79.33/55.23 YES 81.56/55.83 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 81.56/55.83 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 81.56/55.83 81.56/55.83 81.56/55.83 H-Termination with start terms of the given HASKELL could be proven: 81.56/55.83 81.56/55.83 (0) HASKELL 81.56/55.83 (1) LR [EQUIVALENT, 0 ms] 81.56/55.83 (2) HASKELL 81.56/55.83 (3) IFR [EQUIVALENT, 0 ms] 81.56/55.83 (4) HASKELL 81.56/55.83 (5) BR [EQUIVALENT, 0 ms] 81.56/55.83 (6) HASKELL 81.56/55.83 (7) COR [EQUIVALENT, 24 ms] 81.56/55.83 (8) HASKELL 81.56/55.83 (9) LetRed [EQUIVALENT, 0 ms] 81.56/55.83 (10) HASKELL 81.56/55.83 (11) NumRed [SOUND, 0 ms] 81.56/55.83 (12) HASKELL 81.56/55.83 (13) Narrow [SOUND, 0 ms] 81.56/55.83 (14) AND 81.56/55.83 (15) QDP 81.56/55.83 (16) DependencyGraphProof [EQUIVALENT, 0 ms] 81.56/55.83 (17) QDP 81.56/55.83 (18) QDPOrderProof [EQUIVALENT, 21 ms] 81.56/55.83 (19) QDP 81.56/55.83 (20) DependencyGraphProof [EQUIVALENT, 0 ms] 81.56/55.83 (21) QDP 81.56/55.83 (22) QDPSizeChangeProof [EQUIVALENT, 0 ms] 81.56/55.83 (23) YES 81.56/55.83 (24) QDP 81.56/55.83 (25) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (26) QDP 81.56/55.83 (27) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (28) QDP 81.56/55.83 (29) UsableRulesProof [EQUIVALENT, 0 ms] 81.56/55.83 (30) QDP 81.56/55.83 (31) QReductionProof [EQUIVALENT, 0 ms] 81.56/55.83 (32) QDP 81.56/55.83 (33) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (34) QDP 81.56/55.83 (35) DependencyGraphProof [EQUIVALENT, 0 ms] 81.56/55.83 (36) QDP 81.56/55.83 (37) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (38) QDP 81.56/55.83 (39) DependencyGraphProof [EQUIVALENT, 0 ms] 81.56/55.83 (40) QDP 81.56/55.83 (41) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (42) QDP 81.56/55.83 (43) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (44) QDP 81.56/55.83 (45) DependencyGraphProof [EQUIVALENT, 0 ms] 81.56/55.83 (46) TRUE 81.56/55.83 (47) QDP 81.56/55.83 (48) QDPSizeChangeProof [EQUIVALENT, 0 ms] 81.56/55.83 (49) YES 81.56/55.83 (50) QDP 81.56/55.83 (51) DependencyGraphProof [EQUIVALENT, 0 ms] 81.56/55.83 (52) QDP 81.56/55.83 (53) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (54) QDP 81.56/55.83 (55) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (56) QDP 81.56/55.83 (57) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (58) QDP 81.56/55.83 (59) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (60) QDP 81.56/55.83 (61) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (62) QDP 81.56/55.83 (63) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (64) QDP 81.56/55.83 (65) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (66) QDP 81.56/55.83 (67) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (68) QDP 81.56/55.83 (69) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (70) QDP 81.56/55.83 (71) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (72) QDP 81.56/55.83 (73) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (74) QDP 81.56/55.83 (75) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (76) QDP 81.56/55.83 (77) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (78) QDP 81.56/55.83 (79) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (80) QDP 81.56/55.83 (81) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (82) QDP 81.56/55.83 (83) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (84) QDP 81.56/55.83 (85) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (86) QDP 81.56/55.83 (87) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (88) QDP 81.56/55.83 (89) UsableRulesProof [EQUIVALENT, 0 ms] 81.56/55.83 (90) QDP 81.56/55.83 (91) QReductionProof [EQUIVALENT, 0 ms] 81.56/55.83 (92) QDP 81.56/55.83 (93) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (94) QDP 81.56/55.83 (95) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (96) QDP 81.56/55.83 (97) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (98) QDP 81.56/55.83 (99) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (100) QDP 81.56/55.83 (101) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (102) QDP 81.56/55.83 (103) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (104) QDP 81.56/55.83 (105) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (106) QDP 81.56/55.83 (107) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (108) QDP 81.56/55.83 (109) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (110) QDP 81.56/55.83 (111) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (112) QDP 81.56/55.83 (113) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (114) QDP 81.56/55.83 (115) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (116) QDP 81.56/55.83 (117) DependencyGraphProof [EQUIVALENT, 0 ms] 81.56/55.83 (118) AND 81.56/55.83 (119) QDP 81.56/55.83 (120) UsableRulesProof [EQUIVALENT, 0 ms] 81.56/55.83 (121) QDP 81.56/55.83 (122) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (123) QDP 81.56/55.83 (124) UsableRulesProof [EQUIVALENT, 0 ms] 81.56/55.83 (125) QDP 81.56/55.83 (126) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (127) QDP 81.56/55.83 (128) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (129) QDP 81.56/55.83 (130) UsableRulesProof [EQUIVALENT, 0 ms] 81.56/55.83 (131) QDP 81.56/55.83 (132) QReductionProof [EQUIVALENT, 0 ms] 81.56/55.83 (133) QDP 81.56/55.83 (134) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (135) QDP 81.56/55.83 (136) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (137) QDP 81.56/55.83 (138) DependencyGraphProof [EQUIVALENT, 0 ms] 81.56/55.83 (139) TRUE 81.56/55.83 (140) QDP 81.56/55.83 (141) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (142) QDP 81.56/55.83 (143) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (144) QDP 81.56/55.83 (145) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (146) QDP 81.56/55.83 (147) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (148) QDP 81.56/55.83 (149) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (150) QDP 81.56/55.83 (151) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (152) QDP 81.56/55.83 (153) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (154) QDP 81.56/55.83 (155) DependencyGraphProof [EQUIVALENT, 0 ms] 81.56/55.83 (156) AND 81.56/55.83 (157) QDP 81.56/55.83 (158) UsableRulesProof [EQUIVALENT, 0 ms] 81.56/55.83 (159) QDP 81.56/55.83 (160) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (161) QDP 81.56/55.83 (162) UsableRulesProof [EQUIVALENT, 0 ms] 81.56/55.83 (163) QDP 81.56/55.83 (164) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (165) QDP 81.56/55.83 (166) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (167) QDP 81.56/55.83 (168) UsableRulesProof [EQUIVALENT, 0 ms] 81.56/55.83 (169) QDP 81.56/55.83 (170) QReductionProof [EQUIVALENT, 0 ms] 81.56/55.83 (171) QDP 81.56/55.83 (172) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (173) QDP 81.56/55.83 (174) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (175) QDP 81.56/55.83 (176) DependencyGraphProof [EQUIVALENT, 0 ms] 81.56/55.83 (177) TRUE 81.56/55.83 (178) QDP 81.56/55.83 (179) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (180) QDP 81.56/55.83 (181) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (182) QDP 81.56/55.83 (183) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (184) QDP 81.56/55.83 (185) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (186) QDP 81.56/55.83 (187) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (188) QDP 81.56/55.83 (189) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (190) QDP 81.56/55.83 (191) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (192) QDP 81.56/55.83 (193) DependencyGraphProof [EQUIVALENT, 0 ms] 81.56/55.83 (194) QDP 81.56/55.83 (195) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (196) QDP 81.56/55.83 (197) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (198) QDP 81.56/55.83 (199) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (200) QDP 81.56/55.83 (201) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (202) QDP 81.56/55.83 (203) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (204) QDP 81.56/55.83 (205) TransformationProof [EQUIVALENT, 0 ms] 81.56/55.83 (206) QDP 81.56/55.83 (207) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (208) QDP 82.06/55.93 (209) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (210) QDP 82.06/55.93 (211) QDPOrderProof [EQUIVALENT, 46 ms] 82.06/55.93 (212) QDP 82.06/55.93 (213) QDPOrderProof [EQUIVALENT, 0 ms] 82.06/55.93 (214) QDP 82.06/55.93 (215) DependencyGraphProof [EQUIVALENT, 0 ms] 82.06/55.93 (216) AND 82.06/55.93 (217) QDP 82.06/55.93 (218) QDPOrderProof [EQUIVALENT, 0 ms] 82.06/55.93 (219) QDP 82.06/55.93 (220) QDPPairToRuleProof [EQUIVALENT, 0 ms] 82.06/55.93 (221) AND 82.06/55.93 (222) QDP 82.06/55.93 (223) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (224) QDP 82.06/55.93 (225) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (226) QDP 82.06/55.93 (227) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (228) QDP 82.06/55.93 (229) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (230) QDP 82.06/55.93 (231) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (232) QDP 82.06/55.93 (233) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (234) QDP 82.06/55.93 (235) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (236) QDP 82.06/55.93 (237) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (238) QDP 82.06/55.93 (239) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (240) QDP 82.06/55.93 (241) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (242) QDP 82.06/55.93 (243) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (244) QDP 82.06/55.93 (245) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (246) QDP 82.06/55.93 (247) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (248) QDP 82.06/55.93 (249) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (250) QDP 82.06/55.93 (251) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (252) QDP 82.06/55.93 (253) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (254) QDP 82.06/55.93 (255) UsableRulesProof [EQUIVALENT, 0 ms] 82.06/55.93 (256) QDP 82.06/55.93 (257) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (258) QDP 82.06/55.93 (259) InductionCalculusProof [EQUIVALENT, 0 ms] 82.06/55.93 (260) QDP 82.06/55.93 (261) NonInfProof [EQUIVALENT, 3434 ms] 82.06/55.93 (262) AND 82.06/55.93 (263) QDP 82.06/55.93 (264) DependencyGraphProof [EQUIVALENT, 0 ms] 82.06/55.93 (265) TRUE 82.06/55.93 (266) QDP 82.06/55.93 (267) InductionCalculusProof [EQUIVALENT, 0 ms] 82.06/55.93 (268) QDP 82.06/55.93 (269) NonInfProof [EQUIVALENT, 16.6 s] 82.06/55.93 (270) AND 82.06/55.93 (271) QDP 82.06/55.93 (272) DependencyGraphProof [EQUIVALENT, 0 ms] 82.06/55.93 (273) TRUE 82.06/55.93 (274) QDP 82.06/55.93 (275) DependencyGraphProof [EQUIVALENT, 0 ms] 82.06/55.93 (276) TRUE 82.06/55.93 (277) QDP 82.06/55.93 (278) QDPSizeChangeProof [EQUIVALENT, 0 ms] 82.06/55.93 (279) YES 82.06/55.93 (280) QDP 82.06/55.93 (281) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (282) QDP 82.06/55.93 (283) UsableRulesProof [EQUIVALENT, 0 ms] 82.06/55.93 (284) QDP 82.06/55.93 (285) QReductionProof [EQUIVALENT, 3 ms] 82.06/55.93 (286) QDP 82.06/55.93 (287) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (288) QDP 82.06/55.93 (289) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (290) QDP 82.06/55.93 (291) TransformationProof [EQUIVALENT, 0 ms] 82.06/55.93 (292) QDP 82.06/55.93 (293) QDPOrderProof [EQUIVALENT, 0 ms] 82.06/55.93 (294) QDP 82.06/55.93 (295) QDPSizeChangeProof [EQUIVALENT, 0 ms] 82.06/55.93 (296) YES 82.06/55.93 (297) QDP 82.06/55.93 (298) QDPSizeChangeProof [EQUIVALENT, 0 ms] 82.06/55.93 (299) YES 82.06/55.93 (300) QDP 82.06/55.93 (301) QDPSizeChangeProof [EQUIVALENT, 0 ms] 82.06/55.93 (302) YES 82.06/55.93 82.06/55.93 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (0) 82.06/55.93 Obligation: 82.06/55.93 mainModule Main 82.06/55.93 module Main where { 82.06/55.93 import qualified Prelude; 82.06/55.93 } 82.06/55.93 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (1) LR (EQUIVALENT) 82.06/55.93 Lambda Reductions: 82.06/55.93 The following Lambda expression 82.06/55.93 "\(m,_)->m" 82.06/55.93 is transformed to 82.06/55.93 "m0 (m,_) = m; 82.06/55.93 " 82.06/55.93 The following Lambda expression 82.06/55.93 "\(q,_)->q" 82.06/55.93 is transformed to 82.06/55.93 "q1 (q,_) = q; 82.06/55.93 " 82.06/55.93 The following Lambda expression 82.06/55.93 "\(_,r)->r" 82.06/55.93 is transformed to 82.06/55.93 "r0 (_,r) = r; 82.06/55.93 " 82.06/55.93 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (2) 82.06/55.93 Obligation: 82.06/55.93 mainModule Main 82.06/55.93 module Main where { 82.06/55.93 import qualified Prelude; 82.06/55.93 } 82.06/55.93 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (3) IFR (EQUIVALENT) 82.06/55.93 If Reductions: 82.06/55.93 The following If expression 82.06/55.93 "if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero" 82.06/55.93 is transformed to 82.06/55.93 "primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y)); 82.06/55.93 primDivNatS0 x y False = Zero; 82.06/55.93 " 82.06/55.93 The following If expression 82.06/55.93 "if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x" 82.06/55.93 is transformed to 82.06/55.93 "primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y); 82.06/55.93 primModNatS0 x y False = Succ x; 82.06/55.93 " 82.06/55.93 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (4) 82.06/55.93 Obligation: 82.06/55.93 mainModule Main 82.06/55.93 module Main where { 82.06/55.93 import qualified Prelude; 82.06/55.93 } 82.06/55.93 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (5) BR (EQUIVALENT) 82.06/55.93 Replaced joker patterns by fresh variables and removed binding patterns. 82.06/55.93 82.06/55.93 Binding Reductions: 82.06/55.93 The bind variable of the following binding Pattern 82.06/55.93 "frac@(Float vz wu)" 82.06/55.93 is replaced by the following term 82.06/55.93 "Float vz wu" 82.06/55.93 The bind variable of the following binding Pattern 82.06/55.93 "frac@(Double xu xv)" 82.06/55.93 is replaced by the following term 82.06/55.93 "Double xu xv" 82.06/55.93 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (6) 82.06/55.93 Obligation: 82.06/55.93 mainModule Main 82.06/55.93 module Main where { 82.06/55.93 import qualified Prelude; 82.06/55.93 } 82.06/55.93 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (7) COR (EQUIVALENT) 82.06/55.93 Cond Reductions: 82.06/55.93 The following Function with conditions 82.06/55.93 "takeWhile p [] = []; 82.06/55.93 takeWhile p (x : xs)|p xx : takeWhile p xs|otherwise[]; 82.06/55.93 " 82.06/55.93 is transformed to 82.06/55.93 "takeWhile p [] = takeWhile3 p []; 82.06/55.93 takeWhile p (x : xs) = takeWhile2 p (x : xs); 82.06/55.93 " 82.06/55.93 "takeWhile1 p x xs True = x : takeWhile p xs; 82.06/55.93 takeWhile1 p x xs False = takeWhile0 p x xs otherwise; 82.06/55.93 " 82.06/55.93 "takeWhile0 p x xs True = []; 82.06/55.93 " 82.06/55.93 "takeWhile2 p (x : xs) = takeWhile1 p x xs (p x); 82.06/55.93 " 82.06/55.93 "takeWhile3 p [] = []; 82.06/55.93 takeWhile3 xy xz = takeWhile2 xy xz; 82.06/55.93 " 82.06/55.93 The following Function with conditions 82.06/55.93 "undefined |Falseundefined; 82.06/55.93 " 82.06/55.93 is transformed to 82.06/55.93 "undefined = undefined1; 82.06/55.93 " 82.06/55.93 "undefined0 True = undefined; 82.06/55.93 " 82.06/55.93 "undefined1 = undefined0 False; 82.06/55.93 " 82.06/55.93 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (8) 82.06/55.93 Obligation: 82.06/55.93 mainModule Main 82.06/55.93 module Main where { 82.06/55.93 import qualified Prelude; 82.06/55.93 } 82.06/55.93 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (9) LetRed (EQUIVALENT) 82.06/55.93 Let/Where Reductions: 82.06/55.93 The bindings of the following Let/Where expression 82.06/55.93 "m where { 82.06/55.93 m = m0 vu6; 82.06/55.93 ; 82.06/55.93 m0 (m,vv) = m; 82.06/55.93 ; 82.06/55.93 vu6 = properFraction x; 82.06/55.93 } 82.06/55.93 " 82.06/55.93 are unpacked to the following functions on top level 82.06/55.93 "truncateM yu = truncateM0 yu (truncateVu6 yu); 82.06/55.93 " 82.06/55.93 "truncateM0 yu (m,vv) = m; 82.06/55.93 " 82.06/55.93 "truncateVu6 yu = properFraction yu; 82.06/55.93 " 82.06/55.93 The bindings of the following Let/Where expression 82.06/55.93 "(fromIntegral q,r :% y) where { 82.06/55.93 q = q1 vu30; 82.06/55.93 ; 82.06/55.93 q1 (q,vw) = q; 82.06/55.93 ; 82.06/55.93 r = r0 vu30; 82.06/55.93 ; 82.06/55.93 r0 (vx,r) = r; 82.06/55.93 ; 82.06/55.93 vu30 = quotRem x y; 82.06/55.93 } 82.06/55.93 " 82.06/55.93 are unpacked to the following functions on top level 82.06/55.93 "properFractionQ yv yw = properFractionQ1 yv yw (properFractionVu30 yv yw); 82.06/55.93 " 82.06/55.93 "properFractionR0 yv yw (vx,r) = r; 82.06/55.93 " 82.06/55.93 "properFractionQ1 yv yw (q,vw) = q; 82.06/55.93 " 82.06/55.93 "properFractionR yv yw = properFractionR0 yv yw (properFractionVu30 yv yw); 82.06/55.93 " 82.06/55.93 "properFractionVu30 yv yw = quotRem yv yw; 82.06/55.93 " 82.06/55.93 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (10) 82.06/55.93 Obligation: 82.06/55.93 mainModule Main 82.06/55.93 module Main where { 82.06/55.93 import qualified Prelude; 82.06/55.93 } 82.06/55.93 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (11) NumRed (SOUND) 82.06/55.93 Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero. 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (12) 82.06/55.93 Obligation: 82.06/55.93 mainModule Main 82.06/55.93 module Main where { 82.06/55.93 import qualified Prelude; 82.06/55.93 } 82.06/55.93 82.06/55.93 ---------------------------------------- 82.06/55.93 82.06/55.93 (13) Narrow (SOUND) 82.06/55.93 Haskell To QDPs 82.06/55.93 82.06/55.93 digraph dp_graph { 82.06/55.93 node [outthreshold=100, inthreshold=100];1[label="enumFromTo",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 82.06/55.93 3[label="enumFromTo yx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 82.06/55.93 4[label="enumFromTo yx3 yx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 82.06/55.93 5[label="map toEnum (enumFromTo (fromEnum yx3) (fromEnum yx4))",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 82.06/55.93 6[label="map toEnum (numericEnumFromTo (fromEnum yx3) (fromEnum yx4))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 82.06/55.93 7[label="map toEnum (takeWhile (flip (<=) (fromEnum yx4)) (numericEnumFrom (fromEnum yx3)))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 82.06/55.93 8[label="map toEnum (takeWhile (flip (<=) (fromEnum yx4)) (fromEnum yx3 : (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 82.06/55.93 9[label="map toEnum (takeWhile2 (flip (<=) (fromEnum yx4)) (fromEnum yx3 : (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3]; 82.06/55.93 10[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromEnum yx3) (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero))) (flip (<=) (fromEnum yx4) (fromEnum yx3)))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3]; 82.06/55.93 11[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromEnum yx3) (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero))) ((<=) fromEnum yx3 fromEnum yx4))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3]; 82.06/55.93 12[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromEnum yx3) (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero))) (compare (fromEnum yx3) (fromEnum yx4) /= GT))",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3]; 82.06/55.93 13[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromEnum yx3) (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero))) (not (compare (fromEnum yx3) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3]; 82.06/55.93 14[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromEnum yx3) (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero))) (not (primCmpInt (fromEnum yx3) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3]; 82.06/55.93 15[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (truncate yx3) (numericEnumFrom $! truncate yx3 + fromInt (Pos (Succ Zero))) (not (primCmpInt (truncate yx3) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3]; 82.06/55.93 16[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (truncateM yx3) (numericEnumFrom $! truncateM yx3 + fromInt (Pos (Succ Zero))) (not (primCmpInt (truncateM yx3) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3]; 82.06/55.93 17[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (truncateM0 yx3 (truncateVu6 yx3)) (numericEnumFrom $! truncateM0 yx3 (truncateVu6 yx3) + fromInt (Pos (Succ Zero))) (not (primCmpInt (truncateM0 yx3 (truncateVu6 yx3)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3]; 82.06/55.93 18[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (truncateM0 yx3 (properFraction yx3)) (numericEnumFrom $! truncateM0 yx3 (properFraction yx3) + fromInt (Pos (Succ Zero))) (not (primCmpInt (truncateM0 yx3 (properFraction yx3)) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4544[label="yx3/yx30 :% yx31",fontsize=10,color="white",style="solid",shape="box"];18 -> 4544[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4544 -> 19[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 19[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (truncateM0 (yx30 :% yx31) (properFraction (yx30 :% yx31))) (numericEnumFrom $! truncateM0 (yx30 :% yx31) (properFraction (yx30 :% yx31)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (truncateM0 (yx30 :% yx31) (properFraction (yx30 :% yx31))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3]; 82.06/55.93 20[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (truncateM0 (yx30 :% yx31) (fromIntegral (properFractionQ yx30 yx31),properFractionR yx30 yx31 :% yx31)) (numericEnumFrom $! truncateM0 (yx30 :% yx31) (fromIntegral (properFractionQ yx30 yx31),properFractionR yx30 yx31 :% yx31) + fromInt (Pos (Succ Zero))) (not (primCmpInt (truncateM0 (yx30 :% yx31) (fromIntegral (properFractionQ yx30 yx31),properFractionR yx30 yx31 :% yx31)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3]; 82.06/55.93 21[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromIntegral (properFractionQ yx30 yx31)) (numericEnumFrom $! fromIntegral (properFractionQ yx30 yx31) + fromInt (Pos (Succ Zero))) (not (primCmpInt (fromIntegral (properFractionQ yx30 yx31)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3]; 82.06/55.93 22[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromInteger . toInteger) (numericEnumFrom $! fromInteger . toInteger + fromInt (Pos (Succ Zero))) (not (primCmpInt (fromInteger . toInteger) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3]; 82.06/55.93 23[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromInteger (toInteger (properFractionQ yx30 yx31))) (numericEnumFrom $! fromInteger (toInteger (properFractionQ yx30 yx31)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (fromInteger (toInteger (properFractionQ yx30 yx31))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3]; 82.06/55.93 24[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromInteger (Integer (properFractionQ yx30 yx31))) (numericEnumFrom $! fromInteger (Integer (properFractionQ yx30 yx31)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (fromInteger (Integer (properFractionQ yx30 yx31))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];24 -> 25[label="",style="solid", color="black", weight=3]; 82.06/55.93 25[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (properFractionQ yx30 yx31) (numericEnumFrom $! properFractionQ yx30 yx31 + fromInt (Pos (Succ Zero))) (not (primCmpInt (properFractionQ yx30 yx31) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];25 -> 26[label="",style="solid", color="black", weight=3]; 82.06/55.93 26[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (properFractionQ1 yx30 yx31 (properFractionVu30 yx30 yx31)) (numericEnumFrom $! properFractionQ1 yx30 yx31 (properFractionVu30 yx30 yx31) + fromInt (Pos (Succ Zero))) (not (primCmpInt (properFractionQ1 yx30 yx31 (properFractionVu30 yx30 yx31)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3]; 82.06/55.93 27[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (properFractionQ1 yx30 yx31 (quotRem yx30 yx31)) (numericEnumFrom $! properFractionQ1 yx30 yx31 (quotRem yx30 yx31) + fromInt (Pos (Succ Zero))) (not (primCmpInt (properFractionQ1 yx30 yx31 (quotRem yx30 yx31)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];27 -> 28[label="",style="solid", color="black", weight=3]; 82.06/55.93 28[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (properFractionQ1 yx30 yx31 (primQrmInt yx30 yx31)) (numericEnumFrom $! properFractionQ1 yx30 yx31 (primQrmInt yx30 yx31) + fromInt (Pos (Succ Zero))) (not (primCmpInt (properFractionQ1 yx30 yx31 (primQrmInt yx30 yx31)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];28 -> 29[label="",style="solid", color="black", weight=3]; 82.06/55.93 29[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (properFractionQ1 yx30 yx31 (primQuotInt yx30 yx31,primRemInt yx30 yx31)) (numericEnumFrom $! properFractionQ1 yx30 yx31 (primQuotInt yx30 yx31,primRemInt yx30 yx31) + fromInt (Pos (Succ Zero))) (not (primCmpInt (properFractionQ1 yx30 yx31 (primQuotInt yx30 yx31,primRemInt yx30 yx31)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];29 -> 30[label="",style="solid", color="black", weight=3]; 82.06/55.93 30[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt yx30 yx31) (numericEnumFrom $! primQuotInt yx30 yx31 + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt yx30 yx31) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4545[label="yx30/Pos yx300",fontsize=10,color="white",style="solid",shape="box"];30 -> 4545[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4545 -> 31[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4546[label="yx30/Neg yx300",fontsize=10,color="white",style="solid",shape="box"];30 -> 4546[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4546 -> 32[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 31[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) yx31) (numericEnumFrom $! primQuotInt (Pos yx300) yx31 + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) yx31) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4547[label="yx31/Pos yx310",fontsize=10,color="white",style="solid",shape="box"];31 -> 4547[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4547 -> 33[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4548[label="yx31/Neg yx310",fontsize=10,color="white",style="solid",shape="box"];31 -> 4548[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4548 -> 34[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 32[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) yx31) (numericEnumFrom $! primQuotInt (Neg yx300) yx31 + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) yx31) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4549[label="yx31/Pos yx310",fontsize=10,color="white",style="solid",shape="box"];32 -> 4549[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4549 -> 35[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4550[label="yx31/Neg yx310",fontsize=10,color="white",style="solid",shape="box"];32 -> 4550[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4550 -> 36[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 33[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) (Pos yx310)) (numericEnumFrom $! primQuotInt (Pos yx300) (Pos yx310) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) (Pos yx310)) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4551[label="yx310/Succ yx3100",fontsize=10,color="white",style="solid",shape="box"];33 -> 4551[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4551 -> 37[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4552[label="yx310/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 4552[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4552 -> 38[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 34[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) (Neg yx310)) (numericEnumFrom $! primQuotInt (Pos yx300) (Neg yx310) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) (Neg yx310)) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4553[label="yx310/Succ yx3100",fontsize=10,color="white",style="solid",shape="box"];34 -> 4553[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4553 -> 39[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4554[label="yx310/Zero",fontsize=10,color="white",style="solid",shape="box"];34 -> 4554[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4554 -> 40[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 35[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) (Pos yx310)) (numericEnumFrom $! primQuotInt (Neg yx300) (Pos yx310) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) (Pos yx310)) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4555[label="yx310/Succ yx3100",fontsize=10,color="white",style="solid",shape="box"];35 -> 4555[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4555 -> 41[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4556[label="yx310/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 4556[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4556 -> 42[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 36[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) (Neg yx310)) (numericEnumFrom $! primQuotInt (Neg