Zero)))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x76, x77)=new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x78)))))  ==>  new_gcd0Gcd'1(False, x76, x77)_>=_new_gcd0Gcd'0(x76, x77))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x78)))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x78))))))




*We consider the chain new_gcd0Gcd'1(False, x79, x80) -> new_gcd0Gcd'0(x79, x80), new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x81))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x81)))), Neg(Succ(Succ(Zero)))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x79, x80)=new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x81)))))  ==>  new_gcd0Gcd'1(False, x79, x80)_>=_new_gcd0Gcd'0(x79, x80))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x81)))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x81))))))




*We consider the chain new_gcd0Gcd'1(False, x82, x83) -> new_gcd0Gcd'0(x82, x83), new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x84))))), Neg(Succ(Succ(Succ(Succ(x85)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x85))))), Pos(new_primModNatS02(Succ(Succ(x84)), Succ(Succ(x85)), x84, x85))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x82, x83)=new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x84))))), Neg(Succ(Succ(Succ(Succ(x85))))))  ==>  new_gcd0Gcd'1(False, x82, x83)_>=_new_gcd0Gcd'0(x82, x83))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Pos(Succ(Succ(Succ(Succ(x84))))), Neg(Succ(Succ(Succ(Succ(x85))))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x84))))), Neg(Succ(Succ(Succ(Succ(x85)))))))




*We consider the chain new_gcd0Gcd'1(False, x86, x87) -> new_gcd0Gcd'0(x86, x87), new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x88)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x88))))), Pos(Succ(Succ(Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x86, x87)=new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x88))))))  ==>  new_gcd0Gcd'1(False, x86, x87)_>=_new_gcd0Gcd'0(x86, x87))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x88))))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x88)))))))




*We consider the chain new_gcd0Gcd'1(False, x89, x90) -> new_gcd0Gcd'0(x89, x90), new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x91))))), Pos(Succ(Succ(Succ(Succ(x92)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x92))))), Pos(new_primModNatS02(Succ(Succ(x91)), Succ(Succ(x92)), x91, x92))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x89, x90)=new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x91))))), Pos(Succ(Succ(Succ(Succ(x92))))))  ==>  new_gcd0Gcd'1(False, x89, x90)_>=_new_gcd0Gcd'0(x89, x90))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Pos(Succ(Succ(Succ(Succ(x91))))), Pos(Succ(Succ(Succ(Succ(x92))))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x91))))), Pos(Succ(Succ(Succ(Succ(x92)))))))




*We consider the chain new_gcd0Gcd'1(False, x93, x94) -> new_gcd0Gcd'0(x93, x94), new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x95)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x95))))), Pos(Succ(Succ(Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x93, x94)=new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x95))))))  ==>  new_gcd0Gcd'1(False, x93, x94)_>=_new_gcd0Gcd'0(x93, x94))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x95))))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x95)))))))




*We consider the chain new_gcd0Gcd'1(False, x96, x97) -> new_gcd0Gcd'0(x96, x97), new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x98))))), Pos(Succ(Succ(Succ(Succ(x99)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x99))))), Neg(new_primModNatS02(Succ(Succ(x98)), Succ(Succ(x99)), x98, x99))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x96, x97)=new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x98))))), Pos(Succ(Succ(Succ(Succ(x99))))))  ==>  new_gcd0Gcd'1(False, x96, x97)_>=_new_gcd0Gcd'0(x96, x97))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Succ(x98))))), Pos(Succ(Succ(Succ(Succ(x99))))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x98))))), Pos(Succ(Succ(Succ(Succ(x99)))))))




*We consider the chain new_gcd0Gcd'1(False, x100, x101) -> new_gcd0Gcd'0(x100, x101), new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x102)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x102))))), Neg(Succ(Succ(Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x100, x101)=new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x102))))))  ==>  new_gcd0Gcd'1(False, x100, x101)_>=_new_gcd0Gcd'0(x100, x101))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x102))))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x102)))))))




*We consider the chain new_gcd0Gcd'1(False, x103, x104) -> new_gcd0Gcd'0(x103, x104), new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x105))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x105), Succ(Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x103, x104)=new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x105))))), Pos(Succ(Succ(Succ(Zero)))))  ==>  new_gcd0Gcd'1(False, x103, x104)_>=_new_gcd0Gcd'0(x103, x104))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Succ(x105))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x105))))), Pos(Succ(Succ(Succ(Zero))))))




*We consider the chain new_gcd0Gcd'1(False, x106, x107) -> new_gcd0Gcd'0(x106, x107), new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x106, x107)=new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero))))  ==>  new_gcd0Gcd'1(False, x106, x107)_>=_new_gcd0Gcd'0(x106, x107))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))))




*We consider the chain new_gcd0Gcd'1(False, x108, x109) -> new_gcd0Gcd'0(x108, x109), new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x110))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x110, Zero, x110, Zero))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x108, x109)=new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x110))))), Pos(Succ(Succ(Zero))))  ==>  new_gcd0Gcd'1(False, x108, x109)_>=_new_gcd0Gcd'0(x108, x109))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Succ(x110))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x110))))), Pos(Succ(Succ(Zero)))))




*We consider the chain new_gcd0Gcd'1(False, x111, x112) -> new_gcd0Gcd'0(x111, x112), new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x113))))), Neg(Succ(Succ(Succ(Succ(x114)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x114))))), Neg(new_primModNatS02(Succ(Succ(x113)), Succ(Succ(x114)), x113, x114))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x111, x112)=new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x113))))), Neg(Succ(Succ(Succ(Succ(x114))))))  ==>  new_gcd0Gcd'1(False, x111, x112)_>=_new_gcd0Gcd'0(x111, x112))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Succ(x113))))), Neg(Succ(Succ(Succ(Succ(x114))))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x113))))), Neg(Succ(Succ(Succ(Succ(x114)))))))




*We consider the chain new_gcd0Gcd'1(False, x115, x116) -> new_gcd0Gcd'0(x115, x116), new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x117)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x117))))), Neg(Succ(Succ(Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x115, x116)=new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x117))))))  ==>  new_gcd0Gcd'1(False, x115, x116)_>=_new_gcd0Gcd'0(x115, x116))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x117))))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x117)))))))




*We consider the chain new_gcd0Gcd'1(False, x118, x119) -> new_gcd0Gcd'0(x118, x119), new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x120))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x120), Succ(Succ(Zero))))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x118, x119)=new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x120))))), Neg(Succ(Succ(Succ(Zero)))))  ==>  new_gcd0Gcd'1(False, x118, x119)_>=_new_gcd0Gcd'0(x118, x119))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Succ(x120))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x120))))), Neg(Succ(Succ(Succ(Zero))))))




*We consider the chain new_gcd0Gcd'1(False, x121, x122) -> new_gcd0Gcd'0(x121, x122), new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x121, x122)=new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero))))  ==>  new_gcd0Gcd'1(False, x121, x122)_>=_new_gcd0Gcd'0(x121, x122))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))))




*We consider the chain new_gcd0Gcd'1(False, x123, x124) -> new_gcd0Gcd'0(x123, x124), new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x125))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x125, Zero, x125, Zero))) which results in the following constraint:

(1)    (new_gcd0Gcd'0(x123, x124)=new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x125))))), Neg(Succ(Succ(Zero))))  ==>  new_gcd0Gcd'1(False, x123, x124)_>=_new_gcd0Gcd'0(x123, x124))



We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:

(2)    (new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Succ(x125))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x125))))), Neg(Succ(Succ(Zero)))))








For Pair new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x126)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x126))), Pos(Succ(Zero))), new_gcd0Gcd'(x127, Pos(Succ(x128))) -> new_gcd0Gcd'1(False, x127, Pos(Succ(x128))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Neg(Succ(Succ(x126))), Pos(Succ(Zero)))=new_gcd0Gcd'(x127, Pos(Succ(x128)))  ==>  new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x126))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x126))), Pos(Succ(Zero))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x126))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x126))), Pos(Succ(Zero))))








For Pair new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x153)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x153))), Pos(Succ(Zero))), new_gcd0Gcd'(x154, Pos(Succ(x155))) -> new_gcd0Gcd'1(False, x154, Pos(Succ(x155))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Pos(Succ(Succ(x153))), Pos(Succ(Zero)))=new_gcd0Gcd'(x154, Pos(Succ(x155)))  ==>  new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x153))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x153))), Pos(Succ(Zero))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x153))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x153))), Pos(Succ(Zero))))








For Pair new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x185)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x185))), Neg(Succ(Zero))), new_gcd0Gcd'(x186, Neg(Succ(x187))) -> new_gcd0Gcd'1(False, x186, Neg(Succ(x187))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Pos(Succ(Succ(x185))), Neg(Succ(Zero)))=new_gcd0Gcd'(x186, Neg(Succ(x187)))  ==>  new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x185))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x185))), Neg(Succ(Zero))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x185))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x185))), Neg(Succ(Zero))))








For Pair new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0))) the following chains were created:
*We consider the chain new_gcd0Gcd'(x209, Neg(Succ(x210))) -> new_gcd0Gcd'1(False, x209, Neg(Succ(x210))), new_gcd0Gcd'1(False, x211, x212) -> new_gcd0Gcd'0(x211, x212) which results in the following constraint:

(1)    (new_gcd0Gcd'1(False, x209, Neg(Succ(x210)))=new_gcd0Gcd'1(False, x211, x212)  ==>  new_gcd0Gcd'(x209, Neg(Succ(x210)))_>=_new_gcd0Gcd'1(False, x209, Neg(Succ(x210))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'(x209, Neg(Succ(x210)))_>=_new_gcd0Gcd'1(False, x209, Neg(Succ(x210))))








For Pair new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x264)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x264))), Neg(Succ(Zero))), new_gcd0Gcd'(x265, Neg(Succ(x266))) -> new_gcd0Gcd'1(False, x265, Neg(Succ(x266))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Neg(Succ(Succ(x264))), Neg(Succ(Zero)))=new_gcd0Gcd'(x265, Neg(Succ(x266)))  ==>  new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x264))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x264))), Neg(Succ(Zero))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x264))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x264))), Neg(Succ(Zero))))








For Pair new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x286))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x286)))), Pos(Succ(Succ(Zero)))), new_gcd0Gcd'(x287, Pos(Succ(x288))) -> new_gcd0Gcd'1(False, x287, Pos(Succ(x288))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Neg(Succ(Succ(Succ(x286)))), Pos(Succ(Succ(Zero))))=new_gcd0Gcd'(x287, Pos(Succ(x288)))  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x286)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(x286)))), Pos(Succ(Succ(Zero)))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x286)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(x286)))), Pos(Succ(Succ(Zero)))))








For Pair new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero)))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x313))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x313)))), Pos(Succ(Succ(Zero)))), new_gcd0Gcd'(x314, Pos(Succ(x315))) -> new_gcd0Gcd'1(False, x314, Pos(Succ(x315))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Pos(Succ(Succ(Succ(x313)))), Pos(Succ(Succ(Zero))))=new_gcd0Gcd'(x314, Pos(Succ(x315)))  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x313)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(x313)))), Pos(Succ(Succ(Zero)))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x313)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(x313)))), Pos(Succ(Succ(Zero)))))








For Pair new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x345))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x345)))), Neg(Succ(Succ(Zero)))), new_gcd0Gcd'(x346, Neg(Succ(x347))) -> new_gcd0Gcd'1(False, x346, Neg(Succ(x347))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Pos(Succ(Succ(Succ(x345)))), Neg(Succ(Succ(Zero))))=new_gcd0Gcd'(x346, Neg(Succ(x347)))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x345)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(x345)))), Neg(Succ(Succ(Zero)))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x345)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(x345)))), Neg(Succ(Succ(Zero)))))








For Pair new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero)))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x372))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x372)))), Neg(Succ(Succ(Zero)))), new_gcd0Gcd'(x373, Neg(Succ(x374))) -> new_gcd0Gcd'1(False, x373, Neg(Succ(x374))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Neg(Succ(Succ(Succ(x372)))), Neg(Succ(Succ(Zero))))=new_gcd0Gcd'(x373, Neg(Succ(x374)))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x372)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(x372)))), Neg(Succ(Succ(Zero)))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x372)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(x372)))), Neg(Succ(Succ(Zero)))))








For Pair new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x394))))), Neg(Succ(Succ(Succ(Succ(x395)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x395))))), Pos(new_primModNatS02(Succ(Succ(x394)), Succ(Succ(x395)), x394, x395))), new_gcd0Gcd'(x396, Pos(Succ(x397))) -> new_gcd0Gcd'1(False, x396, Pos(Succ(x397))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x395))))), Pos(new_primModNatS02(Succ(Succ(x394)), Succ(Succ(x395)), x394, x395)))=new_gcd0Gcd'(x396, Pos(Succ(x397)))  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x394))))), Neg(Succ(Succ(Succ(Succ(x395))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x395))))), Pos(new_primModNatS02(Succ(Succ(x394)), Succ(Succ(x395)), x394, x395))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Succ(Succ(x394))=x822 & Succ(Succ(x395))=x823 & new_primModNatS02(x822, x823, x394, x395)=Succ(x397)  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x394))))), Neg(Succ(Succ(Succ(Succ(x395))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x395))))), Pos(new_primModNatS02(Succ(Succ(x394)), Succ(Succ(x395)), x394, x395))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x822, x823, x394, x395)=Succ(x397) which results in the following new constraints:

(3)    (new_primModNatS01(x826, x825)=Succ(x397) & Succ(Succ(Succ(x824)))=x826 & Succ(Succ(Zero))=x825  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x824)))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Succ(x824))), Succ(Succ(Zero)), Succ(x824), Zero))))

(4)    (new_primModNatS02(x830, x829, x828, x827)=Succ(x397) & Succ(Succ(Succ(x828)))=x830 & Succ(Succ(Succ(x827)))=x829 & (\/x831:new_primModNatS02(x830, x829, x828, x827)=Succ(x831) & Succ(Succ(x828))=x830 & Succ(Succ(x827))=x829  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x828))))), Neg(Succ(Succ(Succ(Succ(x827))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x827))))), Pos(new_primModNatS02(Succ(Succ(x828)), Succ(Succ(x827)), x828, x827))))  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x828)))))), Neg(Succ(Succ(Succ(Succ(Succ(x827)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(x827)))))), Pos(new_primModNatS02(Succ(Succ(Succ(x828))), Succ(Succ(Succ(x827))), Succ(x828), Succ(x827)))))

(5)    (new_primModNatS01(x833, x832)=Succ(x397) & Succ(Succ(Zero))=x833 & Succ(Succ(Zero))=x832  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))

(6)    (Succ(Succ(x836))=Succ(x397) & Succ(Succ(Zero))=x836 & Succ(Succ(Succ(x834)))=x835  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Succ(x834)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(x834)))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x834))), Zero, Succ(x834)))))



We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x826, x825)=Succ(x397) which results in the following new constraint:

(7)    (new_primModNatS1(new_primMinusNatS2(Succ(x838), Succ(x837)), Succ(x837))=Succ(x397) & Succ(Succ(Succ(x824)))=x838 & Succ(Succ(Zero))=x837  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x824)))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Succ(x824))), Succ(Succ(Zero)), Succ(x824), Zero))))



We simplified constraint (4) using rule (IV) which results in the following new constraint:

(8)    (new_primModNatS02(x830, x829, x828, x827)=Succ(x397) & Succ(Succ(Succ(x828)))=x830 & Succ(Succ(Succ(x827)))=x829  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x828)))))), Neg(Succ(Succ(Succ(Succ(Succ(x827)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(x827)))))), Pos(new_primModNatS02(Succ(Succ(Succ(x828))), Succ(Succ(Succ(x827))), Succ(x828), Succ(x827)))))



We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x833, x832)=Succ(x397) which results in the following new constraint:

(9)    (new_primModNatS1(new_primMinusNatS2(Succ(x857), Succ(x856)), Succ(x856))=Succ(x397) & Succ(Succ(Zero))=x857 & Succ(Succ(Zero))=x856  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))



We simplified constraint (6) using rules (I), (II), (IV) which results in the following new constraint:

(10)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Succ(x834)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(x834)))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x834))), Zero, Succ(x834)))))



We simplified constraint (7) using rules (III), (IV), (VII) which results in the following new constraint:

(11)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x824)))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Succ(x824))), Succ(Succ(Zero)), Succ(x824), Zero))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x830, x829, x828, x827)=Succ(x397) which results in the following new constraints:

(12)    (new_primModNatS01(x845, x844)=Succ(x397) & Succ(Succ(Succ(Succ(x843))))=x845 & Succ(Succ(Succ(Zero)))=x844  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x843))))))), Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(new_primModNatS02(Succ(Succ(Succ(Succ(x843)))), Succ(Succ(Succ(Zero))), Succ(Succ(x843)), Succ(Zero)))))

(13)    (new_primModNatS02(x849, x848, x847, x846)=Succ(x397) & Succ(Succ(Succ(Succ(x847))))=x849 & Succ(Succ(Succ(Succ(x846))))=x848 & (\/x850:new_primModNatS02(x849, x848, x847, x846)=Succ(x850) & Succ(Succ(Succ(x847)))=x849 & Succ(Succ(Succ(x846)))=x848  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x847)))))), Neg(Succ(Succ(Succ(Succ(Succ(x846)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(x846)))))), Pos(new_primModNatS02(Succ(Succ(Succ(x847))), Succ(Succ(Succ(x846))), Succ(x847), Succ(x846)))))  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x847))))))), Neg(Succ(Succ(Succ(Succ(Succ(Succ(x846))))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x846))))))), Pos(new_primModNatS02(Succ(Succ(Succ(Succ(x847)))), Succ(Succ(Succ(Succ(x846)))), Succ(Succ(x847)), Succ(Succ(x846))))))

(14)    (new_primModNatS01(x852, x851)=Succ(x397) & Succ(Succ(Succ(Zero)))=x852 & Succ(Succ(Succ(Zero)))=x851  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), Succ(Zero), Succ(Zero)))))

(15)    (Succ(Succ(x855))=Succ(x397) & Succ(Succ(Succ(Zero)))=x855 & Succ(Succ(Succ(Succ(x853))))=x854  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Succ(Succ(Succ(x853))))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x853))))))), Pos(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x853)))), Succ(Zero), Succ(Succ(x853))))))



We simplified constraint (12) using rules (III), (IV) which results in the following new constraint:

(16)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x843))))))), Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(new_primModNatS02(Succ(Succ(Succ(Succ(x843)))), Succ(Succ(Succ(Zero))), Succ(Succ(x843)), Succ(Zero)))))



We simplified constraint (13) using rules (III), (IV) which results in the following new constraint:

(17)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x847))))))), Neg(Succ(Succ(Succ(Succ(Succ(Succ(x846))))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x846))))))), Pos(new_primModNatS02(Succ(Succ(Succ(Succ(x847)))), Succ(Succ(Succ(Succ(x846)))), Succ(Succ(x847)), Succ(Succ(x846))))))



We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:

(18)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), Succ(Zero), Succ(Zero)))))



We simplified constraint (15) using rules (I), (II), (IV) which results in the following new constraint:

(19)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Succ(Succ(Succ(x853))))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x853))))))), Pos(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x853)))), Succ(Zero), Succ(Succ(x853))))))



We simplified constraint (9) using rules (III), (IV), (VII) which results in the following new constraint:

(20)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))








For Pair new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x446)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x446))))), Pos(Succ(Succ(Succ(Zero))))), new_gcd0Gcd'(x447, Pos(Succ(x448))) -> new_gcd0Gcd'1(False, x447, Pos(Succ(x448))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x446))))), Pos(Succ(Succ(Succ(Zero)))))=new_gcd0Gcd'(x447, Pos(Succ(x448)))  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x446))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x446))))), Pos(Succ(Succ(Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x446))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x446))))), Pos(Succ(Succ(Succ(Zero))))))








For Pair new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x473))))), Pos(Succ(Succ(Succ(Succ(x474)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x474))))), Pos(new_primModNatS02(Succ(Succ(x473)), Succ(Succ(x474)), x473, x474))), new_gcd0Gcd'(x475, Pos(Succ(x476))) -> new_gcd0Gcd'1(False, x475, Pos(Succ(x476))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x474))))), Pos(new_primModNatS02(Succ(Succ(x473)), Succ(Succ(x474)), x473, x474)))=new_gcd0Gcd'(x475, Pos(Succ(x476)))  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x473))))), Pos(Succ(Succ(Succ(Succ(x474))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x474))))), Pos(new_primModNatS02(Succ(Succ(x473)), Succ(Succ(x474)), x473, x474))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Succ(Succ(x473))=x862 & Succ(Succ(x474))=x863 & new_primModNatS02(x862, x863, x473, x474)=Succ(x476)  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x473))))), Pos(Succ(Succ(Succ(Succ(x474))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x474))))), Pos(new_primModNatS02(Succ(Succ(x473)), Succ(Succ(x474)), x473, x474))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x862, x863, x473, x474)=Succ(x476) which results in the following new constraints:

(3)    (new_primModNatS01(x866, x865)=Succ(x476) & Succ(Succ(Succ(x864)))=x866 & Succ(Succ(Zero))=x865  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x864)))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Succ(x864))), Succ(Succ(Zero)), Succ(x864), Zero))))

(4)    (new_primModNatS02(x870, x869, x868, x867)=Succ(x476) & Succ(Succ(Succ(x868)))=x870 & Succ(Succ(Succ(x867)))=x869 & (\/x871:new_primModNatS02(x870, x869, x868, x867)=Succ(x871) & Succ(Succ(x868))=x870 & Succ(Succ(x867))=x869  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x868))))), Pos(Succ(Succ(Succ(Succ(x867))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x867))))), Pos(new_primModNatS02(Succ(Succ(x868)), Succ(Succ(x867)), x868, x867))))  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x868)))))), Pos(Succ(Succ(Succ(Succ(Succ(x867)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(x867)))))), Pos(new_primModNatS02(Succ(Succ(Succ(x868))), Succ(Succ(Succ(x867))), Succ(x868), Succ(x867)))))

(5)    (new_primModNatS01(x873, x872)=Succ(x476) & Succ(Succ(Zero))=x873 & Succ(Succ(Zero))=x872  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))

(6)    (Succ(Succ(x876))=Succ(x476) & Succ(Succ(Zero))=x876 & Succ(Succ(Succ(x874)))=x875  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Succ(x874)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(x874)))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x874))), Zero, Succ(x874)))))



We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x866, x865)=Succ(x476) which results in the following new constraint:

(7)    (new_primModNatS1(new_primMinusNatS2(Succ(x878), Succ(x877)), Succ(x877))=Succ(x476) & Succ(Succ(Succ(x864)))=x878 & Succ(Succ(Zero))=x877  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x864)))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Succ(x864))), Succ(Succ(Zero)), Succ(x864), Zero))))



We simplified constraint (4) using rule (IV) which results in the following new constraint:

(8)    (new_primModNatS02(x870, x869, x868, x867)=Succ(x476) & Succ(Succ(Succ(x868)))=x870 & Succ(Succ(Succ(x867)))=x869  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x868)))))), Pos(Succ(Succ(Succ(Succ(Succ(x867)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(x867)))))), Pos(new_primModNatS02(Succ(Succ(Succ(x868))), Succ(Succ(Succ(x867))), Succ(x868), Succ(x867)))))



We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x873, x872)=Succ(x476) which results in the following new constraint:

(9)    (new_primModNatS1(new_primMinusNatS2(Succ(x897), Succ(x896)), Succ(x896))=Succ(x476) & Succ(Succ(Zero))=x897 & Succ(Succ(Zero))=x896  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))