yx300) (Neg yx310) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) (Neg yx310)) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4557[label="yx310/Succ yx3100",fontsize=10,color="white",style="solid",shape="box"];36 -> 4557[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4557 -> 43[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4558[label="yx310/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 4558[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4558 -> 44[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 37[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) (Pos (Succ yx3100))) (numericEnumFrom $! primQuotInt (Pos yx300) (Pos (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) (Pos (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];37 -> 45[label="",style="solid", color="black", weight=3]; 82.06/55.93 38[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) (Pos Zero)) (numericEnumFrom $! primQuotInt (Pos yx300) (Pos Zero) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) (Pos Zero)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];38 -> 46[label="",style="solid", color="black", weight=3]; 82.06/55.93 39[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) (Neg (Succ yx3100))) (numericEnumFrom $! primQuotInt (Pos yx300) (Neg (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) (Neg (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];39 -> 47[label="",style="solid", color="black", weight=3]; 82.06/55.93 40[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) (Neg Zero)) (numericEnumFrom $! primQuotInt (Pos yx300) (Neg Zero) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) (Neg Zero)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];40 -> 48[label="",style="solid", color="black", weight=3]; 82.06/55.93 41[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) (Pos (Succ yx3100))) (numericEnumFrom $! primQuotInt (Neg yx300) (Pos (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) (Pos (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];41 -> 49[label="",style="solid", color="black", weight=3]; 82.06/55.93 42[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) (Pos Zero)) (numericEnumFrom $! primQuotInt (Neg yx300) (Pos Zero) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) (Pos Zero)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];42 -> 50[label="",style="solid", color="black", weight=3]; 82.06/55.93 43[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) (Neg (Succ yx3100))) (numericEnumFrom $! primQuotInt (Neg yx300) (Neg (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) (Neg (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];43 -> 51[label="",style="solid", color="black", weight=3]; 82.06/55.93 44[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) (Neg Zero)) (numericEnumFrom $! primQuotInt (Neg yx300) (Neg Zero) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) (Neg Zero)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];44 -> 52[label="",style="solid", color="black", weight=3]; 82.06/55.93 45 -> 3690[label="",style="dashed", color="red", weight=0]; 82.06/55.93 45[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Pos (primDivNatS yx300 (Succ yx3100))) (numericEnumFrom $! Pos (primDivNatS yx300 (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (primDivNatS yx300 (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];45 -> 3691[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 45 -> 3692[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 45 -> 3693[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 45 -> 3694[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 45 -> 3695[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 46[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (error []) (numericEnumFrom $! error [] + fromInt (Pos (Succ Zero))) (not (primCmpInt (error []) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="triangle"];46 -> 55[label="",style="solid", color="black", weight=3]; 82.06/55.93 47[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS yx300 (Succ yx3100))) (numericEnumFrom $! Neg (primDivNatS yx300 (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS yx300 (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="triangle"];4559[label="yx300/Succ yx3000",fontsize=10,color="white",style="solid",shape="box"];47 -> 4559[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4559 -> 56[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4560[label="yx300/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 4560[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4560 -> 57[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 48 -> 46[label="",style="dashed", color="red", weight=0]; 82.06/55.93 48[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (error []) (numericEnumFrom $! error [] + fromInt (Pos (Succ Zero))) (not (primCmpInt (error []) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];49 -> 47[label="",style="dashed", color="red", weight=0]; 82.06/55.93 49[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS yx300 (Succ yx3100))) (numericEnumFrom $! Neg (primDivNatS yx300 (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS yx300 (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];49 -> 58[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 49 -> 59[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 50 -> 46[label="",style="dashed", color="red", weight=0]; 82.06/55.93 50[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (error []) (numericEnumFrom $! error [] + fromInt (Pos (Succ Zero))) (not (primCmpInt (error []) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];51 -> 3690[label="",style="dashed", color="red", weight=0]; 82.06/55.93 51[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Pos (primDivNatS yx300 (Succ yx3100))) (numericEnumFrom $! Pos (primDivNatS yx300 (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (primDivNatS yx300 (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];51 -> 3696[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 51 -> 3697[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 51 -> 3698[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 51 -> 3699[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 51 -> 3700[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 52 -> 46[label="",style="dashed", color="red", weight=0]; 82.06/55.93 52[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (error []) (numericEnumFrom $! error [] + fromInt (Pos (Succ Zero))) (not (primCmpInt (error []) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];3691 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 3691[label="primDivNatS yx300 (Succ yx3100)",fontsize=16,color="magenta"];3691 -> 3926[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3691 -> 3927[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3692 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 3692[label="primDivNatS yx300 (Succ yx3100)",fontsize=16,color="magenta"];3692 -> 3928[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3692 -> 3929[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3693 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.93 3693[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3693 -> 3930[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3694 -> 459[label="",style="dashed", color="red", weight=0]; 82.06/55.93 3694[label="fromEnum yx4",fontsize=16,color="magenta"];3694 -> 3931[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3695 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 3695[label="primDivNatS yx300 (Succ yx3100)",fontsize=16,color="magenta"];3695 -> 3932[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3695 -> 3933[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3690[label="map toEnum (takeWhile1 (flip (<=) yx199) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpInt (Pos yx251) yx199 == GT)))",fontsize=16,color="burlywood",shape="triangle"];4561[label="yx251/Succ yx2510",fontsize=10,color="white",style="solid",shape="box"];3690 -> 4561[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4561 -> 3934[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4562[label="yx251/Zero",fontsize=10,color="white",style="solid",shape="box"];3690 -> 4562[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4562 -> 3935[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 55[label="error []",fontsize=16,color="red",shape="box"];56[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS (Succ yx3000) (Succ yx3100))) (numericEnumFrom $! Neg (primDivNatS (Succ yx3000) (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS (Succ yx3000) (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];56 -> 64[label="",style="solid", color="black", weight=3]; 82.06/55.93 57[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS Zero (Succ yx3100))) (numericEnumFrom $! Neg (primDivNatS Zero (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS Zero (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];57 -> 65[label="",style="solid", color="black", weight=3]; 82.06/55.93 58[label="yx3100",fontsize=16,color="green",shape="box"];59[label="yx300",fontsize=16,color="green",shape="box"];3696 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 3696[label="primDivNatS yx300 (Succ yx3100)",fontsize=16,color="magenta"];3696 -> 3936[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3696 -> 3937[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3697 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 3697[label="primDivNatS yx300 (Succ yx3100)",fontsize=16,color="magenta"];3697 -> 3938[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3697 -> 3939[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3698 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.93 3698[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3698 -> 3940[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3699 -> 459[label="",style="dashed", color="red", weight=0]; 82.06/55.93 3699[label="fromEnum yx4",fontsize=16,color="magenta"];3699 -> 3941[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3700 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 3700[label="primDivNatS yx300 (Succ yx3100)",fontsize=16,color="magenta"];3700 -> 3942[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3700 -> 3943[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 3926[label="yx3100",fontsize=16,color="green",shape="box"];3927[label="yx300",fontsize=16,color="green",shape="box"];1689[label="primDivNatS yx400 (Succ yx4100)",fontsize=16,color="burlywood",shape="triangle"];4563[label="yx400/Succ yx4000",fontsize=10,color="white",style="solid",shape="box"];1689 -> 4563[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4563 -> 1725[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4564[label="yx400/Zero",fontsize=10,color="white",style="solid",shape="box"];1689 -> 4564[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4564 -> 1726[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 3928[label="yx3100",fontsize=16,color="green",shape="box"];3929[label="yx300",fontsize=16,color="green",shape="box"];3930[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];576[label="fromInt yx8",fontsize=16,color="black",shape="triangle"];576 -> 650[label="",style="solid", color="black", weight=3]; 82.06/55.93 3931[label="yx4",fontsize=16,color="green",shape="box"];459[label="fromEnum yx7",fontsize=16,color="black",shape="triangle"];459 -> 518[label="",style="solid", color="black", weight=3]; 82.06/55.93 3932[label="yx3100",fontsize=16,color="green",shape="box"];3933[label="yx300",fontsize=16,color="green",shape="box"];3934[label="map toEnum (takeWhile1 (flip (<=) yx199) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpInt (Pos (Succ yx2510)) yx199 == GT)))",fontsize=16,color="burlywood",shape="box"];4565[label="yx199/Pos yx1990",fontsize=10,color="white",style="solid",shape="box"];3934 -> 4565[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4565 -> 3952[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4566[label="yx199/Neg yx1990",fontsize=10,color="white",style="solid",shape="box"];3934 -> 4566[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4566 -> 3953[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 3935[label="map toEnum (takeWhile1 (flip (<=) yx199) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpInt (Pos Zero) yx199 == GT)))",fontsize=16,color="burlywood",shape="box"];4567[label="yx199/Pos yx1990",fontsize=10,color="white",style="solid",shape="box"];3935 -> 4567[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4567 -> 3954[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4568[label="yx199/Neg yx1990",fontsize=10,color="white",style="solid",shape="box"];3935 -> 4568[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4568 -> 3955[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 64[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 yx3000 yx3100 (primGEqNatS yx3000 yx3100))) (numericEnumFrom $! Neg (primDivNatS0 yx3000 yx3100 (primGEqNatS yx3000 yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 yx3000 yx3100 (primGEqNatS yx3000 yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4569[label="yx3000/Succ yx30000",fontsize=10,color="white",style="solid",shape="box"];64 -> 4569[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4569 -> 69[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4570[label="yx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];64 -> 4570[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4570 -> 70[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 65[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="triangle"];65 -> 71[label="",style="solid", color="black", weight=3]; 82.06/55.93 3936[label="yx3100",fontsize=16,color="green",shape="box"];3937[label="yx300",fontsize=16,color="green",shape="box"];3938[label="yx3100",fontsize=16,color="green",shape="box"];3939[label="yx300",fontsize=16,color="green",shape="box"];3940[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3941[label="yx4",fontsize=16,color="green",shape="box"];3942[label="yx3100",fontsize=16,color="green",shape="box"];3943[label="yx300",fontsize=16,color="green",shape="box"];1725[label="primDivNatS (Succ yx4000) (Succ yx4100)",fontsize=16,color="black",shape="box"];1725 -> 1745[label="",style="solid", color="black", weight=3]; 82.06/55.93 1726[label="primDivNatS Zero (Succ yx4100)",fontsize=16,color="black",shape="box"];1726 -> 1746[label="",style="solid", color="black", weight=3]; 82.06/55.93 650[label="yx8",fontsize=16,color="green",shape="box"];518[label="truncate yx7",fontsize=16,color="black",shape="box"];518 -> 578[label="",style="solid", color="black", weight=3]; 82.06/55.93 3952[label="map toEnum (takeWhile1 (flip (<=) (Pos yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpInt (Pos (Succ yx2510)) (Pos yx1990) == GT)))",fontsize=16,color="black",shape="box"];3952 -> 3999[label="",style="solid", color="black", weight=3]; 82.06/55.93 3953[label="map toEnum (takeWhile1 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpInt (Pos (Succ yx2510)) (Neg yx1990) == GT)))",fontsize=16,color="black",shape="box"];3953 -> 4000[label="",style="solid", color="black", weight=3]; 82.06/55.93 3954[label="map toEnum (takeWhile1 (flip (<=) (Pos yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpInt (Pos Zero) (Pos yx1990) == GT)))",fontsize=16,color="burlywood",shape="box"];4571[label="yx1990/Succ yx19900",fontsize=10,color="white",style="solid",shape="box"];3954 -> 4571[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4571 -> 4001[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4572[label="yx1990/Zero",fontsize=10,color="white",style="solid",shape="box"];3954 -> 4572[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4572 -> 4002[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 3955[label="map toEnum (takeWhile1 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpInt (Pos Zero) (Neg yx1990) == GT)))",fontsize=16,color="burlywood",shape="box"];4573[label="yx1990/Succ yx19900",fontsize=10,color="white",style="solid",shape="box"];3955 -> 4573[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4573 -> 4003[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4574[label="yx1990/Zero",fontsize=10,color="white",style="solid",shape="box"];3955 -> 4574[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4574 -> 4004[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 69[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 (Succ yx30000) yx3100 (primGEqNatS (Succ yx30000) yx3100))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx30000) yx3100 (primGEqNatS (Succ yx30000) yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 (Succ yx30000) yx3100 (primGEqNatS (Succ yx30000) yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4575[label="yx3100/Succ yx31000",fontsize=10,color="white",style="solid",shape="box"];69 -> 4575[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4575 -> 77[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4576[label="yx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];69 -> 4576[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4576 -> 78[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 70[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 Zero yx3100 (primGEqNatS Zero yx3100))) (numericEnumFrom $! Neg (primDivNatS0 Zero yx3100 (primGEqNatS Zero yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 Zero yx3100 (primGEqNatS Zero yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4577[label="yx3100/Succ yx31000",fontsize=10,color="white",style="solid",shape="box"];70 -> 4577[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4577 -> 79[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4578[label="yx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];70 -> 4578[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4578 -> 80[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 71[label="map toEnum (takeWhile1 (flip (<=) (truncate yx4)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (truncate yx4) == GT)))",fontsize=16,color="black",shape="box"];71 -> 81[label="",style="solid", color="black", weight=3]; 82.06/55.93 1745 -> 1151[label="",style="dashed", color="red", weight=0]; 82.06/55.93 1745[label="primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)",fontsize=16,color="magenta"];1746[label="Zero",fontsize=16,color="green",shape="box"];578[label="truncateM yx7",fontsize=16,color="black",shape="box"];578 -> 652[label="",style="solid", color="black", weight=3]; 82.06/55.93 3999[label="map toEnum (takeWhile1 (flip (<=) (Pos yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpNat (Succ yx2510) yx1990 == GT)))",fontsize=16,color="burlywood",shape="box"];4579[label="yx1990/Succ yx19900",fontsize=10,color="white",style="solid",shape="box"];3999 -> 4579[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4579 -> 4018[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4580[label="yx1990/Zero",fontsize=10,color="white",style="solid",shape="box"];3999 -> 4580[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4580 -> 4019[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4000[label="map toEnum (takeWhile1 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (GT == GT)))",fontsize=16,color="black",shape="triangle"];4000 -> 4020[label="",style="solid", color="black", weight=3]; 82.06/55.93 4001[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx19900))) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpInt (Pos Zero) (Pos (Succ yx19900)) == GT)))",fontsize=16,color="black",shape="box"];4001 -> 4021[label="",style="solid", color="black", weight=3]; 82.06/55.93 4002[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];4002 -> 4022[label="",style="solid", color="black", weight=3]; 82.06/55.93 4003[label="map toEnum (takeWhile1 (flip (<=) (Neg (Succ yx19900))) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpInt (Pos Zero) (Neg (Succ yx19900)) == GT)))",fontsize=16,color="black",shape="box"];4003 -> 4023[label="",style="solid", color="black", weight=3]; 82.06/55.93 4004[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpInt (Pos Zero) (Neg Zero) == GT)))",fontsize=16,color="black",shape="box"];4004 -> 4024[label="",style="solid", color="black", weight=3]; 82.06/55.93 77[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 (Succ yx30000) (Succ yx31000) (primGEqNatS (Succ yx30000) (Succ yx31000)))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx30000) (Succ yx31000) (primGEqNatS (Succ yx30000) (Succ yx31000))) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 (Succ yx30000) (Succ yx31000) (primGEqNatS (Succ yx30000) (Succ yx31000)))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];77 -> 87[label="",style="solid", color="black", weight=3]; 82.06/55.93 78[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 (Succ yx30000) Zero (primGEqNatS (Succ yx30000) Zero))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx30000) Zero (primGEqNatS (Succ yx30000) Zero)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 (Succ yx30000) Zero (primGEqNatS (Succ yx30000) Zero))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];78 -> 88[label="",style="solid", color="black", weight=3]; 82.06/55.93 79[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 Zero (Succ yx31000) (primGEqNatS Zero (Succ yx31000)))) (numericEnumFrom $! Neg (primDivNatS0 Zero (Succ yx31000) (primGEqNatS Zero (Succ yx31000))) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 Zero (Succ yx31000) (primGEqNatS Zero (Succ yx31000)))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];79 -> 89[label="",style="solid", color="black", weight=3]; 82.06/55.93 80[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 Zero Zero (primGEqNatS Zero Zero))) (numericEnumFrom $! Neg (primDivNatS0 Zero Zero (primGEqNatS Zero Zero)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 Zero Zero (primGEqNatS Zero Zero))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];80 -> 90[label="",style="solid", color="black", weight=3]; 82.06/55.93 81[label="map toEnum (takeWhile1 (flip (<=) (truncateM yx4)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (truncateM yx4) == GT)))",fontsize=16,color="black",shape="box"];81 -> 91[label="",style="solid", color="black", weight=3]; 82.06/55.93 1151[label="primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)",fontsize=16,color="burlywood",shape="triangle"];4581[label="yx4000/Succ yx40000",fontsize=10,color="white",style="solid",shape="box"];1151 -> 4581[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4581 -> 1276[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4582[label="yx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];1151 -> 4582[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4582 -> 1277[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 652[label="truncateM0 yx7 (truncateVu6 yx7)",fontsize=16,color="black",shape="box"];652 -> 821[label="",style="solid", color="black", weight=3]; 82.06/55.93 4018[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx19900))) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpNat (Succ yx2510) (Succ yx19900) == GT)))",fontsize=16,color="black",shape="box"];4018 -> 4030[label="",style="solid", color="black", weight=3]; 82.06/55.93 4019[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpNat (Succ yx2510) Zero == GT)))",fontsize=16,color="black",shape="box"];4019 -> 4031[label="",style="solid", color="black", weight=3]; 82.06/55.93 4020[label="map toEnum (takeWhile1 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not True))",fontsize=16,color="black",shape="box"];4020 -> 4032[label="",style="solid", color="black", weight=3]; 82.06/55.93 4021 -> 4408[label="",style="dashed", color="red", weight=0]; 82.06/55.93 4021[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx19900))) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpNat Zero (Succ yx19900) == GT)))",fontsize=16,color="magenta"];4021 -> 4409[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4021 -> 4410[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4021 -> 4411[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4021 -> 4412[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4021 -> 4413[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4021 -> 4414[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4022[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];4022 -> 4034[label="",style="solid", color="black", weight=3]; 82.06/55.93 4023 -> 4000[label="",style="dashed", color="red", weight=0]; 82.06/55.93 4023[label="map toEnum (takeWhile1 (flip (<=) (Neg (Succ yx19900))) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (GT == GT)))",fontsize=16,color="magenta"];4023 -> 4035[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4024[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];4024 -> 4036[label="",style="solid", color="black", weight=3]; 82.06/55.93 87 -> 2234[label="",style="dashed", color="red", weight=0]; 82.06/55.93 87[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 (Succ yx30000) (Succ yx31000) (primGEqNatS yx30000 yx31000))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx30000) (Succ yx31000) (primGEqNatS yx30000 yx31000)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 (Succ yx30000) (Succ yx31000) (primGEqNatS yx30000 yx31000))) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];87 -> 2235[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 87 -> 2236[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 87 -> 2237[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 87 -> 2238[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 87 -> 2239[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 87 -> 2240[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 87 -> 2241[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 88[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 (Succ yx30000) Zero True)) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx30000) Zero True) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 (Succ yx30000) Zero True)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];88 -> 100[label="",style="solid", color="black", weight=3]; 82.06/55.93 89[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 Zero (Succ yx31000) False)) (numericEnumFrom $! Neg (primDivNatS0 Zero (Succ yx31000) False) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 Zero (Succ yx31000) False)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];89 -> 101[label="",style="solid", color="black", weight=3]; 82.06/55.93 90[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 Zero Zero True)) (numericEnumFrom $! Neg (primDivNatS0 Zero Zero True) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 Zero Zero True)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];90 -> 102[label="",style="solid", color="black", weight=3]; 82.06/55.93 91[label="map toEnum (takeWhile1 (flip (<=) (truncateM0 yx4 (truncateVu6 yx4))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (truncateM0 yx4 (truncateVu6 yx4)) == GT)))",fontsize=16,color="black",shape="box"];91 -> 103[label="",style="solid", color="black", weight=3]; 82.06/55.93 1276[label="primDivNatS0 (Succ yx40000) yx4100 (primGEqNatS (Succ yx40000) yx4100)",fontsize=16,color="burlywood",shape="box"];4583[label="yx4100/Succ yx41000",fontsize=10,color="white",style="solid",shape="box"];1276 -> 4583[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4583 -> 1405[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4584[label="yx4100/Zero",fontsize=10,color="white",style="solid",shape="box"];1276 -> 4584[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4584 -> 1406[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 1277[label="primDivNatS0 Zero yx4100 (primGEqNatS Zero yx4100)",fontsize=16,color="burlywood",shape="box"];4585[label="yx4100/Succ yx41000",fontsize=10,color="white",style="solid",shape="box"];1277 -> 4585[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4585 -> 1407[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4586[label="yx4100/Zero",fontsize=10,color="white",style="solid",shape="box"];1277 -> 4586[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4586 -> 1408[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 821[label="truncateM0 yx7 (properFraction yx7)",fontsize=16,color="burlywood",shape="box"];4587[label="yx7/yx70 :% yx71",fontsize=10,color="white",style="solid",shape="box"];821 -> 4587[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4587 -> 1033[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4030 -> 4408[label="",style="dashed", color="red", weight=0]; 82.06/55.93 4030[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx19900))) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (primCmpNat yx2510 yx19900 == GT)))",fontsize=16,color="magenta"];4030 -> 4415[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4030 -> 4416[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4030 -> 4417[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4030 -> 4418[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4030 -> 4419[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4030 -> 4420[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4031[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not (GT == GT)))",fontsize=16,color="black",shape="box"];4031 -> 