We simplified constraint (6) using rules (I), (II), (IV) which results in the following new constraint:

(10)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Succ(x874)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(x874)))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x874))), Zero, Succ(x874)))))



We simplified constraint (7) using rules (III), (IV), (VII) which results in the following new constraint:

(11)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x864)))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Succ(x864))), Succ(Succ(Zero)), Succ(x864), Zero))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x870, x869, x868, x867)=Succ(x476) which results in the following new constraints:

(12)    (new_primModNatS01(x885, x884)=Succ(x476) & Succ(Succ(Succ(Succ(x883))))=x885 & Succ(Succ(Succ(Zero)))=x884  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x883))))))), Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(new_primModNatS02(Succ(Succ(Succ(Succ(x883)))), Succ(Succ(Succ(Zero))), Succ(Succ(x883)), Succ(Zero)))))

(13)    (new_primModNatS02(x889, x888, x887, x886)=Succ(x476) & Succ(Succ(Succ(Succ(x887))))=x889 & Succ(Succ(Succ(Succ(x886))))=x888 & (\/x890:new_primModNatS02(x889, x888, x887, x886)=Succ(x890) & Succ(Succ(Succ(x887)))=x889 & Succ(Succ(Succ(x886)))=x888  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x887)))))), Pos(Succ(Succ(Succ(Succ(Succ(x886)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(x886)))))), Pos(new_primModNatS02(Succ(Succ(Succ(x887))), Succ(Succ(Succ(x886))), Succ(x887), Succ(x886)))))  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x887))))))), Pos(Succ(Succ(Succ(Succ(Succ(Succ(x886))))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x886))))))), Pos(new_primModNatS02(Succ(Succ(Succ(Succ(x887)))), Succ(Succ(Succ(Succ(x886)))), Succ(Succ(x887)), Succ(Succ(x886))))))

(14)    (new_primModNatS01(x892, x891)=Succ(x476) & Succ(Succ(Succ(Zero)))=x892 & Succ(Succ(Succ(Zero)))=x891  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), Succ(Zero), Succ(Zero)))))

(15)    (Succ(Succ(x895))=Succ(x476) & Succ(Succ(Succ(Zero)))=x895 & Succ(Succ(Succ(Succ(x893))))=x894  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Succ(Succ(Succ(x893))))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x893))))))), Pos(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x893)))), Succ(Zero), Succ(Succ(x893))))))



We simplified constraint (12) using rules (III), (IV) which results in the following new constraint:

(16)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x883))))))), Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(new_primModNatS02(Succ(Succ(Succ(Succ(x883)))), Succ(Succ(Succ(Zero))), Succ(Succ(x883)), Succ(Zero)))))



We simplified constraint (13) using rules (III), (IV) which results in the following new constraint:

(17)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x887))))))), Pos(Succ(Succ(Succ(Succ(Succ(Succ(x886))))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x886))))))), Pos(new_primModNatS02(Succ(Succ(Succ(Succ(x887)))), Succ(Succ(Succ(Succ(x886)))), Succ(Succ(x887)), Succ(Succ(x886))))))



We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:

(18)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), Succ(Zero), Succ(Zero)))))



We simplified constraint (15) using rules (I), (II), (IV) which results in the following new constraint:

(19)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Succ(Succ(Succ(x893))))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x893))))))), Pos(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x893)))), Succ(Zero), Succ(Succ(x893))))))



We simplified constraint (9) using rules (III), (IV), (VII) which results in the following new constraint:

(20)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))








For Pair new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x525)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x525))))), Pos(Succ(Succ(Succ(Zero))))), new_gcd0Gcd'(x526, Pos(Succ(x527))) -> new_gcd0Gcd'1(False, x526, Pos(Succ(x527))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x525))))), Pos(Succ(Succ(Succ(Zero)))))=new_gcd0Gcd'(x526, Pos(Succ(x527)))  ==>  new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x525))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x525))))), Pos(Succ(Succ(Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x525))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x525))))), Pos(Succ(Succ(Succ(Zero))))))








For Pair new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x562))))), Pos(Succ(Succ(Succ(Succ(x563)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x563))))), Neg(new_primModNatS02(Succ(Succ(x562)), Succ(Succ(x563)), x562, x563))), new_gcd0Gcd'(x564, Neg(Succ(x565))) -> new_gcd0Gcd'1(False, x564, Neg(Succ(x565))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x563))))), Neg(new_primModNatS02(Succ(Succ(x562)), Succ(Succ(x563)), x562, x563)))=new_gcd0Gcd'(x564, Neg(Succ(x565)))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x562))))), Pos(Succ(Succ(Succ(Succ(x563))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x563))))), Neg(new_primModNatS02(Succ(Succ(x562)), Succ(Succ(x563)), x562, x563))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Succ(Succ(x562))=x902 & Succ(Succ(x563))=x903 & new_primModNatS02(x902, x903, x562, x563)=Succ(x565)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x562))))), Pos(Succ(Succ(Succ(Succ(x563))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x563))))), Neg(new_primModNatS02(Succ(Succ(x562)), Succ(Succ(x563)), x562, x563))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x902, x903, x562, x563)=Succ(x565) which results in the following new constraints:

(3)    (new_primModNatS01(x906, x905)=Succ(x565) & Succ(Succ(Succ(x904)))=x906 & Succ(Succ(Zero))=x905  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x904)))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Succ(x904))), Succ(Succ(Zero)), Succ(x904), Zero))))

(4)    (new_primModNatS02(x910, x909, x908, x907)=Succ(x565) & Succ(Succ(Succ(x908)))=x910 & Succ(Succ(Succ(x907)))=x909 & (\/x911:new_primModNatS02(x910, x909, x908, x907)=Succ(x911) & Succ(Succ(x908))=x910 & Succ(Succ(x907))=x909  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x908))))), Pos(Succ(Succ(Succ(Succ(x907))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x907))))), Neg(new_primModNatS02(Succ(Succ(x908)), Succ(Succ(x907)), x908, x907))))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x908)))))), Pos(Succ(Succ(Succ(Succ(Succ(x907)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(x907)))))), Neg(new_primModNatS02(Succ(Succ(Succ(x908))), Succ(Succ(Succ(x907))), Succ(x908), Succ(x907)))))

(5)    (new_primModNatS01(x913, x912)=Succ(x565) & Succ(Succ(Zero))=x913 & Succ(Succ(Zero))=x912  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))

(6)    (Succ(Succ(x916))=Succ(x565) & Succ(Succ(Zero))=x916 & Succ(Succ(Succ(x914)))=x915  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Succ(x914)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(x914)))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x914))), Zero, Succ(x914)))))



We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x906, x905)=Succ(x565) which results in the following new constraint:

(7)    (new_primModNatS1(new_primMinusNatS2(Succ(x918), Succ(x917)), Succ(x917))=Succ(x565) & Succ(Succ(Succ(x904)))=x918 & Succ(Succ(Zero))=x917  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x904)))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Succ(x904))), Succ(Succ(Zero)), Succ(x904), Zero))))



We simplified constraint (4) using rule (IV) which results in the following new constraint:

(8)    (new_primModNatS02(x910, x909, x908, x907)=Succ(x565) & Succ(Succ(Succ(x908)))=x910 & Succ(Succ(Succ(x907)))=x909  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x908)))))), Pos(Succ(Succ(Succ(Succ(Succ(x907)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(x907)))))), Neg(new_primModNatS02(Succ(Succ(Succ(x908))), Succ(Succ(Succ(x907))), Succ(x908), Succ(x907)))))



We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x913, x912)=Succ(x565) which results in the following new constraint:

(9)    (new_primModNatS1(new_primMinusNatS2(Succ(x937), Succ(x936)), Succ(x936))=Succ(x565) & Succ(Succ(Zero))=x937 & Succ(Succ(Zero))=x936  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))



We simplified constraint (6) using rules (I), (II), (IV) which results in the following new constraint:

(10)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Succ(x914)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(x914)))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x914))), Zero, Succ(x914)))))



We simplified constraint (7) using rules (III), (IV), (VII) which results in the following new constraint:

(11)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x904)))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Succ(x904))), Succ(Succ(Zero)), Succ(x904), Zero))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x910, x909, x908, x907)=Succ(x565) which results in the following new constraints:

(12)    (new_primModNatS01(x925, x924)=Succ(x565) & Succ(Succ(Succ(Succ(x923))))=x925 & Succ(Succ(Succ(Zero)))=x924  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x923))))))), Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(new_primModNatS02(Succ(Succ(Succ(Succ(x923)))), Succ(Succ(Succ(Zero))), Succ(Succ(x923)), Succ(Zero)))))

(13)    (new_primModNatS02(x929, x928, x927, x926)=Succ(x565) & Succ(Succ(Succ(Succ(x927))))=x929 & Succ(Succ(Succ(Succ(x926))))=x928 & (\/x930:new_primModNatS02(x929, x928, x927, x926)=Succ(x930) & Succ(Succ(Succ(x927)))=x929 & Succ(Succ(Succ(x926)))=x928  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x927)))))), Pos(Succ(Succ(Succ(Succ(Succ(x926)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(x926)))))), Neg(new_primModNatS02(Succ(Succ(Succ(x927))), Succ(Succ(Succ(x926))), Succ(x927), Succ(x926)))))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x927))))))), Pos(Succ(Succ(Succ(Succ(Succ(Succ(x926))))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x926))))))), Neg(new_primModNatS02(Succ(Succ(Succ(Succ(x927)))), Succ(Succ(Succ(Succ(x926)))), Succ(Succ(x927)), Succ(Succ(x926))))))

(14)    (new_primModNatS01(x932, x931)=Succ(x565) & Succ(Succ(Succ(Zero)))=x932 & Succ(Succ(Succ(Zero)))=x931  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), Succ(Zero), Succ(Zero)))))

(15)    (Succ(Succ(x935))=Succ(x565) & Succ(Succ(Succ(Zero)))=x935 & Succ(Succ(Succ(Succ(x933))))=x934  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Succ(Succ(Succ(x933))))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x933))))))), Neg(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x933)))), Succ(Zero), Succ(Succ(x933))))))



We simplified constraint (12) using rules (III), (IV) which results in the following new constraint:

(16)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x923))))))), Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(new_primModNatS02(Succ(Succ(Succ(Succ(x923)))), Succ(Succ(Succ(Zero))), Succ(Succ(x923)), Succ(Zero)))))



We simplified constraint (13) using rules (III), (IV) which results in the following new constraint:

(17)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x927))))))), Pos(Succ(Succ(Succ(Succ(Succ(Succ(x926))))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x926))))))), Neg(new_primModNatS02(Succ(Succ(Succ(Succ(x927)))), Succ(Succ(Succ(Succ(x926)))), Succ(Succ(x927)), Succ(Succ(x926))))))



We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:

(18)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), Succ(Zero), Succ(Zero)))))



We simplified constraint (15) using rules (I), (II), (IV) which results in the following new constraint:

(19)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Succ(Succ(Succ(x933))))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x933))))))), Neg(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x933)))), Succ(Zero), Succ(Succ(x933))))))



We simplified constraint (9) using rules (III), (IV), (VII) which results in the following new constraint:

(20)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))








For Pair new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x609)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x609))))), Neg(Succ(Succ(Succ(Zero))))), new_gcd0Gcd'(x610, Neg(Succ(x611))) -> new_gcd0Gcd'1(False, x610, Neg(Succ(x611))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x609))))), Neg(Succ(Succ(Succ(Zero)))))=new_gcd0Gcd'(x610, Neg(Succ(x611)))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x609))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x609))))), Neg(Succ(Succ(Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x609))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x609))))), Neg(Succ(Succ(Succ(Zero))))))








For Pair new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x636))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x636), Succ(Succ(Zero))))), new_gcd0Gcd'(x637, Neg(Succ(x638))) -> new_gcd0Gcd'1(False, x637, Neg(Succ(x638))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x636), Succ(Succ(Zero)))))=new_gcd0Gcd'(x637, Neg(Succ(x638)))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x636))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x636), Succ(Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Succ(x636)=x942 & Succ(Succ(Zero))=x943 & new_primModNatS1(x942, x943)=Succ(x638)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x636))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x636), Succ(Succ(Zero))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS1(x942, x943)=Succ(x638) which results in the following new constraints:

(3)    (Succ(Zero)=Succ(x638) & Succ(x636)=Succ(Zero) & Succ(Succ(Zero))=Succ(x944)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x636))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x636), Succ(Succ(Zero))))))

(4)    (new_primModNatS1(new_primMinusNatS1, Zero)=Succ(x638) & Succ(x636)=Succ(Zero) & Succ(Succ(Zero))=Zero  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x636))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x636), Succ(Succ(Zero))))))

(5)    (new_primModNatS1(new_primMinusNatS0(x946), Zero)=Succ(x638) & Succ(x636)=Succ(Succ(x946)) & Succ(Succ(Zero))=Zero  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x636))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x636), Succ(Succ(Zero))))))

(6)    (new_primModNatS02(x948, x947, x948, x947)=Succ(x638) & Succ(x636)=Succ(Succ(x948)) & Succ(Succ(Zero))=Succ(x947)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x636))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x636), Succ(Succ(Zero))))))



We simplified constraint (3) using rules (I), (II), (III), (IV) which results in the following new constraint:

(7)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Zero), Succ(Succ(Zero))))))



We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II), (III), (VII) which results in the following new constraint:

(8)    (x948=x949 & x947=x950 & new_primModNatS02(x948, x947, x949, x950)=Succ(x638) & Succ(Zero)=x947  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x948)))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(x948)), Succ(Succ(Zero))))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x948, x947, x949, x950)=Succ(x638) which results in the following new constraints:

(9)    (new_primModNatS01(x953, x952)=Succ(x638) & x953=Succ(x951) & x952=Zero & Succ(Zero)=x952  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x953)))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(x953)), Succ(Succ(Zero))))))

(10)    (new_primModNatS02(x957, x956, x955, x954)=Succ(x638) & x957=Succ(x955) & x956=Succ(x954) & Succ(Zero)=x956 & (\/x958:new_primModNatS02(x957, x956, x955, x954)=Succ(x958) & x957=x955 & x956=x954 & Succ(Zero)=x956  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x957)))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(x957)), Succ(Succ(Zero))))))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x957)))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(x957)), Succ(Succ(Zero))))))

(11)    (new_primModNatS01(x960, x959)=Succ(x638) & x960=Zero & x959=Zero & Succ(Zero)=x959  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x960)))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(x960)), Succ(Succ(Zero))))))

(12)    (Succ(Succ(x963))=Succ(x638) & x963=Zero & x962=Succ(x961) & Succ(Zero)=x962  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x963)))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(x963)), Succ(Succ(Zero))))))



We solved constraint (9) using rules (I), (II), (III).We simplified constraint (10) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(13)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x955))))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(Succ(x955))), Succ(Succ(Zero))))))



We solved constraint (11) using rules (I), (II), (III).We simplified constraint (12) using rules (I), (II), (III), (IV) which results in the following new constraint:

(14)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(Zero)), Succ(Succ(Zero))))))








For Pair new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))), new_gcd0Gcd'(x658, Neg(Succ(x659))) -> new_gcd0Gcd'1(False, x658, Neg(Succ(x659))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))=new_gcd0Gcd'(x658, Neg(Succ(x659)))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))))








For Pair new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x665))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x665, Zero, x665, Zero))), new_gcd0Gcd'(x666, Neg(Succ(x667))) -> new_gcd0Gcd'1(False, x666, Neg(Succ(x667))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x665, Zero, x665, Zero)))=new_gcd0Gcd'(x666, Neg(Succ(x667)))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x665))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x665, Zero, x665, Zero))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Zero=x966 & x665=x967 & Zero=x968 & new_primModNatS02(x665, x966, x967, x968)=Succ(x667)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x665))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x665, Zero, x665, Zero))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x665, x966, x967, x968)=Succ(x667) which results in the following new constraints:

(3)    (new_primModNatS01(x971, x970)=Succ(x667) & Zero=x970 & x971=Succ(x969) & Zero=Zero  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x971))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x971, Zero, x971, Zero))))

(4)    (new_primModNatS02(x975, x974, x973, x972)=Succ(x667) & Zero=x974 & x975=Succ(x973) & Zero=Succ(x972) & (\/x976:new_primModNatS02(x975, x974, x973, x972)=Succ(x976) & Zero=x974 & x975=x973 & Zero=x972  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x975))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x975, Zero, x975, Zero))))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x975))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x975, Zero, x975, Zero))))

(5)    (new_primModNatS01(x978, x977)=Succ(x667) & Zero=x977 & x978=Zero & Zero=Zero  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x978))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x978, Zero, x978, Zero))))

(6)    (Succ(Succ(x981))=Succ(x667) & Zero=x980 & x981=Zero & Zero=Succ(x979)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x981))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x981, Zero, x981, Zero))))



We simplified constraint (3) using rules (I), (II), (III), (VII) which results in the following new constraint:

(7)    (Succ(x969)=x982 & new_primModNatS01(x982, x970)=Succ(x667) & Zero=x970  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x969)))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(Succ(x969), Zero, Succ(x969), Zero))))



We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (III), (VII) which results in the following new constraint:

(8)    (Zero=x989 & new_primModNatS01(x989, x977)=Succ(x667) & Zero=x977  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(Zero, Zero, Zero, Zero))))



We solved constraint (6) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x982, x970)=Succ(x667) which results in the following new constraint:

(9)    (new_primModNatS1(new_primMinusNatS2(Succ(x984), Succ(x983)), Succ(x983))=Succ(x667) & Succ(x969)=x984 & Zero=x983  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x969)))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(Succ(x969), Zero, Succ(x969), Zero))))



We simplified constraint (9) using rules (III), (IV), (VII) which results in the following new constraint:

(10)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x969)))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(Succ(x969), Zero, Succ(x969), Zero))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x989, x977)=Succ(x667) which results in the following new constraint:

(11)    (new_primModNatS1(new_primMinusNatS2(Succ(x991), Succ(x990)), Succ(x990))=Succ(x667) & Zero=x991 & Zero=x990  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(Zero, Zero, Zero, Zero))))



We simplified constraint (11) using rules (III), (IV), (VII) which results in the following new constraint:

(12)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(Zero, Zero, Zero, Zero))))








For Pair new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x697))))), Neg(Succ(Succ(Succ(Succ(x698)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x698))))), Neg(new_primModNatS02(Succ(Succ(x697)), Succ(Succ(x698)), x697, x698))), new_gcd0Gcd'(x699, Neg(Succ(x700))) -> new_gcd0Gcd'1(False, x699, Neg(Succ(x700))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x698))))), Neg(new_primModNatS02(Succ(Succ(x697)), Succ(Succ(x698)), x697, x698)))=new_gcd0Gcd'(x699, Neg(Succ(x700)))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x697))))), Neg(Succ(Succ(Succ(Succ(x698))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x698))))), Neg(new_primModNatS02(Succ(Succ(x697)), Succ(Succ(x698)), x697, x698))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Succ(Succ(x697))=x996 & Succ(Succ(x698))=x997 & new_primModNatS02(x996, x997, x697, x698)=Succ(x700)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x697))))), Neg(Succ(Succ(Succ(Succ(x698))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x698))))), Neg(new_primModNatS02(Succ(Succ(x697)), Succ(Succ(x698)), x697, x698))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x996, x997, x697, x698)=Succ(x700) which results in the following new constraints:

(3)    (new_primModNatS01(x1000, x999)=Succ(x700) & Succ(Succ(Succ(x998)))=x1000 & Succ(Succ(Zero))=x999  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x998)))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Succ(x998))), Succ(Succ(Zero)), Succ(x998), Zero))))

(4)    (new_primModNatS02(x1004, x1003, x1002, x1001)=Succ(x700) & Succ(Succ(Succ(x1002)))=x1004 & Succ(Succ(Succ(x1001)))=x1003 & (\/x1005:new_primModNatS02(x1004, x1003, x1002, x1001)=Succ(x1005) & Succ(Succ(x1002))=x1004 & Succ(Succ(x1001))=x1003  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x1002))))), Neg(Succ(Succ(Succ(Succ(x1001))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x1001))))), Neg(new_primModNatS02(Succ(Succ(x1002)), Succ(Succ(x1001)), x1002, x1001))))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x1002)))))), Neg(Succ(Succ(Succ(Succ(Succ(x1001)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(x1001)))))), Neg(new_primModNatS02(Succ(Succ(Succ(x1002))), Succ(Succ(Succ(x1001))), Succ(x1002), Succ(x1001)))))

(5)    (new_primModNatS01(x1007, x1006)=Succ(x700) & Succ(Succ(Zero))=x1007 & Succ(Succ(Zero))=x1006  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))

(6)    (Succ(Succ(x1010))=Succ(x700) & Succ(Succ(Zero))=x1010 & Succ(Succ(Succ(x1008)))=x1009  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Succ(x1008)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(x1008)))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x1008))), Zero, Succ(x1008)))))



We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x1000, x999)=Succ(x700) which results in the following new constraint:

(7)    (new_primModNatS1(new_primMinusNatS2(Succ(x1012), Succ(x1011)), Succ(x1011))=Succ(x700) & Succ(Succ(Succ(x998)))=x1012 & Succ(Succ(Zero))=x1011  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x998)))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Succ(x998))), Succ(Succ(Zero)), Succ(x998), Zero))))



We simplified constraint (4) using rule (IV) which results in the following new constraint:

(8)    (new_primModNatS02(x1004, x1003, x1002, x1001)=Succ(x700) & Succ(Succ(Succ(x1002)))=x1004 & Succ(Succ(Succ(x1001)))=x1003  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x1002)))))), Neg(Succ(Succ(Succ(Succ(Succ(x1001)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(x1001)))))), Neg(new_primModNatS02(Succ(Succ(Succ(x1002))), Succ(Succ(Succ(x1001))), Succ(x1002), Succ(x1001)))))



We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x1007, x1006)=Succ(x700) which results in the following new constraint:

(9)    (new_primModNatS1(new_primMinusNatS2(Succ(x1031), Succ(x1030)), Succ(x1030))=Succ(x700) & Succ(Succ(Zero))=x1031 & Succ(Succ(Zero))=x1030  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))



We simplified constraint (6) using rules (I), (II), (IV) which results in the following new constraint:

(10)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Succ(x1008)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(x1008)))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x1008))), Zero, Succ(x1008)))))



We simplified constraint (7) using rules (III), (IV), (VII) which results in the following new constraint:

(11)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x998)))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Succ(x998))), Succ(Succ(Zero)), Succ(x998), Zero))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x1004, x1003, x1002, x1001)=Succ(x700) which results in the following new constraints:

(12)    (new_primModNatS01(x1019, x1018)=Succ(x700) & Succ(Succ(Succ(Succ(x1017))))=x1019 & Succ(Succ(Succ(Zero)))=x1018  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1017))))))), Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(new_primModNatS02(Succ(Succ(Succ(Succ(x1017)))), Succ(Succ(Succ(Zero))), Succ(Succ(x1017)), Succ(Zero)))))

(13)    (new_primModNatS02(x1023, x1022, x1021, x1020)=Succ(x700) & Succ(Succ(Succ(Succ(x1021))))=x1023 & Succ(Succ(Succ(Succ(x1020))))=x1022 & (\/x1024:new_primModNatS02(x1023, x1022, x1021, x1020)=Succ(x1024) & Succ(Succ(Succ(x1021)))=x1023 & Succ(Succ(Succ(x1020)))=x1022  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x1021)))))), Neg(Succ(Succ(Succ(Succ(Succ(x1020)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(x1020)))))), Neg(new_primModNatS02(Succ(Succ(Succ(x1021))), Succ(Succ(Succ(x1020))), Succ(x1021), Succ(x1020)))))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1021))))))), Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1020))))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1020))))))), Neg(new_primModNatS02(Succ(Succ(Succ(Succ(x1021)))), Succ(Succ(Succ(Succ(x1020)))), Succ(Succ(x1021)), Succ(Succ(x1020))))))