4045[label="",style="solid", color="black", weight=3]; 82.06/55.93 4032[label="map toEnum (takeWhile1 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) False)",fontsize=16,color="black",shape="box"];4032 -> 4046[label="",style="solid", color="black", weight=3]; 82.06/55.93 4409[label="yx249",fontsize=16,color="green",shape="box"];4410[label="Zero",fontsize=16,color="green",shape="box"];4411[label="Succ yx19900",fontsize=16,color="green",shape="box"];4412[label="yx19900",fontsize=16,color="green",shape="box"];4413[label="yx236",fontsize=16,color="green",shape="box"];4414[label="yx250",fontsize=16,color="green",shape="box"];4408[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not (primCmpNat yx283 yx284 == GT)))",fontsize=16,color="burlywood",shape="triangle"];4588[label="yx283/Succ yx2830",fontsize=10,color="white",style="solid",shape="box"];4408 -> 4588[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4588 -> 4469[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4589[label="yx283/Zero",fontsize=10,color="white",style="solid",shape="box"];4408 -> 4589[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4589 -> 4470[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4034[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not False))",fontsize=16,color="black",shape="box"];4034 -> 4048[label="",style="solid", color="black", weight=3]; 82.06/55.93 4035[label="Succ yx19900",fontsize=16,color="green",shape="box"];4036[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not False))",fontsize=16,color="black",shape="box"];4036 -> 4049[label="",style="solid", color="black", weight=3]; 82.06/55.93 2235[label="yx31000",fontsize=16,color="green",shape="box"];2236[label="yx31000",fontsize=16,color="green",shape="box"];2237 -> 459[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2237[label="fromEnum yx4",fontsize=16,color="magenta"];2237 -> 2312[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2238[label="yx30000",fontsize=16,color="green",shape="box"];2239 -> 459[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2239[label="fromEnum yx4",fontsize=16,color="magenta"];2239 -> 2313[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2240 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2240[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];2240 -> 2314[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2241[label="yx30000",fontsize=16,color="green",shape="box"];2234[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS yx210 yx211))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS yx210 yx211)) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS yx210 yx211))) yx213 == GT)))",fontsize=16,color="burlywood",shape="triangle"];4590[label="yx210/Succ yx2100",fontsize=10,color="white",style="solid",shape="box"];2234 -> 4590[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4590 -> 2315[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4591[label="yx210/Zero",fontsize=10,color="white",style="solid",shape="box"];2234 -> 4591[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4591 -> 2316[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 100 -> 2772[label="",style="dashed", color="red", weight=0]; 82.06/55.93 100[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (Succ (primDivNatS (primMinusNatS (Succ yx30000) Zero) (Succ Zero)))) (numericEnumFrom $! Neg (Succ (primDivNatS (primMinusNatS (Succ yx30000) Zero) (Succ Zero))) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (Succ (primDivNatS (primMinusNatS (Succ yx30000) Zero) (Succ Zero)))) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];100 -> 2773[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 100 -> 2774[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 100 -> 2775[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 100 -> 2776[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 100 -> 2777[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 100 -> 2778[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 101 -> 65[label="",style="dashed", color="red", weight=0]; 82.06/55.93 101[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];102 -> 2772[label="",style="dashed", color="red", weight=0]; 82.06/55.93 102[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero)))) (numericEnumFrom $! Neg (Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero)))) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];102 -> 2779[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 102 -> 2780[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 102 -> 2781[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 102 -> 2782[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 102 -> 2783[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 102 -> 2784[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 103[label="map toEnum (takeWhile1 (flip (<=) (truncateM0 yx4 (properFraction yx4))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (truncateM0 yx4 (properFraction yx4)) == GT)))",fontsize=16,color="burlywood",shape="box"];4592[label="yx4/yx40 :% yx41",fontsize=10,color="white",style="solid",shape="box"];103 -> 4592[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4592 -> 117[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 1405[label="primDivNatS0 (Succ yx40000) (Succ yx41000) (primGEqNatS (Succ yx40000) (Succ yx41000))",fontsize=16,color="black",shape="box"];1405 -> 1541[label="",style="solid", color="black", weight=3]; 82.06/55.93 1406[label="primDivNatS0 (Succ yx40000) Zero (primGEqNatS (Succ yx40000) Zero)",fontsize=16,color="black",shape="box"];1406 -> 1542[label="",style="solid", color="black", weight=3]; 82.06/55.93 1407[label="primDivNatS0 Zero (Succ yx41000) (primGEqNatS Zero (Succ yx41000))",fontsize=16,color="black",shape="box"];1407 -> 1543[label="",style="solid", color="black", weight=3]; 82.06/55.93 1408[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];1408 -> 1544[label="",style="solid", color="black", weight=3]; 82.06/55.93 1033[label="truncateM0 (yx70 :% yx71) (properFraction (yx70 :% yx71))",fontsize=16,color="black",shape="box"];1033 -> 1174[label="",style="solid", color="black", weight=3]; 82.06/55.93 4415[label="yx249",fontsize=16,color="green",shape="box"];4416[label="yx2510",fontsize=16,color="green",shape="box"];4417[label="yx19900",fontsize=16,color="green",shape="box"];4418[label="yx19900",fontsize=16,color="green",shape="box"];4419[label="yx236",fontsize=16,color="green",shape="box"];4420[label="yx250",fontsize=16,color="green",shape="box"];4045[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) (not True))",fontsize=16,color="black",shape="box"];4045 -> 4058[label="",style="solid", color="black", weight=3]; 82.06/55.93 4046[label="map toEnum (takeWhile0 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) otherwise)",fontsize=16,color="black",shape="box"];4046 -> 4059[label="",style="solid", color="black", weight=3]; 82.06/55.93 4469[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not (primCmpNat (Succ yx2830) yx284 == GT)))",fontsize=16,color="burlywood",shape="box"];4593[label="yx284/Succ yx2840",fontsize=10,color="white",style="solid",shape="box"];4469 -> 4593[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4593 -> 4471[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4594[label="yx284/Zero",fontsize=10,color="white",style="solid",shape="box"];4469 -> 4594[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4594 -> 4472[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4470[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not (primCmpNat Zero yx284 == GT)))",fontsize=16,color="burlywood",shape="box"];4595[label="yx284/Succ yx2840",fontsize=10,color="white",style="solid",shape="box"];4470 -> 4595[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4595 -> 4473[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4596[label="yx284/Zero",fontsize=10,color="white",style="solid",shape="box"];4470 -> 4596[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4596 -> 4474[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4048[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) True)",fontsize=16,color="black",shape="box"];4048 -> 4061[label="",style="solid", color="black", weight=3]; 82.06/55.93 4049[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) True)",fontsize=16,color="black",shape="box"];4049 -> 4062[label="",style="solid", color="black", weight=3]; 82.06/55.93 2312[label="yx4",fontsize=16,color="green",shape="box"];2313[label="yx4",fontsize=16,color="green",shape="box"];2314[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];2315[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) yx211))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) yx211)) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) yx211))) yx213 == GT)))",fontsize=16,color="burlywood",shape="box"];4597[label="yx211/Succ yx2110",fontsize=10,color="white",style="solid",shape="box"];2315 -> 4597[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4597 -> 2345[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4598[label="yx211/Zero",fontsize=10,color="white",style="solid",shape="box"];2315 -> 4598[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4598 -> 2346[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 2316[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero yx211))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero yx211)) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero yx211))) yx213 == GT)))",fontsize=16,color="burlywood",shape="box"];4599[label="yx211/Succ yx2110",fontsize=10,color="white",style="solid",shape="box"];2316 -> 4599[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4599 -> 2347[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4600[label="yx211/Zero",fontsize=10,color="white",style="solid",shape="box"];2316 -> 4600[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4600 -> 2348[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 2773 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2773[label="primDivNatS (primMinusNatS (Succ yx30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];2773 -> 2992[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2773 -> 2993[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2774 -> 459[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2774[label="fromEnum yx4",fontsize=16,color="magenta"];2774 -> 2994[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2775 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2775[label="primDivNatS (primMinusNatS (Succ yx30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];2775 -> 2995[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2775 -> 2996[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2776 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2776[label="primDivNatS (primMinusNatS (Succ yx30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];2776 -> 2997[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2776 -> 2998[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2777 -> 459[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2777[label="fromEnum yx4",fontsize=16,color="magenta"];2777 -> 2999[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2778 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2778[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];2778 -> 3000[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2772[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpInt (Neg (Succ yx219)) yx213 == GT)))",fontsize=16,color="burlywood",shape="triangle"];4601[label="yx213/Pos yx2130",fontsize=10,color="white",style="solid",shape="box"];2772 -> 4601[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4601 -> 3001[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4602[label="yx213/Neg yx2130",fontsize=10,color="white",style="solid",shape="box"];2772 -> 4602[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4602 -> 3002[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 2779 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2779[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];2779 -> 3003[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2779 -> 3004[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2780 -> 459[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2780[label="fromEnum yx4",fontsize=16,color="magenta"];2780 -> 3005[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2781 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2781[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];2781 -> 3006[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2781 -> 3007[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2782 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2782[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];2782 -> 3008[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2782 -> 3009[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2783 -> 459[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2783[label="fromEnum yx4",fontsize=16,color="magenta"];2783 -> 3010[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2784 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2784[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];2784 -> 3011[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 117[label="map toEnum (takeWhile1 (flip (<=) (truncateM0 (yx40 :% yx41) (properFraction (yx40 :% yx41)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (truncateM0 (yx40 :% yx41) (properFraction (yx40 :% yx41))) == GT)))",fontsize=16,color="black",shape="box"];117 -> 131[label="",style="solid", color="black", weight=3]; 82.06/55.93 1541 -> 3958[label="",style="dashed", color="red", weight=0]; 82.06/55.93 1541[label="primDivNatS0 (Succ yx40000) (Succ yx41000) (primGEqNatS yx40000 yx41000)",fontsize=16,color="magenta"];1541 -> 3959[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 1541 -> 3960[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 1541 -> 3961[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 1541 -> 3962[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 1542[label="primDivNatS0 (Succ yx40000) Zero True",fontsize=16,color="black",shape="box"];1542 -> 1752[label="",style="solid", color="black", weight=3]; 82.06/55.93 1543[label="primDivNatS0 Zero (Succ yx41000) False",fontsize=16,color="black",shape="box"];1543 -> 1753[label="",style="solid", color="black", weight=3]; 82.06/55.93 1544[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];1544 -> 1754[label="",style="solid", color="black", weight=3]; 82.06/55.93 1174[label="truncateM0 (yx70 :% yx71) (fromIntegral (properFractionQ yx70 yx71),properFractionR yx70 yx71 :% yx71)",fontsize=16,color="black",shape="box"];1174 -> 1302[label="",style="solid", color="black", weight=3]; 82.06/55.93 4058[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) False)",fontsize=16,color="black",shape="box"];4058 -> 4068[label="",style="solid", color="black", weight=3]; 82.06/55.93 4059[label="map toEnum (takeWhile0 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) True)",fontsize=16,color="black",shape="box"];4059 -> 4069[label="",style="solid", color="black", weight=3]; 82.06/55.93 4471[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not (primCmpNat (Succ yx2830) (Succ yx2840) == GT)))",fontsize=16,color="black",shape="box"];4471 -> 4475[label="",style="solid", color="black", weight=3]; 82.06/55.93 4472[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not (primCmpNat (Succ yx2830) Zero == GT)))",fontsize=16,color="black",shape="box"];4472 -> 4476[label="",style="solid", color="black", weight=3]; 82.06/55.93 4473[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not (primCmpNat Zero (Succ yx2840) == GT)))",fontsize=16,color="black",shape="box"];4473 -> 4477[label="",style="solid", color="black", weight=3]; 82.06/55.93 4474[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not (primCmpNat Zero Zero == GT)))",fontsize=16,color="black",shape="box"];4474 -> 4478[label="",style="solid", color="black", weight=3]; 82.06/55.93 4061[label="map toEnum (Pos yx249 : takeWhile (flip (<=) (Pos Zero)) (numericEnumFrom $! Pos yx250 + yx236))",fontsize=16,color="black",shape="box"];4061 -> 4071[label="",style="solid", color="black", weight=3]; 82.06/55.93 4062[label="map toEnum (Pos yx249 : takeWhile (flip (<=) (Neg Zero)) (numericEnumFrom $! Pos yx250 + yx236))",fontsize=16,color="black",shape="box"];4062 -> 4072[label="",style="solid", color="black", weight=3]; 82.06/55.93 2345[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) (Succ yx2110)))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) (Succ yx2110))) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) (Succ yx2110)))) yx213 == GT)))",fontsize=16,color="black",shape="box"];2345 -> 2382[label="",style="solid", color="black", weight=3]; 82.06/55.93 2346[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) Zero))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) Zero)) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) Zero))) yx213 == GT)))",fontsize=16,color="black",shape="box"];2346 -> 2383[label="",style="solid", color="black", weight=3]; 82.06/55.93 2347[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero (Succ yx2110)))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero (Succ yx2110))) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero (Succ yx2110)))) yx213 == GT)))",fontsize=16,color="black",shape="box"];2347 -> 2384[label="",style="solid", color="black", weight=3]; 82.06/55.93 2348[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero Zero))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero Zero)) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero Zero))) yx213 == GT)))",fontsize=16,color="black",shape="box"];2348 -> 2385[label="",style="solid", color="black", weight=3]; 82.06/55.93 2992[label="Zero",fontsize=16,color="green",shape="box"];2993 -> 1978[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2993[label="primMinusNatS (Succ yx30000) Zero",fontsize=16,color="magenta"];2994[label="yx4",fontsize=16,color="green",shape="box"];2995[label="Zero",fontsize=16,color="green",shape="box"];2996 -> 1978[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2996[label="primMinusNatS (Succ yx30000) Zero",fontsize=16,color="magenta"];2997[label="Zero",fontsize=16,color="green",shape="box"];2998 -> 1978[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2998[label="primMinusNatS (Succ yx30000) Zero",fontsize=16,color="magenta"];2999[label="yx4",fontsize=16,color="green",shape="box"];3000[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3001[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpInt (Neg (Succ yx219)) (Pos yx2130) == GT)))",fontsize=16,color="black",shape="box"];3001 -> 3046[label="",style="solid", color="black", weight=3]; 82.06/55.93 3002[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpInt (Neg (Succ yx219)) (Neg yx2130) == GT)))",fontsize=16,color="black",shape="box"];3002 -> 3047[label="",style="solid", color="black", weight=3]; 82.06/55.93 3003[label="Zero",fontsize=16,color="green",shape="box"];3004 -> 1730[label="",style="dashed", color="red", weight=0]; 82.06/55.93 3004[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];3005[label="yx4",fontsize=16,color="green",shape="box"];3006[label="Zero",fontsize=16,color="green",shape="box"];3007 -> 1730[label="",style="dashed", color="red", weight=0]; 82.06/55.93 3007[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];3008[label="Zero",fontsize=16,color="green",shape="box"];3009 -> 1730[label="",style="dashed", color="red", weight=0]; 82.06/55.93 3009[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];3010[label="yx4",fontsize=16,color="green",shape="box"];3011[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];131[label="map toEnum (takeWhile1 (flip (<=) (truncateM0 (yx40 :% yx41) (fromIntegral (properFractionQ yx40 yx41),properFractionR yx40 yx41 :% yx41))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (truncateM0 (yx40 :% yx41) (fromIntegral (properFractionQ yx40 yx41),properFractionR yx40 yx41 :% yx41)) == GT)))",fontsize=16,color="black",shape="box"];131 -> 147[label="",style="solid", color="black", weight=3]; 82.06/55.93 3959[label="yx41000",fontsize=16,color="green",shape="box"];3960[label="yx40000",fontsize=16,color="green",shape="box"];3961[label="yx41000",fontsize=16,color="green",shape="box"];3962[label="yx40000",fontsize=16,color="green",shape="box"];3958[label="primDivNatS0 (Succ yx255) (Succ yx256) (primGEqNatS yx257 yx258)",fontsize=16,color="burlywood",shape="triangle"];4603[label="yx257/Succ yx2570",fontsize=10,color="white",style="solid",shape="box"];3958 -> 4603[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4603 -> 4005[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4604[label="yx257/Zero",fontsize=10,color="white",style="solid",shape="box"];3958 -> 4604[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4604 -> 4006[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 1752[label="Succ (primDivNatS (primMinusNatS (Succ yx40000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];1752 -> 2069[label="",style="dashed", color="green", weight=3]; 82.06/55.93 1753[label="Zero",fontsize=16,color="green",shape="box"];1754[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];1754 -> 2070[label="",style="dashed", color="green", weight=3]; 82.06/55.93 1302[label="fromIntegral (properFractionQ yx70 yx71)",fontsize=16,color="black",shape="box"];1302 -> 1447[label="",style="solid", color="black", weight=3]; 82.06/55.93 4068[label="map toEnum (takeWhile0 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) otherwise)",fontsize=16,color="black",shape="box"];4068 -> 4080[label="",style="solid", color="black", weight=3]; 82.06/55.93 4069 -> 3189[label="",style="dashed", color="red", weight=0]; 82.06/55.93 4069[label="map toEnum []",fontsize=16,color="magenta"];4475 -> 4408[label="",style="dashed", color="red", weight=0]; 82.06/55.93 4475[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not (primCmpNat yx2830 yx2840 == GT)))",fontsize=16,color="magenta"];4475 -> 4479[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4475 -> 4480[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 4476[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not (GT == GT)))",fontsize=16,color="black",shape="box"];4476 -> 4481[label="",style="solid", color="black", weight=3]; 82.06/55.93 4477[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not (LT == GT)))",fontsize=16,color="black",shape="box"];4477 -> 4482[label="",style="solid", color="black", weight=3]; 82.06/55.93 4478[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];4478 -> 4483[label="",style="solid", color="black", weight=3]; 82.06/55.93 4071[label="toEnum (Pos yx249) : map toEnum (takeWhile (flip (<=) (Pos Zero)) (numericEnumFrom $! Pos yx250 + yx236))",fontsize=16,color="green",shape="box"];4071 -> 4082[label="",style="dashed", color="green", weight=3]; 82.06/55.93 4071 -> 4083[label="",style="dashed", color="green", weight=3]; 82.06/55.93 4072[label="toEnum (Pos yx249) : map toEnum (takeWhile (flip (<=) (Neg Zero)) (numericEnumFrom $! Pos yx250 + yx236))",fontsize=16,color="green",shape="box"];4072 -> 4084[label="",style="dashed", color="green", weight=3]; 82.06/55.93 4072 -> 4085[label="",style="dashed", color="green", weight=3]; 82.06/55.93 2382 -> 2234[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2382[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS yx2100 yx2110))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS yx2100 yx2110)) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS yx2100 yx2110))) yx213 == GT)))",fontsize=16,color="magenta"];2382 -> 2418[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2382 -> 2419[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2383[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) True)) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) True) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) True)) yx213 == GT)))",fontsize=16,color="black",shape="triangle"];2383 -> 2420[label="",style="solid", color="black", weight=3]; 82.06/55.93 2384[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) False)) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) False) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) False)) yx213 == GT)))",fontsize=16,color="black",shape="box"];2384 -> 2421[label="",style="solid", color="black", weight=3]; 82.06/55.93 2385 -> 2383[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2385[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) True)) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) True) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) True)) yx213 == GT)))",fontsize=16,color="magenta"];1978[label="primMinusNatS (Succ yx30000) Zero",fontsize=16,color="black",shape="triangle"];1978 -> 2041[label="",style="solid", color="black", weight=3]; 82.06/55.93 3046[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (LT == GT)))",fontsize=16,color="black",shape="triangle"];3046 -> 3081[label="",style="solid", color="black", weight=3]; 82.06/55.93 3047[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat yx2130 (Succ yx219) == GT)))",fontsize=16,color="burlywood",shape="box"];4605[label="yx2130/Succ yx21300",fontsize=10,color="white",style="solid",shape="box"];3047 -> 4605[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4605 -> 3082[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4606[label="yx2130/Zero",fontsize=10,color="white",style="solid",shape="box"];3047 -> 4606[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4606 -> 3083[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 1730[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];1730 -> 1749[label="",style="solid", color="black", weight=3]; 82.06/55.93 147[label="map toEnum (takeWhile1 (flip (<=) (fromIntegral (properFractionQ yx40 yx41))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (fromIntegral (properFractionQ yx40 yx41)) == GT)))",fontsize=16,color="black",shape="box"];147 -> 165[label="",style="solid", color="black", weight=3]; 82.06/55.93 4005[label="primDivNatS0 (Succ yx255) (Succ yx256) (primGEqNatS (Succ yx2570) yx258)",fontsize=16,color="burlywood",shape="box"];4607[label="yx258/Succ yx2580",fontsize=10,color="white",style="solid",shape="box"];4005 -> 4607[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4607 -> 4025[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4608[label="yx258/Zero",fontsize=10,color="white",style="solid",shape="box"];4005 -> 4608[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4608 -> 4026[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4006[label="primDivNatS0 (Succ yx255) (Succ yx256) (primGEqNatS Zero yx258)",fontsize=16,color="burlywood",shape="box"];4609[label="yx258/Succ yx2580",fontsize=10,color="white",style="solid",shape="box"];4006 -> 4609[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4609 -> 4027[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 4610[label="yx258/Zero",fontsize=10,color="white",style="solid",shape="box"];4006 -> 4610[label="",style="solid", color="burlywood", weight=9]; 82.06/55.93 4610 -> 4028[label="",style="solid", color="burlywood", weight=3]; 82.06/55.93 2069 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2069[label="primDivNatS (primMinusNatS (Succ yx40000) Zero) (Succ Zero)",fontsize=16,color="magenta"];2069 -> 2097[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2069 -> 2098[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2070 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2070[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];2070 -> 2099[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 2070 -> 2100[label="",style="dashed", color="magenta", weight=3]; 82.06/55.93 1447[label="fromInteger . toInteger",fontsize=16,color="black",shape="box"];1447 -> 1584[label="",style="solid", color="black", weight=3]; 82.06/55.93 4080[label="map toEnum (takeWhile0 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx236) True)",fontsize=16,color="black",shape="box"];4080 -> 4094[label="",style="solid", color="black", weight=3]; 82.06/55.93 3189[label="map toEnum []",fontsize=16,color="black",shape="triangle"];3189 -> 3224[label="",style="solid", color="black", weight=3]; 82.06/55.93 4479[label="yx2830",fontsize=16,color="green",shape="box"];4480[label="yx2840",fontsize=16,color="green",shape="box"];4481[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not True))",fontsize=16,color="black",shape="box"];4481 -> 4484[label="",style="solid", color="black", weight=3]; 82.06/55.93 4482[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not False))",fontsize=16,color="black",shape="triangle"];4482 -> 4485[label="",style="solid", color="black", weight=3]; 82.06/55.93 4483 -> 4482[label="",style="dashed", color="red", weight=0]; 82.06/55.93 4483[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) (not False))",fontsize=16,color="magenta"];4082[label="toEnum (Pos yx249)",fontsize=16,color="blue",shape="box"];4611[label="toEnum :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];4082 -> 4611[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4611 -> 4097[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4612[label="toEnum :: Int -> ()",fontsize=10,color="white",style="solid",shape="box"];4082 -> 4612[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4612 -> 4098[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4613[label="toEnum :: Int -> Char",fontsize=10,color="white",style="solid",shape="box"];4082 -> 4613[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4613 -> 4099[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4614[label="toEnum :: Int -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4082 -> 4614[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4614 -> 4100[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4615[label="toEnum :: Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];4082 -> 4615[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4615 -> 4101[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4616[label="toEnum :: Int -> Double",fontsize=10,color="white",style="solid",shape="box"];4082 -> 4616[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4616 -> 4102[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4617[label="toEnum :: Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];4082 -> 4617[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4617 -> 4103[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4618[label="toEnum :: Int -> Float",fontsize=10,color="white",style="solid",shape="box"];4082 -> 4618[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4618 -> 4104[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4619[label="toEnum :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4082 -> 4619[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4619 -> 4105[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4083[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (numericEnumFrom $! Pos yx250 + yx236))",fontsize=16,color="black",shape="box"];4083 -> 4106[label="",style="solid", color="black", weight=3]; 82.06/55.93 4084[label="toEnum (Pos yx249)",fontsize=16,color="blue",shape="box"];4620[label="toEnum :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];4084 -> 4620[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4620 -> 4107[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4621[label="toEnum :: Int -> ()",fontsize=10,color="white",style="solid",shape="box"];4084 -> 4621[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4621 -> 4108[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4622[label="toEnum :: Int -> Char",fontsize=10,color="white",style="solid",shape="box"];4084 -> 4622[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4622 -> 4109[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4623[label="toEnum :: Int -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4084 -> 4623[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4623 -> 4110[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4624[label="toEnum :: Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];4084 -> 4624[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4624 -> 4111[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4625[label="toEnum :: Int -> Double",fontsize=10,color="white",style="solid",shape="box"];4084 -> 4625[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4625 -> 4112[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4626[label="toEnum :: Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];4084 -> 4626[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4626 -> 4113[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4627[label="toEnum :: Int -> Float",fontsize=10,color="white",style="solid",shape="box"];4084 -> 4627[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4627 -> 4114[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4628[label="toEnum :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4084 -> 4628[label="",style="solid", color="blue", weight=9]; 82.06/55.93 4628 -> 4115[label="",style="solid", color="blue", weight=3]; 82.06/55.93 4085[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (numericEnumFrom $! Pos yx250 + yx236))",fontsize=16,color="black",shape="box"];4085 -> 4116[label="",style="solid", color="black", weight=3]; 82.06/55.93 2418[label="yx2110",fontsize=16,color="green",shape="box"];2419[label="yx2100",fontsize=16,color="green",shape="box"];2420 -> 2772[label="",style="dashed", color="red", weight=0]; 82.06/55.93 2420[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ (primDivNatS (primMinusNatS (Succ yx208) (Succ yx209)) (Succ (Succ yx209))))) (numericEnumFrom $! Neg (Succ (primDivNatS (primMinusNatS (Succ yx208) (Succ yx209)) (Succ (Succ yx209)))) + yx212) (not (primCmpInt (Neg (Succ (primDivNatS (primMinusNatS (Succ yx208) (Succ yx209)) (Succ (Succ yx209))))) yx213 == GT)))",fontsize=16,color="magenta"];2420 -> 2821[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 2420 -> 2822[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 2420 -> 2823[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 2421[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (primCmpInt (Neg Zero) yx213 == GT)))",fontsize=16,color="burlywood",shape="triangle"];4629[label="yx213/Pos yx2130",fontsize=10,color="white",style="solid",shape="box"];2421 -> 4629[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4629 -> 3014[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4630[label="yx213/Neg yx2130",fontsize=10,color="white",style="solid",shape="box"];2421 -> 4630[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4630 -> 3015[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 2041[label="Succ yx30000",fontsize=16,color="green",shape="box"];3081[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not False))",fontsize=16,color="black",shape="triangle"];3081 -> 3112[label="",style="solid", color="black", weight=3]; 82.06/55.94 3082[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat (Succ yx21300) (Succ yx219) == GT)))",fontsize=16,color="black",shape="box"];3082 -> 3113[label="",style="solid", color="black", weight=3]; 82.06/55.94 3083[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat Zero (Succ yx219) == GT)))",fontsize=16,color="black",shape="box"];3083 -> 3114[label="",style="solid", color="black", weight=3]; 82.06/55.94 1749[label="Zero",fontsize=16,color="green",shape="box"];165[label="map toEnum (takeWhile1 (flip (<=) (fromInteger . toInteger)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (fromInteger . toInteger) == GT)))",fontsize=16,color="black",shape="box"];165 -> 183[label="",style="solid", color="black", weight=3]; 82.06/55.94 4025[label="primDivNatS0 (Succ yx255) (Succ yx256) (primGEqNatS (Succ yx2570) (Succ yx2580))",fontsize=16,color="black",shape="box"];4025 -> 4037[label="",style="solid", color="black", weight=3]; 82.06/55.94 4026[label="primDivNatS0 (Succ yx255) (Succ yx256) (primGEqNatS (Succ yx2570) Zero)",fontsize=16,color="black",shape="box"];4026 -> 4038[label="",style="solid", color="black", weight=3]; 82.06/55.94 4027[label="primDivNatS0 (Succ yx255) (Succ yx256) (primGEqNatS Zero (Succ yx2580))",fontsize=16,color="black",shape="box"];4027 -> 4039[label="",style="solid", color="black", weight=3]; 82.06/55.94 4028[label="primDivNatS0 (Succ yx255) (Succ yx256) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];4028 -> 4040[label="",style="solid", color="black", weight=3]; 82.06/55.94 2097[label="Zero",fontsize=16,color="green",shape="box"];2098 -> 1978[label="",style="dashed", color="red", weight=0]; 82.06/55.94 2098[label="primMinusNatS (Succ yx40000) Zero",fontsize=16,color="magenta"];2098 -> 2120[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 2099[label="Zero",fontsize=16,color="green",shape="box"];2100 -> 1730[label="",style="dashed", color="red", weight=0]; 82.06/55.94 2100[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];1584[label="fromInteger (toInteger (properFractionQ yx70 yx71))",fontsize=16,color="blue",shape="box"];4631[label="toInteger :: Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];1584 -> 4631[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4631 -> 2034[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4632[label="toInteger :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];1584 -> 4632[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4632 -> 2035[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4094 -> 3189[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4094[label="map toEnum []",fontsize=16,color="magenta"];3224[label="[]",fontsize=16,color="green",shape="box"];4484[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) False)",fontsize=16,color="black",shape="box"];4484 -> 4486[label="",style="solid", color="black", weight=3]; 82.06/55.94 4485[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) True)",fontsize=16,color="black",shape="box"];4485 -> 4487[label="",style="solid", color="black", weight=3]; 82.06/55.94 4097[label="toEnum (Pos yx249)",fontsize=16,color="black",shape="triangle"];4097 -> 4132[label="",style="solid", color="black", weight=3]; 82.06/55.94 4098[label="toEnum (Pos yx249)",fontsize=16,color="black",shape="triangle"];4098 -> 4133[label="",style="solid", color="black", weight=3]; 82.06/55.94 4099[label="toEnum (Pos yx249)",fontsize=16,color="black",shape="triangle"];4099 -> 4134[label="",style="solid", color="black", weight=3]; 82.06/55.94 4100[label="toEnum (Pos yx249)",fontsize=16,color="black",shape="triangle"];4100 -> 4135[label="",style="solid", color="black", weight=3]; 82.06/55.94 4101[label="toEnum (Pos yx249)",fontsize=16,color="black",shape="triangle"];4101 -> 4136[label="",style="solid", color="black", weight=3]; 82.06/55.94 4102[label="toEnum (Pos yx249)",fontsize=16,color="black",shape="triangle"];4102 -> 4137[label="",style="solid", color="black", weight=3]; 82.06/55.94 4103[label="toEnum (Pos yx249)",fontsize=16,color="black",shape="triangle"];4103 -> 4138[label="",style="solid", color="black", weight=3]; 82.06/55.94 4104[label="toEnum (Pos yx249)",fontsize=16,color="black",shape="triangle"];4104 -> 4139[label="",style="solid", color="black", weight=3]; 82.06/55.94 4105[label="toEnum (Pos yx249)",fontsize=16,color="black",shape="triangle"];4105 -> 4140[label="",style="solid", color="black", weight=3]; 82.06/55.94 4106[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (Pos yx250 + yx236 `seq` numericEnumFrom (Pos yx250 + yx236)))",fontsize=16,color="black",shape="box"];4106 -> 4141[label="",style="solid", color="black", weight=3]; 82.06/55.94 4107 -> 4097[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4107[label="toEnum (Pos yx249)",fontsize=16,color="magenta"];4108 -> 4098[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4108[label="toEnum (Pos yx249)",fontsize=16,color="magenta"];4109 -> 4099[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4109[label="toEnum (Pos yx249)",fontsize=16,color="magenta"];4110 -> 4100[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4110[label="toEnum (Pos yx249)",fontsize=16,color="magenta"];4111 -> 4101[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4111[label="toEnum (Pos yx249)",fontsize=16,color="magenta"];4112 -> 4102[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4112[label="toEnum (Pos yx249)",fontsize=16,color="magenta"];4113 -> 4103[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4113[label="toEnum (Pos yx249)",fontsize=16,color="magenta"];4114 -> 4104[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4114[label="toEnum (Pos yx249)",fontsize=16,color="magenta"];4115 -> 4105[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4115[label="toEnum (Pos yx249)",fontsize=16,color="magenta"];4116[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (Pos yx250 + yx236 `seq` numericEnumFrom (Pos yx250 + yx236)))",fontsize=16,color="black",shape="box"];4116 -> 4142[label="",style="solid", color="black", weight=3]; 82.06/55.94 2821 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.94 2821[label="primDivNatS (primMinusNatS (Succ yx208) (Succ yx209)) (Succ (Succ yx209))",fontsize=16,color="magenta"];2821 -> 3026[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 2821 -> 3027[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 2822 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.94 2822[label="primDivNatS (primMinusNatS (Succ yx208) (Succ yx209)) (Succ (Succ yx209))",fontsize=16,color="magenta"];2822 -> 3028[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 2822 -> 3029[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 2823 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.94 2823[label="primDivNatS (primMinusNatS (Succ yx208) (Succ yx209)) (Succ (Succ yx209))",fontsize=16,color="magenta"];2823 -> 3030[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 2823 -> 3031[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3014[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (primCmpInt (Neg Zero) (Pos yx2130) == GT)))",fontsize=16,color="burlywood",shape="box"];4633[label="yx2130/Succ yx21300",fontsize=10,color="white",style="solid",shape="box"];3014 -> 4633[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4633 -> 3051[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4634[label="yx2130/Zero",fontsize=10,color="white",style="solid",shape="box"];3014 -> 4634[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4634 -> 3052[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3015[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (primCmpInt (Neg Zero) (Neg yx2130) == GT)))",fontsize=16,color="burlywood",shape="box"];4635[label="yx2130/Succ yx21300",fontsize=10,color="white",style="solid",shape="box"];3015 -> 4635[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4635 -> 3053[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4636[label="yx2130/Zero",fontsize=10,color="white",style="solid",shape="box"];3015 -> 4636[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4636 -> 3054[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3112[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) True)",fontsize=16,color="black",shape="box"];3112 -> 3146[label="",style="solid", color="black", weight=3]; 82.06/55.94 3113[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat yx21300 yx219 == GT)))",fontsize=16,color="burlywood",shape="triangle"];4637[label="yx21300/Succ yx213000",fontsize=10,color="white",style="solid",shape="box"];3113 -> 4637[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4637 -> 3147[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4638[label="yx21300/Zero",fontsize=10,color="white",style="solid",shape="box"];3113 -> 4638[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4638 -> 3148[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3114 -> 3046[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3114[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (LT == GT)))",fontsize=16,color="magenta"];183[label="map toEnum (takeWhile1 (flip (<=) (fromInteger (toInteger (properFractionQ yx40 yx41)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (fromInteger (toInteger (properFractionQ yx40 yx41))) == GT)))",fontsize=16,color="black",shape="box"];183 -> 203[label="",style="solid", color="black", weight=3]; 82.06/55.94 4037 -> 3958[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4037[label="primDivNatS0 (Succ yx255) (Succ yx256) (primGEqNatS yx2570 yx2580)",fontsize=16,color="magenta"];4037 -> 4050[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4037 -> 4051[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4038[label="primDivNatS0 (Succ yx255) (Succ yx256) True",fontsize=16,color="black",shape="triangle"];4038 -> 4052[label="",style="solid", color="black", weight=3]; 82.06/55.94 4039[label="primDivNatS0 (Succ yx255) (Succ yx256) False",fontsize=16,color="black",shape="box"];4039 -> 4053[label="",style="solid", color="black", weight=3]; 82.06/55.94 4040 -> 4038[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4040[label="primDivNatS0 (Succ yx255) (Succ yx256) True",fontsize=16,color="magenta"];2120[label="yx40000",fontsize=16,color="green",shape="box"];2034[label="fromInteger (toInteger (properFractionQ yx70 yx71))",fontsize=16,color="black",shape="box"];2034 -> 2321[label="",style="solid", color="black", weight=3]; 82.06/55.94 2035[label="fromInteger (toInteger (properFractionQ yx70 yx71))",fontsize=16,color="black",shape="box"];2035 -> 2322[label="",style="solid", color="black", weight=3]; 82.06/55.94 4486[label="map toEnum (takeWhile0 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) otherwise)",fontsize=16,color="black",shape="box"];4486 -> 4488[label="",style="solid", color="black", weight=3]; 82.06/55.94 4487[label="map toEnum (Pos yx280 : takeWhile (flip (<=) (Pos (Succ yx279))) (numericEnumFrom $! Pos yx281 + yx282))",fontsize=16,color="black",shape="box"];4487 -> 4489[label="",style="solid", color="black", weight=3]; 82.06/55.94 4132[label="error []",fontsize=16,color="red",shape="box"];4133[label="error []",fontsize=16,color="red",shape="box"];4134[label="error []",fontsize=16,color="red",shape="box"];4135[label="fromInt (Pos yx249)",fontsize=16,color="black",shape="box"];4135 -> 4150[label="",style="solid", color="black", weight=3]; 82.06/55.94 4136[label="error []",fontsize=16,color="red",shape="box"];4137[label="error []",fontsize=16,color="red",shape="box"];4138[label="error []",fontsize=16,color="red",shape="box"];4139[label="error []",fontsize=16,color="red",shape="box"];4140[label="error []",fontsize=16,color="red",shape="box"];4141[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (Pos yx250 + yx236)) (numericEnumFrom (Pos yx250 + yx236))))",fontsize=16,color="black",shape="box"];4141 -> 4151[label="",style="solid", color="black", weight=3]; 82.06/55.94 4142[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (enforceWHNF (WHNF (Pos yx250 + yx236)) (numericEnumFrom (Pos yx250 + yx236))))",fontsize=16,color="black",shape="box"];4142 -> 4152[label="",style="solid", color="black", weight=3]; 82.06/55.94 3026[label="Succ yx209",fontsize=16,color="green",shape="box"];3027 -> 3017[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3027[label="primMinusNatS (Succ yx208) (Succ yx209)",fontsize=16,color="magenta"];3027 -> 3060[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3027 -> 3061[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3028[label="Succ yx209",fontsize=16,color="green",shape="box"];3029 -> 3017[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3029[label="primMinusNatS (Succ yx208) (Succ yx209)",fontsize=16,color="magenta"];3029 -> 3062[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3029 -> 3063[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3030[label="Succ yx209",fontsize=16,color="green",shape="box"];3031 -> 3017[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3031[label="primMinusNatS (Succ yx208) (Succ yx209)",fontsize=16,color="magenta"];3031 -> 3064[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3031 -> 3065[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3051[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (primCmpInt (Neg Zero) (Pos (Succ yx21300)) == GT)))",fontsize=16,color="black",shape="box"];3051 -> 3087[label="",style="solid", color="black", weight=3]; 82.06/55.94 3052[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];3052 -> 3088[label="",style="solid", color="black", weight=3]; 82.06/55.94 3053[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (primCmpInt (Neg Zero) (Neg (Succ yx21300)) == GT)))",fontsize=16,color="black",shape="box"];3053 -> 3089[label="",style="solid", color="black", weight=3]; 82.06/55.94 3054[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (primCmpInt (Neg Zero) (Neg Zero) == GT)))",fontsize=16,color="black",shape="box"];3054 -> 3090[label="",style="solid", color="black", weight=3]; 82.06/55.94 3146[label="map toEnum (Neg (Succ yx217) : takeWhile (flip (<=) yx207) (numericEnumFrom $! Neg (Succ yx218) + yx212))",fontsize=16,color="black",shape="box"];3146 -> 3180[label="",style="solid", color="black", weight=3]; 82.06/55.94 3147[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat (Succ yx213000) yx219 == GT)))",fontsize=16,color="burlywood",shape="box"];4639[label="yx219/Succ yx2190",fontsize=10,color="white",style="solid",shape="box"];3147 -> 4639[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4639 -> 3181[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4640[label="yx219/Zero",fontsize=10,color="white",style="solid",shape="box"];3147 -> 4640[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4640 -> 3182[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3148[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat Zero yx219 == GT)))",fontsize=16,color="burlywood",shape="box"];4641[label="yx219/Succ yx2190",fontsize=10,color="white",style="solid",shape="box"];3148 -> 4641[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4641 -> 3183[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4642[label="yx219/Zero",fontsize=10,color="white",style="solid",shape="box"];3148 -> 4642[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4642 -> 3184[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 203[label="map toEnum (takeWhile1 (flip (<=) (fromInteger (Integer (properFractionQ yx40 yx41)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (fromInteger (Integer (properFractionQ yx40 yx41))) == GT)))",fontsize=16,color="black",shape="box"];203 -> 225[label="",style="solid", color="black", weight=3]; 82.06/55.94 4050[label="yx2580",fontsize=16,color="green",shape="box"];4051[label="yx2570",fontsize=16,color="green",shape="box"];4052[label="Succ (primDivNatS (primMinusNatS (Succ yx255) (Succ yx256)) (Succ (Succ yx256)))",fontsize=16,color="green",shape="box"];4052 -> 4063[label="",style="dashed", color="green", weight=3]; 82.06/55.94 4053[label="Zero",fontsize=16,color="green",shape="box"];2321[label="fromInteger (properFractionQ yx70 yx71)",fontsize=16,color="black",shape="box"];2321 -> 2353[label="",style="solid", color="black", weight=3]; 82.06/55.94 2322[label="fromInteger (Integer (properFractionQ yx70 yx71))",fontsize=16,color="black",shape="box"];2322 -> 2354[label="",style="solid", color="black", weight=3]; 82.06/55.94 4488[label="map toEnum (takeWhile0 (flip (<=) (Pos (Succ yx279))) (Pos yx280) (numericEnumFrom $! Pos yx281 + yx282) True)",fontsize=16,color="black",shape="box"];4488 -> 4490[label="",style="solid", color="black", weight=3]; 82.06/55.94 4489[label="toEnum (Pos yx280) : map toEnum (takeWhile (flip (<=) (Pos (Succ yx279))) (numericEnumFrom $! Pos yx281 + yx282))",fontsize=16,color="green",shape="box"];4489 -> 4491[label="",style="dashed", color="green", weight=3]; 82.06/55.94 4489 -> 4492[label="",style="dashed", color="green", weight=3]; 82.06/55.94 4150[label="intToRatio (Pos yx249)",fontsize=16,color="black",shape="box"];4150 -> 4161[label="",style="solid", color="black", weight=3]; 82.06/55.94 4151[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (primPlusInt (Pos yx250) yx236)) (numericEnumFrom (primPlusInt (Pos yx250) yx236))))",fontsize=16,color="burlywood",shape="box"];4643[label="yx236/Pos yx2360",fontsize=10,color="white",style="solid",shape="box"];4151 -> 4643[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4643 -> 4162[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4644[label="yx236/Neg yx2360",fontsize=10,color="white",style="solid",shape="box"];4151 -> 4644[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4644 -> 4163[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4152[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (enforceWHNF (WHNF (primPlusInt (Pos yx250) yx236)) (numericEnumFrom (primPlusInt (Pos yx250) yx236))))",fontsize=16,color="burlywood",shape="box"];4645[label="yx236/Pos yx2360",fontsize=10,color="white",style="solid",shape="box"];4152 -> 4645[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4645 -> 4164[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4646[label="yx236/Neg yx2360",fontsize=10,color="white",style="solid",shape="box"];4152 -> 4646[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4646 -> 4165[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3060[label="yx209",fontsize=16,color="green",shape="box"];3061[label="yx208",fontsize=16,color="green",shape="box"];3017[label="primMinusNatS (Succ yx200) (Succ yx201)",fontsize=16,color="black",shape="triangle"];3017 -> 3055[label="",style="solid", color="black", weight=3]; 82.06/55.94 3062[label="yx209",fontsize=16,color="green",shape="box"];3063[label="yx208",fontsize=16,color="green",shape="box"];3064[label="yx209",fontsize=16,color="green",shape="box"];3065[label="yx208",fontsize=16,color="green",shape="box"];3087[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (LT == GT)))",fontsize=16,color="black",shape="box"];3087 -> 3118[label="",style="solid", color="black", weight=3]; 82.06/55.94 3088[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (EQ == GT)))",fontsize=16,color="black",shape="triangle"];3088 -> 3119[label="",style="solid", color="black", weight=3]; 82.06/55.94 3089[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (primCmpNat (Succ yx21300) Zero == GT)))",fontsize=16,color="black",shape="box"];3089 -> 3120[label="",style="solid", color="black", weight=3]; 82.06/55.94 3090 -> 3088[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3090[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (EQ == GT)))",fontsize=16,color="magenta"];3180[label="toEnum (Neg (Succ yx217)) : map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom $! Neg (Succ yx218) + yx212))",fontsize=16,color="green",shape="box"];3180 -> 3214[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3180 -> 3215[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3181[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat (Succ yx213000) (Succ yx2190) == GT)))",fontsize=16,color="black",shape="box"];3181 -> 3216[label="",style="solid", color="black", weight=3]; 82.06/55.94 3182[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat (Succ yx213000) Zero == GT)))",fontsize=16,color="black",shape="box"];3182 -> 3217[label="",style="solid", color="black", weight=3]; 82.06/55.94 3183[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat Zero (Succ yx2190) == GT)))",fontsize=16,color="black",shape="box"];3183 -> 3218[label="",style="solid", color="black", weight=3]; 82.06/55.94 3184[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat Zero Zero == GT)))",fontsize=16,color="black",shape="box"];3184 -> 3219[label="",style="solid", color="black", weight=3]; 82.06/55.94 225[label="map toEnum (takeWhile1 (flip (<=) (properFractionQ yx40 yx41)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (properFractionQ yx40 yx41) == GT)))",fontsize=16,color="black",shape="box"];225 -> 247[label="",style="solid", color="black", weight=3]; 82.06/55.94 4063 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4063[label="primDivNatS (primMinusNatS (Succ yx255) (Succ yx256)) (Succ (Succ yx256))",fontsize=16,color="magenta"];4063 -> 4073[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4063 -> 4074[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 2353[label="fromInteger (properFractionQ1 yx70 yx71 (properFractionVu30 yx70 yx71))",fontsize=16,color="black",shape="box"];2353 -> 2390[label="",style="solid", color="black", weight=3]; 82.06/55.94 2354[label="properFractionQ yx70 yx71",fontsize=16,color="black",shape="box"];2354 -> 2391[label="",style="solid", color="black", weight=3]; 82.06/55.94 4490 -> 3189[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4490[label="map toEnum []",fontsize=16,color="magenta"];4491[label="toEnum (Pos yx280)",fontsize=16,color="blue",shape="box"];4647[label="toEnum :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];4491 -> 4647[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4647 -> 4493[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4648[label="toEnum :: Int -> ()",fontsize=10,color="white",style="solid",shape="box"];4491 -> 4648[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4648 -> 4494[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4649[label="toEnum :: Int -> Char",fontsize=10,color="white",style="solid",shape="box"];4491 -> 4649[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4649 -> 4495[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4650[label="toEnum :: Int -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4491 -> 4650[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4650 -> 4496[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4651[label="toEnum :: Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];4491 -> 4651[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4651 -> 4497[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4652[label="toEnum :: Int -> Double",fontsize=10,color="white",style="solid",shape="box"];4491 -> 4652[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4652 -> 4498[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4653[label="toEnum :: Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];4491 -> 4653[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4653 -> 4499[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4654[label="toEnum :: Int -> Float",fontsize=10,color="white",style="solid",shape="box"];4491 -> 4654[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4654 -> 4500[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4655[label="toEnum :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4491 -> 4655[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4655 -> 