(14)    (new_primModNatS01(x1026, x1025)=Succ(x700) & Succ(Succ(Succ(Zero)))=x1026 & Succ(Succ(Succ(Zero)))=x1025  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), Succ(Zero), Succ(Zero)))))

(15)    (Succ(Succ(x1029))=Succ(x700) & Succ(Succ(Succ(Zero)))=x1029 & Succ(Succ(Succ(Succ(x1027))))=x1028  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1027))))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1027))))))), Neg(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1027)))), Succ(Zero), Succ(Succ(x1027))))))



We simplified constraint (12) using rules (III), (IV) which results in the following new constraint:

(16)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1017))))))), Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(new_primModNatS02(Succ(Succ(Succ(Succ(x1017)))), Succ(Succ(Succ(Zero))), Succ(Succ(x1017)), Succ(Zero)))))



We simplified constraint (13) using rules (III), (IV) which results in the following new constraint:

(17)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1021))))))), Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1020))))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1020))))))), Neg(new_primModNatS02(Succ(Succ(Succ(Succ(x1021)))), Succ(Succ(Succ(Succ(x1020)))), Succ(Succ(x1021)), Succ(Succ(x1020))))))



We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:

(18)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), Succ(Zero), Succ(Zero)))))



We simplified constraint (15) using rules (I), (II), (IV) which results in the following new constraint:

(19)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1027))))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1027))))))), Neg(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1027)))), Succ(Zero), Succ(Succ(x1027))))))



We simplified constraint (9) using rules (III), (IV), (VII) which results in the following new constraint:

(20)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))








For Pair new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x744)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x744))))), Neg(Succ(Succ(Succ(Zero))))), new_gcd0Gcd'(x745, Neg(Succ(x746))) -> new_gcd0Gcd'1(False, x745, Neg(Succ(x746))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x744))))), Neg(Succ(Succ(Succ(Zero)))))=new_gcd0Gcd'(x745, Neg(Succ(x746)))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x744))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x744))))), Neg(Succ(Succ(Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x744))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x744))))), Neg(Succ(Succ(Succ(Zero))))))








For Pair new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero))))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x771))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x771), Succ(Succ(Zero))))), new_gcd0Gcd'(x772, Neg(Succ(x773))) -> new_gcd0Gcd'1(False, x772, Neg(Succ(x773))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x771), Succ(Succ(Zero)))))=new_gcd0Gcd'(x772, Neg(Succ(x773)))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x771))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x771), Succ(Succ(Zero))))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Succ(x771)=x1036 & Succ(Succ(Zero))=x1037 & new_primModNatS1(x1036, x1037)=Succ(x773)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x771))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x771), Succ(Succ(Zero))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS1(x1036, x1037)=Succ(x773) which results in the following new constraints:

(3)    (Succ(Zero)=Succ(x773) & Succ(x771)=Succ(Zero) & Succ(Succ(Zero))=Succ(x1038)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x771))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x771), Succ(Succ(Zero))))))

(4)    (new_primModNatS1(new_primMinusNatS1, Zero)=Succ(x773) & Succ(x771)=Succ(Zero) & Succ(Succ(Zero))=Zero  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x771))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x771), Succ(Succ(Zero))))))

(5)    (new_primModNatS1(new_primMinusNatS0(x1040), Zero)=Succ(x773) & Succ(x771)=Succ(Succ(x1040)) & Succ(Succ(Zero))=Zero  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x771))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x771), Succ(Succ(Zero))))))

(6)    (new_primModNatS02(x1042, x1041, x1042, x1041)=Succ(x773) & Succ(x771)=Succ(Succ(x1042)) & Succ(Succ(Zero))=Succ(x1041)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x771))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x771), Succ(Succ(Zero))))))



We simplified constraint (3) using rules (I), (II), (III), (IV) which results in the following new constraint:

(7)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Zero), Succ(Succ(Zero))))))



We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II), (III), (VII) which results in the following new constraint:

(8)    (x1042=x1043 & x1041=x1044 & new_primModNatS02(x1042, x1041, x1043, x1044)=Succ(x773) & Succ(Zero)=x1041  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x1042)))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(x1042)), Succ(Succ(Zero))))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x1042, x1041, x1043, x1044)=Succ(x773) which results in the following new constraints:

(9)    (new_primModNatS01(x1047, x1046)=Succ(x773) & x1047=Succ(x1045) & x1046=Zero & Succ(Zero)=x1046  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x1047)))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(x1047)), Succ(Succ(Zero))))))

(10)    (new_primModNatS02(x1051, x1050, x1049, x1048)=Succ(x773) & x1051=Succ(x1049) & x1050=Succ(x1048) & Succ(Zero)=x1050 & (\/x1052:new_primModNatS02(x1051, x1050, x1049, x1048)=Succ(x1052) & x1051=x1049 & x1050=x1048 & Succ(Zero)=x1050  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x1051)))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(x1051)), Succ(Succ(Zero))))))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x1051)))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(x1051)), Succ(Succ(Zero))))))

(11)    (new_primModNatS01(x1054, x1053)=Succ(x773) & x1054=Zero & x1053=Zero & Succ(Zero)=x1053  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x1054)))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(x1054)), Succ(Succ(Zero))))))

(12)    (Succ(Succ(x1057))=Succ(x773) & x1057=Zero & x1056=Succ(x1055) & Succ(Zero)=x1056  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x1057)))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(x1057)), Succ(Succ(Zero))))))



We solved constraint (9) using rules (I), (II), (III).We simplified constraint (10) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(13)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1049))))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(Succ(x1049))), Succ(Succ(Zero))))))



We solved constraint (11) using rules (I), (II), (III).We simplified constraint (12) using rules (I), (II), (III), (IV) which results in the following new constraint:

(14)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(Zero)), Succ(Succ(Zero))))))








For Pair new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))), new_gcd0Gcd'(x793, Neg(Succ(x794))) -> new_gcd0Gcd'1(False, x793, Neg(Succ(x794))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero)))=new_gcd0Gcd'(x793, Neg(Succ(x794)))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))))



We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:

(2)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))))








For Pair new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero))) the following chains were created:
*We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x800))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x800, Zero, x800, Zero))), new_gcd0Gcd'(x801, Neg(Succ(x802))) -> new_gcd0Gcd'1(False, x801, Neg(Succ(x802))) which results in the following constraint:

(1)    (new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x800, Zero, x800, Zero)))=new_gcd0Gcd'(x801, Neg(Succ(x802)))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x800))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x800, Zero, x800, Zero))))



We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint:

(2)    (Zero=x1060 & x800=x1061 & Zero=x1062 & new_primModNatS02(x800, x1060, x1061, x1062)=Succ(x802)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x800))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x800, Zero, x800, Zero))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS02(x800, x1060, x1061, x1062)=Succ(x802) which results in the following new constraints:

(3)    (new_primModNatS01(x1065, x1064)=Succ(x802) & Zero=x1064 & x1065=Succ(x1063) & Zero=Zero  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x1065))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x1065, Zero, x1065, Zero))))

(4)    (new_primModNatS02(x1069, x1068, x1067, x1066)=Succ(x802) & Zero=x1068 & x1069=Succ(x1067) & Zero=Succ(x1066) & (\/x1070:new_primModNatS02(x1069, x1068, x1067, x1066)=Succ(x1070) & Zero=x1068 & x1069=x1067 & Zero=x1066  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x1069))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x1069, Zero, x1069, Zero))))  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x1069))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x1069, Zero, x1069, Zero))))

(5)    (new_primModNatS01(x1072, x1071)=Succ(x802) & Zero=x1071 & x1072=Zero & Zero=Zero  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x1072))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x1072, Zero, x1072, Zero))))

(6)    (Succ(Succ(x1075))=Succ(x802) & Zero=x1074 & x1075=Zero & Zero=Succ(x1073)  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x1075))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x1075, Zero, x1075, Zero))))



We simplified constraint (3) using rules (I), (II), (III), (VII) which results in the following new constraint:

(7)    (Succ(x1063)=x1076 & new_primModNatS01(x1076, x1064)=Succ(x802) & Zero=x1064  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x1063)))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(Succ(x1063), Zero, Succ(x1063), Zero))))



We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (III), (VII) which results in the following new constraint:

(8)    (Zero=x1083 & new_primModNatS01(x1083, x1071)=Succ(x802) & Zero=x1071  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(Zero, Zero, Zero, Zero))))



We solved constraint (6) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x1076, x1064)=Succ(x802) which results in the following new constraint:

(9)    (new_primModNatS1(new_primMinusNatS2(Succ(x1078), Succ(x1077)), Succ(x1077))=Succ(x802) & Succ(x1063)=x1078 & Zero=x1077  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x1063)))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(Succ(x1063), Zero, Succ(x1063), Zero))))



We simplified constraint (9) using rules (III), (IV), (VII) which results in the following new constraint:

(10)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x1063)))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(Succ(x1063), Zero, Succ(x1063), Zero))))



We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primModNatS01(x1083, x1071)=Succ(x802) which results in the following new constraint:

(11)    (new_primModNatS1(new_primMinusNatS2(Succ(x1085), Succ(x1084)), Succ(x1084))=Succ(x802) & Zero=x1085 & Zero=x1084  ==>  new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(Zero, Zero, Zero, Zero))))



We simplified constraint (11) using rules (III), (IV), (VII) which results in the following new constraint:

(12)    (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(Zero, Zero, Zero, Zero))))








To summarize, we get the following constraints P__>=_ for the following pairs.

*new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))

*(new_gcd0Gcd'(x2, Pos(Succ(x3)))_>=_new_gcd0Gcd'1(False, x2, Pos(Succ(x3))))




*new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)

*(new_gcd0Gcd'1(False, Pos(Succ(Zero)), Neg(Succ(Succ(x58))))_>=_new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x58)))))


*(new_gcd0Gcd'1(False, Pos(Succ(Zero)), Pos(Succ(Succ(x61))))_>=_new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x61)))))


*(new_gcd0Gcd'1(False, Neg(Succ(Zero)), Pos(Succ(Succ(x64))))_>=_new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x64)))))


*(new_gcd0Gcd'1(False, Neg(Succ(Zero)), Neg(Succ(Succ(x69))))_>=_new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x69)))))


*(new_gcd0Gcd'1(False, Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x72)))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x72))))))


*(new_gcd0Gcd'1(False, Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x75)))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x75))))))


*(new_gcd0Gcd'1(False, Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x78)))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x78))))))


*(new_gcd0Gcd'1(False, Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x81)))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x81))))))


*(new_gcd0Gcd'1(False, Pos(Succ(Succ(Succ(Succ(x84))))), Neg(Succ(Succ(Succ(Succ(x85))))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x84))))), Neg(Succ(Succ(Succ(Succ(x85)))))))


*(new_gcd0Gcd'1(False, Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x88))))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x88)))))))


*(new_gcd0Gcd'1(False, Pos(Succ(Succ(Succ(Succ(x91))))), Pos(Succ(Succ(Succ(Succ(x92))))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x91))))), Pos(Succ(Succ(Succ(Succ(x92)))))))


*(new_gcd0Gcd'1(False, Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x95))))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x95)))))))


*(new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Succ(x98))))), Pos(Succ(Succ(Succ(Succ(x99))))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x98))))), Pos(Succ(Succ(Succ(Succ(x99)))))))


*(new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x102))))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x102)))))))


*(new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Succ(x105))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x105))))), Pos(Succ(Succ(Succ(Zero))))))


*(new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))))


*(new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Succ(x110))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x110))))), Pos(Succ(Succ(Zero)))))


*(new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Succ(x113))))), Neg(Succ(Succ(Succ(Succ(x114))))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x113))))), Neg(Succ(Succ(Succ(Succ(x114)))))))


*(new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x117))))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x117)))))))


*(new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Succ(x120))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x120))))), Neg(Succ(Succ(Succ(Zero))))))


*(new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))))


*(new_gcd0Gcd'1(False, Neg(Succ(Succ(Succ(Succ(x125))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x125))))), Neg(Succ(Succ(Zero)))))




*new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))

*(new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x126))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x126))), Pos(Succ(Zero))))




*new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))

*(new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x153))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x153))), Pos(Succ(Zero))))




*new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))

*(new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x185))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(x185))), Neg(Succ(Zero))))




*new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))

*(new_gcd0Gcd'(x209, Neg(Succ(x210)))_>=_new_gcd0Gcd'1(False, x209, Neg(Succ(x210))))




*new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))

*(new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x264))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(x264))), Neg(Succ(Zero))))




*new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))

*(new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x286)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(x286)))), Pos(Succ(Succ(Zero)))))




*new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))

*(new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x313)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(x313)))), Pos(Succ(Succ(Zero)))))




*new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))

*(new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x345)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(x345)))), Neg(Succ(Succ(Zero)))))




*new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))

*(new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x372)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(x372)))), Neg(Succ(Succ(Zero)))))




*new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))

*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Succ(Succ(Succ(x853))))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x853))))))), Pos(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x853)))), Succ(Zero), Succ(Succ(x853))))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Succ(x834)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(x834)))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x834))), Zero, Succ(x834)))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x824)))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Succ(x824))), Succ(Succ(Zero)), Succ(x824), Zero))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x843))))))), Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(new_primModNatS02(Succ(Succ(Succ(Succ(x843)))), Succ(Succ(Succ(Zero))), Succ(Succ(x843)), Succ(Zero)))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x847))))))), Neg(Succ(Succ(Succ(Succ(Succ(Succ(x846))))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x846))))))), Pos(new_primModNatS02(Succ(Succ(Succ(Succ(x847)))), Succ(Succ(Succ(Succ(x846)))), Succ(Succ(x847)), Succ(Succ(x846))))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), Succ(Zero), Succ(Zero)))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))




*new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))

*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x446))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x446))))), Pos(Succ(Succ(Succ(Zero))))))




*new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))

*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Succ(Succ(Succ(x893))))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x893))))))), Pos(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x893)))), Succ(Zero), Succ(Succ(x893))))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Succ(x874)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(x874)))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x874))), Zero, Succ(x874)))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(x864)))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Succ(x864))), Succ(Succ(Zero)), Succ(x864), Zero))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x883))))))), Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(new_primModNatS02(Succ(Succ(Succ(Succ(x883)))), Succ(Succ(Succ(Zero))), Succ(Succ(x883)), Succ(Zero)))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x887))))))), Pos(Succ(Succ(Succ(Succ(Succ(Succ(x886))))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x886))))))), Pos(new_primModNatS02(Succ(Succ(Succ(Succ(x887)))), Succ(Succ(Succ(Succ(x886)))), Succ(Succ(x887)), Succ(Succ(x886))))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), Succ(Zero), Succ(Zero)))))


*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Pos(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))




*new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))

*(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x525))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x525))))), Pos(Succ(Succ(Succ(Zero))))))




*new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))

*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Succ(Succ(Succ(x933))))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x933))))))), Neg(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x933)))), Succ(Zero), Succ(Succ(x933))))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Succ(x914)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(x914)))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x914))), Zero, Succ(x914)))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x904)))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Succ(x904))), Succ(Succ(Zero)), Succ(x904), Zero))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x923))))))), Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(new_primModNatS02(Succ(Succ(Succ(Succ(x923)))), Succ(Succ(Succ(Zero))), Succ(Succ(x923)), Succ(Zero)))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x927))))))), Pos(Succ(Succ(Succ(Succ(Succ(Succ(x926))))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x926))))))), Neg(new_primModNatS02(Succ(Succ(Succ(Succ(x927)))), Succ(Succ(Succ(Succ(x926)))), Succ(Succ(x927)), Succ(Succ(x926))))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), Succ(Zero), Succ(Zero)))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))




*new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))

*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x609))))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x609))))), Neg(Succ(Succ(Succ(Zero))))))




*new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))

*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Zero), Succ(Succ(Zero))))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(Zero)), Succ(Succ(Zero))))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x955))))))), Pos(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(Succ(x955))), Succ(Succ(Zero))))))




*new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))

*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero))))




*new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))

*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x969)))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(Succ(x969), Zero, Succ(x969), Zero))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Pos(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(Zero, Zero, Zero, Zero))))




*new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))

*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1027))))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1027))))))), Neg(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1027)))), Succ(Zero), Succ(Succ(x1027))))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Succ(x1008)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(x1008)))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Succ(x1008))), Zero, Succ(x1008)))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x998)))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Succ(x998))), Succ(Succ(Zero)), Succ(x998), Zero))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1017))))))), Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(new_primModNatS02(Succ(Succ(Succ(Succ(x1017)))), Succ(Succ(Succ(Zero))), Succ(Succ(x1017)), Succ(Zero)))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1021))))))), Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1020))))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1020))))))), Neg(new_primModNatS02(Succ(Succ(Succ(Succ(x1021)))), Succ(Succ(Succ(Succ(x1020)))), Succ(Succ(x1021)), Succ(Succ(x1020))))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(new_primModNatS02(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), Succ(Zero), Succ(Zero)))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Succ(Zero))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(new_primModNatS02(Succ(Succ(Zero)), Succ(Succ(Zero)), Zero, Zero))))




*new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))

*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x744))))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x744))))), Neg(Succ(Succ(Succ(Zero))))))




*new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))

*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Zero), Succ(Succ(Zero))))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Zero)))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(Zero)), Succ(Succ(Zero))))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(Succ(x1049))))))), Neg(Succ(Succ(Succ(Zero)))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(Succ(Succ(x1049))), Succ(Succ(Zero))))))




*new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero)))

*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero))))




*new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))

*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Succ(x1063)))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(Succ(x1063), Zero, Succ(x1063), Zero))))


*(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(Zero))))), Neg(Succ(Succ(Zero))))_>=_new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(Zero, Zero, Zero, Zero))))








The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
----------------------------------------

(730)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gcd0Gcd'(y0, Pos(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Pos(Succ(x0)))
   new_gcd0Gcd'1(False, vuz47, vuz46) -> new_gcd0Gcd'0(vuz47, vuz46)
   new_gcd0Gcd'0(Pos(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Pos(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Pos(Succ(Succ(x0)))) -> new_gcd0Gcd'(Pos(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'(y0, Neg(Succ(x0))) -> new_gcd0Gcd'1(False, y0, Neg(Succ(x0)))
   new_gcd0Gcd'0(Neg(Succ(Zero)), Neg(Succ(Succ(x0)))) -> new_gcd0Gcd'(Neg(Succ(Succ(x0))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Pos(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Pos(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Neg(Succ(Succ(Succ(x2))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(x2)))), Neg(Succ(Succ(Zero))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Pos(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Pos(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Pos(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Pos(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Pos(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Pos(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Succ(x3)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x3))))), Neg(new_primModNatS02(Succ(Succ(x2)), Succ(Succ(x3)), x2, x3)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Succ(Succ(x2)))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x2))))), Neg(Succ(Succ(Succ(Zero))))) -> new_gcd0Gcd'(Neg(Succ(Succ(Succ(Zero)))), Neg(new_primModNatS1(Succ(x2), Succ(Succ(Zero)))))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Zero)))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(Succ(Zero)))
   new_gcd0Gcd'0(Neg(Succ(Succ(Succ(Succ(x0))))), Neg(Succ(Succ(Zero)))) -> new_gcd0Gcd'(Neg(Succ(Succ(Zero))), Neg(new_primModNatS02(x0, Zero, x0, Zero)))

The TRS R consists of the following rules:

   new_primModNatS1(Succ(Zero), Succ(vuz43000)) -> Succ(Zero)
   new_primModNatS1(Zero, vuz4300) -> Zero
   new_primModNatS1(Succ(Zero), Zero) -> new_primModNatS1(new_primMinusNatS1, Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Zero) -> new_primModNatS1(new_primMinusNatS0(vuz44000), Zero)
   new_primModNatS1(Succ(Succ(vuz44000)), Succ(vuz43000)) -> new_primModNatS02(vuz44000, vuz43000, vuz44000, vuz43000)
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS01(vuz89, vuz90) -> new_primModNatS1(new_primMinusNatS2(Succ(vuz89), Succ(vuz90)), Succ(vuz90))
   new_primModNatS02(vuz89, vuz90, Succ(vuz910), Succ(vuz920)) -> new_primModNatS02(vuz89, vuz90, vuz910, vuz920)
   new_primModNatS02(vuz89, vuz90, Zero, Zero) -> new_primModNatS01(vuz89, vuz90)
   new_primModNatS02(vuz89, vuz90, Zero, Succ(vuz920)) -> Succ(Succ(vuz89))
   new_primMinusNatS2(Succ(vuz690), Succ(vuz700)) -> new_primMinusNatS2(vuz690, vuz700)
   new_primMinusNatS2(Zero, Zero) -> Zero
   new_primMinusNatS2(Zero, Succ(vuz700)) -> Zero
   new_primMinusNatS2(Succ(vuz690), Zero) -> Succ(vuz690)
   new_primMinusNatS1 -> Zero
   new_primMinusNatS0(vuz22000) -> Succ(vuz22000)