4501[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4492[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx279))) (numericEnumFrom $! Pos yx281 + yx282))",fontsize=16,color="black",shape="box"];4492 -> 4502[label="",style="solid", color="black", weight=3]; 82.06/55.94 4161[label="fromInt (Pos yx249) :% fromInt (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];4161 -> 4173[label="",style="dashed", color="green", weight=3]; 82.06/55.94 4161 -> 4174[label="",style="dashed", color="green", weight=3]; 82.06/55.94 4162[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (primPlusInt (Pos yx250) (Pos yx2360))) (numericEnumFrom (primPlusInt (Pos yx250) (Pos yx2360)))))",fontsize=16,color="black",shape="box"];4162 -> 4175[label="",style="solid", color="black", weight=3]; 82.06/55.94 4163[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (primPlusInt (Pos yx250) (Neg yx2360))) (numericEnumFrom (primPlusInt (Pos yx250) (Neg yx2360)))))",fontsize=16,color="black",shape="box"];4163 -> 4176[label="",style="solid", color="black", weight=3]; 82.06/55.94 4164[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (enforceWHNF (WHNF (primPlusInt (Pos yx250) (Pos yx2360))) (numericEnumFrom (primPlusInt (Pos yx250) (Pos yx2360)))))",fontsize=16,color="black",shape="box"];4164 -> 4177[label="",style="solid", color="black", weight=3]; 82.06/55.94 4165[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (enforceWHNF (WHNF (primPlusInt (Pos yx250) (Neg yx2360))) (numericEnumFrom (primPlusInt (Pos yx250) (Neg yx2360)))))",fontsize=16,color="black",shape="box"];4165 -> 4178[label="",style="solid", color="black", weight=3]; 82.06/55.94 3055[label="primMinusNatS yx200 yx201",fontsize=16,color="burlywood",shape="triangle"];4656[label="yx200/Succ yx2000",fontsize=10,color="white",style="solid",shape="box"];3055 -> 4656[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4656 -> 3091[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4657[label="yx200/Zero",fontsize=10,color="white",style="solid",shape="box"];3055 -> 4657[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4657 -> 3092[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3118[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not False))",fontsize=16,color="black",shape="triangle"];3118 -> 3154[label="",style="solid", color="black", weight=3]; 82.06/55.94 3119 -> 3118[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3119[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not False))",fontsize=16,color="magenta"];3120[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (GT == GT)))",fontsize=16,color="black",shape="box"];3120 -> 3155[label="",style="solid", color="black", weight=3]; 82.06/55.94 3214[label="toEnum (Neg (Succ yx217))",fontsize=16,color="blue",shape="box"];4658[label="toEnum :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];3214 -> 4658[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4658 -> 3248[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4659[label="toEnum :: Int -> ()",fontsize=10,color="white",style="solid",shape="box"];3214 -> 4659[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4659 -> 3249[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4660[label="toEnum :: Int -> Char",fontsize=10,color="white",style="solid",shape="box"];3214 -> 4660[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4660 -> 3250[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4661[label="toEnum :: Int -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];3214 -> 4661[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4661 -> 3251[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4662[label="toEnum :: Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];3214 -> 4662[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4662 -> 3252[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4663[label="toEnum :: Int -> Double",fontsize=10,color="white",style="solid",shape="box"];3214 -> 4663[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4663 -> 3253[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4664[label="toEnum :: Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];3214 -> 4664[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4664 -> 3254[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4665[label="toEnum :: Int -> Float",fontsize=10,color="white",style="solid",shape="box"];3214 -> 4665[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4665 -> 3255[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4666[label="toEnum :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];3214 -> 4666[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4666 -> 3256[label="",style="solid", color="blue", weight=3]; 82.06/55.94 3215[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom $! Neg (Succ yx218) + yx212))",fontsize=16,color="black",shape="box"];3215 -> 3257[label="",style="solid", color="black", weight=3]; 82.06/55.94 3216 -> 3113[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3216[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat yx213000 yx2190 == GT)))",fontsize=16,color="magenta"];3216 -> 3258[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3216 -> 3259[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3217[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (GT == GT)))",fontsize=16,color="black",shape="box"];3217 -> 3260[label="",style="solid", color="black", weight=3]; 82.06/55.94 3218 -> 3046[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3218[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (LT == GT)))",fontsize=16,color="magenta"];3219[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];3219 -> 3261[label="",style="solid", color="black", weight=3]; 82.06/55.94 247[label="map toEnum (takeWhile1 (flip (<=) (properFractionQ1 yx40 yx41 (properFractionVu30 yx40 yx41))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (properFractionQ1 yx40 yx41 (properFractionVu30 yx40 yx41)) == GT)))",fontsize=16,color="black",shape="box"];247 -> 271[label="",style="solid", color="black", weight=3]; 82.06/55.94 4073[label="Succ yx256",fontsize=16,color="green",shape="box"];4074 -> 3055[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4074[label="primMinusNatS (Succ yx255) (Succ yx256)",fontsize=16,color="magenta"];4074 -> 4086[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4074 -> 4087[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 2390[label="fromInteger (properFractionQ1 yx70 yx71 (quotRem yx70 yx71))",fontsize=16,color="burlywood",shape="box"];4667[label="yx70/Integer yx700",fontsize=10,color="white",style="solid",shape="box"];2390 -> 4667[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4667 -> 3032[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 2391[label="properFractionQ1 yx70 yx71 (properFractionVu30 yx70 yx71)",fontsize=16,color="black",shape="box"];2391 -> 3033[label="",style="solid", color="black", weight=3]; 82.06/55.94 4493 -> 4097[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4493[label="toEnum (Pos yx280)",fontsize=16,color="magenta"];4493 -> 4503[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4494 -> 4098[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4494[label="toEnum (Pos yx280)",fontsize=16,color="magenta"];4494 -> 4504[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4495 -> 4099[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4495[label="toEnum (Pos yx280)",fontsize=16,color="magenta"];4495 -> 4505[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4496 -> 4100[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4496[label="toEnum (Pos yx280)",fontsize=16,color="magenta"];4496 -> 4506[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4497 -> 4101[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4497[label="toEnum (Pos yx280)",fontsize=16,color="magenta"];4497 -> 4507[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4498 -> 4102[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4498[label="toEnum (Pos yx280)",fontsize=16,color="magenta"];4498 -> 4508[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4499 -> 4103[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4499[label="toEnum (Pos yx280)",fontsize=16,color="magenta"];4499 -> 4509[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4500 -> 4104[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4500[label="toEnum (Pos yx280)",fontsize=16,color="magenta"];4500 -> 4510[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4501 -> 4105[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4501[label="toEnum (Pos yx280)",fontsize=16,color="magenta"];4501 -> 4511[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4502[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx279))) (Pos yx281 + yx282 `seq` numericEnumFrom (Pos yx281 + yx282)))",fontsize=16,color="black",shape="box"];4502 -> 4512[label="",style="solid", color="black", weight=3]; 82.06/55.94 4173[label="fromInt (Pos yx249)",fontsize=16,color="blue",shape="box"];4668[label="fromInt :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4173 -> 4668[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4668 -> 4187[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4669[label="fromInt :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];4173 -> 4669[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4669 -> 4188[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4174[label="fromInt (Pos (Succ Zero))",fontsize=16,color="blue",shape="box"];4670[label="fromInt :: -> Int Int",fontsize=10,color="white",style="solid",shape="box"];4174 -> 4670[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4670 -> 4189[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4671[label="fromInt :: -> Int Integer",fontsize=10,color="white",style="solid",shape="box"];4174 -> 4671[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4671 -> 4190[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4175 -> 4191[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4175[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (Pos (primPlusNat yx250 yx2360))) (numericEnumFrom (Pos (primPlusNat yx250 yx2360)))))",fontsize=16,color="magenta"];4175 -> 4192[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4175 -> 4193[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4176 -> 3453[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4176[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (primMinusNat yx250 yx2360)) (numericEnumFrom (primMinusNat yx250 yx2360))))",fontsize=16,color="magenta"];4176 -> 4194[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4176 -> 4195[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4176 -> 4196[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4177 -> 4197[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4177[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (enforceWHNF (WHNF (Pos (primPlusNat yx250 yx2360))) (numericEnumFrom (Pos (primPlusNat yx250 yx2360)))))",fontsize=16,color="magenta"];4177 -> 4198[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4177 -> 4199[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4178 -> 3453[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4178[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (enforceWHNF (WHNF (primMinusNat yx250 yx2360)) (numericEnumFrom (primMinusNat yx250 yx2360))))",fontsize=16,color="magenta"];4178 -> 4200[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4178 -> 4201[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4178 -> 4202[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3091[label="primMinusNatS (Succ yx2000) yx201",fontsize=16,color="burlywood",shape="box"];4672[label="yx201/Succ yx2010",fontsize=10,color="white",style="solid",shape="box"];3091 -> 4672[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4672 -> 3121[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4673[label="yx201/Zero",fontsize=10,color="white",style="solid",shape="box"];3091 -> 4673[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4673 -> 3122[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3092[label="primMinusNatS Zero yx201",fontsize=16,color="burlywood",shape="box"];4674[label="yx201/Succ yx2010",fontsize=10,color="white",style="solid",shape="box"];3092 -> 4674[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4674 -> 3123[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4675[label="yx201/Zero",fontsize=10,color="white",style="solid",shape="box"];3092 -> 4675[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4675 -> 3124[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3154[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) True)",fontsize=16,color="black",shape="box"];3154 -> 3190[label="",style="solid", color="black", weight=3]; 82.06/55.94 3155[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not True))",fontsize=16,color="black",shape="box"];3155 -> 3191[label="",style="solid", color="black", weight=3]; 82.06/55.94 3248[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3248 -> 3289[label="",style="solid", color="black", weight=3]; 82.06/55.94 3249[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3249 -> 3290[label="",style="solid", color="black", weight=3]; 82.06/55.94 3250[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3250 -> 3291[label="",style="solid", color="black", weight=3]; 82.06/55.94 3251[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3251 -> 3292[label="",style="solid", color="black", weight=3]; 82.06/55.94 3252[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3252 -> 3293[label="",style="solid", color="black", weight=3]; 82.06/55.94 3253[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3253 -> 3294[label="",style="solid", color="black", weight=3]; 82.06/55.94 3254[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3254 -> 3295[label="",style="solid", color="black", weight=3]; 82.06/55.94 3255[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3255 -> 3296[label="",style="solid", color="black", weight=3]; 82.06/55.94 3256[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3256 -> 3297[label="",style="solid", color="black", weight=3]; 82.06/55.94 3257[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (Succ yx218) + yx212 `seq` numericEnumFrom (Neg (Succ yx218) + yx212)))",fontsize=16,color="black",shape="box"];3257 -> 3298[label="",style="solid", color="black", weight=3]; 82.06/55.94 3258[label="yx2190",fontsize=16,color="green",shape="box"];3259[label="yx213000",fontsize=16,color="green",shape="box"];3260[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not True))",fontsize=16,color="black",shape="box"];3260 -> 3299[label="",style="solid", color="black", weight=3]; 82.06/55.94 3261 -> 3081[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3261[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not False))",fontsize=16,color="magenta"];271[label="map toEnum (takeWhile1 (flip (<=) (properFractionQ1 yx40 yx41 (quotRem yx40 yx41))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (properFractionQ1 yx40 yx41 (quotRem yx40 yx41)) == GT)))",fontsize=16,color="black",shape="box"];271 -> 297[label="",style="solid", color="black", weight=3]; 82.06/55.94 4086[label="Succ yx256",fontsize=16,color="green",shape="box"];4087[label="Succ yx255",fontsize=16,color="green",shape="box"];3032[label="fromInteger (properFractionQ1 (Integer yx700) yx71 (quotRem (Integer yx700) yx71))",fontsize=16,color="burlywood",shape="box"];4676[label="yx71/Integer yx710",fontsize=10,color="white",style="solid",shape="box"];3032 -> 4676[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4676 -> 3066[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3033[label="properFractionQ1 yx70 yx71 (quotRem yx70 yx71)",fontsize=16,color="black",shape="box"];3033 -> 3067[label="",style="solid", color="black", weight=3]; 82.06/55.94 4503[label="yx280",fontsize=16,color="green",shape="box"];4504[label="yx280",fontsize=16,color="green",shape="box"];4505[label="yx280",fontsize=16,color="green",shape="box"];4506[label="yx280",fontsize=16,color="green",shape="box"];4507[label="yx280",fontsize=16,color="green",shape="box"];4508[label="yx280",fontsize=16,color="green",shape="box"];4509[label="yx280",fontsize=16,color="green",shape="box"];4510[label="yx280",fontsize=16,color="green",shape="box"];4511[label="yx280",fontsize=16,color="green",shape="box"];4512[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx279))) (enforceWHNF (WHNF (Pos yx281 + yx282)) (numericEnumFrom (Pos yx281 + yx282))))",fontsize=16,color="black",shape="box"];4512 -> 4513[label="",style="solid", color="black", weight=3]; 82.06/55.94 4187 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4187[label="fromInt (Pos yx249)",fontsize=16,color="magenta"];4187 -> 4216[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4188[label="fromInt (Pos yx249)",fontsize=16,color="black",shape="triangle"];4188 -> 4217[label="",style="solid", color="black", weight=3]; 82.06/55.94 4189 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4189[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4189 -> 4218[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4190 -> 4188[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4190[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4190 -> 4219[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4192 -> 3673[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4192[label="primPlusNat yx250 yx2360",fontsize=16,color="magenta"];4192 -> 4220[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4192 -> 4221[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4193 -> 3673[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4193[label="primPlusNat yx250 yx2360",fontsize=16,color="magenta"];4193 -> 4222[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4193 -> 4223[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4191[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (Pos yx260)) (numericEnumFrom (Pos yx259))))",fontsize=16,color="black",shape="triangle"];4191 -> 4224[label="",style="solid", color="black", weight=3]; 82.06/55.94 4194[label="Pos Zero",fontsize=16,color="green",shape="box"];4195[label="yx2360",fontsize=16,color="green",shape="box"];4196[label="yx250",fontsize=16,color="green",shape="box"];3453[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat yx21200 yx218)) (numericEnumFrom (primMinusNat yx21200 yx218))))",fontsize=16,color="burlywood",shape="triangle"];4677[label="yx21200/Succ yx212000",fontsize=10,color="white",style="solid",shape="box"];3453 -> 4677[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4677 -> 3482[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4678[label="yx21200/Zero",fontsize=10,color="white",style="solid",shape="box"];3453 -> 4678[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4678 -> 3483[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4198 -> 3673[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4198[label="primPlusNat yx250 yx2360",fontsize=16,color="magenta"];4198 -> 4225[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4198 -> 4226[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4199 -> 3673[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4199[label="primPlusNat yx250 yx2360",fontsize=16,color="magenta"];4199 -> 4227[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4199 -> 4228[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4197[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (enforceWHNF (WHNF (Pos yx262)) (numericEnumFrom (Pos yx261))))",fontsize=16,color="black",shape="triangle"];4197 -> 4229[label="",style="solid", color="black", weight=3]; 82.06/55.94 4200[label="Neg Zero",fontsize=16,color="green",shape="box"];4201[label="yx2360",fontsize=16,color="green",shape="box"];4202[label="yx250",fontsize=16,color="green",shape="box"];3121[label="primMinusNatS (Succ yx2000) (Succ yx2010)",fontsize=16,color="black",shape="box"];3121 -> 3156[label="",style="solid", color="black", weight=3]; 82.06/55.94 3122[label="primMinusNatS (Succ yx2000) Zero",fontsize=16,color="black",shape="box"];3122 -> 3157[label="",style="solid", color="black", weight=3]; 82.06/55.94 3123[label="primMinusNatS Zero (Succ yx2010)",fontsize=16,color="black",shape="box"];3123 -> 3158[label="",style="solid", color="black", weight=3]; 82.06/55.94 3124[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];3124 -> 3159[label="",style="solid", color="black", weight=3]; 82.06/55.94 3190[label="map toEnum (Neg Zero : takeWhile (flip (<=) yx207) (numericEnumFrom $! Neg Zero + yx212))",fontsize=16,color="black",shape="box"];3190 -> 3225[label="",style="solid", color="black", weight=3]; 82.06/55.94 3191[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) False)",fontsize=16,color="black",shape="box"];3191 -> 3226[label="",style="solid", color="black", weight=3]; 82.06/55.94 3289[label="error []",fontsize=16,color="red",shape="box"];3290[label="error []",fontsize=16,color="red",shape="box"];3291[label="error []",fontsize=16,color="red",shape="box"];3292[label="fromInt (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3292 -> 3329[label="",style="solid", color="black", weight=3]; 82.06/55.94 3293[label="error []",fontsize=16,color="red",shape="box"];3294[label="error []",fontsize=16,color="red",shape="box"];3295[label="error []",fontsize=16,color="red",shape="box"];3296[label="error []",fontsize=16,color="red",shape="box"];3297[label="error []",fontsize=16,color="red",shape="box"];3298[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Neg (Succ yx218) + yx212)) (numericEnumFrom (Neg (Succ yx218) + yx212))))",fontsize=16,color="black",shape="box"];3298 -> 3330[label="",style="solid", color="black", weight=3]; 82.06/55.94 3299[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) False)",fontsize=16,color="black",shape="box"];3299 -> 3331[label="",style="solid", color="black", weight=3]; 82.06/55.94 297[label="map toEnum (takeWhile1 (flip (<=) (properFractionQ1 yx40 yx41 (primQrmInt yx40 yx41))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (properFractionQ1 yx40 yx41 (primQrmInt yx40 yx41)) == GT)))",fontsize=16,color="black",shape="box"];297 -> 323[label="",style="solid", color="black", weight=3]; 82.06/55.94 3066[label="fromInteger (properFractionQ1 (Integer yx700) (Integer yx710) (quotRem (Integer yx700) (Integer yx710)))",fontsize=16,color="black",shape="box"];3066 -> 3096[label="",style="solid", color="black", weight=3]; 82.06/55.94 3067[label="properFractionQ1 yx70 yx71 (primQrmInt yx70 yx71)",fontsize=16,color="black",shape="box"];3067 -> 3097[label="",style="solid", color="black", weight=3]; 82.06/55.94 4513[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx279))) (enforceWHNF (WHNF (primPlusInt (Pos yx281) yx282)) (numericEnumFrom (primPlusInt (Pos yx281) yx282))))",fontsize=16,color="burlywood",shape="box"];4679[label="yx282/Pos yx2820",fontsize=10,color="white",style="solid",shape="box"];4513 -> 4679[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4679 -> 4514[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4680[label="yx282/Neg yx2820",fontsize=10,color="white",style="solid",shape="box"];4513 -> 4680[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4680 -> 4515[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4216[label="Pos yx249",fontsize=16,color="green",shape="box"];4217[label="Integer (Pos yx249)",fontsize=16,color="green",shape="box"];4218[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];4219[label="Succ Zero",fontsize=16,color="green",shape="box"];4220[label="yx250",fontsize=16,color="green",shape="box"];4221[label="yx2360",fontsize=16,color="green",shape="box"];3673[label="primPlusNat yx218 yx21200",fontsize=16,color="burlywood",shape="triangle"];4681[label="yx218/Succ yx2180",fontsize=10,color="white",style="solid",shape="box"];3673 -> 4681[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4681 -> 3686[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4682[label="yx218/Zero",fontsize=10,color="white",style="solid",shape="box"];3673 -> 4682[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4682 -> 3687[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4222[label="yx250",fontsize=16,color="green",shape="box"];4223[label="yx2360",fontsize=16,color="green",shape="box"];4224[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (numericEnumFrom (Pos yx259)))",fontsize=16,color="black",shape="box"];4224 -> 4240[label="",style="solid", color="black", weight=3]; 82.06/55.94 3482[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat (Succ yx212000) yx218)) (numericEnumFrom (primMinusNat (Succ yx212000) yx218))))",fontsize=16,color="burlywood",shape="box"];4683[label="yx218/Succ yx2180",fontsize=10,color="white",style="solid",shape="box"];3482 -> 4683[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4683 -> 3490[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4684[label="yx218/Zero",fontsize=10,color="white",style="solid",shape="box"];3482 -> 4684[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4684 -> 3491[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3483[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat Zero yx218)) (numericEnumFrom (primMinusNat Zero yx218))))",fontsize=16,color="burlywood",shape="box"];4685[label="yx218/Succ yx2180",fontsize=10,color="white",style="solid",shape="box"];3483 -> 4685[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4685 -> 3492[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4686[label="yx218/Zero",fontsize=10,color="white",style="solid",shape="box"];3483 -> 4686[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4686 -> 3493[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4225[label="yx250",fontsize=16,color="green",shape="box"];4226[label="yx2360",fontsize=16,color="green",shape="box"];4227[label="yx250",fontsize=16,color="green",shape="box"];4228[label="yx2360",fontsize=16,color="green",shape="box"];4229[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (numericEnumFrom (Pos yx261)))",fontsize=16,color="black",shape="box"];4229 -> 4241[label="",style="solid", color="black", weight=3]; 82.06/55.94 3156 -> 3055[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3156[label="primMinusNatS yx2000 yx2010",fontsize=16,color="magenta"];3156 -> 3192[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3156 -> 3193[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3157[label="Succ yx2000",fontsize=16,color="green",shape="box"];3158[label="Zero",fontsize=16,color="green",shape="box"];3159[label="Zero",fontsize=16,color="green",shape="box"];3225[label="toEnum (Neg Zero) : map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom $! Neg Zero + yx212))",fontsize=16,color="green",shape="box"];3225 -> 3263[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3225 -> 3264[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3226[label="map toEnum (takeWhile0 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) otherwise)",fontsize=16,color="black",shape="box"];3226 -> 3265[label="",style="solid", color="black", weight=3]; 82.06/55.94 3329[label="intToRatio (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3329 -> 3369[label="",style="solid", color="black", weight=3]; 82.06/55.94 3330[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primPlusInt (Neg (Succ yx218)) yx212)) (numericEnumFrom (primPlusInt (Neg (Succ yx218)) yx212))))",fontsize=16,color="burlywood",shape="box"];4687[label="yx212/Pos yx2120",fontsize=10,color="white",style="solid",shape="box"];3330 -> 4687[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4687 -> 3370[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4688[label="yx212/Neg yx2120",fontsize=10,color="white",style="solid",shape="box"];3330 -> 4688[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4688 -> 3371[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3331[label="map toEnum (takeWhile0 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) otherwise)",fontsize=16,color="black",shape="box"];3331 -> 3372[label="",style="solid", color="black", weight=3]; 82.06/55.94 323[label="map toEnum (takeWhile1 (flip (<=) (properFractionQ1 yx40 yx41 (primQuotInt yx40 yx41,primRemInt yx40 yx41))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (properFractionQ1 yx40 yx41 (primQuotInt yx40 yx41,primRemInt yx40 yx41)) == GT)))",fontsize=16,color="black",shape="box"];323 -> 353[label="",style="solid", color="black", weight=3]; 82.06/55.94 3096[label="fromInteger (properFractionQ1 (Integer yx700) (Integer yx710) (Integer (primQuotInt yx700 yx710),Integer (primRemInt yx700 yx710)))",fontsize=16,color="black",shape="box"];3096 -> 3128[label="",style="solid", color="black", weight=3]; 82.06/55.94 3097[label="properFractionQ1 yx70 yx71 (primQuotInt yx70 yx71,primRemInt yx70 yx71)",fontsize=16,color="black",shape="box"];3097 -> 3129[label="",style="solid", color="black", weight=3]; 82.06/55.94 4514[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx279))) (enforceWHNF (WHNF (primPlusInt (Pos yx281) (Pos yx2820))) (numericEnumFrom (primPlusInt (Pos yx281) (Pos yx2820)))))",fontsize=16,color="black",shape="box"];4514 -> 4516[label="",style="solid", color="black", weight=3]; 82.06/55.94 4515[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx279))) (enforceWHNF (WHNF (primPlusInt (Pos yx281) (Neg yx2820))) (numericEnumFrom (primPlusInt (Pos yx281) (Neg yx2820)))))",fontsize=16,color="black",shape="box"];4515 -> 4517[label="",style="solid", color="black", weight=3]; 