The set Q consists of the following terms:

   new_primModNatS02(x0, x1, Succ(x2), Zero)
   new_primMinusNatS0(x0)
   new_primMinusNatS1
   new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
   new_primMinusNatS2(Zero, Succ(x0))
   new_primMinusNatS2(Succ(x0), Zero)
   new_primMinusNatS2(Succ(x0), Succ(x1))
   new_primModNatS1(Zero, x0)
   new_primModNatS02(x0, x1, Zero, Succ(x2))
   new_primMinusNatS2(Zero, Zero)
   new_primModNatS1(Succ(Zero), Succ(x0))
   new_primModNatS1(Succ(Zero), Zero)
   new_primModNatS01(x0, x1)
   new_primModNatS1(Succ(Succ(x0)), Succ(x1))
   new_primModNatS1(Succ(Succ(x0)), Zero)
   new_primModNatS02(x0, x1, Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(731) Narrow (COMPLETE)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="(*)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="(*) vuz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="(*) vuz3 vuz4",fontsize=16,color="blue",shape="box"];1023[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4 -> 1023[label="",style="solid", color="blue", weight=9];
1023 -> 5[label="",style="solid", color="blue", weight=3];
1024[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4 -> 1024[label="",style="solid", color="blue", weight=9];
1024 -> 6[label="",style="solid", color="blue", weight=3];
1025[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4 -> 1025[label="",style="solid", color="blue", weight=9];
1025 -> 7[label="",style="solid", color="blue", weight=3];
1026[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4 -> 1026[label="",style="solid", color="blue", weight=9];
1026 -> 8[label="",style="solid", color="blue", weight=3];
1027[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4 -> 1027[label="",style="solid", color="blue", weight=9];
1027 -> 9[label="",style="solid", color="blue", weight=3];
5[label="(*) vuz3 vuz4",fontsize=16,color="burlywood",shape="box"];1028[label="vuz3/vuz30 :% vuz31",fontsize=10,color="white",style="solid",shape="box"];5 -> 1028[label="",style="solid", color="burlywood", weight=9];
1028 -> 10[label="",style="solid", color="burlywood", weight=3];
6[label="(*) vuz3 vuz4",fontsize=16,color="black",shape="box"];6 -> 11[label="",style="solid", color="black", weight=3];
7[label="(*) vuz3 vuz4",fontsize=16,color="black",shape="triangle"];7 -> 12[label="",style="solid", color="black", weight=3];
8[label="(*) vuz3 vuz4",fontsize=16,color="burlywood",shape="triangle"];1029[label="vuz3/Integer vuz30",fontsize=10,color="white",style="solid",shape="box"];8 -> 1029[label="",style="solid", color="burlywood", weight=9];
1029 -> 13[label="",style="solid", color="burlywood", weight=3];
9[label="(*) vuz3 vuz4",fontsize=16,color="black",shape="box"];9 -> 14[label="",style="solid", color="black", weight=3];
10[label="(*) (vuz30 :% vuz31) vuz4",fontsize=16,color="burlywood",shape="box"];1030[label="vuz4/vuz40 :% vuz41",fontsize=10,color="white",style="solid",shape="box"];10 -> 1030[label="",style="solid", color="burlywood", weight=9];
1030 -> 15[label="",style="solid", color="burlywood", weight=3];
11[label="primMulDouble vuz3 vuz4",fontsize=16,color="burlywood",shape="box"];1031[label="vuz3/Double vuz30 vuz31",fontsize=10,color="white",style="solid",shape="box"];11 -> 1031[label="",style="solid", color="burlywood", weight=9];
1031 -> 16[label="",style="solid", color="burlywood", weight=3];
12[label="primMulInt vuz3 vuz4",fontsize=16,color="burlywood",shape="triangle"];1032[label="vuz3/Pos vuz30",fontsize=10,color="white",style="solid",shape="box"];12 -> 1032[label="",style="solid", color="burlywood", weight=9];
1032 -> 17[label="",style="solid", color="burlywood", weight=3];
1033[label="vuz3/Neg vuz30",fontsize=10,color="white",style="solid",shape="box"];12 -> 1033[label="",style="solid", color="burlywood", weight=9];
1033 -> 18[label="",style="solid", color="burlywood", weight=3];
13[label="(*) Integer vuz30 vuz4",fontsize=16,color="burlywood",shape="box"];1034[label="vuz4/Integer vuz40",fontsize=10,color="white",style="solid",shape="box"];13 -> 1034[label="",style="solid", color="burlywood", weight=9];
1034 -> 19[label="",style="solid", color="burlywood", weight=3];
14[label="primMulFloat vuz3 vuz4",fontsize=16,color="burlywood",shape="box"];1035[label="vuz3/Float vuz30 vuz31",fontsize=10,color="white",style="solid",shape="box"];14 -> 1035[label="",style="solid", color="burlywood", weight=9];
1035 -> 20[label="",style="solid", color="burlywood", weight=3];
15[label="(*) (vuz30 :% vuz31) (vuz40 :% vuz41)",fontsize=16,color="black",shape="box"];15 -> 21[label="",style="solid", color="black", weight=3];
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19[label="(*) Integer vuz30 Integer vuz40",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
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26[label="primMulInt (Neg vuz30) (Neg vuz40)",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
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30 -> 39[label="",style="dashed", color="green", weight=3];
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32[label="Neg (primMulNat vuz30 vuz40)",fontsize=16,color="green",shape="box"];32 -> 41[label="",style="dashed", color="green", weight=3];
33[label="Neg (primMulNat vuz30 vuz40)",fontsize=16,color="green",shape="box"];33 -> 42[label="",style="dashed", color="green", weight=3];
34[label="Pos (primMulNat vuz30 vuz40)",fontsize=16,color="green",shape="box"];34 -> 43[label="",style="dashed", color="green", weight=3];
35 -> 12[label="",style="dashed", color="red", weight=0];
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35 -> 45[label="",style="dashed", color="magenta", weight=3];
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36 -> 47[label="",style="dashed", color="green", weight=3];
37 -> 48[label="",style="dashed", color="red", weight=0];
37[label="reduce2Reduce1 (vuz30 * vuz40) (vuz31 * vuz41) (vuz30 * vuz40) (vuz31 * vuz41) (vuz31 * vuz41 == fromInt (Pos Zero))",fontsize=16,color="magenta"];37 -> 49[label="",style="dashed", color="magenta", weight=3];
37 -> 50[label="",style="dashed", color="magenta", weight=3];
37 -> 51[label="",style="dashed", color="magenta", weight=3];
37 -> 52[label="",style="dashed", color="magenta", weight=3];
37 -> 53[label="",style="dashed", color="magenta", weight=3];
38 -> 7[label="",style="dashed", color="red", weight=0];
38[label="vuz30 * vuz40",fontsize=16,color="magenta"];38 -> 54[label="",style="dashed", color="magenta", weight=3];
38 -> 55[label="",style="dashed", color="magenta", weight=3];
39 -> 7[label="",style="dashed", color="red", weight=0];
39[label="vuz31 * vuz41",fontsize=16,color="magenta"];39 -> 56[label="",style="dashed", color="magenta", weight=3];
39 -> 57[label="",style="dashed", color="magenta", weight=3];
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1042 -> 58[label="",style="solid", color="burlywood", weight=3];
1043[label="vuz30/Zero",fontsize=10,color="white",style="solid",shape="box"];40 -> 1043[label="",style="solid", color="burlywood", weight=9];
1043 -> 59[label="",style="solid", color="burlywood", weight=3];
41 -> 40[label="",style="dashed", color="red", weight=0];
41[label="primMulNat vuz30 vuz40",fontsize=16,color="magenta"];41 -> 60[label="",style="dashed", color="magenta", weight=3];
42 -> 40[label="",style="dashed", color="red", weight=0];
42[label="primMulNat vuz30 vuz40",fontsize=16,color="magenta"];42 -> 61[label="",style="dashed", color="magenta", weight=3];
43 -> 40[label="",style="dashed", color="red", weight=0];
43[label="primMulNat vuz30 vuz40",fontsize=16,color="magenta"];43 -> 62[label="",style="dashed", color="magenta", weight=3];
43 -> 63[label="",style="dashed", color="magenta", weight=3];
44[label="vuz40",fontsize=16,color="green",shape="box"];45[label="vuz30",fontsize=16,color="green",shape="box"];46 -> 7[label="",style="dashed", color="red", weight=0];
46[label="vuz30 * vuz40",fontsize=16,color="magenta"];46 -> 64[label="",style="dashed", color="magenta", weight=3];
46 -> 65[label="",style="dashed", color="magenta", weight=3];
47 -> 7[label="",style="dashed", color="red", weight=0];
47[label="vuz31 * vuz41",fontsize=16,color="magenta"];47 -> 66[label="",style="dashed", color="magenta", weight=3];
47 -> 67[label="",style="dashed", color="magenta", weight=3];
49[label="vuz41",fontsize=16,color="green",shape="box"];50[label="vuz31 * vuz41 == fromInt (Pos Zero)",fontsize=16,color="blue",shape="box"];1044[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];50 -> 1044[label="",style="solid", color="blue", weight=9];
1044 -> 68[label="",style="solid", color="blue", weight=3];
1045[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];50 -> 1045[label="",style="solid", color="blue", weight=9];
1045 -> 69[label="",style="solid", color="blue", weight=3];
51[label="vuz40",fontsize=16,color="green",shape="box"];52[label="vuz30",fontsize=16,color="green",shape="box"];53[label="vuz31",fontsize=16,color="green",shape="box"];48[label="reduce2Reduce1 (vuz11 * vuz12) (vuz13 * vuz14) (vuz11 * vuz12) (vuz13 * vuz14) vuz15",fontsize=16,color="burlywood",shape="triangle"];1046[label="vuz15/False",fontsize=10,color="white",style="solid",shape="box"];48 -> 1046[label="",style="solid", color="burlywood", weight=9];
1046 -> 70[label="",style="solid", color="burlywood", weight=3];
1047[label="vuz15/True",fontsize=10,color="white",style="solid",shape="box"];48 -> 1047[label="",style="solid", color="burlywood", weight=9];
1047 -> 71[label="",style="solid", color="burlywood", weight=3];
54[label="vuz40",fontsize=16,color="green",shape="box"];55[label="vuz30",fontsize=16,color="green",shape="box"];56[label="vuz41",fontsize=16,color="green",shape="box"];57[label="vuz31",fontsize=16,color="green",shape="box"];58[label="primMulNat (Succ vuz300) vuz40",fontsize=16,color="burlywood",shape="box"];1048[label="vuz40/Succ vuz400",fontsize=10,color="white",style="solid",shape="box"];58 -> 1048[label="",style="solid", color="burlywood", weight=9];
1048 -> 72[label="",style="solid", color="burlywood", weight=3];
1049[label="vuz40/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 1049[label="",style="solid", color="burlywood", weight=9];
1049 -> 73[label="",style="solid", color="burlywood", weight=3];
59[label="primMulNat Zero vuz40",fontsize=16,color="burlywood",shape="box"];1050[label="vuz40/Succ vuz400",fontsize=10,color="white",style="solid",shape="box"];59 -> 1050[label="",style="solid", color="burlywood", weight=9];
1050 -> 74[label="",style="solid", color="burlywood", weight=3];
1051[label="vuz40/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 1051[label="",style="solid", color="burlywood", weight=9];
1051 -> 75[label="",style="solid", color="burlywood", weight=3];
60[label="vuz40",fontsize=16,color="green",shape="box"];61[label="vuz30",fontsize=16,color="green",shape="box"];62[label="vuz30",fontsize=16,color="green",shape="box"];63[label="vuz40",fontsize=16,color="green",shape="box"];64[label="vuz40",fontsize=16,color="green",shape="box"];65[label="vuz30",fontsize=16,color="green",shape="box"];66[label="vuz41",fontsize=16,color="green",shape="box"];67[label="vuz31",fontsize=16,color="green",shape="box"];68 -> 76[label="",style="dashed", color="red", weight=0];
68[label="vuz31 * vuz41 == fromInt (Pos Zero)",fontsize=16,color="magenta"];68 -> 77[label="",style="dashed", color="magenta", weight=3];
69 -> 78[label="",style="dashed", color="red", weight=0];
69[label="vuz31 * vuz41 == fromInt (Pos Zero)",fontsize=16,color="magenta"];69 -> 79[label="",style="dashed", color="magenta", weight=3];
70[label="reduce2Reduce1 (vuz11 * vuz12) (vuz13 * vuz14) (vuz11 * vuz12) (vuz13 * vuz14) False",fontsize=16,color="black",shape="box"];70 -> 80[label="",style="solid", color="black", weight=3];
71[label="reduce2Reduce1 (vuz11 * vuz12) (vuz13 * vuz14) (vuz11 * vuz12) (vuz13 * vuz14) True",fontsize=16,color="black",shape="box"];71 -> 81[label="",style="solid", color="black", weight=3];
72[label="primMulNat (Succ vuz300) (Succ vuz400)",fontsize=16,color="black",shape="box"];72 -> 82[label="",style="solid", color="black", weight=3];
73[label="primMulNat (Succ vuz300) Zero",fontsize=16,color="black",shape="box"];73 -> 83[label="",style="solid", color="black", weight=3];
74[label="primMulNat Zero (Succ vuz400)",fontsize=16,color="black",shape="box"];74 -> 84[label="",style="solid", color="black", weight=3];
75[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];75 -> 85[label="",style="solid", color="black", weight=3];
77 -> 8[label="",style="dashed", color="red", weight=0];
77[label="vuz31 * vuz41",fontsize=16,color="magenta"];77 -> 86[label="",style="dashed", color="magenta", weight=3];
77 -> 87[label="",style="dashed", color="magenta", weight=3];
76[label="vuz16 == fromInt (Pos Zero)",fontsize=16,color="burlywood",shape="triangle"];1052[label="vuz16/Integer vuz160",fontsize=10,color="white",style="solid",shape="box"];76 -> 1052[label="",style="solid", color="burlywood", weight=9];
1052 -> 88[label="",style="solid", color="burlywood", weight=3];
79 -> 7[label="",style="dashed", color="red", weight=0];
79[label="vuz31 * vuz41",fontsize=16,color="magenta"];79 -> 89[label="",style="dashed", color="magenta", weight=3];
79 -> 90[label="",style="dashed", color="magenta", weight=3];
78[label="vuz17 == fromInt (Pos Zero)",fontsize=16,color="black",shape="triangle"];78 -> 91[label="",style="solid", color="black", weight=3];
80[label="reduce2Reduce0 (vuz11 * vuz12) (vuz13 * vuz14) (vuz11 * vuz12) (vuz13 * vuz14) otherwise",fontsize=16,color="black",shape="box"];80 -> 92[label="",style="solid", color="black", weight=3];
81[label="error []",fontsize=16,color="black",shape="box"];81 -> 93[label="",style="solid", color="black", weight=3];
82 -> 94[label="",style="dashed", color="red", weight=0];
82[label="primPlusNat (primMulNat vuz300 (Succ vuz400)) (Succ vuz400)",fontsize=16,color="magenta"];82 -> 95[label="",style="dashed", color="magenta", weight=3];
83[label="Zero",fontsize=16,color="green",shape="box"];84[label="Zero",fontsize=16,color="green",shape="box"];85[label="Zero",fontsize=16,color="green",shape="box"];86[label="vuz41",fontsize=16,color="green",shape="box"];87[label="vuz31",fontsize=16,color="green",shape="box"];88[label="Integer vuz160 == fromInt (Pos Zero)",fontsize=16,color="black",shape="box"];88 -> 96[label="",style="solid", color="black", weight=3];
89[label="vuz41",fontsize=16,color="green",shape="box"];90[label="vuz31",fontsize=16,color="green",shape="box"];91[label="primEqInt vuz17 (fromInt (Pos Zero))",fontsize=16,color="burlywood",shape="box"];1053[label="vuz17/Pos vuz170",fontsize=10,color="white",style="solid",shape="box"];91 -> 1053[label="",style="solid", color="burlywood", weight=9];
1053 -> 97[label="",style="solid", color="burlywood", weight=3];
1054[label="vuz17/Neg vuz170",fontsize=10,color="white",style="solid",shape="box"];91 -> 1054[label="",style="solid", color="burlywood", weight=9];
1054 -> 98[label="",style="solid", color="burlywood", weight=3];
92[label="reduce2Reduce0 (vuz11 * vuz12) (vuz13 * vuz14) (vuz11 * vuz12) (vuz13 * vuz14) True",fontsize=16,color="black",shape="box"];92 -> 99[label="",style="solid", color="black", weight=3];
93[label="error []",fontsize=16,color="red",shape="box"];95 -> 40[label="",style="dashed", color="red", weight=0];
95[label="primMulNat vuz300 (Succ vuz400)",fontsize=16,color="magenta"];95 -> 100[label="",style="dashed", color="magenta", weight=3];
95 -> 101[label="",style="dashed", color="magenta", weight=3];
94[label="primPlusNat vuz18 (Succ vuz400)",fontsize=16,color="burlywood",shape="triangle"];1055[label="vuz18/Succ vuz180",fontsize=10,color="white",style="solid",shape="box"];94 -> 1055[label="",style="solid", color="burlywood", weight=9];
1055 -> 102[label="",style="solid", color="burlywood", weight=3];
1056[label="vuz18/Zero",fontsize=10,color="white",style="solid",shape="box"];94 -> 1056[label="",style="solid", color="burlywood", weight=9];
1056 -> 103[label="",style="solid", color="burlywood", weight=3];
96[label="Integer vuz160 == Integer (Pos Zero)",fontsize=16,color="black",shape="box"];96 -> 104[label="",style="solid", color="black", weight=3];
97[label="primEqInt (Pos vuz170) (fromInt (Pos Zero))",fontsize=16,color="burlywood",shape="box"];1057[label="vuz170/Succ vuz1700",fontsize=10,color="white",style="solid",shape="box"];97 -> 1057[label="",style="solid", color="burlywood", weight=9];
1057 -> 105[label="",style="solid", color="burlywood", weight=3];
1058[label="vuz170/Zero",fontsize=10,color="white",style="solid",shape="box"];97 -> 1058[label="",style="solid", color="burlywood", weight=9];
1058 -> 106[label="",style="solid", color="burlywood", weight=3];
98[label="primEqInt (Neg vuz170) (fromInt (Pos Zero))",fontsize=16,color="burlywood",shape="box"];1059[label="vuz170/Succ vuz1700",fontsize=10,color="white",style="solid",shape="box"];98 -> 1059[label="",style="solid", color="burlywood", weight=9];
1059 -> 107[label="",style="solid", color="burlywood", weight=3];
1060[label="vuz170/Zero",fontsize=10,color="white",style="solid",shape="box"];98 -> 1060[label="",style="solid", color="burlywood", weight=9];
1060 -> 108[label="",style="solid", color="burlywood", weight=3];
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99 -> 110[label="",style="dashed", color="green", weight=3];
100[label="vuz300",fontsize=16,color="green",shape="box"];101[label="Succ vuz400",fontsize=16,color="green",shape="box"];102[label="primPlusNat (Succ vuz180) (Succ vuz400)",fontsize=16,color="black",shape="box"];102 -> 111[label="",style="solid", color="black", weight=3];
103[label="primPlusNat Zero (Succ vuz400)",fontsize=16,color="black",shape="box"];103 -> 112[label="",style="solid", color="black", weight=3];
104[label="primEqInt vuz160 (Pos Zero)",fontsize=16,color="burlywood",shape="triangle"];1061[label="vuz160/Pos vuz1600",fontsize=10,color="white",style="solid",shape="box"];104 -> 1061[label="",style="solid", color="burlywood", weight=9];
1061 -> 113[label="",style="solid", color="burlywood", weight=3];
1062[label="vuz160/Neg vuz1600",fontsize=10,color="white",style="solid",shape="box"];104 -> 1062[label="",style="solid", color="burlywood", weight=9];
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105[label="primEqInt (Pos (Succ vuz1700)) (fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];105 -> 115[label="",style="solid", color="black", weight=3];
106[label="primEqInt (Pos Zero) (fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];106 -> 116[label="",style="solid", color="black", weight=3];
107[label="primEqInt (Neg (Succ vuz1700)) (fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];107 -> 117[label="",style="solid", color="black", weight=3];
108[label="primEqInt (Neg Zero) (fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];108 -> 118[label="",style="solid", color="black", weight=3];
109[label="vuz11 * vuz12 `quot` reduce2D (vuz11 * vuz12) (vuz13 * vuz14)",fontsize=16,color="blue",shape="box"];1063[label="`quot` :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];109 -> 1063[label="",style="solid", color="blue", weight=9];
1063 -> 119[label="",style="solid", color="blue", weight=3];
1064[label="`quot` :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];109 -> 1064[label="",style="solid", color="blue", weight=9];
1064 -> 120[label="",style="solid", color="blue", weight=3];
110[label="vuz13 * vuz14 `quot` reduce2D (vuz11 * vuz12) (vuz13 * vuz14)",fontsize=16,color="blue",shape="box"];1065[label="`quot` :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];110 -> 1065[label="",style="solid", color="blue", weight=9];
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1066[label="`quot` :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];110 -> 1066[label="",style="solid", color="blue", weight=9];
1066 -> 122[label="",style="solid", color="blue", weight=3];
111[label="Succ (Succ (primPlusNat vuz180 vuz400))",fontsize=16,color="green",shape="box"];111 -> 123[label="",style="dashed", color="green", weight=3];
112[label="Succ vuz400",fontsize=16,color="green",shape="box"];113[label="primEqInt (Pos vuz1600) (Pos Zero)",fontsize=16,color="burlywood",shape="box"];1067[label="vuz1600/Succ vuz16000",fontsize=10,color="white",style="solid",shape="box"];113 -> 1067[label="",style="solid", color="burlywood", weight=9];
1067 -> 124[label="",style="solid", color="burlywood", weight=3];
1068[label="vuz1600/Zero",fontsize=10,color="white",style="solid",shape="box"];113 -> 1068[label="",style="solid", color="burlywood", weight=9];
1068 -> 125[label="",style="solid", color="burlywood", weight=3];
114[label="primEqInt (Neg vuz1600) (Pos Zero)",fontsize=16,color="burlywood",shape="box"];1069[label="vuz1600/Succ vuz16000",fontsize=10,color="white",style="solid",shape="box"];114 -> 1069[label="",style="solid", color="burlywood", weight=9];
1069 -> 126[label="",style="solid", color="burlywood", weight=3];
1070[label="vuz1600/Zero",fontsize=10,color="white",style="solid",shape="box"];114 -> 1070[label="",style="solid", color="burlywood", weight=9];
1070 -> 127[label="",style="solid", color="burlywood", weight=3];
115 -> 104[label="",style="dashed", color="red", weight=0];
115[label="primEqInt (Pos (Succ vuz1700)) (Pos Zero)",fontsize=16,color="magenta"];115 -> 128[label="",style="dashed", color="magenta", weight=3];
116 -> 104[label="",style="dashed", color="red", weight=0];
116[label="primEqInt (Pos Zero) (Pos Zero)",fontsize=16,color="magenta"];116 -> 129[label="",style="dashed", color="magenta", weight=3];
117 -> 104[label="",style="dashed", color="red", weight=0];
117[label="primEqInt (Neg (Succ vuz1700)) (Pos Zero)",fontsize=16,color="magenta"];117 -> 130[label="",style="dashed", color="magenta", weight=3];
118 -> 104[label="",style="dashed", color="red", weight=0];
118[label="primEqInt (Neg Zero) (Pos Zero)",fontsize=16,color="magenta"];118 -> 131[label="",style="dashed", color="magenta", weight=3];
119 -> 132[label="",style="dashed", color="red", weight=0];
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119 -> 134[label="",style="dashed", color="magenta", weight=3];
119 -> 135[label="",style="dashed", color="magenta", weight=3];
120 -> 139[label="",style="dashed", color="red", weight=0];
120[label="vuz11 * vuz12 `quot` reduce2D (vuz11 * vuz12) (vuz13 * vuz14)",fontsize=16,color="magenta"];120 -> 140[label="",style="dashed", color="magenta", weight=3];
120 -> 141[label="",style="dashed", color="magenta", weight=3];
120 -> 142[label="",style="dashed", color="magenta", weight=3];
121 -> 132[label="",style="dashed", color="red", weight=0];
121[label="vuz13 * vuz14 `quot` reduce2D (vuz11 * vuz12) (vuz13 * vuz14)",fontsize=16,color="magenta"];121 -> 136[label="",style="dashed", color="magenta", weight=3];
121 -> 137[label="",style="dashed", color="magenta", weight=3];
121 -> 138[label="",style="dashed", color="magenta", weight=3];
122 -> 139[label="",style="dashed", color="red", weight=0];
122[label="vuz13 * vuz14 `quot` reduce2D (vuz11 * vuz12) (vuz13 * vuz14)",fontsize=16,color="magenta"];122 -> 143[label="",style="dashed", color="magenta", weight=3];
122 -> 144[label="",style="dashed", color="magenta", weight=3];
122 -> 145[label="",style="dashed", color="magenta", weight=3];
123[label="primPlusNat vuz180 vuz400",fontsize=16,color="burlywood",shape="triangle"];1071[label="vuz180/Succ vuz1800",fontsize=10,color="white",style="solid",shape="box"];123 -> 1071[label="",style="solid", color="burlywood", weight=9];
1071 -> 146[label="",style="solid", color="burlywood", weight=3];
1072[label="vuz180/Zero",fontsize=10,color="white",style="solid",shape="box"];123 -> 1072[label="",style="solid", color="burlywood", weight=9];
1072 -> 147[label="",style="solid", color="burlywood", weight=3];
124[label="primEqInt (Pos (Succ vuz16000)) (Pos Zero)",fontsize=16,color="black",shape="box"];124 -> 148[label="",style="solid", color="black", weight=3];
125[label="primEqInt (Pos Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];125 -> 149[label="",style="solid", color="black", weight=3];
126[label="primEqInt (Neg (Succ vuz16000)) (Pos Zero)",fontsize=16,color="black",shape="box"];126 -> 150[label="",style="solid", color="black", weight=3];
127[label="primEqInt (Neg Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];127 -> 151[label="",style="solid", color="black", weight=3];
128[label="Pos (Succ vuz1700)",fontsize=16,color="green",shape="box"];129[label="Pos Zero",fontsize=16,color="green",shape="box"];130[label="Neg (Succ vuz1700)",fontsize=16,color="green",shape="box"];131[label="Neg Zero",fontsize=16,color="green",shape="box"];133 -> 8[label="",style="dashed", color="red", weight=0];
133[label="vuz11 * vuz12",fontsize=16,color="magenta"];133 -> 152[label="",style="dashed", color="magenta", weight=3];
133 -> 153[label="",style="dashed", color="magenta", weight=3];
134 -> 8[label="",style="dashed", color="red", weight=0];
134[label="vuz11 * vuz12",fontsize=16,color="magenta"];134 -> 154[label="",style="dashed", color="magenta", weight=3];
134 -> 155[label="",style="dashed", color="magenta", weight=3];
135 -> 8[label="",style="dashed", color="red", weight=0];
135[label="vuz13 * vuz14",fontsize=16,color="magenta"];135 -> 156[label="",style="dashed", color="magenta", weight=3];
135 -> 157[label="",style="dashed", color="magenta", weight=3];
132[label="vuz19 `quot` reduce2D vuz21 vuz20",fontsize=16,color="burlywood",shape="triangle"];1073[label="vuz19/Integer vuz190",fontsize=10,color="white",style="solid",shape="box"];132 -> 1073[label="",style="solid", color="burlywood", weight=9];
1073 -> 158[label="",style="solid", color="burlywood", weight=3];
140 -> 7[label="",style="dashed", color="red", weight=0];
140[label="vuz11 * vuz12",fontsize=16,color="magenta"];140 -> 159[label="",style="dashed", color="magenta", weight=3];
140 -> 160[label="",style="dashed", color="magenta", weight=3];
141 -> 7[label="",style="dashed", color="red", weight=0];
141[label="vuz11 * vuz12",fontsize=16,color="magenta"];141 -> 161[label="",style="dashed", color="magenta", weight=3];
141 -> 162[label="",style="dashed", color="magenta", weight=3];
142 -> 7[label="",style="dashed", color="red", weight=0];
142[label="vuz13 * vuz14",fontsize=16,color="magenta"];142 -> 163[label="",style="dashed", color="magenta", weight=3];
142 -> 164[label="",style="dashed", color="magenta", weight=3];
139[label="vuz22 `quot` reduce2D vuz24 vuz23",fontsize=16,color="black",shape="triangle"];139 -> 165[label="",style="solid", color="black", weight=3];
136 -> 8[label="",style="dashed", color="red", weight=0];
136[label="vuz13 * vuz14",fontsize=16,color="magenta"];136 -> 166[label="",style="dashed", color="magenta", weight=3];
136 -> 167[label="",style="dashed", color="magenta", weight=3];
137 -> 8[label="",style="dashed", color="red", weight=0];
137[label="vuz11 * vuz12",fontsize=16,color="magenta"];137 -> 168[label="",style="dashed", color="magenta", weight=3];
137 -> 169[label="",style="dashed", color="magenta", weight=3];
138 -> 8[label="",style="dashed", color="red", weight=0];
138[label="vuz13 * vuz14",fontsize=16,color="magenta"];138 -> 170[label="",style="dashed", color="magenta", weight=3];
138 -> 171[label="",style="dashed", color="magenta", weight=3];
143 -> 7[label="",style="dashed", color="red", weight=0];
143[label="vuz13 * vuz14",fontsize=16,color="magenta"];143 -> 172[label="",style="dashed", color="magenta", weight=3];
143 -> 173[label="",style="dashed", color="magenta", weight=3];
144 -> 7[label="",style="dashed", color="red", weight=0];
144[label="vuz11 * vuz12",fontsize=16,color="magenta"];144 -> 174[label="",style="dashed", color="magenta", weight=3];
144 -> 175[label="",style="dashed", color="magenta", weight=3];
145 -> 7[label="",style="dashed", color="red", weight=0];
145[label="vuz13 * vuz14",fontsize=16,color="magenta"];145 -> 176[label="",style="dashed", color="magenta", weight=3];
145 -> 177[label="",style="dashed", color="magenta", weight=3];
146[label="primPlusNat (Succ vuz1800) vuz400",fontsize=16,color="burlywood",shape="box"];1074[label="vuz400/Succ vuz4000",fontsize=10,color="white",style="solid",shape="box"];146 -> 1074[label="",style="solid", color="burlywood", weight=9];
1074 -> 178[label="",style="solid", color="burlywood", weight=3];
1075[label="vuz400/Zero",fontsize=10,color="white",style="solid",shape="box"];146 -> 1075[label="",style="solid", color="burlywood", weight=9];
1075 -> 179[label="",style="solid", color="burlywood", weight=3];
147[label="primPlusNat Zero vuz400",fontsize=16,color="burlywood",shape="box"];1076[label="vuz400/Succ vuz4000",fontsize=10,color="white",style="solid",shape="box"];147 -> 1076[label="",style="solid", color="burlywood", weight=9];
1076 -> 180[label="",style="solid", color="burlywood", weight=3];
1077[label="vuz400/Zero",fontsize=10,color="white",style="solid",shape="box"];147 -> 1077[label="",style="solid", color="burlywood", weight=9];
1077 -> 181[label="",style="solid", color="burlywood", weight=3];
148[label="False",fontsize=16,color="green",shape="box"];149[label="True",fontsize=16,color="green",shape="box"];150[label="False",fontsize=16,color="green",shape="box"];151[label="True",fontsize=16,color="green",shape="box"];152[label="vuz12",fontsize=16,color="green",shape="box"];153[label="vuz11",fontsize=16,color="green",shape="box"];154[label="vuz12",fontsize=16,color="green",shape="box"];155[label="vuz11",fontsize=16,color="green",shape="box"];156[label="vuz14",fontsize=16,color="green",shape="box"];157[label="vuz13",fontsize=16,color="green",shape="box"];158[label="Integer vuz190 `quot` reduce2D vuz21 vuz20",fontsize=16,color="black",shape="box"];158 -> 182[label="",style="solid", color="black", weight=3];
159[label="vuz12",fontsize=16,color="green",shape="box"];160[label="vuz11",fontsize=16,color="green",shape="box"];161[label="vuz12",fontsize=16,color="green",shape="box"];162[label="vuz11",fontsize=16,color="green",shape="box"];163[label="vuz14",fontsize=16,color="green",shape="box"];164[label="vuz13",fontsize=16,color="green",shape="box"];165[label="primQuotInt vuz22 (reduce2D vuz24 vuz23)",fontsize=16,color="burlywood",shape="box"];1078[label="vuz22/Pos vuz220",fontsize=10,color="white",style="solid",shape="box"];165 -> 1078[label="",style="solid", color="burlywood", weight=9];
1078 -> 183[label="",style="solid", color="burlywood", weight=3];
1079[label="vuz22/Neg vuz220",fontsize=10,color="white",style="solid",shape="box"];165 -> 1079[label="",style="solid", color="burlywood", weight=9];
1079 -> 184[label="",style="solid", color="burlywood", weight=3];
166[label="vuz14",fontsize=16,color="green",shape="box"];167[label="vuz13",fontsize=16,color="green",shape="box"];168[label="vuz12",fontsize=16,color="green",shape="box"];169[label="vuz11",fontsize=16,color="green",shape="box"];170[label="vuz14",fontsize=16,color="green",shape="box"];171[label="vuz13",fontsize=16,color="green",shape="box"];172[label="vuz14",fontsize=16,color="green",shape="box"];173[label="vuz13",fontsize=16,color="green",shape="box"];174[label="vuz12",fontsize=16,color="green",shape="box"];175[label="vuz11",fontsize=16,color="green",shape="box"];176[label="vuz14",fontsize=16,color="green",shape="box"];177[label="vuz13",fontsize=16,color="green",shape="box"];178[label="primPlusNat (Succ vuz1800) (Succ vuz4000)",fontsize=16,color="black",shape="box"];178 -> 185[label="",style="solid", color="black", weight=3];
179[label="primPlusNat (Succ vuz1800) Zero",fontsize=16,color="black",shape="box"];179 -> 186[label="",style="solid", color="black", weight=3];
180[label="primPlusNat Zero (Succ vuz4000)",fontsize=16,color="black",shape="box"];180 -> 187[label="",style="solid", color="black", weight=3];
181[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];181 -> 188[label="",style="solid", color="black", weight=3];
182 -> 276[label="",style="dashed", color="red", weight=0];
182[label="Integer vuz190 `quot` gcd vuz21 vuz20",fontsize=16,color="magenta"];182 -> 277[label="",style="dashed", color="magenta", weight=3];
183[label="primQuotInt (Pos vuz220) (reduce2D vuz24 vuz23)",fontsize=16,color="black",shape="box"];183 -> 190[label="",style="solid", color="black", weight=3];
184[label="primQuotInt (Neg vuz220) (reduce2D vuz24 vuz23)",fontsize=16,color="black",shape="box"];184 -> 191[label="",style="solid", color="black", weight=3];
185[label="Succ (Succ (primPlusNat vuz1800 vuz4000))",fontsize=16,color="green",shape="box"];185 -> 192[label="",style="dashed", color="green", weight=3];
186[label="Succ vuz1800",fontsize=16,color="green",shape="box"];187[label="Succ vuz4000",fontsize=16,color="green",shape="box"];188[label="Zero",fontsize=16,color="green",shape="box"];277[label="gcd vuz21 vuz20",fontsize=16,color="black",shape="box"];277 -> 293[label="",style="solid", color="black", weight=3];
276[label="Integer vuz190 `quot` vuz36",fontsize=16,color="burlywood",shape="triangle"];1080[label="vuz36/Integer vuz360",fontsize=10,color="white",style="solid",shape="box"];276 -> 1080[label="",style="solid", color="burlywood", weight=9];
1080 -> 294[label="",style="solid", color="burlywood", weight=3];
190 -> 312[label="",style="dashed", color="red", weight=0];
190[label="primQuotInt (Pos vuz220) (gcd vuz24 vuz23)",fontsize=16,color="magenta"];190 -> 313[label="",style="dashed", color="magenta", weight=3];
191[label="primQuotInt (Neg vuz220) (gcd vuz24 vuz23)",fontsize=16,color="black",shape="box"];191 -> 195[label="",style="solid", color="black", weight=3];
192 -> 123[label="",style="dashed", color="red", weight=0];
192[label="primPlusNat vuz1800 vuz4000",fontsize=16,color="magenta"];192 -> 196[label="",style="dashed", color="magenta", weight=3];
192 -> 197[label="",style="dashed", color="magenta", weight=3];
293[label="gcd3 vuz21 vuz20",fontsize=16,color="black",shape="box"];293 -> 300[label="",style="solid", color="black", weight=3];
294[label="Integer vuz190 `quot` Integer vuz360",fontsize=16,color="black",shape="box"];294 -> 301[label="",style="solid", color="black", weight=3];
313[label="gcd vuz24 vuz23",fontsize=16,color="black",shape="box"];313 -> 329[label="",style="solid", color="black", weight=3];
312[label="primQuotInt (Pos vuz220) vuz40",fontsize=16,color="burlywood",shape="triangle"];1081[label="vuz40/Pos vuz400",fontsize=10,color="white",style="solid",shape="box"];312 -> 1081[label="",style="solid", color="burlywood", weight=9];
1081 -> 330[label="",style="solid", color="burlywood", weight=3];
1082[label="vuz40/Neg vuz400",fontsize=10,color="white",style="solid",shape="box"];312 -> 1082[label="",style="solid", color="burlywood", weight=9];
1082 -> 331[label="",style="solid", color="burlywood", weight=3];
195[label="primQuotInt (Neg vuz220) (gcd3 vuz24 vuz23)",fontsize=16,color="black",shape="box"];195 -> 201[label="",style="solid", color="black", weight=3];
196[label="vuz4000",fontsize=16,color="green",shape="box"];197[label="vuz1800",fontsize=16,color="green",shape="box"];300 -> 306[label="",style="dashed", color="red", weight=0];
300[label="gcd2 (vuz21 == fromInt (Pos Zero)) vuz21 vuz20",fontsize=16,color="magenta"];300 -> 307[label="",style="dashed", color="magenta", weight=3];
301[label="Integer (primQuotInt vuz190 vuz360)",fontsize=16,color="green",shape="box"];301 -> 308[label="",style="dashed", color="green", weight=3];
329[label="gcd3 vuz24 vuz23",fontsize=16,color="black",shape="box"];329 -> 341[label="",style="solid", color="black", weight=3];
330[label="primQuotInt (Pos vuz220) (Pos vuz400)",fontsize=16,color="burlywood",shape="box"];1083[label="vuz400/Succ vuz4000",fontsize=10,color="white",style="solid",shape="box"];330 -> 1083[label="",style="solid", color="burlywood", weight=9];
1083 -> 342[label="",style="solid", color="burlywood", weight=3];
1084[label="vuz400/Zero",fontsize=10,color="white",style="solid",shape="box"];330 -> 1084[label="",style="solid", color="burlywood", weight=9];
1084 -> 343[label="",style="solid", color="burlywood", weight=3];
331[label="primQuotInt (Pos vuz220) (Neg vuz400)",fontsize=16,color="burlywood",shape="box"];1085[label="vuz400/Succ vuz4000",fontsize=10,color="white",style="solid",shape="box"];331 -> 1085[label="",style="solid", color="burlywood", weight=9];
1085 -> 344[label="",style="solid", color="burlywood", weight=3];
1086[label="vuz400/Zero",fontsize=10,color="white",style="solid",shape="box"];331 -> 1086[label="",style="solid", color="burlywood", weight=9];
1086 -> 345[label="",style="solid", color="burlywood", weight=3];
201 -> 207[label="",style="dashed", color="red", weight=0];
201[label="primQuotInt (Neg vuz220) (gcd2 (vuz24 == fromInt (Pos Zero)) vuz24 vuz23)",fontsize=16,color="magenta"];201 -> 208[label="",style="dashed", color="magenta", weight=3];
307 -> 76[label="",style="dashed", color="red", weight=0];
307[label="vuz21 == fromInt (Pos Zero)",fontsize=16,color="magenta"];307 -> 309[label="",style="dashed", color="magenta", weight=3];
306[label="gcd2 vuz39 vuz21 vuz20",fontsize=16,color="burlywood",shape="triangle"];1087[label="vuz39/False",fontsize=10,color="white",style="solid",shape="box"];306 -> 1087[label="",style="solid", color="burlywood", weight=9];
1087 -> 310[label="",style="solid", color="burlywood", weight=3];
1088[label="vuz39/True",fontsize=10,color="white",style="solid",shape="box"];306 -> 1088[label="",style="solid", color="burlywood", weight=9];
1088 -> 311[label="",style="solid", color="burlywood", weight=3];
308[label="primQuotInt vuz190 vuz360",fontsize=16,color="burlywood",shape="triangle"];1089[label="vuz190/Pos vuz1900",fontsize=10,color="white",style="solid",shape="box"];308 -> 1089[label="",style="solid", color="burlywood", weight=9];
1089 -> 332[label="",style="solid", color="burlywood", weight=3];
1090[label="vuz190/Neg vuz1900",fontsize=10,color="white",style="solid",shape="box"];308 -> 1090[label="",style="solid", color="burlywood", weight=9];
1090 -> 333[label="",style="solid", color="burlywood", weight=3];
341 -> 353[label="",style="dashed", color="red", weight=0];
341[label="gcd2 (vuz24 == fromInt (Pos Zero)) vuz24 vuz23",fontsize=16,color="magenta"];341 -> 354[label="",style="dashed", color="magenta", weight=3];
342[label="primQuotInt (Pos vuz220) (Pos (Succ vuz4000))",fontsize=16,color="black",shape="box"];342 -> 355[label="",style="solid", color="black", weight=3];
343[label="primQuotInt (Pos vuz220) (Pos Zero)",fontsize=16,color="black",shape="box"];343 -> 356[label="",style="solid", color="black", weight=3];
344[label="primQuotInt (Pos vuz220) (Neg (Succ vuz4000))",fontsize=16,color="black",shape="box"];344 -> 357[label="",style="solid", color="black", weight=3];
345[label="primQuotInt (Pos vuz220) (Neg Zero)",fontsize=16,color="black",shape="box"];345 -> 358[label="",style="solid", color="black", weight=3];
208 -> 78[label="",style="dashed", color="red", weight=0];
208[label="vuz24 == fromInt (Pos Zero)",fontsize=16,color="magenta"];208 -> 214[label="",style="dashed", color="magenta", weight=3];
207[label="primQuotInt (Neg vuz220) (gcd2 vuz27 vuz24 vuz23)",fontsize=16,color="burlywood",shape="triangle"];1091[label="vuz27/False",fontsize=10,color="white",style="solid",shape="box"];207 -> 1091[label="",style="solid", color="burlywood", weight=9];
1091 -> 215[label="",style="solid", color="burlywood", weight=3];
1092[label="vuz27/True",fontsize=10,color="white",style="solid",shape="box"];207 -> 1092[label="",style="solid", color="burlywood", weight=9];
1092 -> 216[label="",style="solid", color="burlywood", weight=3];
309[label="vuz21",fontsize=16,color="green",shape="box"];310[label="gcd2 False vuz21 vuz20",fontsize=16,color="black",shape="box"];310 -> 334[label="",style="solid", color="black", weight=3];
311[label="gcd2 True vuz21 vuz20",fontsize=16,color="black",shape="box"];311 -> 335[label="",style="solid", color="black", weight=3];
332[label="primQuotInt (Pos vuz1900) vuz360",fontsize=16,color="burlywood",shape="box"];1093[label="vuz360/Pos vuz3600",fontsize=10,color="white",style="solid",shape="box"];332 -> 1093[label="",style="solid", color="burlywood", weight=9];
1093 -> 346[label="",style="solid", color="burlywood", weight=3];
1094[label="vuz360/Neg vuz3600",fontsize=10,color="white",style="solid",shape="box"];332 -> 1094[label="",style="solid", color="burlywood", weight=9];
1094 -> 347[label="",style="solid", color="burlywood", weight=3];
333[label="primQuotInt (Neg vuz1900) vuz360",fontsize=16,color="burlywood",shape="box"];1095[label="vuz360/Pos vuz3600",fontsize=10,color="white",style="solid",shape="box"];333 -> 1095[label="",style="solid", color="burlywood", weight=9];
1095 -> 348[label="",style="solid", color="burlywood", weight=3];
1096[label="vuz360/Neg vuz3600",fontsize=10,color="white",style="solid",shape="box"];333 -> 1096[label="",style="solid", color="burlywood", weight=9];
1096 -> 349[label="",style="solid", color="burlywood", weight=3];
354 -> 78[label="",style="dashed", color="red", weight=0];
354[label="vuz24 == fromInt (Pos Zero)",fontsize=16,color="magenta"];354 -> 359[label="",style="dashed", color="magenta", weight=3];
353[label="gcd2 vuz42 vuz24 vuz23",fontsize=16,color="burlywood",shape="triangle"];1097[label="vuz42/False",fontsize=10,color="white",style="solid",shape="box"];353 -> 1097[label="",style="solid", color="burlywood", weight=9];
1097 -> 360[label="",style="solid", color="burlywood", weight=3];
1098[label="vuz42/True",fontsize=10,color="white",style="solid",shape="box"];353 -> 1098[label="",style="solid", color="burlywood", weight=9];
1098 -> 361[label="",style="solid", color="burlywood", weight=3];
355[label="Pos (primDivNatS vuz220 (Succ vuz4000))",fontsize=16,color="green",shape="box"];355 -> 377[label="",style="dashed", color="green", weight=3];
356[label="error []",fontsize=16,color="black",shape="triangle"];356 -> 378[label="",style="solid", color="black", weight=3];
357[label="Neg (primDivNatS vuz220 (Succ vuz4000))",fontsize=16,color="green",shape="box"];357 -> 379[label="",style="dashed", color="green", weight=3];
358 -> 356[label="",style="dashed", color="red", weight=0];
358[label="error []",fontsize=16,color="magenta"];214[label="vuz24",fontsize=16,color="green",shape="box"];215[label="primQuotInt (Neg vuz220) (gcd2 False vuz24 vuz23)",fontsize=16,color="black",shape="box"];215 -> 222[label="",style="solid", color="black", weight=3];
216[label="primQuotInt (Neg vuz220) (gcd2 True vuz24 vuz23)",fontsize=16,color="black",shape="box"];216 -> 223[label="",style="solid", color="black", weight=3];
334[label="gcd0 vuz21 vuz20",fontsize=16,color="black",shape="triangle"];334 -> 350[label="",style="solid", color="black", weight=3];
335 -> 351[label="",style="dashed", color="red", weight=0];
335[label="gcd1 (vuz20 == fromInt (Pos Zero)) vuz21 vuz20",fontsize=16,color="magenta"];335 -> 352[label="",style="dashed", color="magenta", weight=3];
346[label="primQuotInt (Pos vuz1900) (Pos vuz3600)",fontsize=16,color="burlywood",shape="box"];1099[label="vuz3600/Succ vuz36000",fontsize=10,color="white",style="solid",shape="box"];346 -> 1099[label="",style="solid", color="burlywood", weight=9];
1099 -> 362[label="",style="solid", color="burlywood", weight=3];
1100[label="vuz3600/Zero",fontsize=10,color="white",style="solid",shape="box"];346 -> 1100[label="",style="solid", color="burlywood", weight=9];
1100 -> 363[label="",style="solid", color="burlywood", weight=3];
347[label="primQuotInt (Pos vuz1900) (Neg vuz3600)",fontsize=16,color="burlywood",shape="box"];1101[label="vuz3600/Succ vuz36000",fontsize=10,color="white",style="solid",shape="box"];347 -> 1101[label="",style="solid", color="burlywood", weight=9];
1101 -> 364[label="",style="solid", color="burlywood", weight=3];
1102[label="vuz3600/Zero",fontsize=10,color="white",style="solid",shape="box"];347 -> 1102[label="",style="solid", color="burlywood", weight=9];
1102 -> 365[label="",style="solid", color="burlywood", weight=3];
348[label="primQuotInt (Neg vuz1900) (Pos vuz3600)",fontsize=16,color="burlywood",shape="box"];1103[label="vuz3600/Succ vuz36000",fontsize=10,color="white",style="solid",shape="box"];348 -> 1103[label="",style="solid", color="burlywood", weight=9];
1103 -> 366[label="",style="solid", color="burlywood", weight=3];
1104[label="vuz3600/Zero",fontsize=10,color="white",style="solid",shape="box"];348 -> 1104[label="",style="solid", color="burlywood", weight=9];
1104 -> 367[label="",style="solid", color="burlywood", weight=3];
349[label="primQuotInt (Neg vuz1900) (Neg vuz3600)",fontsize=16,color="burlywood",shape="box"];1105[label="vuz3600/Succ vuz36000",fontsize=10,color="white",style="solid",shape="box"];349 -> 1105[label="",style="solid", color="burlywood", weight=9];
1105 -> 368[label="",style="solid", color="burlywood", weight=3];
1106[label="vuz3600/Zero",fontsize=10,color="white",style="solid",shape="box"];349 -> 1106[label="",style="solid", color="burlywood", weight=9];
1106 -> 369[label="",style="solid", color="burlywood", weight=3];
359[label="vuz24",fontsize=16,color="green",shape="box"];360[label="gcd2 False vuz24 vuz23",fontsize=16,color="black",shape="box"];360 -> 380[label="",style="solid", color="black", weight=3];
361[label="gcd2 True vuz24 vuz23",fontsize=16,color="black",shape="box"];361 -> 381[label="",style="solid", color="black", weight=3];
377[label="primDivNatS vuz220 (Succ vuz4000)",fontsize=16,color="burlywood",shape="triangle"];1107[label="vuz220/Succ vuz2200",fontsize=10,color="white",style="solid",shape="box"];377 -> 1107[label="",style="solid", color="burlywood", weight=9];
1107 -> 397[label="",style="solid", color="burlywood", weight=3];
1108[label="vuz220/Zero",fontsize=10,color="white",style="solid",shape="box"];377 -> 1108[label="",style="solid", color="burlywood", weight=9];
1108 -> 398[label="",style="solid", color="burlywood", weight=3];
378[label="error []",fontsize=16,color="red",shape="box"];379 -> 377[label="",style="dashed", color="red", weight=0];
379[label="primDivNatS vuz220 (Succ vuz4000)",fontsize=16,color="magenta"];379 -> 399[label="",style="dashed", color="magenta", weight=3];
222[label="primQuotInt (Neg vuz220) (gcd0 vuz24 vuz23)",fontsize=16,color="black",shape="triangle"];222 -> 231[label="",style="solid", color="black", weight=3];
223 -> 232[label="",style="dashed", color="red", weight=0];
223[label="primQuotInt (Neg vuz220) (gcd1 (vuz23 == fromInt (Pos Zero)) vuz24 vuz23)",fontsize=16,color="magenta"];223 -> 233[label="",style="dashed", color="magenta", weight=3];
350[label="gcd0Gcd' (abs vuz21) (abs vuz20)",fontsize=16,color="black",shape="box"];350 -> 370[label="",style="solid", color="black", weight=3];
352 -> 76[label="",style="dashed", color="red", weight=0];
352[label="vuz20 == fromInt (Pos Zero)",fontsize=16,color="magenta"];352 -> 371[label="",style="dashed", color="magenta", weight=3];
351[label="gcd1 vuz41 vuz21 vuz20",fontsize=16,color="burlywood",shape="triangle"];1109[label="vuz41/False",fontsize=10,color="white",style="solid",shape="box"];351 -> 1109[label="",style="solid", color="burlywood", weight=9];
1109 -> 372[label="",style="solid", color="burlywood", weight=3];
1110[label="vuz41/True",fontsize=10,color="white",style="solid",shape="box"];351 -> 1110[label="",style="solid", color="burlywood", weight=9];
1110 -> 373[label="",style="solid", color="burlywood", weight=3];
362[label="primQuotInt (Pos vuz1900) (Pos (Succ vuz36000))",fontsize=16,color="black",shape="box"];362 -> 382[label="",style="solid", color="black", weight=3];
363[label="primQuotInt (Pos vuz1900) (Pos Zero)",fontsize=16,color="black",shape="box"];363 -> 383[label="",style="solid", color="black", weight=3];
364[label="primQuotInt (Pos vuz1900) (Neg (Succ vuz36000))",fontsize=16,color="black",shape="box"];364 -> 384[label="",style="solid", color="black", weight=3];
365[label="primQuotInt (Pos vuz1900) (Neg Zero)",fontsize=16,color="black",shape="box"];365 -> 385[label="",style="solid", color="black", weight=3];
366[label="primQuotInt (Neg vuz1900) (Pos (Succ vuz36000))",fontsize=16,color="black",shape="box"];366 -> 386[label="",style="solid", color="black", weight=3];
367[label="primQuotInt (Neg vuz1900) (Pos Zero)",fontsize=16,color="black",shape="box"];367 -> 387[label="",style="solid", color="black", weight=3];
368[label="primQuotInt (Neg vuz1900) (Neg (Succ vuz36000))",fontsize=16,color="black",shape="box"];368 -> 388[label="",style="solid", color="black", weight=3];
369[label="primQuotInt (Neg vuz1900) (Neg Zero)",fontsize=16,color="black",shape="box"];369 -> 389[label="",style="solid", color="black", weight=3];
380[label="gcd0 vuz24 vuz23",fontsize=16,color="black",shape="triangle"];380 -> 400[label="",style="solid", color="black", weight=3];
381 -> 401[label="",style="dashed", color="red", weight=0];
381[label="gcd1 (vuz23 == fromInt (Pos Zero)) vuz24 vuz23",fontsize=16,color="magenta"];381 -> 402[label="",style="dashed", color="magenta", weight=3];
397[label="primDivNatS (Succ vuz2200) (Succ vuz4000)",fontsize=16,color="black",shape="box"];397 -> 403[label="",style="solid", color="black", weight=3];
398[label="primDivNatS Zero (Succ vuz4000)",fontsize=16,color="black",shape="box"];398 -> 404[label="",style="solid", color="black", weight=3];
399[label="vuz4000",fontsize=16,color="green",shape="box"];231[label="primQuotInt (Neg vuz220) (gcd0Gcd' (abs vuz24) (abs vuz23))",fontsize=16,color="black",shape="box"];231 -> 241[label="",style="solid", color="black", weight=3];
233 -> 78[label="",style="dashed", color="red", weight=0];
233[label="vuz23 == fromInt (Pos Zero)",fontsize=16,color="magenta"];233 -> 242[label="",style="dashed", color="magenta", weight=3];
232[label="primQuotInt (Neg vuz220) (gcd1 vuz30 vuz24 vuz23)",fontsize=16,color="burlywood",shape="triangle"];1111[label="vuz30/False",fontsize=10,color="white",style="solid",shape="box"];232 -> 1111[label="",style="solid", color="burlywood", weight=9];
1111 -> 243[label="",style="solid", color="burlywood", weight=3];
1112[label="vuz30/True",fontsize=10,color="white",style="solid",shape="box"];232 -> 1112[label="",style="solid", color="burlywood", weight=9];
1112 -> 244[label="",style="solid", color="burlywood", weight=3];
370[label="gcd0Gcd'2 (abs vuz21) (abs vuz20)",fontsize=16,color="black",shape="box"];370 -> 390[label="",style="solid", color="black", weight=3];
371[label="vuz20",fontsize=16,color="green",shape="box"];372[label="gcd1 False vuz21 vuz20",fontsize=16,color="black",shape="box"];372 -> 391[label="",style="solid", color="black", weight=3];
373[label="gcd1 True vuz21 vuz20",fontsize=16,color="black",shape="box"];373 -> 392[label="",style="solid", color="black", weight=3];
382[label="Pos (primDivNatS vuz1900 (Succ vuz36000))",fontsize=16,color="green",shape="box"];382 -> 405[label="",style="dashed", color="green", weight=3];
383 -> 356[label="",style="dashed", color="red", weight=0];
383[label="error []",fontsize=16,color="magenta"];384[label="Neg (primDivNatS vuz1900 (Succ vuz36000))",fontsize=16,color="green",shape="box"];384 -> 406[label="",style="dashed", color="green", weight=3];
385 -> 356[label="",style="dashed", color="red", weight=0];
385[label="error []",fontsize=16,color="magenta"];386[label="Neg (primDivNatS vuz1900 (Succ vuz36000))",fontsize=16,color="green",shape="box"];386 -> 407[label="",style="dashed", color="green", weight=3];
387 -> 356[label="",style="dashed", color="red", weight=0];
387[label="error []",fontsize=16,color="magenta"];388[label="Pos (primDivNatS vuz1900 (Succ vuz36000))",fontsize=16,color="green",shape="box"];388 -> 408[label="",style="dashed", color="green", weight=3];
389 -> 356[label="",style="dashed", color="red", weight=0];
389[label="error []",fontsize=16,color="magenta"];400 -> 409[label="",style="dashed", color="red", weight=0];
400[label="gcd0Gcd' (abs vuz24) (abs vuz23)",fontsize=16,color="magenta"];400 -> 410[label="",style="dashed", color="magenta", weight=3];
400 -> 411[label="",style="dashed", color="magenta", weight=3];
402 -> 78[label="",style="dashed", color="red", weight=0];
402[label="vuz23 == fromInt (Pos Zero)",fontsize=16,color="magenta"];402 -> 414[label="",style="dashed", color="magenta", weight=3];
401[label="gcd1 vuz45 vuz24 vuz23",fontsize=16,color="burlywood",shape="triangle"];1113[label="vuz45/False",fontsize=10,color="white",style="solid",shape="box"];401 -> 1113[label="",style="solid", color="burlywood", weight=9];
1113 -> 415[label="",style="solid", color="burlywood", weight=3];
1114[label="vuz45/True",fontsize=10,color="white",style="solid",shape="box"];401 -> 1114[label="",style="solid", color="burlywood", weight=9];
1114 -> 416[label="",style="solid", color="burlywood", weight=3];
403[label="primDivNatS0 vuz2200 vuz4000 (primGEqNatS vuz2200 vuz4000)",fontsize=16,color="burlywood",shape="box"];1115[label="vuz2200/Succ vuz22000",fontsize=10,color="white",style="solid",shape="box"];403 -> 1115[label="",style="solid", color="burlywood", weight=9];
1115 -> 417[label="",style="solid", color="burlywood", weight=3];
1116[label="vuz2200/Zero",fontsize=10,color="white",style="solid",shape="box"];403 -> 1116[label="",style="solid", color="burlywood", weight=9];
1116 -> 418[label="",style="solid", color="burlywood", weight=3];
404[label="Zero",fontsize=16,color="green",shape="box"];241[label="primQuotInt (Neg vuz220) (gcd0Gcd'2 (abs vuz24) (abs vuz23))",fontsize=16,color="black",shape="box"];241 -> 251[label="",style="solid", color="black", weight=3];
242[label="vuz23",fontsize=16,color="green",shape="box"];243[label="primQuotInt (Neg vuz220) (gcd1 False vuz24 vuz23)",fontsize=16,color="black",shape="box"];243 -> 252[label="",style="solid", color="black", weight=3];
244[label="primQuotInt (Neg vuz220) (gcd1 True vuz24 vuz23)",fontsize=16,color="black",shape="box"];244 -> 253[label="",style="solid", color="black", weight=3];
390 -> 419[label="",style="dashed", color="red", weight=0];
390[label="gcd0Gcd'1 (abs vuz20 == fromInt (Pos Zero)) (abs vuz21) (abs vuz20)",fontsize=16,color="magenta"];390 -> 420[label="",style="dashed", color="magenta", weight=3];
391 -> 334[label="",style="dashed", color="red", weight=0];
391[label="gcd0 vuz21 vuz20",fontsize=16,color="magenta"];392[label="error []",fontsize=16,color="black",shape="box"];392 -> 421[label="",style="solid", color="black", weight=3];
405 -> 377[label="",style="dashed", color="red", weight=0];
405[label="primDivNatS vuz1900 (Succ vuz36000)",fontsize=16,color="magenta"];405 -> 422[label="",style="dashed", color="magenta", weight=3];
405 -> 423[label="",style="dashed", color="magenta", weight=3];
406 -> 377[label="",style="dashed", color="red", weight=0];
406[label="primDivNatS vuz1900 (Succ vuz36000)",fontsize=16,color="magenta"];406 -> 424[label="",style="dashed", color="magenta", weight=3];
406 -> 425[label="",style="dashed", color="magenta", weight=3];
407 -> 377[label="",style="dashed", color="red", weight=0];
407[label="primDivNatS vuz1900 (Succ vuz36000)",fontsize=16,color="magenta"];407 -> 426[label="",style="dashed", color="magenta", weight=3];
407 -> 427[label="",style="dashed", color="magenta", weight=3];
408 -> 377[label="",style="dashed", color="red", weight=0];
408[label="primDivNatS vuz1900 (Succ vuz36000)",fontsize=16,color="magenta"];408 -> 428[label="",style="dashed", color="magenta", weight=3];
408 -> 429[label="",style="dashed", color="magenta", weight=3];
410 -> 266[label="",style="dashed", color="red", weight=0];
410[label="abs vuz23",fontsize=16,color="magenta"];411 -> 266[label="",style="dashed", color="red", weight=0];
411[label="abs vuz24",fontsize=16,color="magenta"];411 -> 430[label="",style="dashed", color="magenta", weight=3];
409[label="gcd0Gcd' vuz47 vuz46",fontsize=16,color="black",shape="triangle"];409 -> 431[label="",style="solid", color="black", weight=3];
414[label="vuz23",fontsize=16,color="green",shape="box"];415[label="gcd1 False vuz24 vuz23",fontsize=16,color="black",shape="box"];415 -> 432[label="",style="solid", color="black", weight=3];
416[label="gcd1 True vuz24 vuz23",fontsize=16,color="black",shape="box"];416 -> 433[label="",style="solid", color="black", weight=3];
417[label="primDivNatS0 (Succ vuz22000) vuz4000 (primGEqNatS (Succ vuz22000) vuz4000)",fontsize=16,color="burlywood",shape="box"];1117[label="vuz4000/Succ vuz40000",fontsize=10,color="white",style="solid",shape="box"];417 -> 1117[label="",style="solid", color="burlywood", weight=9];
1117 -> 434[label="",style="solid", color="burlywood", weight=3];
1118[label="vuz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];417 -> 1118[label="",style="solid", color="burlywood", weight=9];
1118 -> 435[label="",style="solid", color="burlywood", weight=3];
418[label="primDivNatS0 Zero vuz4000 (primGEqNatS Zero vuz4000)",fontsize=16,color="burlywood",shape="box"];1119[label="vuz4000/Succ vuz40000",fontsize=10,color="white",style="solid",shape="box"];418 -> 1119[label="",style="solid", color="burlywood", weight=9];
1119 -> 436[label="",style="solid", color="burlywood", weight=3];
1120[label="vuz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];418 -> 1120[label="",style="solid", color="burlywood", weight=9];
1120 -> 437[label="",style="solid", color="burlywood", weight=3];
251 -> 260[label="",style="dashed", color="red", weight=0];
251[label="primQuotInt (Neg vuz220) (gcd0Gcd'1 (abs vuz23 == fromInt (Pos Zero)) (abs vuz24) (abs vuz23))",fontsize=16,color="magenta"];251 -> 261[label="",style="dashed", color="magenta", weight=3];
252 -> 222[label="",style="dashed", color="red", weight=0];
252[label="primQuotInt (Neg vuz220) (gcd0 vuz24 vuz23)",fontsize=16,color="magenta"];253[label="primQuotInt (Neg vuz220) (error [])",fontsize=16,color="black",shape="box"];253 -> 262[label="",style="solid", color="black", weight=3];
420 -> 76[label="",style="dashed", color="red", weight=0];
420[label="abs vuz20 == fromInt (Pos Zero)",fontsize=16,color="magenta"];420 -> 438[label="",style="dashed", color="magenta", weight=3];
419[label="gcd0Gcd'1 vuz48 (abs vuz21) (abs vuz20)",fontsize=16,color="burlywood",shape="triangle"];1121[label="vuz48/False",fontsize=10,color="white",style="solid",shape="box"];419 -> 1121[label="",style="solid", color="burlywood", weight=9];
1121 -> 439[label="",style="solid", color="burlywood", weight=3];
1122[label="vuz48/True",fontsize=10,color="white",style="solid",shape="box"];419 -> 1122[label="",style="solid", color="burlywood", weight=9];
1122 -> 440[label="",style="solid", color="burlywood", weight=3];
421[label="error []",fontsize=16,color="red",shape="box"];422[label="vuz1900",fontsize=16,color="green",shape="box"];423[label="vuz36000",fontsize=16,color="green",shape="box"];424[label="vuz1900",fontsize=16,color="green",shape="box"];425[label="vuz36000",fontsize=16,color="green",shape="box"];426[label="vuz1900",fontsize=16,color="green",shape="box"];427[label="vuz36000",fontsize=16,color="green",shape="box"];428[label="vuz1900",fontsize=16,color="green",shape="box"];429[label="vuz36000",fontsize=16,color="green",shape="box"];266[label="abs vuz23",fontsize=16,color="black",shape="triangle"];266 -> 295[label="",style="solid", color="black", weight=3];
430[label="vuz24",fontsize=16,color="green",shape="box"];431[label="gcd0Gcd'2 vuz47 vuz46",fontsize=16,color="black",shape="box"];431 -> 443[label="",style="solid", color="black", weight=3];
432 -> 380[label="",style="dashed", color="red", weight=0];
432[label="gcd0 vuz24 vuz23",fontsize=16,color="magenta"];433 -> 356[label="",style="dashed", color="red", weight=0];
433[label="error []",fontsize=16,color="magenta"];434[label="primDivNatS0 (Succ vuz22000) (Succ vuz40000) (primGEqNatS (Succ vuz22000) (Succ vuz40000))",fontsize=16,color="black",shape="box"];434 -> 444[label="",style="solid", color="black", weight=3];
435[label="primDivNatS0 (Succ vuz22000) Zero (primGEqNatS (Succ vuz22000) Zero)",fontsize=16,color="black",shape="box"];435 -> 445[label="",style="solid", color="black", weight=3];
436[label="primDivNatS0 Zero (Succ vuz40000) (primGEqNatS Zero (Succ vuz40000))",fontsize=16,color="black",shape="box"];436 -> 446[label="",style="solid", color="black", weight=3];
437[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];437 -> 447[label="",style="solid", color="black", weight=3];
261 -> 78[label="",style="dashed", color="red", weight=0];
261[label="abs vuz23 == fromInt (Pos Zero)",fontsize=16,color="magenta"];261 -> 269[label="",style="dashed", color="magenta", weight=3];
260[label="primQuotInt (Neg vuz220) (gcd0Gcd'1 vuz33 (abs vuz24) (abs vuz23))",fontsize=16,color="burlywood",shape="triangle"];1123[label="vuz33/False",fontsize=10,color="white",style="solid",shape="box"];260 -> 1123[label="",style="solid", color="burlywood", weight=9];
1123 -> 270[label="",style="solid", color="burlywood", weight=3];
1124[label="vuz33/True",fontsize=10,color="white",style="solid",shape="box"];260 -> 1124[label="",style="solid", color="burlywood", weight=9];
1124 -> 271[label="",style="solid", color="burlywood", weight=3];
262[label="error []",fontsize=16,color="red",shape="box"];438[label="abs vuz20",fontsize=16,color="black",shape="triangle"];438 -> 448[label="",style="solid", color="black", weight=3];
439[label="gcd0Gcd'1 False (abs vuz21) (abs vuz20)",fontsize=16,color="black",shape="box"];439 -> 449[label="",style="solid", color="black", weight=3];
440[label="gcd0Gcd'1 True (abs vuz21) (abs vuz20)",fontsize=16,color="black",shape="box"];440 -> 450[label="",style="solid", color="black", weight=3];
295[label="absReal vuz23",fontsize=16,color="black",shape="box"];295 -> 302[label="",style="solid", color="black", weight=3];
443 -> 454[label="",style="dashed", color="red", weight=0];
443[label="gcd0Gcd'1 (vuz46 == fromInt (Pos Zero)) vuz47 vuz46",fontsize=16,color="magenta"];443 -> 455[label="",style="dashed", color="magenta", weight=3];
444 -> 791[label="",style="dashed", color="red", weight=0];
444[label="primDivNatS0 (Succ vuz22000) (Succ vuz40000) (primGEqNatS vuz22000 vuz40000)",fontsize=16,color="magenta"];444 -> 792[label="",style="dashed", color="magenta", weight=3];
444 -> 793[label="",style="dashed", color="magenta", weight=3];
444 -> 794[label="",style="dashed", color="magenta", weight=3];
444 -> 795[label="",style="dashed", color="magenta", weight=3];
445[label="primDivNatS0 (Succ vuz22000) Zero True",fontsize=16,color="black",shape="box"];445 -> 458[label="",style="solid", color="black", weight=3];
446[label="primDivNatS0 Zero (Succ vuz40000) False",fontsize=16,color="black",shape="box"];446 -> 459[label="",style="solid", color="black", weight=3];
447[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];447 -> 460[label="",style="solid", color="black", weight=3];
269 -> 266[label="",style="dashed", color="red", weight=0];
269[label="abs vuz23",fontsize=16,color="magenta"];270[label="primQuotInt (Neg vuz220) (gcd0Gcd'1 False (abs vuz24) (abs vuz23))",fontsize=16,color="black",shape="box"];270 -> 298[label="",style="solid", color="black", weight=3];
271[label="primQuotInt (Neg vuz220) (gcd0Gcd'1 True (abs vuz24) (abs vuz23))",fontsize=16,color="black",shape="box"];271 -> 299[label="",style="solid", color="black", weight=3];
448[label="absReal vuz20",fontsize=16,color="black",shape="box"];448 -> 461[label="",style="solid", color="black", weight=3];
449 -> 462[label="",style="dashed", color="red", weight=0];
449[label="gcd0Gcd'0 (abs vuz21) (abs vuz20)",fontsize=16,color="magenta"];449 -> 463[label="",style="dashed", color="magenta", weight=3];
449 -> 464[label="",style="dashed", color="magenta", weight=3];
450 -> 438[label="",style="dashed", color="red", weight=0];
450[label="abs vuz21",fontsize=16,color="magenta"];450 -> 465[label="",style="dashed", color="magenta", weight=3];
302[label="absReal2 vuz23",fontsize=16,color="black",shape="box"];302 -> 340[label="",style="solid", color="black", weight=3];
455 -> 78[label="",style="dashed", color="red", weight=0];
455[label="vuz46 == fromInt (Pos Zero)",fontsize=16,color="magenta"];455 -> 466[label="",style="dashed", color="magenta", weight=3];
454[label="gcd0Gcd'1 vuz49 vuz47 vuz46",fontsize=16,color="burlywood",shape="triangle"];1125[label="vuz49/False",fontsize=10,color="white",style="solid",shape="box"];454 -> 1125[label="",style="solid", color="burlywood", weight=9];
1125 -> 467[label="",style="solid", color="burlywood", weight=3];
1126[label="vuz49/True",fontsize=10,color="white",style="solid",shape="box"];454 -> 1126[label="",style="solid", color="burlywood", weight=9];
1126 -> 468[label="",style="solid", color="burlywood", weight=3];
792[label="vuz40000",fontsize=16,color="green",shape="box"];793[label="vuz22000",fontsize=16,color="green",shape="box"];794[label="vuz22000",fontsize=16,color="green",shape="box"];795[label="vuz40000",fontsize=16,color="green",shape="box"];791[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS vuz71 vuz72)",fontsize=16,color="burlywood",shape="triangle"];1127[label="vuz71/Succ vuz710",fontsize=10,color="white",style="solid",shape="box"];791 -> 1127[label="",style="solid", color="burlywood", weight=9];
1127 -> 824[label="",style="solid", color="burlywood", weight=3];
1128[label="vuz71/Zero",fontsize=10,color="white",style="solid",shape="box"];791 -> 1128[label="",style="solid", color="burlywood", weight=9];
1128 -> 825[label="",style="solid", color="burlywood", weight=3];
458[label="Succ (primDivNatS (primMinusNatS (Succ vuz22000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];458 -> 473[label="",style="dashed", color="green", weight=3];
459[label="Zero",fontsize=16,color="green",shape="box"];460[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];460 -> 474[label="",style="dashed", color="green", weight=3];
298 -> 308[label="",style="dashed", color="red", weight=0];
298[label="primQuotInt (Neg vuz220) (gcd0Gcd'0 (abs vuz24) (abs vuz23))",fontsize=16,color="magenta"];298 -> 336[label="",style="dashed", color="magenta", weight=3];
298 -> 337[label="",style="dashed", color="magenta", weight=3];
299 -> 308[label="",style="dashed", color="red", weight=0];
299[label="primQuotInt (Neg vuz220) (abs vuz24)",fontsize=16,color="magenta"];299 -> 338[label="",style="dashed", color="magenta", weight=3];
299 -> 339[label="",style="dashed", color="magenta", weight=3];
461[label="absReal2 vuz20",fontsize=16,color="black",shape="box"];461 -> 475[label="",style="solid", color="black", weight=3];
463 -> 438[label="",style="dashed", color="red", weight=0];
463[label="abs vuz20",fontsize=16,color="magenta"];464 -> 438[label="",style="dashed", color="red", weight=0];
464[label="abs vuz21",fontsize=16,color="magenta"];464 -> 476[label="",style="dashed", color="magenta", weight=3];
462[label="gcd0Gcd'0 vuz51 vuz50",fontsize=16,color="black",shape="triangle"];462 -> 477[label="",style="solid", color="black", weight=3];
465[label="vuz21",fontsize=16,color="green",shape="box"];340[label="absReal1 vuz23 (vuz23 >= fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];340 -> 396[label="",style="solid", color="black", weight=3];
466[label="vuz46",fontsize=16,color="green",shape="box"];467[label="gcd0Gcd'1 False vuz47 vuz46",fontsize=16,color="black",shape="box"];467 -> 484[label="",style="solid", color="black", weight=3];
468[label="gcd0Gcd'1 True vuz47 vuz46",fontsize=16,color="black",shape="box"];468 -> 485[label="",style="solid", color="black", weight=3];
824[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS (Succ vuz710) vuz72)",fontsize=16,color="burlywood",shape="box"];1129[label="vuz72/Succ vuz720",fontsize=10,color="white",style="solid",shape="box"];824 -> 1129[label="",style="solid", color="burlywood", weight=9];
1129 -> 832[label="",style="solid", color="burlywood", weight=3];
1130[label="vuz72/Zero",fontsize=10,color="white",style="solid",shape="box"];824 -> 1130[label="",style="solid", color="burlywood", weight=9];
1130 -> 833[label="",style="solid", color="burlywood", weight=3];
825[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS Zero vuz72)",fontsize=16,color="burlywood",shape="box"];1131[label="vuz72/Succ vuz720",fontsize=10,color="white",style="solid",shape="box"];825 -> 1131[label="",style="solid", color="burlywood", weight=9];
1131 -> 834[label="",style="solid", color="burlywood", weight=3];
1132[label="vuz72/Zero",fontsize=10,color="white",style="solid",shape="box"];825 -> 1132[label="",style="solid", color="burlywood", weight=9];
1132 -> 835[label="",style="solid", color="burlywood", weight=3];
473 -> 377[label="",style="dashed", color="red", weight=0];
473[label="primDivNatS (primMinusNatS (Succ vuz22000) Zero) (Succ Zero)",fontsize=16,color="magenta"];473 -> 490[label="",style="dashed", color="magenta", weight=3];
473 -> 491[label="",style="dashed", color="magenta", weight=3];
474 -> 377[label="",style="dashed", color="red", weight=0];
474[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];474 -> 492[label="",style="dashed", color="magenta", weight=3];
474 -> 493[label="",style="dashed", color="magenta", weight=3];
336 -> 374[label="",style="dashed", color="red", weight=0];
336[label="gcd0Gcd'0 (abs vuz24) (abs vuz23)",fontsize=16,color="magenta"];336 -> 375[label="",style="dashed", color="magenta", weight=3];
336 -> 376[label="",style="dashed", color="magenta", weight=3];
337[label="Neg vuz220",fontsize=16,color="green",shape="box"];338 -> 266[label="",style="dashed", color="red", weight=0];
338[label="abs vuz24",fontsize=16,color="magenta"];338 -> 393[label="",style="dashed", color="magenta", weight=3];
339[label="Neg vuz220",fontsize=16,color="green",shape="box"];475[label="absReal1 vuz20 (vuz20 >= fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];475 -> 494[label="",style="solid", color="black", weight=3];
476[label="vuz21",fontsize=16,color="green",shape="box"];477[label="gcd0Gcd' vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="black",shape="box"];477 -> 495[label="",style="solid", color="black", weight=3];
396[label="absReal1 vuz23 (compare vuz23 (fromInt (Pos Zero)) /= LT)",fontsize=16,color="black",shape="box"];396 -> 441[label="",style="solid", color="black", weight=3];
484 -> 374[label="",style="dashed", color="red", weight=0];
484[label="gcd0Gcd'0 vuz47 vuz46",fontsize=16,color="magenta"];484 -> 508[label="",style="dashed", color="magenta", weight=3];
484 -> 509[label="",style="dashed", color="magenta", weight=3];
485[label="vuz47",fontsize=16,color="green",shape="box"];832[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS (Succ vuz710) (Succ vuz720))",fontsize=16,color="black",shape="box"];832 -> 844[label="",style="solid", color="black", weight=3];
833[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS (Succ vuz710) Zero)",fontsize=16,color="black",shape="box"];833 -> 845[label="",style="solid", color="black", weight=3];
834[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS Zero (Succ vuz720))",fontsize=16,color="black",shape="box"];834 -> 846[label="",style="solid", color="black", weight=3];
835[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];835 -> 847[label="",style="solid", color="black", weight=3];
490[label="primMinusNatS (Succ vuz22000) Zero",fontsize=16,color="black",shape="triangle"];490 -> 515[label="",style="solid", color="black", weight=3];
491[label="Zero",fontsize=16,color="green",shape="box"];492[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];492 -> 516[label="",style="solid", color="black", weight=3];
493[label="Zero",fontsize=16,color="green",shape="box"];375 -> 266[label="",style="dashed", color="red", weight=0];
375[label="abs vuz24",fontsize=16,color="magenta"];375 -> 394[label="",style="dashed", color="magenta", weight=3];
376 -> 266[label="",style="dashed", color="red", weight=0];
376[label="abs vuz23",fontsize=16,color="magenta"];374[label="gcd0Gcd'0 vuz44 vuz43",fontsize=16,color="black",shape="triangle"];374 -> 395[label="",style="solid", color="black", weight=3];
393[label="vuz24",fontsize=16,color="green",shape="box"];494[label="absReal1 vuz20 (compare vuz20 (fromInt (Pos Zero)) /= LT)",fontsize=16,color="black",shape="box"];494 -> 517[label="",style="solid", color="black", weight=3];
495[label="gcd0Gcd'2 vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="black",shape="box"];495 -> 518[label="",style="solid", color="black", weight=3];
441[label="absReal1 vuz23 (not (compare vuz23 (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];441 -> 451[label="",style="solid", color="black", weight=3];
508[label="vuz47",fontsize=16,color="green",shape="box"];509[label="vuz46",fontsize=16,color="green",shape="box"];844 -> 791[label="",style="dashed", color="red", weight=0];
844[label="primDivNatS0 (Succ vuz69) (Succ vuz70) (primGEqNatS vuz710 vuz720)",fontsize=16,color="magenta"];844 -> 854[label="",style="dashed", color="magenta", weight=3];
844 -> 855[label="",style="dashed", color="magenta", weight=3];
845[label="primDivNatS0 (Succ vuz69) (Succ vuz70) True",fontsize=16,color="black",shape="triangle"];845 -> 856[label="",style="solid", color="black", weight=3];
846[label="primDivNatS0 (Succ vuz69) (Succ vuz70) False",fontsize=16,color="black",shape="box"];846 -> 857[label="",style="solid", color="black", weight=3];
847 -> 845[label="",style="dashed", color="red", weight=0];
847[label="primDivNatS0 (Succ vuz69) (Succ vuz70) True",fontsize=16,color="magenta"];515[label="Succ vuz22000",fontsize=16,color="green",shape="box"];516[label="Zero",fontsize=16,color="green",shape="box"];394[label="vuz24",fontsize=16,color="green",shape="box"];395 -> 409[label="",style="dashed", color="red", weight=0];
395[label="gcd0Gcd' vuz43 (vuz44 `rem` vuz43)",fontsize=16,color="magenta"];395 -> 412[label="",style="dashed", color="magenta", weight=3];
395 -> 413[label="",style="dashed", color="magenta", weight=3];
517[label="absReal1 vuz20 (not (compare vuz20 (fromInt (Pos Zero)) == LT))",fontsize=16,color="burlywood",shape="box"];1133[label="vuz20/Integer vuz200",fontsize=10,color="white",style="solid",shape="box"];517 -> 1133[label="",style="solid", color="burlywood", weight=9];
1133 -> 537[label="",style="solid", color="burlywood", weight=3];
518 -> 538[label="",style="dashed", color="red", weight=0];
518[label="gcd0Gcd'1 (vuz51 `rem` vuz50 == fromInt (Pos Zero)) vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="magenta"];518 -> 539[label="",style="dashed", color="magenta", weight=3];
451[label="absReal1 vuz23 (not (primCmpInt vuz23 (fromInt (Pos Zero)) == LT))",fontsize=16,color="burlywood",shape="box"];1134[label="vuz23/Pos vuz230",fontsize=10,color="white",style="solid",shape="box"];451 -> 1134[label="",style="solid", color="burlywood", weight=9];
1134 -> 478[label="",style="solid", color="burlywood", weight=3];
1135[label="vuz23/Neg vuz230",fontsize=10,color="white",style="solid",shape="box"];451 -> 1135[label="",style="solid", color="burlywood", weight=9];
1135 -> 479[label="",style="solid", color="burlywood", weight=3];
854[label="vuz710",fontsize=16,color="green",shape="box"];855[label="vuz720",fontsize=16,color="green",shape="box"];856[label="Succ (primDivNatS (primMinusNatS (Succ vuz69) (Succ vuz70)) (Succ (Succ vuz70)))",fontsize=16,color="green",shape="box"];856 -> 864[label="",style="dashed", color="green", weight=3];
857[label="Zero",fontsize=16,color="green",shape="box"];412[label="vuz44 `rem` vuz43",fontsize=16,color="black",shape="box"];412 -> 442[label="",style="solid", color="black", weight=3];
413[label="vuz43",fontsize=16,color="green",shape="box"];537[label="absReal1 (Integer vuz200) (not (compare (Integer vuz200) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];537 -> 548[label="",style="solid", color="black", weight=3];
539 -> 76[label="",style="dashed", color="red", weight=0];
539[label="vuz51 `rem` vuz50 == fromInt (Pos Zero)",fontsize=16,color="magenta"];539 -> 549[label="",style="dashed", color="magenta", weight=3];
538[label="gcd0Gcd'1 vuz52 vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="burlywood",shape="triangle"];1136[label="vuz52/False",fontsize=10,color="white",style="solid",shape="box"];538 -> 1136[label="",style="solid", color="burlywood", weight=9];
1136 -> 550[label="",style="solid", color="burlywood", weight=3];
1137[label="vuz52/True",fontsize=10,color="white",style="solid",shape="box"];538 -> 1137[label="",style="solid", color="burlywood", weight=9];
1137 -> 551[label="",style="solid", color="burlywood", weight=3];
478[label="absReal1 (Pos vuz230) (not (primCmpInt (Pos vuz230) (fromInt (Pos Zero)) == LT))",fontsize=16,color="burlywood",shape="box"];1138[label="vuz230/Succ vuz2300",fontsize=10,color="white",style="solid",shape="box"];478 -> 1138[label="",style="solid", color="burlywood", weight=9];
1138 -> 496[label="",style="solid", color="burlywood", weight=3];
1139[label="vuz230/Zero",fontsize=10,color="white",style="solid",shape="box"];478 -> 1139[label="",style="solid", color="burlywood", weight=9];
1139 -> 497[label="",style="solid", color="burlywood", weight=3];
479[label="absReal1 (Neg vuz230) (not (primCmpInt (Neg vuz230) (fromInt (Pos Zero)) == LT))",fontsize=16,color="burlywood",shape="box"];1140[label="vuz230/Succ vuz2300",fontsize=10,color="white",style="solid",shape="box"];479 -> 1140[label="",style="solid", color="burlywood", weight=9];
1140 -> 498[label="",style="solid", color="burlywood", weight=3];
1141[label="vuz230/Zero",fontsize=10,color="white",style="solid",shape="box"];479 -> 1141[label="",style="solid", color="burlywood", weight=9];
1141 -> 499[label="",style="solid", color="burlywood", weight=3];
864 -> 377[label="",style="dashed", color="red", weight=0];
864[label="primDivNatS (primMinusNatS (Succ vuz69) (Succ vuz70)) (Succ (Succ vuz70))",fontsize=16,color="magenta"];864 -> 873[label="",style="dashed", color="magenta", weight=3];
864 -> 874[label="",style="dashed", color="magenta", weight=3];
442[label="primRemInt vuz44 vuz43",fontsize=16,color="burlywood",shape="triangle"];1142[label="vuz44/Pos vuz440",fontsize=10,color="white",style="solid",shape="box"];442 -> 1142[label="",style="solid", color="burlywood", weight=9];
1142 -> 452[label="",style="solid", color="burlywood", weight=3];
1143[label="vuz44/Neg vuz440",fontsize=10,color="white",style="solid",shape="box"];442 -> 1143[label="",style="solid", color="burlywood", weight=9];
1143 -> 453[label="",style="solid", color="burlywood", weight=3];
548[label="absReal1 (Integer vuz200) (not (compare (Integer vuz200) (Integer (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];548 -> 567[label="",style="solid", color="black", weight=3];
549[label="vuz51 `rem` vuz50",fontsize=16,color="burlywood",shape="triangle"];1144[label="vuz51/Integer vuz510",fontsize=10,color="white",style="solid",shape="box"];549 -> 1144[label="",style="solid", color="burlywood", weight=9];
1144 -> 568[label="",style="solid", color="burlywood", weight=3];
550[label="gcd0Gcd'1 False vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="black",shape="box"];550 -> 569[label="",style="solid", color="black", weight=3];
551[label="gcd0Gcd'1 True vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="black",shape="box"];551 -> 570[label="",style="solid", color="black", weight=3];
496[label="absReal1 (Pos (Succ vuz2300)) (not (primCmpInt (Pos (Succ vuz2300)) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];496 -> 519[label="",style="solid", color="black", weight=3];
497[label="absReal1 (Pos Zero) (not (primCmpInt (Pos Zero) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];497 -> 520[label="",style="solid", color="black", weight=3];
498[label="absReal1 (Neg (Succ vuz2300)) (not (primCmpInt (Neg (Succ vuz2300)) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];498 -> 521[label="",style="solid", color="black", weight=3];
499[label="absReal1 (Neg Zero) (not (primCmpInt (Neg Zero) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];499 -> 522[label="",style="solid", color="black", weight=3];
873[label="primMinusNatS (Succ vuz69) (Succ vuz70)",fontsize=16,color="black",shape="triangle"];873 -> 881[label="",style="solid", color="black", weight=3];
874[label="Succ vuz70",fontsize=16,color="green",shape="box"];452[label="primRemInt (Pos vuz440) vuz43",fontsize=16,color="burlywood",shape="box"];1145[label="vuz43/Pos vuz430",fontsize=10,color="white",style="solid",shape="box"];452 -> 1145[label="",style="solid", color="burlywood", weight=9];
1145 -> 480[label="",style="solid", color="burlywood", weight=3];
1146[label="vuz43/Neg vuz430",fontsize=10,color="white",style="solid",shape="box"];452 -> 1146[label="",style="solid", color="burlywood", weight=9];
1146 -> 481[label="",style="solid", color="burlywood", weight=3];
453[label="primRemInt (Neg vuz440) vuz43",fontsize=16,color="burlywood",shape="box"];1147[label="vuz43/Pos vuz430",fontsize=10,color="white",style="solid",shape="box"];453 -> 1147[label="",style="solid", color="burlywood", weight=9];
1147 -> 482[label="",style="solid", color="burlywood", weight=3];
1148[label="vuz43/Neg vuz430",fontsize=10,color="white",style="solid",shape="box"];453 -> 1148[label="",style="solid", color="burlywood", weight=9];
1148 -> 483[label="",style="solid", color="burlywood", weight=3];
567[label="absReal1 (Integer vuz200) (not (primCmpInt vuz200 (Pos Zero) == LT))",fontsize=16,color="burlywood",shape="box"];1149[label="vuz200/Pos vuz2000",fontsize=10,color="white",style="solid",shape="box"];567 -> 1149[label="",style="solid", color="burlywood", weight=9];
1149 -> 588[label="",style="solid", color="burlywood", weight=3];
1150[label="vuz200/Neg vuz2000",fontsize=10,color="white",style="solid",shape="box"];567 -> 1150[label="",style="solid", color="burlywood", weight=9];
1150 -> 589[label="",style="solid", color="burlywood", weight=3];
568[label="Integer vuz510 `rem` vuz50",fontsize=16,color="burlywood",shape="box"];1151[label="vuz50/Integer vuz500",fontsize=10,color="white",style="solid",shape="box"];568 -> 1151[label="",style="solid", color="burlywood", weight=9];
1151 -> 590[label="",style="solid", color="burlywood", weight=3];
569 -> 462[label="",style="dashed", color="red", weight=0];
569[label="gcd0Gcd'0 vuz50 (vuz51 `rem` vuz50)",fontsize=16,color="magenta"];569 -> 591[label="",style="dashed", color="magenta", weight=3];
569 -> 592[label="",style="dashed", color="magenta", weight=3];
570[label="vuz50",fontsize=16,color="green",shape="box"];519[label="absReal1 (Pos (Succ vuz2300)) (not (primCmpInt (Pos (Succ vuz2300)) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];519 -> 552[label="",style="solid", color="black", weight=3];
520[label="absReal1 (Pos Zero) (not (primCmpInt (Pos Zero) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];520 -> 553[label="",style="solid", color="black", weight=3];
521[label="absReal1 (Neg (Succ vuz2300)) (not (primCmpInt (Neg (Succ vuz2300)) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];521 -> 554[label="",style="solid", color="black", weight=3];
522[label="absReal1 (Neg Zero) (not (primCmpInt (Neg Zero) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];522 -> 555[label="",style="solid", color="black", weight=3];
881[label="primMinusNatS vuz69 vuz70",fontsize=16,color="burlywood",shape="triangle"];1152[label="vuz69/Succ vuz690",fontsize=10,color="white",style="solid",shape="box"];881 -> 1152[label="",style="solid", color="burlywood", weight=9];
1152 -> 891[label="",style="solid", color="burlywood", weight=3];
1153[label="vuz69/Zero",fontsize=10,color="white",style="solid",shape="box"];881 -> 1153[label="",style="solid", color="burlywood", weight=9];
1153 -> 892[label="",style="solid", color="burlywood", weight=3];
480[label="primRemInt (Pos vuz440) (Pos vuz430)",fontsize=16,color="burlywood",shape="box"];1154[label="vuz430/Succ vuz4300",fontsize=10,color="white",style="solid",shape="box"];480 -> 1154[label="",style="solid", color="burlywood", weight=9];
1154 -> 500[label="",style="solid", color="burlywood", weight=3];
1155[label="vuz430/Zero",fontsize=10,color="white",style="solid",shape="box"];480 -> 1155[label="",style="solid", color="burlywood", weight=9];
1155 -> 501[label="",style="solid", color="burlywood", weight=3];
481[label="primRemInt (Pos vuz440) (Neg vuz430)",fontsize=16,color="burlywood",shape="box"];1156[label="vuz430/Succ vuz4300",fontsize=10,color="white",style="solid",shape="box"];481 -> 1156[label="",style="solid", color="burlywood", weight=9];
1156 -> 502[label="",style="solid", color="burlywood", weight=3];
1157[label="vuz430/Zero",fontsize=10,color="white",style="solid",shape="box"];481 -> 1157[label="",style="solid", color="burlywood", weight=9];
1157 -> 503[label="",style="solid", color="burlywood", weight=3];
482[label="primRemInt (Neg vuz440) (Pos vuz430)",fontsize=16,color="burlywood",shape="box"];1158[label="vuz430/Succ vuz4300",fontsize=10,color="white",style="solid",shape="box"];482 -> 1158[label="",style="solid", color="burlywood", weight=9];
1158 -> 504[label="",style="solid", color="burlywood", weight=3];
1159[label="vuz430/Zero",fontsize=10,color="white",style="solid",shape="box"];482 -> 1159[label="",style="solid", color="burlywood", weight=9];
1159 -> 505[label="",style="solid", color="burlywood", weight=3];
483[label="primRemInt (Neg vuz440) (Neg vuz430)",fontsize=16,color="burlywood",shape="box"];1160[label="vuz430/Succ vuz4300",fontsize=10,color="white",style="solid",shape="box"];483 -> 1160[label="",style="solid", color="burlywood", weight=9];
1160 -> 506[label="",style="solid", color="burlywood", weight=3];
1161[label="vuz430/Zero",fontsize=10,color="white",style="solid",shape="box"];483 -> 1161[label="",style="solid", color="burlywood", weight=9];
1161 -> 507[label="",style="solid", color="burlywood", weight=3];
588[label="absReal1 (Integer (Pos vuz2000)) (not (primCmpInt (Pos vuz2000) (Pos Zero) == LT))",fontsize=16,color="burlywood",shape="box"];1162[label="vuz2000/Succ vuz20000",fontsize=10,color="white",style="solid",shape="box"];588 -> 1162[label="",style="solid", color="burlywood", weight=9];
1162 -> 607[label="",style="solid", color="burlywood", weight=3];
1163[label="vuz2000/Zero",fontsize=10,color="white",style="solid",shape="box"];588 -> 1163[label="",style="solid", color="burlywood", weight=9];
1163 -> 608[label="",style="solid", color="burlywood", weight=3];
589[label="absReal1 (Integer (Neg vuz2000)) (not (primCmpInt (Neg vuz2000) (Pos Zero) == LT))",fontsize=16,color="burlywood",shape="box"];1164[label="vuz2000/Succ vuz20000",fontsize=10,color="white",style="solid",shape="box"];589 -> 1164[label="",style="solid", color="burlywood", weight=9];
1164 -> 609[label="",style="solid", color="burlywood", weight=3];
1165[label="vuz2000/Zero",fontsize=10,color="white",style="solid",shape="box"];589 -> 1165[label="",style="solid", color="burlywood", weight=9];
1165 -> 610[label="",style="solid", color="burlywood", weight=3];
590[label="Integer vuz510 `rem` Integer vuz500",fontsize=16,color="black",shape="box"];590 -> 611[label="",style="solid", color="black", weight=3];
591 -> 549[label="",style="dashed", color="red", weight=0];
591[label="vuz51 `rem` vuz50",fontsize=16,color="magenta"];592[label="vuz50",fontsize=16,color="green",shape="box"];552[label="absReal1 (Pos (Succ vuz2300)) (not (primCmpNat (Succ vuz2300) Zero == LT))",fontsize=16,color="black",shape="box"];552 -> 571[label="",style="solid", color="black", weight=3];
553[label="absReal1 (Pos Zero) (not (EQ == LT))",fontsize=16,color="black",shape="box"];553 -> 572[label="",style="solid", color="black", weight=3];
554[label="absReal1 (Neg (Succ vuz2300)) (not (LT == LT))",fontsize=16,color="black",shape="box"];554 -> 573[label="",style="solid", color="black", weight=3];
555[label="absReal1 (Neg Zero) (not (EQ == LT))",fontsize=16,color="black",shape="box"];555 -> 574[label="",style="solid", color="black", weight=3];
891[label="primMinusNatS (Succ vuz690) vuz70",fontsize=16,color="burlywood",shape="box"];1166[label="vuz70/Succ vuz700",fontsize=10,color="white",style="solid",shape="box"];891 -> 1166[label="",style="solid", color="burlywood", weight=9];
1166 -> 901[label="",style="solid", color="burlywood", weight=3];
1167[label="vuz70/Zero",fontsize=10,color="white",style="solid",shape="box"];891 -> 1167[label="",style="solid", color="burlywood", weight=9];
1167 -> 902[label="",style="solid", color="burlywood", weight=3];
892[label="primMinusNatS Zero vuz70",fontsize=16,color="burlywood",shape="box"];1168[label="vuz70/Succ vuz700",fontsize=10,color="white",style="solid",shape="box"];892 -> 1168[label="",style="solid", color="burlywood", weight=9];
1168 -> 903[label="",style="solid", color="burlywood", weight=3];
1169[label="vuz70/Zero",fontsize=10,color="white",style="solid",shape="box"];892 -> 1169[label="",style="solid", color="burlywood", weight=9];
1169 -> 904[label="",style="solid", color="burlywood", weight=3];
500[label="primRemInt (Pos vuz440) (Pos (Succ vuz4300))",fontsize=16,color="black",shape="box"];500 -> 523[label="",style="solid", color="black", weight=3];
501[label="primRemInt (Pos vuz440) (Pos Zero)",fontsize=16,color="black",shape="box"];501 -> 524[label="",style="solid", color="black", weight=3];
502[label="primRemInt (Pos vuz440) (Neg (Succ vuz4300))",fontsize=16,color="black",shape="box"];502 -> 525[label="",style="solid", color="black", weight=3];
503[label="primRemInt (Pos vuz440) (Neg Zero)",fontsize=16,color="black",shape="box"];503 -> 526[label="",style="solid", color="black", weight=3];
504[label="primRemInt (Neg vuz440) (Pos (Succ vuz4300))",fontsize=16,color="black",shape="box"];504 -> 527[label="",style="solid", color="black", weight=3];
505[label="primRemInt (Neg vuz440) (Pos Zero)",fontsize=16,color="black",shape="box"];505 -> 528[label="",style="solid", color="black", weight=3];
506[label="primRemInt (Neg vuz440) (Neg (Succ vuz4300))",fontsize=16,color="black",shape="box"];506 -> 529[label="",style="solid", color="black", weight=3];
507[label="primRemInt (Neg vuz440) (Neg Zero)",fontsize=16,color="black",shape="box"];507 -> 530[label="",style="solid", color="black", weight=3];
607[label="absReal1 (Integer (Pos (Succ vuz20000))) (not (primCmpInt (Pos (Succ vuz20000)) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];607 -> 625[label="",style="solid", color="black", weight=3];
608[label="absReal1 (Integer (Pos Zero)) (not (primCmpInt (Pos Zero) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];608 -> 626[label="",style="solid", color="black", weight=3];
609[label="absReal1 (Integer (Neg (Succ vuz20000))) (not (primCmpInt (Neg (Succ vuz20000)) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];609 -> 627[label="",style="solid", color="black", weight=3];
610[label="absReal1 (Integer (Neg Zero)) (not (primCmpInt (Neg Zero) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];610 -> 628[label="",style="solid", color="black", weight=3];
611[label="Integer (primRemInt vuz510 vuz500)",fontsize=16,color="green",shape="box"];611 -> 629[label="",style="dashed", color="green", weight=3];
571[label="absReal1 (Pos (Succ vuz2300)) (not (GT == LT))",fontsize=16,color="black",shape="box"];571 -> 593[label="",style="solid", color="black", weight=3];
572[label="absReal1 (Pos Zero) (not False)",fontsize=16,color="black",shape="box"];572 -> 594[label="",style="solid", color="black", weight=3];
573[label="absReal1 (Neg (Succ vuz2300)) (not True)",fontsize=16,color="black",shape="box"];573 -> 595[label="",style="solid", color="black", weight=3];
574[label="absReal1 (Neg Zero) (not False)",fontsize=16,color="black",shape="box"];574 -> 596[label="",style="solid", color="black", weight=3];
901[label="primMinusNatS (Succ vuz690) (Succ vuz700)",fontsize=16,color="black",shape="box"];901 -> 913[label="",style="solid", color="black", weight=3];
902[label="primMinusNatS (Succ vuz690) Zero",fontsize=16,color="black",shape="box"];902 -> 914[label="",style="solid", color="black", weight=3];
903[label="primMinusNatS Zero (Succ vuz700)",fontsize=16,color="black",shape="box"];903 -> 915[label="",style="solid", color="black", weight=3];
904[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];904 -> 916[label="",style="solid", color="black", weight=3];
523[label="Pos (primModNatS vuz440 (Succ vuz4300))",fontsize=16,color="green",shape="box"];523 -> 556[label="",style="dashed", color="green", weight=3];
524 -> 356[label="",style="dashed", color="red", weight=0];
524[label="error []",fontsize=16,color="magenta"];525[label="Pos (primModNatS vuz440 (Succ vuz4300))",fontsize=16,color="green",shape="box"];525 -> 557[label="",style="dashed", color="green", weight=3];
526 -> 356[label="",style="dashed", color="red", weight=0];
526[label="error []",fontsize=16,color="magenta"];527[label="Neg (primModNatS vuz440 (Succ vuz4300))",fontsize=16,color="green",shape="box"];527 -> 558[label="",style="dashed", color="green", weight=3];
528 -> 356[label="",style="dashed", color="red", weight=0];
528[label="error []",fontsize=16,color="magenta"];529[label="Neg (primModNatS vuz440 (Succ vuz4300))",fontsize=16,color="green",shape="box"];529 -> 559[label="",style="dashed", color="green", weight=3];
530 -> 356[label="",style="dashed", color="red", weight=0];
530[label="error []",fontsize=16,color="magenta"];625[label="absReal1 (Integer (Pos (Succ vuz20000))) (not (primCmpNat (Succ vuz20000) Zero == LT))",fontsize=16,color="black",shape="box"];625 -> 643[label="",style="solid", color="black", weight=3];
626[label="absReal1 (Integer (Pos Zero)) (not (EQ == LT))",fontsize=16,color="black",shape="box"];626 -> 644[label="",style="solid", color="black", weight=3];
627[label="absReal1 (Integer (Neg (Succ vuz20000))) (not (LT == LT))",fontsize=16,color="black",shape="box"];627 -> 645[label="",style="solid", color="black", weight=3];
628[label="absReal1 (Integer (Neg Zero)) (not (EQ == LT))",fontsize=16,color="black",shape="box"];628 -> 646[label="",style="solid", color="black", weight=3];
629 -> 442[label="",style="dashed", color="red", weight=0];
629[label="primRemInt vuz510 vuz500",fontsize=16,color="magenta"];629 -> 647[label="",style="dashed", color="magenta", weight=3];
629 -> 648[label="",style="dashed", color="magenta", weight=3];
593[label="absReal1 (Pos (Succ vuz2300)) (not False)",fontsize=16,color="black",shape="box"];593 -> 612[label="",style="solid", color="black", weight=3];
594[label="absReal1 (Pos Zero) True",fontsize=16,color="black",shape="box"];594 -> 613[label="",style="solid", color="black", weight=3];
595[label="absReal1 (Neg (Succ vuz2300)) False",fontsize=16,color="black",shape="box"];595 -> 614[label="",style="solid", color="black", weight=3];
596[label="absReal1 (Neg Zero) True",fontsize=16,color="black",shape="box"];596 -> 615[label="",style="solid", color="black", weight=3];
913 -> 881[label="",style="dashed", color="red", weight=0];
913[label="primMinusNatS vuz690 vuz700",fontsize=16,color="magenta"];913 -> 921[label="",style="dashed", color="magenta", weight=3];
913 -> 922[label="",style="dashed", color="magenta", weight=3];
914[label="Succ vuz690",fontsize=16,color="green",shape="box"];915[label="Zero",fontsize=16,color="green",shape="box"];916[label="Zero",fontsize=16,color="green",shape="box"];556[label="primModNatS vuz440 (Succ vuz4300)",fontsize=16,color="burlywood",shape="triangle"];1170[label="vuz440/Succ vuz4400",fontsize=10,color="white",style="solid",shape="box"];556 -> 1170[label="",style="solid", color="burlywood", weight=9];
1170 -> 575[label="",style="solid", color="burlywood", weight=3];
1171[label="vuz440/Zero",fontsize=10,color="white",style="solid",shape="box"];556 -> 1171[label="",style="solid", color="burlywood", weight=9];
1171 -> 576[label="",style="solid", color="burlywood", weight=3];
557 -> 556[label="",style="dashed", color="red", weight=0];
557[label="primModNatS vuz440 (Succ vuz4300)",fontsize=16,color="magenta"];557 -> 577[label="",style="dashed", color="magenta", weight=3];
558 -> 556[label="",style="dashed", color="red", weight=0];
558[label="primModNatS vuz440 (Succ vuz4300)",fontsize=16,color="magenta"];558 -> 578[label="",style="dashed", color="magenta", weight=3];
559 -> 556[label="",style="dashed", color="red", weight=0];
559[label="primModNatS vuz440 (Succ vuz4300)",fontsize=16,color="magenta"];559 -> 579[label="",style="dashed", color="magenta", weight=3];
559 -> 580[label="",style="dashed", color="magenta", weight=3];
643[label="absReal1 (Integer (Pos (Succ vuz20000))) (not (GT == LT))",fontsize=16,color="black",shape="box"];643 -> 662[label="",style="solid", color="black", weight=3];
644[label="absReal1 (Integer (Pos Zero)) (not False)",fontsize=16,color="black",shape="box"];644 -> 663[label="",style="solid", color="black", weight=3];
645[label="absReal1 (Integer (Neg (Succ vuz20000))) (not True)",fontsize=16,color="black",shape="box"];645 -> 664[label="",style="solid", color="black", weight=3];
646[label="absReal1 (Integer (Neg Zero)) (not False)",fontsize=16,color="black",shape="box"];646 -> 665[label="",style="solid", color="black", weight=3];
647[label="vuz510",fontsize=16,color="green",shape="box"];648[label="vuz500",fontsize=16,color="green",shape="box"];612[label="absReal1 (Pos (Succ vuz2300)) True",fontsize=16,color="black",shape="box"];612 -> 630[label="",style="solid", color="black", weight=3];
613[label="Pos Zero",fontsize=16,color="green",shape="box"];614[label="absReal0 (Neg (Succ vuz2300)) otherwise",fontsize=16,color="black",shape="box"];614 -> 631[label="",style="solid", color="black", weight=3];
615[label="Neg Zero",fontsize=16,color="green",shape="box"];921[label="vuz700",fontsize=16,color="green",shape="box"];922[label="vuz690",fontsize=16,color="green",shape="box"];575[label="primModNatS (Succ vuz4400) (Succ vuz4300)",fontsize=16,color="black",shape="box"];575 -> 597[label="",style="solid", color="black", weight=3];
576[label="primModNatS Zero (Succ vuz4300)",fontsize=16,color="black",shape="box"];576 -> 598[label="",style="solid", color="black", weight=3];
577[label="vuz4300",fontsize=16,color="green",shape="box"];578[label="vuz440",fontsize=16,color="green",shape="box"];579[label="vuz4300",fontsize=16,color="green",shape="box"];580[label="vuz440",fontsize=16,color="green",shape="box"];662[label="absReal1 (Integer (Pos (Succ vuz20000))) (not False)",fontsize=16,color="black",shape="box"];662 -> 679[label="",style="solid", color="black", weight=3];
663[label="absReal1 (Integer (Pos Zero)) True",fontsize=16,color="black",shape="box"];663 -> 680[label="",style="solid", color="black", weight=3];
664[label="absReal1 (Integer (Neg (Succ vuz20000))) False",fontsize=16,color="black",shape="box"];664 -> 681[label="",style="solid", color="black", weight=3];
665[label="absReal1 (Integer (Neg Zero)) True",fontsize=16,color="black",shape="box"];665 -> 682[label="",style="solid", color="black", weight=3];
630[label="Pos (Succ vuz2300)",fontsize=16,color="green",shape="box"];631[label="absReal0 (Neg (Succ vuz2300)) True",fontsize=16,color="black",shape="box"];631 -> 649[label="",style="solid", color="black", weight=3];
597[label="primModNatS0 vuz4400 vuz4300 (primGEqNatS vuz4400 vuz4300)",fontsize=16,color="burlywood",shape="box"];1172[label="vuz4400/Succ vuz44000",fontsize=10,color="white",style="solid",shape="box"];597 -> 1172[label="",style="solid", color="burlywood", weight=9];
1172 -> 616[label="",style="solid", color="burlywood", weight=3];
1173[label="vuz4400/Zero",fontsize=10,color="white",style="solid",shape="box"];597 -> 1173[label="",style="solid", color="burlywood", weight=9];
1173 -> 617[label="",style="solid", color="burlywood", weight=3];
598[label="Zero",fontsize=16,color="green",shape="box"];679[label="absReal1 (Integer (Pos (Succ vuz20000))) True",fontsize=16,color="black",shape="box"];679 -> 699[label="",style="solid", color="black", weight=3];
680[label="Integer (Pos Zero)",fontsize=16,color="green",shape="box"];681[label="absReal0 (Integer (Neg (Succ vuz20000))) otherwise",fontsize=16,color="black",shape="box"];681 -> 700[label="",style="solid", color="black", weight=3];
682[label="Integer (Neg Zero)",fontsize=16,color="green",shape="box"];649[label="`negate` Neg (Succ vuz2300)",fontsize=16,color="black",shape="box"];649 -> 666[label="",style="solid", color="black", weight=3];
616[label="primModNatS0 (Succ vuz44000) vuz4300 (primGEqNatS (Succ vuz44000) vuz4300)",fontsize=16,color="burlywood",shape="box"];1174[label="vuz4300/Succ vuz43000",fontsize=10,color="white",style="solid",shape="box"];616 -> 1174[label="",style="solid", color="burlywood", weight=9];
1174 -> 632[label="",style="solid", color="burlywood", weight=3];
1175[label="vuz4300/Zero",fontsize=10,color="white",style="solid",shape="box"];616 -> 1175[label="",style="solid", color="burlywood", weight=9];
1175 -> 633[label="",style="solid", color="burlywood", weight=3];
617[label="primModNatS0 Zero vuz4300 (primGEqNatS Zero vuz4300)",fontsize=16,color="burlywood",shape="box"];1176[label="vuz4300/Succ vuz43000",fontsize=10,color="white",style="solid",shape="box"];617 -> 1176[label="",style="solid", color="burlywood", weight=9];
1176 -> 634[label="",style="solid", color="burlywood", weight=3];
1177[label="vuz4300/Zero",fontsize=10,color="white",style="solid",shape="box"];617 -> 1177[label="",style="solid", color="burlywood", weight=9];
1177 -> 635[label="",style="solid", color="burlywood", weight=3];
699[label="Integer (Pos (Succ vuz20000))",fontsize=16,color="green",shape="box"];700[label="absReal0 (Integer (Neg (Succ vuz20000))) True",fontsize=16,color="black",shape="box"];700 -> 714[label="",style="solid", color="black", weight=3];
666[label="primNegInt (Neg (Succ vuz2300))",fontsize=16,color="black",shape="triangle"];666 -> 683[label="",style="solid", color="black", weight=3];
632[label="primModNatS0 (Succ vuz44000) (Succ vuz43000) (primGEqNatS (Succ vuz44000) (Succ vuz43000))",fontsize=16,color="black",shape="box"];632 -> 650[label="",style="solid", color="black", weight=3];
633[label="primModNatS0 (Succ vuz44000) Zero (primGEqNatS (Succ vuz44000) Zero)",fontsize=16,color="black",shape="box"];633 -> 651[label="",style="solid", color="black", weight=3];
634[label="primModNatS0 Zero (Succ vuz43000) (primGEqNatS Zero (Succ vuz43000))",fontsize=16,color="black",shape="box"];634 -> 652[label="",style="solid", color="black", weight=3];
635[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];635 -> 653[label="",style="solid", color="black", weight=3];
714[label="`negate` Integer (Neg (Succ vuz20000))",fontsize=16,color="black",shape="box"];714 -> 734[label="",style="solid", color="black", weight=3];
683[label="Pos (Succ vuz2300)",fontsize=16,color="green",shape="box"];650 -> 972[label="",style="dashed", color="red", weight=0];
650[label="primModNatS0 (Succ vuz44000) (Succ vuz43000) (primGEqNatS vuz44000 vuz43000)",fontsize=16,color="magenta"];650 -> 973[label="",style="dashed", color="magenta", weight=3];
650 -> 974[label="",style="dashed", color="magenta", weight=3];
650 -> 975[label="",style="dashed", color="magenta", weight=3];
650 -> 976[label="",style="dashed", color="magenta", weight=3];
651[label="primModNatS0 (Succ vuz44000) Zero True",fontsize=16,color="black",shape="box"];651 -> 669[label="",style="solid", color="black", weight=3];
652[label="primModNatS0 Zero (Succ vuz43000) False",fontsize=16,color="black",shape="box"];652 -> 670[label="",style="solid", color="black", weight=3];
653[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];653 -> 671[label="",style="solid", color="black", weight=3];
734[label="Integer (primNegInt (Neg (Succ vuz20000)))",fontsize=16,color="green",shape="box"];734 -> 764[label="",style="dashed", color="green", weight=3];
973[label="vuz44000",fontsize=16,color="green",shape="box"];974[label="vuz43000",fontsize=16,color="green",shape="box"];975[label="vuz44000",fontsize=16,color="green",shape="box"];976[label="vuz43000",fontsize=16,color="green",shape="box"];972[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS vuz91 vuz92)",fontsize=16,color="burlywood",shape="triangle"];1178[label="vuz91/Succ vuz910",fontsize=10,color="white",style="solid",shape="box"];972 -> 1178[label="",style="solid", color="burlywood", weight=9];
1178 -> 1005[label="",style="solid", color="burlywood", weight=3];
1179[label="vuz91/Zero",fontsize=10,color="white",style="solid",shape="box"];972 -> 1179[label="",style="solid", color="burlywood", weight=9];
1179 -> 1006[label="",style="solid", color="burlywood", weight=3];
669 -> 556[label="",style="dashed", color="red", weight=0];
669[label="primModNatS (primMinusNatS (Succ vuz44000) Zero) (Succ Zero)",fontsize=16,color="magenta"];669 -> 688[label="",style="dashed", color="magenta", weight=3];
669 -> 689[label="",style="dashed", color="magenta", weight=3];
670[label="Succ Zero",fontsize=16,color="green",shape="box"];671 -> 556[label="",style="dashed", color="red", weight=0];
671[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];671 -> 690[label="",style="dashed", color="magenta", weight=3];
671 -> 691[label="",style="dashed", color="magenta", weight=3];
764 -> 666[label="",style="dashed", color="red", weight=0];
764[label="primNegInt (Neg (Succ vuz20000))",fontsize=16,color="magenta"];764 -> 778[label="",style="dashed", color="magenta", weight=3];
1005[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS (Succ vuz910) vuz92)",fontsize=16,color="burlywood",shape="box"];1180[label="vuz92/Succ vuz920",fontsize=10,color="white",style="solid",shape="box"];1005 -> 1180[label="",style="solid", color="burlywood", weight=9];
1180 -> 1007[label="",style="solid", color="burlywood", weight=3];
1181[label="vuz92/Zero",fontsize=10,color="white",style="solid",shape="box"];1005 -> 1181[label="",style="solid", color="burlywood", weight=9];
1181 -> 1008[label="",style="solid", color="burlywood", weight=3];
1006[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS Zero vuz92)",fontsize=16,color="burlywood",shape="box"];1182[label="vuz92/Succ vuz920",fontsize=10,color="white",style="solid",shape="box"];1006 -> 1182[label="",style="solid", color="burlywood", weight=9];
1182 -> 1009[label="",style="solid", color="burlywood", weight=3];
1183[label="vuz92/Zero",fontsize=10,color="white",style="solid",shape="box"];1006 -> 1183[label="",style="solid", color="burlywood", weight=9];
1183 -> 1010[label="",style="solid", color="burlywood", weight=3];
688[label="Zero",fontsize=16,color="green",shape="box"];689 -> 490[label="",style="dashed", color="red", weight=0];
689[label="primMinusNatS (Succ vuz44000) Zero",fontsize=16,color="magenta"];689 -> 705[label="",style="dashed", color="magenta", weight=3];
690[label="Zero",fontsize=16,color="green",shape="box"];691 -> 492[label="",style="dashed", color="red", weight=0];
691[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];778[label="vuz20000",fontsize=16,color="green",shape="box"];1007[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS (Succ vuz910) (Succ vuz920))",fontsize=16,color="black",shape="box"];1007 -> 1011[label="",style="solid", color="black", weight=3];
1008[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS (Succ vuz910) Zero)",fontsize=16,color="black",shape="box"];1008 -> 1012[label="",style="solid", color="black", weight=3];
1009[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS Zero (Succ vuz920))",fontsize=16,color="black",shape="box"];1009 -> 1013[label="",style="solid", color="black", weight=3];
1010[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];1010 -> 1014[label="",style="solid", color="black", weight=3];
705[label="vuz44000",fontsize=16,color="green",shape="box"];1011 -> 972[label="",style="dashed", color="red", weight=0];
1011[label="primModNatS0 (Succ vuz89) (Succ vuz90) (primGEqNatS vuz910 vuz920)",fontsize=16,color="magenta"];1011 -> 1015[label="",style="dashed", color="magenta", weight=3];
1011 -> 1016[label="",style="dashed", color="magenta", weight=3];
1012[label="primModNatS0 (Succ vuz89) (Succ vuz90) True",fontsize=16,color="black",shape="triangle"];1012 -> 1017[label="",style="solid", color="black", weight=3];
1013[label="primModNatS0 (Succ vuz89) (Succ vuz90) False",fontsize=16,color="black",shape="box"];1013 -> 1018[label="",style="solid", color="black", weight=3];
1014 -> 1012[label="",style="dashed", color="red", weight=0];
1014[label="primModNatS0 (Succ vuz89) (Succ vuz90) True",fontsize=16,color="magenta"];1015[label="vuz910",fontsize=16,color="green",shape="box"];1016[label="vuz920",fontsize=16,color="green",shape="box"];1017 -> 556[label="",style="dashed", color="red", weight=0];
1017[label="primModNatS (primMinusNatS (Succ vuz89) (Succ vuz90)) (Succ (Succ vuz90))",fontsize=16,color="magenta"];1017 -> 1019[label="",style="dashed", color="magenta", weight=3];
1017 -> 1020[label="",style="dashed", color="magenta", weight=3];
1018[label="Succ (Succ vuz89)",fontsize=16,color="green",shape="box"];1019[label="Succ vuz90",fontsize=16,color="green",shape="box"];1020 -> 881[label="",style="dashed", color="red", weight=0];
1020[label="primMinusNatS (Succ vuz89) (Succ vuz90)",fontsize=16,color="magenta"];1020 -> 1021[label="",style="dashed", color="magenta", weight=3];
1020 -> 1022[label="",style="dashed", color="magenta", weight=3];
1021[label="Succ vuz90",fontsize=16,color="green",shape="box"];1022[label="Succ vuz89",fontsize=16,color="green",shape="box"];}

----------------------------------------

(732)
Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(733) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(734)
YES