82.06/55.94 3686[label="primPlusNat (Succ yx2180) yx21200",fontsize=16,color="burlywood",shape="box"];4689[label="yx21200/Succ yx212000",fontsize=10,color="white",style="solid",shape="box"];3686 -> 4689[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4689 -> 3946[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4690[label="yx21200/Zero",fontsize=10,color="white",style="solid",shape="box"];3686 -> 4690[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4690 -> 3947[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3687[label="primPlusNat Zero yx21200",fontsize=16,color="burlywood",shape="box"];4691[label="yx21200/Succ yx212000",fontsize=10,color="white",style="solid",shape="box"];3687 -> 4691[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4691 -> 3948[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4692[label="yx21200/Zero",fontsize=10,color="white",style="solid",shape="box"];3687 -> 4692[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4692 -> 3949[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4240 -> 4249[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4240[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (Pos yx259 : (numericEnumFrom $! Pos yx259 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];4240 -> 4250[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3490[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat (Succ yx212000) (Succ yx2180))) (numericEnumFrom (primMinusNat (Succ yx212000) (Succ yx2180)))))",fontsize=16,color="black",shape="box"];3490 -> 3515[label="",style="solid", color="black", weight=3]; 82.06/55.94 3491[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat (Succ yx212000) Zero)) (numericEnumFrom (primMinusNat (Succ yx212000) Zero))))",fontsize=16,color="black",shape="box"];3491 -> 3516[label="",style="solid", color="black", weight=3]; 82.06/55.94 3492[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat Zero (Succ yx2180))) (numericEnumFrom (primMinusNat Zero (Succ yx2180)))))",fontsize=16,color="black",shape="box"];3492 -> 3517[label="",style="solid", color="black", weight=3]; 82.06/55.94 3493[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat Zero Zero)) (numericEnumFrom (primMinusNat Zero Zero))))",fontsize=16,color="black",shape="box"];3493 -> 3518[label="",style="solid", color="black", weight=3]; 82.06/55.94 4241 -> 4251[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4241[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (Pos yx261 : (numericEnumFrom $! Pos yx261 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];4241 -> 4252[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3192[label="yx2010",fontsize=16,color="green",shape="box"];3193[label="yx2000",fontsize=16,color="green",shape="box"];3263[label="toEnum (Neg Zero)",fontsize=16,color="blue",shape="box"];4693[label="toEnum :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];3263 -> 4693[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4693 -> 3301[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4694[label="toEnum :: Int -> ()",fontsize=10,color="white",style="solid",shape="box"];3263 -> 4694[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4694 -> 3302[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4695[label="toEnum :: Int -> Char",fontsize=10,color="white",style="solid",shape="box"];3263 -> 4695[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4695 -> 3303[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4696[label="toEnum :: Int -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];3263 -> 4696[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4696 -> 3304[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4697[label="toEnum :: Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];3263 -> 4697[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4697 -> 3305[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4698[label="toEnum :: Int -> Double",fontsize=10,color="white",style="solid",shape="box"];3263 -> 4698[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4698 -> 3306[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4699[label="toEnum :: Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];3263 -> 4699[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4699 -> 3307[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4700[label="toEnum :: Int -> Float",fontsize=10,color="white",style="solid",shape="box"];3263 -> 4700[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4700 -> 3308[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4701[label="toEnum :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];3263 -> 4701[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4701 -> 3309[label="",style="solid", color="blue", weight=3]; 82.06/55.94 3264[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom $! Neg Zero + yx212))",fontsize=16,color="black",shape="box"];3264 -> 3310[label="",style="solid", color="black", weight=3]; 82.06/55.94 3265[label="map toEnum (takeWhile0 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) True)",fontsize=16,color="black",shape="box"];3265 -> 3311[label="",style="solid", color="black", weight=3]; 82.06/55.94 3369[label="fromInt (Neg (Succ yx217)) :% fromInt (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];3369 -> 3392[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3369 -> 3393[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3370[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primPlusInt (Neg (Succ yx218)) (Pos yx2120))) (numericEnumFrom (primPlusInt (Neg (Succ yx218)) (Pos yx2120)))))",fontsize=16,color="black",shape="box"];3370 -> 3394[label="",style="solid", color="black", weight=3]; 82.06/55.94 3371[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primPlusInt (Neg (Succ yx218)) (Neg yx2120))) (numericEnumFrom (primPlusInt (Neg (Succ yx218)) (Neg yx2120)))))",fontsize=16,color="black",shape="box"];3371 -> 3395[label="",style="solid", color="black", weight=3]; 82.06/55.94 3372[label="map toEnum (takeWhile0 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) True)",fontsize=16,color="black",shape="box"];3372 -> 3396[label="",style="solid", color="black", weight=3]; 82.06/55.94 353[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt yx40 yx41)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt yx40 yx41) == GT)))",fontsize=16,color="burlywood",shape="box"];4702[label="yx40/Pos yx400",fontsize=10,color="white",style="solid",shape="box"];353 -> 4702[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4702 -> 385[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4703[label="yx40/Neg yx400",fontsize=10,color="white",style="solid",shape="box"];353 -> 4703[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4703 -> 386[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3128[label="fromInteger (Integer (primQuotInt yx700 yx710))",fontsize=16,color="black",shape="box"];3128 -> 3162[label="",style="solid", color="black", weight=3]; 82.06/55.94 3129[label="primQuotInt yx70 yx71",fontsize=16,color="burlywood",shape="triangle"];4704[label="yx70/Pos yx700",fontsize=10,color="white",style="solid",shape="box"];3129 -> 4704[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4704 -> 3163[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4705[label="yx70/Neg yx700",fontsize=10,color="white",style="solid",shape="box"];3129 -> 4705[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4705 -> 3164[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4516 -> 4518[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4516[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx279))) (enforceWHNF (WHNF (Pos (primPlusNat yx281 yx2820))) (numericEnumFrom (Pos (primPlusNat yx281 yx2820)))))",fontsize=16,color="magenta"];4516 -> 4519[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4516 -> 4520[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4517 -> 3453[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4517[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx279))) (enforceWHNF (WHNF (primMinusNat yx281 yx2820)) (numericEnumFrom (primMinusNat yx281 yx2820))))",fontsize=16,color="magenta"];4517 -> 4521[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4517 -> 4522[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4517 -> 4523[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3946[label="primPlusNat (Succ yx2180) (Succ yx212000)",fontsize=16,color="black",shape="box"];3946 -> 4014[label="",style="solid", color="black", weight=3]; 82.06/55.94 3947[label="primPlusNat (Succ yx2180) Zero",fontsize=16,color="black",shape="box"];3947 -> 4015[label="",style="solid", color="black", weight=3]; 82.06/55.94 3948[label="primPlusNat Zero (Succ yx212000)",fontsize=16,color="black",shape="box"];3948 -> 4016[label="",style="solid", color="black", weight=3]; 82.06/55.94 3949[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];3949 -> 4017[label="",style="solid", color="black", weight=3]; 82.06/55.94 4250 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4250[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4250 -> 4262[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4249[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (Pos yx259 : (numericEnumFrom $! Pos yx259 + yx265)))",fontsize=16,color="black",shape="triangle"];4249 -> 4263[label="",style="solid", color="black", weight=3]; 82.06/55.94 3515 -> 3453[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3515[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat yx212000 yx2180)) (numericEnumFrom (primMinusNat yx212000 yx2180))))",fontsize=16,color="magenta"];3515 -> 3536[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3515 -> 3537[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3516[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Pos (Succ yx212000))) (numericEnumFrom (Pos (Succ yx212000)))))",fontsize=16,color="black",shape="triangle"];3516 -> 3538[label="",style="solid", color="black", weight=3]; 82.06/55.94 3517 -> 3454[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3517[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Neg (Succ yx2180))) (numericEnumFrom (Neg (Succ yx2180)))))",fontsize=16,color="magenta"];3517 -> 3539[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3518[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Pos Zero)) (numericEnumFrom (Pos Zero))))",fontsize=16,color="black",shape="triangle"];3518 -> 3540[label="",style="solid", color="black", weight=3]; 82.06/55.94 4252 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4252[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4252 -> 4264[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4251[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (Pos yx261 : (numericEnumFrom $! Pos yx261 + yx266)))",fontsize=16,color="black",shape="triangle"];4251 -> 4265[label="",style="solid", color="black", weight=3]; 82.06/55.94 3301[label="toEnum (Neg Zero)",fontsize=16,color="black",shape="box"];3301 -> 3333[label="",style="solid", color="black", weight=3]; 82.06/55.94 3302[label="toEnum (Neg Zero)",fontsize=16,color="black",shape="box"];3302 -> 3334[label="",style="solid", color="black", weight=3]; 82.06/55.94 3303[label="toEnum (Neg Zero)",fontsize=16,color="black",shape="box"];3303 -> 3335[label="",style="solid", color="black", weight=3]; 82.06/55.94 3304[label="toEnum (Neg Zero)",fontsize=16,color="black",shape="box"];3304 -> 3336[label="",style="solid", color="black", weight=3]; 82.06/55.94 3305[label="toEnum (Neg Zero)",fontsize=16,color="black",shape="box"];3305 -> 3337[label="",style="solid", color="black", weight=3]; 82.06/55.94 3306[label="toEnum (Neg Zero)",fontsize=16,color="black",shape="box"];3306 -> 3338[label="",style="solid", color="black", weight=3]; 82.06/55.94 3307[label="toEnum (Neg Zero)",fontsize=16,color="black",shape="box"];3307 -> 3339[label="",style="solid", color="black", weight=3]; 82.06/55.94 3308[label="toEnum (Neg Zero)",fontsize=16,color="black",shape="box"];3308 -> 3340[label="",style="solid", color="black", weight=3]; 82.06/55.94 3309[label="toEnum (Neg Zero)",fontsize=16,color="black",shape="box"];3309 -> 3341[label="",style="solid", color="black", weight=3]; 82.06/55.94 3310[label="map toEnum (takeWhile (flip (<=) yx207) (Neg Zero + yx212 `seq` numericEnumFrom (Neg Zero + yx212)))",fontsize=16,color="black",shape="box"];3310 -> 3342[label="",style="solid", color="black", weight=3]; 82.06/55.94 3311 -> 3189[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3311[label="map toEnum []",fontsize=16,color="magenta"];3392[label="fromInt (Neg (Succ yx217))",fontsize=16,color="blue",shape="box"];4706[label="fromInt :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];3392 -> 4706[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4706 -> 3419[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4707[label="fromInt :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];3392 -> 4707[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4707 -> 3420[label="",style="solid", color="blue", weight=3]; 82.06/55.94 3393[label="fromInt (Pos (Succ Zero))",fontsize=16,color="blue",shape="box"];4708[label="fromInt :: -> Int Int",fontsize=10,color="white",style="solid",shape="box"];3393 -> 4708[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4708 -> 3421[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4709[label="fromInt :: -> Int Integer",fontsize=10,color="white",style="solid",shape="box"];3393 -> 4709[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4709 -> 3422[label="",style="solid", color="blue", weight=3]; 82.06/55.94 3394[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat yx2120 (Succ yx218))) (numericEnumFrom (primMinusNat yx2120 (Succ yx218)))))",fontsize=16,color="burlywood",shape="box"];4710[label="yx2120/Succ yx21200",fontsize=10,color="white",style="solid",shape="box"];3394 -> 4710[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4710 -> 3423[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4711[label="yx2120/Zero",fontsize=10,color="white",style="solid",shape="box"];3394 -> 4711[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4711 -> 3424[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3395[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Neg (primPlusNat (Succ yx218) yx2120))) (numericEnumFrom (Neg (primPlusNat (Succ yx218) yx2120)))))",fontsize=16,color="black",shape="box"];3395 -> 3425[label="",style="solid", color="black", weight=3]; 82.06/55.94 3396 -> 3189[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3396[label="map toEnum []",fontsize=16,color="magenta"];385[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Pos yx400) yx41)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Pos yx400) yx41) == GT)))",fontsize=16,color="burlywood",shape="box"];4712[label="yx41/Pos yx410",fontsize=10,color="white",style="solid",shape="box"];385 -> 4712[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4712 -> 422[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4713[label="yx41/Neg yx410",fontsize=10,color="white",style="solid",shape="box"];385 -> 4713[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4713 -> 423[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 386[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Neg yx400) yx41)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Neg yx400) yx41) == GT)))",fontsize=16,color="burlywood",shape="box"];4714[label="yx41/Pos yx410",fontsize=10,color="white",style="solid",shape="box"];386 -> 4714[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4714 -> 424[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4715[label="yx41/Neg yx410",fontsize=10,color="white",style="solid",shape="box"];386 -> 4715[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4715 -> 425[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3162 -> 3129[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3162[label="primQuotInt yx700 yx710",fontsize=16,color="magenta"];3162 -> 3196[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3162 -> 3197[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3163[label="primQuotInt (Pos yx700) yx71",fontsize=16,color="burlywood",shape="box"];4716[label="yx71/Pos yx710",fontsize=10,color="white",style="solid",shape="box"];3163 -> 4716[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4716 -> 3198[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4717[label="yx71/Neg yx710",fontsize=10,color="white",style="solid",shape="box"];3163 -> 4717[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4717 -> 3199[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3164[label="primQuotInt (Neg yx700) yx71",fontsize=16,color="burlywood",shape="box"];4718[label="yx71/Pos yx710",fontsize=10,color="white",style="solid",shape="box"];3164 -> 4718[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4718 -> 3200[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4719[label="yx71/Neg yx710",fontsize=10,color="white",style="solid",shape="box"];3164 -> 4719[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4719 -> 3201[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4519 -> 3673[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4519[label="primPlusNat yx281 yx2820",fontsize=16,color="magenta"];4519 -> 4524[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4519 -> 4525[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4520 -> 3673[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4520[label="primPlusNat yx281 yx2820",fontsize=16,color="magenta"];4520 -> 4526[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4520 -> 4527[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4518[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx279))) (enforceWHNF (WHNF (Pos yx286)) (numericEnumFrom (Pos yx285))))",fontsize=16,color="black",shape="triangle"];4518 -> 4528[label="",style="solid", color="black", weight=3]; 82.06/55.94 4521[label="Pos (Succ yx279)",fontsize=16,color="green",shape="box"];4522[label="yx2820",fontsize=16,color="green",shape="box"];4523[label="yx281",fontsize=16,color="green",shape="box"];4014[label="Succ (Succ (primPlusNat yx2180 yx212000))",fontsize=16,color="green",shape="box"];4014 -> 4029[label="",style="dashed", color="green", weight=3]; 82.06/55.94 4015[label="Succ yx2180",fontsize=16,color="green",shape="box"];4016[label="Succ yx212000",fontsize=16,color="green",shape="box"];4017[label="Zero",fontsize=16,color="green",shape="box"];4262[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];4263[label="map toEnum (takeWhile2 (flip (<=) (Pos Zero)) (Pos yx259 : (numericEnumFrom $! Pos yx259 + yx265)))",fontsize=16,color="black",shape="box"];4263 -> 4273[label="",style="solid", color="black", weight=3]; 82.06/55.94 3536[label="yx2180",fontsize=16,color="green",shape="box"];3537[label="yx212000",fontsize=16,color="green",shape="box"];3538[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom (Pos (Succ yx212000))))",fontsize=16,color="black",shape="box"];3538 -> 3561[label="",style="solid", color="black", weight=3]; 82.06/55.94 3539[label="yx2180",fontsize=16,color="green",shape="box"];3454[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Neg (Succ yx218))) (numericEnumFrom (Neg (Succ yx218)))))",fontsize=16,color="black",shape="triangle"];3454 -> 3484[label="",style="solid", color="black", weight=3]; 82.06/55.94 3540[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom (Pos Zero)))",fontsize=16,color="black",shape="box"];3540 -> 3562[label="",style="solid", color="black", weight=3]; 82.06/55.94 4264[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];4265[label="map toEnum (takeWhile2 (flip (<=) (Neg Zero)) (Pos yx261 : (numericEnumFrom $! Pos yx261 + yx266)))",fontsize=16,color="black",shape="box"];4265 -> 4274[label="",style="solid", color="black", weight=3]; 82.06/55.94 3333[label="error []",fontsize=16,color="red",shape="box"];3334[label="error []",fontsize=16,color="red",shape="box"];3335[label="error []",fontsize=16,color="red",shape="box"];3336[label="fromInt (Neg Zero)",fontsize=16,color="black",shape="box"];3336 -> 3375[label="",style="solid", color="black", weight=3]; 82.06/55.94 3337[label="error []",fontsize=16,color="red",shape="box"];3338[label="error []",fontsize=16,color="red",shape="box"];3339[label="error []",fontsize=16,color="red",shape="box"];3340[label="error []",fontsize=16,color="red",shape="box"];3341[label="error []",fontsize=16,color="red",shape="box"];3342[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Neg Zero + yx212)) (numericEnumFrom (Neg Zero + yx212))))",fontsize=16,color="black",shape="box"];3342 -> 3376[label="",style="solid", color="black", weight=3]; 82.06/55.94 3419 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3419[label="fromInt (Neg (Succ yx217))",fontsize=16,color="magenta"];3419 -> 3449[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3420[label="fromInt (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3420 -> 3450[label="",style="solid", color="black", weight=3]; 82.06/55.94 3421 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3421[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3421 -> 3451[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3422[label="fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="triangle"];3422 -> 3452[label="",style="solid", color="black", weight=3]; 82.06/55.94 3423[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat (Succ yx21200) (Succ yx218))) (numericEnumFrom (primMinusNat (Succ yx21200) (Succ yx218)))))",fontsize=16,color="black",shape="box"];3423 -> 3453[label="",style="solid", color="black", weight=3]; 82.06/55.94 3424[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat Zero (Succ yx218))) (numericEnumFrom (primMinusNat Zero (Succ yx218)))))",fontsize=16,color="black",shape="box"];3424 -> 3454[label="",style="solid", color="black", weight=3]; 82.06/55.94 3425[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom (Neg (primPlusNat (Succ yx218) yx2120))))",fontsize=16,color="black",shape="box"];3425 -> 3455[label="",style="solid", color="black", weight=3]; 82.06/55.94 422[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Pos yx400) (Pos yx410))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Pos yx400) (Pos yx410)) == GT)))",fontsize=16,color="burlywood",shape="box"];4720[label="yx410/Succ yx4100",fontsize=10,color="white",style="solid",shape="box"];422 -> 4720[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4720 -> 475[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4721[label="yx410/Zero",fontsize=10,color="white",style="solid",shape="box"];422 -> 4721[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4721 -> 476[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 423[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Pos yx400) (Neg yx410))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Pos yx400) (Neg yx410)) == GT)))",fontsize=16,color="burlywood",shape="box"];4722[label="yx410/Succ yx4100",fontsize=10,color="white",style="solid",shape="box"];423 -> 4722[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4722 -> 477[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4723[label="yx410/Zero",fontsize=10,color="white",style="solid",shape="box"];423 -> 4723[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4723 -> 478[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 424[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Neg yx400) (Pos yx410))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Neg yx400) (Pos yx410)) == GT)))",fontsize=16,color="burlywood",shape="box"];4724[label="yx410/Succ yx4100",fontsize=10,color="white",style="solid",shape="box"];424 -> 4724[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4724 -> 479[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4725[label="yx410/Zero",fontsize=10,color="white",style="solid",shape="box"];424 -> 4725[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4725 -> 480[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 425[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Neg yx400) (Neg yx410))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Neg yx400) (Neg yx410)) == GT)))",fontsize=16,color="burlywood",shape="box"];4726[label="yx410/Succ yx4100",fontsize=10,color="white",style="solid",shape="box"];425 -> 4726[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4726 -> 481[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4727[label="yx410/Zero",fontsize=10,color="white",style="solid",shape="box"];425 -> 4727[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4727 -> 482[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3196[label="yx700",fontsize=16,color="green",shape="box"];3197[label="yx710",fontsize=16,color="green",shape="box"];3198[label="primQuotInt (Pos yx700) (Pos yx710)",fontsize=16,color="burlywood",shape="box"];4728[label="yx710/Succ yx7100",fontsize=10,color="white",style="solid",shape="box"];3198 -> 4728[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4728 -> 3230[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4729[label="yx710/Zero",fontsize=10,color="white",style="solid",shape="box"];3198 -> 4729[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4729 -> 3231[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3199[label="primQuotInt (Pos yx700) (Neg yx710)",fontsize=16,color="burlywood",shape="box"];4730[label="yx710/Succ yx7100",fontsize=10,color="white",style="solid",shape="box"];3199 -> 4730[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4730 -> 3232[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4731[label="yx710/Zero",fontsize=10,color="white",style="solid",shape="box"];3199 -> 4731[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4731 -> 3233[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3200[label="primQuotInt (Neg yx700) (Pos yx710)",fontsize=16,color="burlywood",shape="box"];4732[label="yx710/Succ yx7100",fontsize=10,color="white",style="solid",shape="box"];3200 -> 4732[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4732 -> 3234[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4733[label="yx710/Zero",fontsize=10,color="white",style="solid",shape="box"];3200 -> 4733[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4733 -> 3235[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3201[label="primQuotInt (Neg yx700) (Neg yx710)",fontsize=16,color="burlywood",shape="box"];4734[label="yx710/Succ yx7100",fontsize=10,color="white",style="solid",shape="box"];3201 -> 4734[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4734 -> 3236[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4735[label="yx710/Zero",fontsize=10,color="white",style="solid",shape="box"];3201 -> 4735[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4735 -> 3237[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4524[label="yx281",fontsize=16,color="green",shape="box"];4525[label="yx2820",fontsize=16,color="green",shape="box"];4526[label="yx281",fontsize=16,color="green",shape="box"];4527[label="yx2820",fontsize=16,color="green",shape="box"];4528[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx279))) (numericEnumFrom (Pos yx285)))",fontsize=16,color="black",shape="box"];4528 -> 4529[label="",style="solid", color="black", weight=3]; 82.06/55.94 4029 -> 3673[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4029[label="primPlusNat yx2180 yx212000",fontsize=16,color="magenta"];4029 -> 4041[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4029 -> 4042[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4273[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx259) (numericEnumFrom $! Pos yx259 + yx265) (flip (<=) (Pos Zero) (Pos yx259)))",fontsize=16,color="black",shape="box"];4273 -> 4282[label="",style="solid", color="black", weight=3]; 82.06/55.94 3561 -> 3582[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3561[label="map toEnum (takeWhile (flip (<=) yx207) (Pos (Succ yx212000) : (numericEnumFrom $! Pos (Succ yx212000) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3561 -> 3583[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3484[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom (Neg (Succ yx218))))",fontsize=16,color="black",shape="box"];3484 -> 3494[label="",style="solid", color="black", weight=3]; 82.06/55.94 3562 -> 3587[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3562[label="map toEnum (takeWhile (flip (<=) yx207) (Pos Zero : (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3562 -> 3588[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4274[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx261) (numericEnumFrom $! Pos yx261 + yx266) (flip (<=) (Neg Zero) (Pos yx261)))",fontsize=16,color="black",shape="box"];4274 -> 4283[label="",style="solid", color="black", weight=3]; 82.06/55.94 3375[label="intToRatio (Neg Zero)",fontsize=16,color="black",shape="box"];3375 -> 3407[label="",style="solid", color="black", weight=3]; 82.06/55.94 3376[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primPlusInt (Neg Zero) yx212)) (numericEnumFrom (primPlusInt (Neg Zero) yx212))))",fontsize=16,color="burlywood",shape="box"];4736[label="yx212/Pos yx2120",fontsize=10,color="white",style="solid",shape="box"];3376 -> 4736[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4736 -> 3408[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4737[label="yx212/Neg yx2120",fontsize=10,color="white",style="solid",shape="box"];3376 -> 4737[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4737 -> 3409[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3449[label="Neg (Succ yx217)",fontsize=16,color="green",shape="box"];3450[label="Integer (Neg (Succ yx217))",fontsize=16,color="green",shape="box"];3451[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3452[label="Integer (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];3455 -> 3485[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3455[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120) : (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3455 -> 3486[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 475[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Pos yx400) (Pos (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Pos yx400) (Pos (Succ yx4100))) == GT)))",fontsize=16,color="black",shape="box"];475 -> 534[label="",style="solid", color="black", weight=3]; 82.06/55.94 476[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Pos yx400) (Pos Zero))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Pos yx400) (Pos Zero)) == GT)))",fontsize=16,color="black",shape="box"];476 -> 535[label="",style="solid", color="black", weight=3]; 82.06/55.94 477[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Pos yx400) (Neg (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Pos yx400) (Neg (Succ yx4100))) == GT)))",fontsize=16,color="black",shape="box"];477 -> 536[label="",style="solid", color="black", weight=3]; 82.06/55.94 478[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Pos yx400) (Neg Zero))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Pos yx400) (Neg Zero)) == GT)))",fontsize=16,color="black",shape="box"];478 -> 537[label="",style="solid", color="black", weight=3]; 82.06/55.94 479[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Neg yx400) (Pos (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Neg yx400) (Pos (Succ yx4100))) == GT)))",fontsize=16,color="black",shape="box"];479 -> 538[label="",style="solid", color="black", weight=3]; 82.06/55.94 480[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Neg yx400) (Pos Zero))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Neg yx400) (Pos Zero)) == GT)))",fontsize=16,color="black",shape="box"];480 -> 539[label="",style="solid", color="black", weight=3]; 82.06/55.94 481[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Neg yx400) (Neg (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Neg yx400) (Neg (Succ yx4100))) == GT)))",fontsize=16,color="black",shape="box"];481 -> 540[label="",style="solid", color="black", weight=3]; 82.06/55.94 482[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Neg yx400) (Neg Zero))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Neg yx400) (Neg Zero)) == GT)))",fontsize=16,color="black",shape="box"];482 -> 541[label="",style="solid", color="black", weight=3]; 82.06/55.94 3230[label="primQuotInt (Pos yx700) (Pos (Succ yx7100))",fontsize=16,color="black",shape="box"];3230 -> 3276[label="",style="solid", color="black", weight=3]; 82.06/55.94 3231[label="primQuotInt (Pos yx700) (Pos Zero)",fontsize=16,color="black",shape="box"];3231 -> 3277[label="",style="solid", color="black", weight=3]; 82.06/55.94 3232[label="primQuotInt (Pos yx700) (Neg (Succ yx7100))",fontsize=16,color="black",shape="box"];3232 -> 3278[label="",style="solid", color="black", weight=3]; 82.06/55.94 3233[label="primQuotInt (Pos yx700) (Neg Zero)",fontsize=16,color="black",shape="box"];3233 -> 3279[label="",style="solid", color="black", weight=3]; 82.06/55.94 3234[label="primQuotInt (Neg yx700) (Pos (Succ yx7100))",fontsize=16,color="black",shape="box"];3234 -> 3280[label="",style="solid", color="black", weight=3]; 82.06/55.94 3235[label="primQuotInt (Neg yx700) (Pos Zero)",fontsize=16,color="black",shape="box"];3235 -> 3281[label="",style="solid", color="black", weight=3]; 82.06/55.94 3236[label="primQuotInt (Neg yx700) (Neg (Succ yx7100))",fontsize=16,color="black",shape="box"];3236 -> 3282[label="",style="solid", color="black", weight=3]; 82.06/55.94 3237[label="primQuotInt (Neg yx700) (Neg Zero)",fontsize=16,color="black",shape="box"];3237 -> 3283[label="",style="solid", color="black", weight=3]; 82.06/55.94 4529 -> 4530[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4529[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx279))) (Pos yx285 : (numericEnumFrom $! Pos yx285 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];4529 -> 4531[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4041[label="yx2180",fontsize=16,color="green",shape="box"];4042[label="yx212000",fontsize=16,color="green",shape="box"];4282[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx259) (numericEnumFrom $! Pos yx259 + yx265) ((<=) Pos yx259 Pos Zero))",fontsize=16,color="black",shape="box"];4282 -> 4292[label="",style="solid", color="black", weight=3]; 82.06/55.94 3583 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3583[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3583 -> 3599[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3582[label="map toEnum (takeWhile (flip (<=) yx207) (Pos (Succ yx212000) : (numericEnumFrom $! Pos (Succ yx212000) + yx237)))",fontsize=16,color="black",shape="triangle"];3582 -> 3600[label="",style="solid", color="black", weight=3]; 82.06/55.94 3494 -> 3519[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3494[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (Succ yx218) : (numericEnumFrom $! Neg (Succ yx218) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3494 -> 3520[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3588 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3588[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3588 -> 3601[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3587[label="map toEnum (takeWhile (flip (<=) yx207) (Pos Zero : (numericEnumFrom $! Pos Zero + yx238)))",fontsize=16,color="black",shape="triangle"];3587 -> 3602[label="",style="solid", color="black", weight=3]; 82.06/55.94 4283[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx261) (numericEnumFrom $! Pos yx261 + yx266) ((<=) Pos yx261 Neg Zero))",fontsize=16,color="black",shape="box"];4283 -> 4293[label="",style="solid", color="black", weight=3]; 82.06/55.94 3407[label="fromInt (Neg Zero) :% fromInt (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];3407 -> 3436[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3407 -> 3437[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3408[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primPlusInt (Neg Zero) (Pos yx2120))) (numericEnumFrom (primPlusInt (Neg Zero) (Pos yx2120)))))",fontsize=16,color="black",shape="box"];3408 -> 3438[label="",style="solid", color="black", weight=3]; 82.06/55.94 3409[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primPlusInt (Neg Zero) (Neg yx2120))) (numericEnumFrom (primPlusInt (Neg Zero) (Neg yx2120)))))",fontsize=16,color="black",shape="box"];3409 -> 3439[label="",style="solid", color="black", weight=3]; 82.06/55.94 3486 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3486[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3486 -> 3495[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3485[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120) : (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx231)))",fontsize=16,color="black",shape="triangle"];3485 -> 3496[label="",style="solid", color="black", weight=3]; 82.06/55.94 534[label="map toEnum (takeWhile1 (flip (<=) (Pos (primDivNatS yx400 (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Pos (primDivNatS yx400 (Succ yx4100))) == GT)))",fontsize=16,color="burlywood",shape="triangle"];4738[label="yx400/Succ yx4000",fontsize=10,color="white",style="solid",shape="box"];534 -> 4738[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4738 -> 592[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4739[label="yx400/Zero",fontsize=10,color="white",style="solid",shape="box"];534 -> 4739[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4739 -> 593[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 535[label="map toEnum (takeWhile1 (flip (<=) (error [])) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (error []) == GT)))",fontsize=16,color="black",shape="triangle"];535 -> 594[label="",style="solid", color="black", weight=3]; 82.06/55.94 536[label="map toEnum (takeWhile1 (flip (<=) (Neg (primDivNatS yx400 (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Neg (primDivNatS yx400 (Succ yx4100))) == GT)))",fontsize=16,color="burlywood",shape="triangle"];4740[label="yx400/Succ yx4000",fontsize=10,color="white",style="solid",shape="box"];536 -> 4740[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4740 -> 595[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4741[label="yx400/Zero",fontsize=10,color="white",style="solid",shape="box"];536 -> 4741[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4741 -> 596[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 537 -> 535[label="",style="dashed", color="red", weight=0]; 82.06/55.94 537[label="map toEnum (takeWhile1 (flip (<=) (error [])) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (error []) == GT)))",fontsize=16,color="magenta"];538 -> 536[label="",style="dashed", color="red", weight=0]; 82.06/55.94 538[label="map toEnum (takeWhile1 (flip (<=) (Neg (primDivNatS yx400 (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Neg (primDivNatS yx400 (Succ yx4100))) == GT)))",fontsize=16,color="magenta"];538 -> 597[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 538 -> 598[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 539 -> 535[label="",style="dashed", color="red", weight=0]; 82.06/55.94 539[label="map toEnum (takeWhile1 (flip (<=) (error [])) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (error []) == GT)))",fontsize=16,color="magenta"];540 -> 534[label="",style="dashed", color="red", weight=0]; 82.06/55.94 540[label="map toEnum (takeWhile1 (flip (<=) (Pos (primDivNatS yx400 (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Pos (primDivNatS yx400 (Succ yx4100))) == GT)))",fontsize=16,color="magenta"];540 -> 599[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 540 -> 600[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 541 -> 535[label="",style="dashed", color="red", weight=0]; 82.06/55.94 541[label="map toEnum (takeWhile1 (flip (<=) (error [])) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (error []) == GT)))",fontsize=16,color="magenta"];3276[label="Pos (primDivNatS yx700 (Succ yx7100))",fontsize=16,color="green",shape="box"];3276 -> 3322[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3277[label="error []",fontsize=16,color="black",shape="triangle"];3277 -> 3323[label="",style="solid", color="black", weight=3]; 82.06/55.94 3278[label="Neg (primDivNatS yx700 (Succ yx7100))",fontsize=16,color="green",shape="box"];3278 -> 3324[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3279 -> 3277[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3279[label="error []",fontsize=16,color="magenta"];3280[label="Neg (primDivNatS yx700 (Succ yx7100))",fontsize=16,color="green",shape="box"];3280 -> 3325[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3281 -> 3277[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3281[label="error []",fontsize=16,color="magenta"];3282[label="Pos (primDivNatS yx700 (Succ yx7100))",fontsize=16,color="green",shape="box"];3282 -> 3326[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3283 -> 3277[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3283[label="error []",fontsize=16,color="magenta"];4531 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4531[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4531 -> 4532[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4530[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx279))) (Pos yx285 : (numericEnumFrom $! Pos yx285 + yx287)))",fontsize=16,color="black",shape="triangle"];4530 -> 4533[label="",style="solid", color="black", weight=3]; 82.06/55.94 4292[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx259) (numericEnumFrom $! Pos yx259 + yx265) (compare (Pos yx259) (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];4292 -> 4321[label="",style="solid", color="black", weight=3]; 82.06/55.94 3599[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3600[label="map toEnum (takeWhile2 (flip (<=) yx207) (Pos (Succ yx212000) : (numericEnumFrom $! Pos (Succ yx212000) + yx237)))",fontsize=16,color="black",shape="box"];3600 -> 3613[label="",style="solid", color="black", weight=3]; 82.06/55.94 3520 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3520[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3520 -> 3541[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3519[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (Succ yx218) : (numericEnumFrom $! Neg (Succ yx218) + yx233)))",fontsize=16,color="black",shape="triangle"];3519 -> 3542[label="",style="solid", color="black", weight=3]; 82.06/55.94 3601[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3602[label="map toEnum (takeWhile2 (flip (<=) yx207) (Pos Zero : (numericEnumFrom $! Pos Zero + yx238)))",fontsize=16,color="black",shape="box"];3602 -> 3614[label="",style="solid", color="black", weight=3]; 82.06/55.94 4293[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx261) (numericEnumFrom $! Pos yx261 + yx266) (compare (Pos yx261) (Neg Zero) /= GT))",fontsize=16,color="black",shape="box"];4293 -> 4322[label="",style="solid", color="black", weight=3]; 82.06/55.94 3436[label="fromInt (Neg Zero)",fontsize=16,color="blue",shape="box"];4742[label="fromInt :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];3436 -> 4742[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4742 -> 3458[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4743[label="fromInt :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];3436 -> 4743[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4743 -> 3459[label="",style="solid", color="blue", weight=3]; 82.06/55.94 3437[label="fromInt (Pos (Succ Zero))",fontsize=16,color="blue",shape="box"];4744[label="fromInt :: -> Int Int",fontsize=10,color="white",style="solid",shape="box"];3437 -> 4744[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4744 -> 3460[label="",style="solid", color="blue", weight=3]; 82.06/55.94 4745[label="fromInt :: -> Int Integer",fontsize=10,color="white",style="solid",shape="box"];3437 -> 4745[label="",style="solid", color="blue", weight=9]; 82.06/55.94 4745 -> 3461[label="",style="solid", color="blue", weight=3]; 82.06/55.94 3438[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat yx2120 Zero)) (numericEnumFrom (primMinusNat yx2120 Zero))))",fontsize=16,color="burlywood",shape="box"];4746[label="yx2120/Succ yx21200",fontsize=10,color="white",style="solid",shape="box"];3438 -> 4746[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4746 -> 3462[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4747[label="yx2120/Zero",fontsize=10,color="white",style="solid",shape="box"];3438 -> 4747[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4747 -> 3463[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3439[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Neg (primPlusNat Zero yx2120))) (numericEnumFrom (Neg (primPlusNat Zero yx2120)))))",fontsize=16,color="black",shape="box"];3439 -> 3464[label="",style="solid", color="black", weight=3]; 82.06/55.94 3495[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3496[label="map toEnum (takeWhile2 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120) : (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx231)))",fontsize=16,color="black",shape="box"];3496 -> 3527[label="",style="solid", color="black", weight=3]; 82.06/55.94 592[label="map toEnum (takeWhile1 (flip (<=) (Pos (primDivNatS (Succ yx4000) (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Pos (primDivNatS (Succ yx4000) (Succ yx4100))) == GT)))",fontsize=16,color="black",shape="box"];592 -> 677[label="",style="solid", color="black", weight=3]; 82.06/55.94 593[label="map toEnum (takeWhile1 (flip (<=) (Pos (primDivNatS Zero (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Pos (primDivNatS Zero (Succ yx4100))) == GT)))",fontsize=16,color="black",shape="box"];593 -> 678[label="",style="solid", color="black", weight=3]; 82.06/55.94 594[label="error []",fontsize=16,color="red",shape="box"];595[label="map toEnum (takeWhile1 (flip (<=) (Neg (primDivNatS (Succ yx4000) (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Neg (primDivNatS (Succ yx4000) (Succ yx4100))) == GT)))",fontsize=16,color="black",shape="box"];595 -> 679[label="",style="solid", color="black", weight=3]; 82.06/55.94 596[label="map toEnum (takeWhile1 (flip (<=) (Neg (primDivNatS Zero (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Neg (primDivNatS Zero (Succ yx4100))) == GT)))",fontsize=16,color="black",shape="box"];596 -> 680[label="",style="solid", color="black", weight=3]; 82.06/55.94 597[label="yx4100",fontsize=16,color="green",shape="box"];598[label="yx400",fontsize=16,color="green",shape="box"];599[label="yx4100",fontsize=16,color="green",shape="box"];600[label="yx400",fontsize=16,color="green",shape="box"];3322 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3322[label="primDivNatS yx700 (Succ yx7100)",fontsize=16,color="magenta"];3322 -> 3345[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3322 -> 3346[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3323[label="error []",fontsize=16,color="red",shape="box"];3324 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3324[label="primDivNatS yx700 (Succ yx7100)",fontsize=16,color="magenta"];3324 -> 3347[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3324 -> 3348[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3325 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3325[label="primDivNatS yx700 (Succ yx7100)",fontsize=16,color="magenta"];3325 -> 3349[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3325 -> 3350[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3326 -> 1689[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3326[label="primDivNatS yx700 (Succ yx7100)",fontsize=16,color="magenta"];3326 -> 3351[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3326 -> 3352[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4532[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];4533[label="map toEnum (takeWhile2 (flip (<=) (Pos (Succ yx279))) (Pos yx285 : (numericEnumFrom $! Pos yx285 + yx287)))",fontsize=16,color="black",shape="box"];4533 -> 4534[label="",style="solid", color="black", weight=3]; 82.06/55.94 4321[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx259) (numericEnumFrom $! Pos yx259 + yx265) (not (compare (Pos yx259) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];4321 -> 4330[label="",style="solid", color="black", weight=3]; 82.06/55.94 3613[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos (Succ yx212000)) (numericEnumFrom $! Pos (Succ yx212000) + yx237) (flip (<=) yx207 (Pos (Succ yx212000))))",fontsize=16,color="black",shape="box"];3613 -> 3623[label="",style="solid", color="black", weight=3]; 82.06/55.94 3541[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3542[label="map toEnum (takeWhile2 (flip (<=) yx207) (Neg (Succ yx218) : (numericEnumFrom $! Neg (Succ yx218) + yx233)))",fontsize=16,color="black",shape="box"];3542 -> 3563[label="",style="solid", color="black", weight=3]; 82.06/55.94 3614[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos Zero) (numericEnumFrom $! Pos Zero + yx238) (flip (<=) yx207 (Pos Zero)))",fontsize=16,color="black",shape="box"];3614 -> 3624[label="",style="solid", color="black", weight=3]; 82.06/55.94 4322[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx261) (numericEnumFrom $! Pos yx261 + yx266) (not (compare (Pos yx261) (Neg Zero) == GT)))",fontsize=16,color="black",shape="box"];4322 -> 4331[label="",style="solid", color="black", weight=3]; 82.06/55.94 3458 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3458[label="fromInt (Neg Zero)",fontsize=16,color="magenta"];3458 -> 3497[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3459[label="fromInt (Neg Zero)",fontsize=16,color="black",shape="box"];3459 -> 3498[label="",style="solid", color="black", weight=3]; 82.06/55.94 3460 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3460[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3460 -> 3499[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3461 -> 3422[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3461[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3462[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat (Succ yx21200) Zero)) (numericEnumFrom (primMinusNat (Succ yx21200) Zero))))",fontsize=16,color="black",shape="box"];3462 -> 3500[label="",style="solid", color="black", weight=3]; 82.06/55.94 3463[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat Zero Zero)) (numericEnumFrom (primMinusNat Zero Zero))))",fontsize=16,color="black",shape="box"];3463 -> 3501[label="",style="solid", color="black", weight=3]; 82.06/55.94 3464[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom (Neg (primPlusNat Zero yx2120))))",fontsize=16,color="black",shape="box"];3464 -> 3502[label="",style="solid", color="black", weight=3]; 82.06/55.94 3527[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120)) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx231) (flip (<=) yx207 (Neg (primPlusNat (Succ yx218) yx2120))))",fontsize=16,color="black",shape="box"];3527 -> 3555[label="",style="solid", color="black", weight=3]; 82.06/55.94 677 -> 856[label="",style="dashed", color="red", weight=0]; 82.06/55.94 677[label="map toEnum (takeWhile1 (flip (<=) (Pos (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Pos (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))) == GT)))",fontsize=16,color="magenta"];677 -> 857[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 678 -> 858[label="",style="dashed", color="red", weight=0]; 82.06/55.94 678[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="magenta"];678 -> 859[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 679 -> 860[label="",style="dashed", color="red", weight=0]; 82.06/55.94 679[label="map toEnum (takeWhile1 (flip (<=) (Neg (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Neg (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))) == GT)))",fontsize=16,color="magenta"];679 -> 861[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 680 -> 862[label="",style="dashed", color="red", weight=0]; 82.06/55.94 680[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Neg Zero) == GT)))",fontsize=16,color="magenta"];680 -> 863[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3345[label="yx7100",fontsize=16,color="green",shape="box"];3346[label="yx700",fontsize=16,color="green",shape="box"];3347[label="yx7100",fontsize=16,color="green",shape="box"];3348[label="yx700",fontsize=16,color="green",shape="box"];3349[label="yx7100",fontsize=16,color="green",shape="box"];3350[label="yx700",fontsize=16,color="green",shape="box"];3351[label="yx7100",fontsize=16,color="green",shape="box"];3352[label="yx700",fontsize=16,color="green",shape="box"];4534[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx285) (numericEnumFrom $! Pos yx285 + yx287) (flip (<=) (Pos (Succ yx279)) (Pos yx285)))",fontsize=16,color="black",shape="box"];4534 -> 4535[label="",style="solid", color="black", weight=3]; 82.06/55.94 4330 -> 3690[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4330[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx259) (numericEnumFrom $! Pos yx259 + yx265) (not (primCmpInt (Pos yx259) (Pos Zero) == GT)))",fontsize=16,color="magenta"];4330 -> 4340[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4330 -> 4341[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4330 -> 4342[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4330 -> 4343[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4330 -> 4344[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3623[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos (Succ yx212000)) (numericEnumFrom $! Pos (Succ yx212000) + yx237) ((<=) Pos (Succ yx212000) yx207))",fontsize=16,color="black",shape="box"];3623 -> 3642[label="",style="solid", color="black", weight=3]; 82.06/55.94 3563[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx218)) (numericEnumFrom $! Neg (Succ yx218) + yx233) (flip (<=) yx207 (Neg (Succ yx218))))",fontsize=16,color="black",shape="box"];3563 -> 3593[label="",style="solid", color="black", weight=3]; 82.06/55.94 3624[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos Zero) (numericEnumFrom $! Pos Zero + yx238) ((<=) Pos Zero yx207))",fontsize=16,color="black",shape="box"];3624 -> 3643[label="",style="solid", color="black", weight=3]; 82.06/55.94 4331 -> 3690[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4331[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx261) (numericEnumFrom $! Pos yx261 + yx266) (not (primCmpInt (Pos yx261) (Neg Zero) == GT)))",fontsize=16,color="magenta"];4331 -> 4345[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4331 -> 4346[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4331 -> 4347[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4331 -> 4348[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4331 -> 4349[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3497[label="Neg Zero",fontsize=16,color="green",shape="box"];3498[label="Integer (Neg Zero)",fontsize=16,color="green",shape="box"];3499[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3500 -> 3516[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3500[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Pos (Succ yx21200))) (numericEnumFrom (Pos (Succ yx21200)))))",fontsize=16,color="magenta"];3500 -> 3546[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3501 -> 3518[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3501[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Pos Zero)) (numericEnumFrom (Pos Zero))))",fontsize=16,color="magenta"];3502 -> 3547[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3502[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (primPlusNat Zero yx2120) : (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3502 -> 3548[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3555[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120)) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx231) ((<=) Neg (primPlusNat (Succ yx218) yx2120) yx207))",fontsize=16,color="black",shape="box"];3555 -> 3579[label="",style="solid", color="black", weight=3]; 82.06/55.94 857 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 857[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];857 -> 1062[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 856 -> 803[label="",style="dashed", color="red", weight=0]; 82.06/55.94 856[label="map toEnum (takeWhile1 (flip (<=) (Pos (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + yx64) (not (primCmpInt (Neg Zero) (Pos (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))) == GT)))",fontsize=16,color="magenta"];856 -> 1063[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 856 -> 1064[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 856 -> 1065[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 859 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 859[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];859 -> 1066[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 858 -> 803[label="",style="dashed", color="red", weight=0]; 82.06/55.94 858[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Neg Zero) (numericEnumFrom $! Neg Zero + yx65) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="magenta"];858 -> 1067[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 858 -> 1068[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 858 -> 1069[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 861 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 861[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];861 -> 1070[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 860 -> 803[label="",style="dashed", color="red", weight=0]; 82.06/55.94 860[label="map toEnum (takeWhile1 (flip (<=) (Neg (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + yx66) (not (primCmpInt (Neg Zero) (Neg (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))) == GT)))",fontsize=16,color="magenta"];860 -> 1071[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 860 -> 1072[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 860 -> 1073[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 863 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 863[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];863 -> 1074[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 862 -> 803[label="",style="dashed", color="red", weight=0]; 82.06/55.94 862[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Neg Zero) (numericEnumFrom $! Neg Zero + yx67) (not (primCmpInt (Neg Zero) (Neg Zero) == GT)))",fontsize=16,color="magenta"];862 -> 1075[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 862 -> 1076[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 862 -> 1077[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4535[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx285) (numericEnumFrom $! Pos yx285 + yx287) ((<=) Pos yx285 Pos (Succ yx279)))",fontsize=16,color="black",shape="box"];4535 -> 4536[label="",style="solid", color="black", weight=3]; 82.06/55.94 4340[label="yx259",fontsize=16,color="green",shape="box"];4341[label="yx259",fontsize=16,color="green",shape="box"];4342[label="yx265",fontsize=16,color="green",shape="box"];4343[label="Pos Zero",fontsize=16,color="green",shape="box"];4344[label="yx259",fontsize=16,color="green",shape="box"];3642[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos (Succ yx212000)) (numericEnumFrom $! Pos (Succ yx212000) + yx237) (compare (Pos (Succ yx212000)) yx207 /= GT))",fontsize=16,color="black",shape="box"];3642 -> 3667[label="",style="solid", color="black", weight=3]; 82.06/55.94 3593[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx218)) (numericEnumFrom $! Neg (Succ yx218) + yx233) ((<=) Neg (Succ yx218) yx207))",fontsize=16,color="black",shape="box"];3593 -> 3608[label="",style="solid", color="black", weight=3]; 82.06/55.94 3643[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos Zero) (numericEnumFrom $! Pos Zero + yx238) (compare (Pos Zero) yx207 /= GT))",fontsize=16,color="black",shape="box"];3643 -> 3668[label="",style="solid", color="black", weight=3]; 82.06/55.94 4345[label="yx261",fontsize=16,color="green",shape="box"];4346[label="yx261",fontsize=16,color="green",shape="box"];4347[label="yx266",fontsize=16,color="green",shape="box"];4348[label="Neg Zero",fontsize=16,color="green",shape="box"];4349[label="yx261",fontsize=16,color="green",shape="box"];3546[label="yx21200",fontsize=16,color="green",shape="box"];3548 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3548[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3548 -> 3566[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3547[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (primPlusNat Zero yx2120) : (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx234)))",fontsize=16,color="black",shape="triangle"];3547 -> 3567[label="",style="solid", color="black", weight=3]; 82.06/55.94 3579[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120)) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx231) (compare (Neg (primPlusNat (Succ yx218) yx2120)) yx207 /= GT))",fontsize=16,color="black",shape="box"];3579 -> 3594[label="",style="solid", color="black", weight=3]; 82.06/55.94 1062[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];1063[label="Pos (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))",fontsize=16,color="green",shape="box"];1063 -> 1217[label="",style="dashed", color="green", weight=3]; 82.06/55.94 1064[label="Pos (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))",fontsize=16,color="green",shape="box"];1064 -> 1218[label="",style="dashed", color="green", weight=3]; 82.06/55.94 1065[label="yx64",fontsize=16,color="green",shape="box"];803[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) yx27 == GT)))",fontsize=16,color="burlywood",shape="triangle"];4748[label="yx27/Pos yx270",fontsize=10,color="white",style="solid",shape="box"];803 -> 4748[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4748 -> 972[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4749[label="yx27/Neg yx270",fontsize=10,color="white",style="solid",shape="box"];803 -> 4749[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4749 -> 973[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 1066[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];1067[label="Pos Zero",fontsize=16,color="green",shape="box"];1068[label="Pos Zero",fontsize=16,color="green",shape="box"];1069[label="yx65",fontsize=16,color="green",shape="box"];1070[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];1071[label="Neg (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))",fontsize=16,color="green",shape="box"];1071 -> 1219[label="",style="dashed", color="green", weight=3]; 82.06/55.94 1072[label="Neg (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))",fontsize=16,color="green",shape="box"];1072 -> 1220[label="",style="dashed", color="green", weight=3]; 82.06/55.94 1073[label="yx66",fontsize=16,color="green",shape="box"];1074[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];1075[label="Neg Zero",fontsize=16,color="green",shape="box"];1076[label="Neg Zero",fontsize=16,color="green",shape="box"];1077[label="yx67",fontsize=16,color="green",shape="box"];4536[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx285) (numericEnumFrom $! Pos yx285 + yx287) (compare (Pos yx285) (Pos (Succ yx279)) /= GT))",fontsize=16,color="black",shape="box"];4536 -> 4537[label="",style="solid", color="black", weight=3]; 82.06/55.94 3667[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos (Succ yx212000)) (numericEnumFrom $! Pos (Succ yx212000) + yx237) (not (compare (Pos (Succ yx212000)) yx207 == GT)))",fontsize=16,color="black",shape="box"];3667 -> 3681[label="",style="solid", color="black", weight=3]; 82.06/55.94 3608[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx218)) (numericEnumFrom $! Neg (Succ yx218) + yx233) (compare (Neg (Succ yx218)) yx207 /= GT))",fontsize=16,color="black",shape="box"];3608 -> 3617[label="",style="solid", color="black", weight=3]; 82.06/55.94 3668[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos Zero) (numericEnumFrom $! Pos Zero + yx238) (not (compare (Pos Zero) yx207 == GT)))",fontsize=16,color="black",shape="box"];3668 -> 3682[label="",style="solid", color="black", weight=3]; 82.06/55.94 3566[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3567[label="map toEnum (takeWhile2 (flip (<=) yx207) (Neg (primPlusNat Zero yx2120) : (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx234)))",fontsize=16,color="black",shape="box"];3567 -> 3598[label="",style="solid", color="black", weight=3]; 82.06/55.94 3594[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120)) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx231) (not (compare (Neg (primPlusNat (Succ yx218) yx2120)) yx207 == GT)))",fontsize=16,color="black",shape="box"];3594 -> 3609[label="",style="solid", color="black", weight=3]; 82.06/55.94 1217 -> 1151[label="",style="dashed", color="red", weight=0]; 82.06/55.94 1217[label="primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)",fontsize=16,color="magenta"];1218 -> 1151[label="",style="dashed", color="red", weight=0]; 82.06/55.94 1218[label="primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)",fontsize=16,color="magenta"];972[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) (Pos yx270) == GT)))",fontsize=16,color="burlywood",shape="box"];4750[label="yx270/Succ yx2700",fontsize=10,color="white",style="solid",shape="box"];972 -> 4750[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4750 -> 1161[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4751[label="yx270/Zero",fontsize=10,color="white",style="solid",shape="box"];972 -> 4751[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4751 -> 1162[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 973[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) (Neg yx270) == GT)))",fontsize=16,color="burlywood",shape="box"];4752[label="yx270/Succ yx2700",fontsize=10,color="white",style="solid",shape="box"];973 -> 4752[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4752 -> 1163[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4753[label="yx270/Zero",fontsize=10,color="white",style="solid",shape="box"];973 -> 4753[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4753 -> 1164[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 1219 -> 1151[label="",style="dashed", color="red", weight=0]; 82.06/55.94 1219[label="primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)",fontsize=16,color="magenta"];1219 -> 1342[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 1220 -> 1151[label="",style="dashed", color="red", weight=0]; 82.06/55.94 1220[label="primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)",fontsize=16,color="magenta"];1220 -> 1343[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4537[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx285) (numericEnumFrom $! Pos yx285 + yx287) (not (compare (Pos yx285) (Pos (Succ yx279)) == GT)))",fontsize=16,color="black",shape="box"];4537 -> 4538[label="",style="solid", color="black", weight=3]; 82.06/55.94 3681 -> 3690[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3681[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos (Succ yx212000)) (numericEnumFrom $! Pos (Succ yx212000) + yx237) (not (primCmpInt (Pos (Succ yx212000)) yx207 == GT)))",fontsize=16,color="magenta"];3681 -> 3908[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3681 -> 3909[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3681 -> 3910[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3681 -> 3911[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3681 -> 3912[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3617[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx218)) (numericEnumFrom $! Neg (Succ yx218) + yx233) (not (compare (Neg (Succ yx218)) yx207 == GT)))",fontsize=16,color="black",shape="box"];3617 -> 3636[label="",style="solid", color="black", weight=3]; 82.06/55.94 3682 -> 3690[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3682[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos Zero) (numericEnumFrom $! Pos Zero + yx238) (not (primCmpInt (Pos Zero) yx207 == GT)))",fontsize=16,color="magenta"];3682 -> 3913[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3682 -> 3914[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3682 -> 3915[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3682 -> 3916[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3682 -> 3917[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3598[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero yx2120)) (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx234) (flip (<=) yx207 (Neg (primPlusNat Zero yx2120))))",fontsize=16,color="black",shape="box"];3598 -> 3612[label="",style="solid", color="black", weight=3]; 82.06/55.94 3609[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120)) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx231) (not (primCmpInt (Neg (primPlusNat (Succ yx218) yx2120)) yx207 == GT)))",fontsize=16,color="burlywood",shape="box"];4754[label="yx2120/Succ yx21200",fontsize=10,color="white",style="solid",shape="box"];3609 -> 4754[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4754 -> 3618[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4755[label="yx2120/Zero",fontsize=10,color="white",style="solid",shape="box"];3609 -> 4755[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4755 -> 3619[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 1161[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) (Pos (Succ yx2700)) == GT)))",fontsize=16,color="black",shape="box"];1161 -> 1288[label="",style="solid", color="black", weight=3]; 82.06/55.94 1162[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];1162 -> 1289[label="",style="solid", color="black", weight=3]; 82.06/55.94 1163[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) (Neg (Succ yx2700)) == GT)))",fontsize=16,color="black",shape="box"];1163 -> 1290[label="",style="solid", color="black", weight=3]; 82.06/55.94 1164[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) (Neg Zero) == GT)))",fontsize=16,color="black",shape="box"];1164 -> 1291[label="",style="solid", color="black", weight=3]; 82.06/55.94 1342[label="yx4100",fontsize=16,color="green",shape="box"];1343[label="yx4100",fontsize=16,color="green",shape="box"];4538 -> 3690[label="",style="dashed", color="red", weight=0]; 82.06/55.94 4538[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx279))) (Pos yx285) (numericEnumFrom $! Pos yx285 + yx287) (not (primCmpInt (Pos yx285) (Pos (Succ yx279)) == GT)))",fontsize=16,color="magenta"];4538 -> 4539[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4538 -> 4540[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4538 -> 4541[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4538 -> 4542[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 4538 -> 4543[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3908[label="Succ yx212000",fontsize=16,color="green",shape="box"];3909[label="Succ yx212000",fontsize=16,color="green",shape="box"];3910[label="yx237",fontsize=16,color="green",shape="box"];3911[label="yx207",fontsize=16,color="green",shape="box"];3912[label="Succ yx212000",fontsize=16,color="green",shape="box"];3636 -> 2772[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3636[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx218)) (numericEnumFrom $! Neg (Succ yx218) + yx233) (not (primCmpInt (Neg (Succ yx218)) yx207 == GT)))",fontsize=16,color="magenta"];3636 -> 3651[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3636 -> 3652[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3636 -> 3653[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3636 -> 3654[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3913[label="Zero",fontsize=16,color="green",shape="box"];3914[label="Zero",fontsize=16,color="green",shape="box"];3915[label="yx238",fontsize=16,color="green",shape="box"];3916[label="yx207",fontsize=16,color="green",shape="box"];3917[label="Zero",fontsize=16,color="green",shape="box"];3612[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero yx2120)) (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx234) ((<=) Neg (primPlusNat Zero yx2120) yx207))",fontsize=16,color="black",shape="box"];3612 -> 3622[label="",style="solid", color="black", weight=3]; 82.06/55.94 3618[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) (Succ yx21200))) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) (Succ yx21200)) + yx231) (not (primCmpInt (Neg (primPlusNat (Succ yx218) (Succ yx21200))) yx207 == GT)))",fontsize=16,color="black",shape="box"];3618 -> 3637[label="",style="solid", color="black", weight=3]; 82.06/55.94 3619[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) Zero)) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) Zero) + yx231) (not (primCmpInt (Neg (primPlusNat (Succ yx218) Zero)) yx207 == GT)))",fontsize=16,color="black",shape="box"];3619 -> 3638[label="",style="solid", color="black", weight=3]; 82.06/55.94 1288[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (LT == GT)))",fontsize=16,color="black",shape="box"];1288 -> 1432[label="",style="solid", color="black", weight=3]; 82.06/55.94 1289[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (EQ == GT)))",fontsize=16,color="black",shape="triangle"];1289 -> 1433[label="",style="solid", color="black", weight=3]; 82.06/55.94 1290[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpNat (Succ yx2700) Zero == GT)))",fontsize=16,color="black",shape="box"];1290 -> 1434[label="",style="solid", color="black", weight=3]; 82.06/55.94 1291 -> 1289[label="",style="dashed", color="red", weight=0]; 82.06/55.94 1291[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (EQ == GT)))",fontsize=16,color="magenta"];4539[label="yx285",fontsize=16,color="green",shape="box"];4540[label="yx285",fontsize=16,color="green",shape="box"];4541[label="yx287",fontsize=16,color="green",shape="box"];4542[label="Pos (Succ yx279)",fontsize=16,color="green",shape="box"];4543[label="yx285",fontsize=16,color="green",shape="box"];3651[label="yx218",fontsize=16,color="green",shape="box"];3652[label="yx218",fontsize=16,color="green",shape="box"];3653[label="yx207",fontsize=16,color="green",shape="box"];3654[label="yx233",fontsize=16,color="green",shape="box"];3622[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero yx2120)) (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx234) (compare (Neg (primPlusNat Zero yx2120)) yx207 /= GT))",fontsize=16,color="black",shape="box"];3622 -> 3641[label="",style="solid", color="black", weight=3]; 82.06/55.94 3637 -> 2772[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3637[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ (Succ (primPlusNat yx218 yx21200)))) (numericEnumFrom $! Neg (Succ (Succ (primPlusNat yx218 yx21200))) + yx231) (not (primCmpInt (Neg (Succ (Succ (primPlusNat yx218 yx21200)))) yx207 == GT)))",fontsize=16,color="magenta"];3637 -> 3655[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3637 -> 3656[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3637 -> 3657[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3637 -> 3658[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3637 -> 3659[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3638 -> 2772[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3638[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx218)) (numericEnumFrom $! Neg (Succ yx218) + yx231) (not (primCmpInt (Neg (Succ yx218)) yx207 == GT)))",fontsize=16,color="magenta"];3638 -> 3660[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3638 -> 3661[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3638 -> 3662[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3638 -> 3663[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 1432[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not False))",fontsize=16,color="black",shape="triangle"];1432 -> 1570[label="",style="solid", color="black", weight=3]; 82.06/55.94 1433 -> 1432[label="",style="dashed", color="red", weight=0]; 82.06/55.94 1433[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not False))",fontsize=16,color="magenta"];1434[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (GT == GT)))",fontsize=16,color="black",shape="box"];1434 -> 1571[label="",style="solid", color="black", weight=3]; 82.06/55.94 3641[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero yx2120)) (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx234) (not (compare (Neg (primPlusNat Zero yx2120)) yx207 == GT)))",fontsize=16,color="black",shape="box"];3641 -> 3666[label="",style="solid", color="black", weight=3]; 82.06/55.94 3655[label="Succ (primPlusNat yx218 yx21200)",fontsize=16,color="green",shape="box"];3655 -> 3673[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3656[label="Succ (primPlusNat yx218 yx21200)",fontsize=16,color="green",shape="box"];3656 -> 3674[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3657[label="Succ (primPlusNat yx218 yx21200)",fontsize=16,color="green",shape="box"];3657 -> 3675[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3658[label="yx207",fontsize=16,color="green",shape="box"];3659[label="yx231",fontsize=16,color="green",shape="box"];3660[label="yx218",fontsize=16,color="green",shape="box"];3661[label="yx218",fontsize=16,color="green",shape="box"];3662[label="yx207",fontsize=16,color="green",shape="box"];3663[label="yx231",fontsize=16,color="green",shape="box"];1570[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) True)",fontsize=16,color="black",shape="box"];1570 -> 1781[label="",style="solid", color="black", weight=3]; 82.06/55.94 1571[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not True))",fontsize=16,color="black",shape="box"];1571 -> 1782[label="",style="solid", color="black", weight=3]; 82.06/55.94 3666[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero yx2120)) (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx234) (not (primCmpInt (Neg (primPlusNat Zero yx2120)) yx207 == GT)))",fontsize=16,color="burlywood",shape="box"];4756[label="yx2120/Succ yx21200",fontsize=10,color="white",style="solid",shape="box"];3666 -> 4756[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4756 -> 3679[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4757[label="yx2120/Zero",fontsize=10,color="white",style="solid",shape="box"];3666 -> 4757[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4757 -> 3680[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3674 -> 3673[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3674[label="primPlusNat yx218 yx21200",fontsize=16,color="magenta"];3675 -> 3673[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3675[label="primPlusNat yx218 yx21200",fontsize=16,color="magenta"];1781[label="map toEnum (Neg Zero : takeWhile (flip (<=) yx26) (numericEnumFrom $! Neg Zero + yx57))",fontsize=16,color="black",shape="box"];1781 -> 2343[label="",style="solid", color="black", weight=3]; 82.06/55.94 1782[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) False)",fontsize=16,color="black",shape="box"];1782 -> 2344[label="",style="solid", color="black", weight=3]; 82.06/55.94 3679[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero (Succ yx21200))) (numericEnumFrom $! Neg (primPlusNat Zero (Succ yx21200)) + yx234) (not (primCmpInt (Neg (primPlusNat Zero (Succ yx21200))) yx207 == GT)))",fontsize=16,color="black",shape="box"];3679 -> 3944[label="",style="solid", color="black", weight=3]; 82.06/55.94 3680[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero Zero)) (numericEnumFrom $! Neg (primPlusNat Zero Zero) + yx234) (not (primCmpInt (Neg (primPlusNat Zero Zero)) yx207 == GT)))",fontsize=16,color="black",shape="box"];3680 -> 3945[label="",style="solid", color="black", weight=3]; 82.06/55.94 2343[label="toEnum (Neg Zero) : map toEnum (takeWhile (flip (<=) yx26) (numericEnumFrom $! Neg Zero + yx57))",fontsize=16,color="green",shape="box"];2343 -> 2379[label="",style="dashed", color="green", weight=3]; 82.06/55.94 2343 -> 2380[label="",style="dashed", color="green", weight=3]; 82.06/55.94 2344[label="map toEnum (takeWhile0 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) otherwise)",fontsize=16,color="black",shape="box"];2344 -> 2381[label="",style="solid", color="black", weight=3]; 82.06/55.94 3944 -> 2772[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3944[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx21200)) (numericEnumFrom $! Neg (Succ yx21200) + yx234) (not (primCmpInt (Neg (Succ yx21200)) yx207 == GT)))",fontsize=16,color="magenta"];3944 -> 4007[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3944 -> 4008[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3944 -> 4009[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3944 -> 4010[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3944 -> 4011[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3945 -> 2421[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3945[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx234) (not (primCmpInt (Neg Zero) yx207 == GT)))",fontsize=16,color="magenta"];3945 -> 4012[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3945 -> 4013[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 2379[label="toEnum (Neg Zero)",fontsize=16,color="black",shape="box"];2379 -> 2415[label="",style="solid", color="black", weight=3]; 82.06/55.94 2380[label="map toEnum (takeWhile (flip (<=) yx26) (numericEnumFrom $! Neg Zero + yx57))",fontsize=16,color="black",shape="box"];2380 -> 2416[label="",style="solid", color="black", weight=3]; 82.06/55.94 2381[label="map toEnum (takeWhile0 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) True)",fontsize=16,color="black",shape="box"];2381 -> 2417[label="",style="solid", color="black", weight=3]; 82.06/55.94 4007[label="yx21200",fontsize=16,color="green",shape="box"];4008[label="yx21200",fontsize=16,color="green",shape="box"];4009[label="yx21200",fontsize=16,color="green",shape="box"];4010[label="yx207",fontsize=16,color="green",shape="box"];4011[label="yx234",fontsize=16,color="green",shape="box"];4012[label="yx207",fontsize=16,color="green",shape="box"];4013[label="yx234",fontsize=16,color="green",shape="box"];2415[label="fromInt (Neg Zero)",fontsize=16,color="black",shape="box"];2415 -> 3044[label="",style="solid", color="black", weight=3]; 82.06/55.94 2416[label="map toEnum (takeWhile (flip (<=) yx26) (Neg Zero + yx57 `seq` numericEnumFrom (Neg Zero + yx57)))",fontsize=16,color="black",shape="box"];2416 -> 3045[label="",style="solid", color="black", weight=3]; 82.06/55.94 2417 -> 2375[label="",style="dashed", color="red", weight=0]; 82.06/55.94 2417[label="map toEnum []",fontsize=16,color="magenta"];3044[label="intToRatio (Neg Zero)",fontsize=16,color="black",shape="box"];3044 -> 3079[label="",style="solid", color="black", weight=3]; 82.06/55.94 3045[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (Neg Zero + yx57)) (numericEnumFrom (Neg Zero + yx57))))",fontsize=16,color="black",shape="box"];3045 -> 3080[label="",style="solid", color="black", weight=3]; 82.06/55.94 2375[label="map toEnum []",fontsize=16,color="black",shape="triangle"];2375 -> 2412[label="",style="solid", color="black", weight=3]; 82.06/55.94 3079[label="fromInt (Neg Zero) :% fromInt (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];3079 -> 3108[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3079 -> 3109[label="",style="dashed", color="green", weight=3]; 82.06/55.94 3080[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (primPlusInt (Neg Zero) yx57)) (numericEnumFrom (primPlusInt (Neg Zero) yx57))))",fontsize=16,color="burlywood",shape="box"];4758[label="yx57/Pos yx570",fontsize=10,color="white",style="solid",shape="box"];3080 -> 4758[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4758 -> 3110[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4759[label="yx57/Neg yx570",fontsize=10,color="white",style="solid",shape="box"];3080 -> 4759[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4759 -> 3111[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 2412[label="[]",fontsize=16,color="green",shape="box"];3108 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3108[label="fromInt (Neg Zero)",fontsize=16,color="magenta"];3108 -> 3142[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3109 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3109[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3109 -> 3143[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3110[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (primPlusInt (Neg Zero) (Pos yx570))) (numericEnumFrom (primPlusInt (Neg Zero) (Pos yx570)))))",fontsize=16,color="black",shape="box"];3110 -> 3144[label="",style="solid", color="black", weight=3]; 82.06/55.94 3111[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (primPlusInt (Neg Zero) (Neg yx570))) (numericEnumFrom (primPlusInt (Neg Zero) (Neg yx570)))))",fontsize=16,color="black",shape="box"];3111 -> 3145[label="",style="solid", color="black", weight=3]; 82.06/55.94 3142[label="Neg Zero",fontsize=16,color="green",shape="box"];3143[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3144[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (primMinusNat yx570 Zero)) (numericEnumFrom (primMinusNat yx570 Zero))))",fontsize=16,color="burlywood",shape="box"];4760[label="yx570/Succ yx5700",fontsize=10,color="white",style="solid",shape="box"];3144 -> 4760[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4760 -> 3177[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 4761[label="yx570/Zero",fontsize=10,color="white",style="solid",shape="box"];3144 -> 4761[label="",style="solid", color="burlywood", weight=9]; 82.06/55.94 4761 -> 3178[label="",style="solid", color="burlywood", weight=3]; 82.06/55.94 3145[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (Neg (primPlusNat Zero yx570))) (numericEnumFrom (Neg (primPlusNat Zero yx570)))))",fontsize=16,color="black",shape="box"];3145 -> 3179[label="",style="solid", color="black", weight=3]; 82.06/55.94 3177[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (primMinusNat (Succ yx5700) Zero)) (numericEnumFrom (primMinusNat (Succ yx5700) Zero))))",fontsize=16,color="black",shape="box"];3177 -> 3211[label="",style="solid", color="black", weight=3]; 82.06/55.94 3178[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (primMinusNat Zero Zero)) (numericEnumFrom (primMinusNat Zero Zero))))",fontsize=16,color="black",shape="box"];3178 -> 3212[label="",style="solid", color="black", weight=3]; 82.06/55.94 3179[label="map toEnum (takeWhile (flip (<=) yx26) (numericEnumFrom (Neg (primPlusNat Zero yx570))))",fontsize=16,color="black",shape="box"];3179 -> 3213[label="",style="solid", color="black", weight=3]; 82.06/55.94 3211[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (Pos (Succ yx5700))) (numericEnumFrom (Pos (Succ yx5700)))))",fontsize=16,color="black",shape="box"];3211 -> 3365[label="",style="solid", color="black", weight=3]; 82.06/55.94 3212 -> 3210[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3212[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (Pos Zero)) (numericEnumFrom (Pos Zero))))",fontsize=16,color="magenta"];3212 -> 3366[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3213 -> 3547[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3213[label="map toEnum (takeWhile (flip (<=) yx26) (Neg (primPlusNat Zero yx570) : (numericEnumFrom $! Neg (primPlusNat Zero yx570) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3213 -> 3549[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3213 -> 3550[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3213 -> 3551[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3365[label="map toEnum (takeWhile (flip (<=) yx26) (numericEnumFrom (Pos (Succ yx5700))))",fontsize=16,color="black",shape="box"];3365 -> 3389[label="",style="solid", color="black", weight=3]; 82.06/55.94 3366[label="yx26",fontsize=16,color="green",shape="box"];3210[label="map toEnum (takeWhile (flip (<=) yx18) (enforceWHNF (WHNF (Pos Zero)) (numericEnumFrom (Pos Zero))))",fontsize=16,color="black",shape="triangle"];3210 -> 3364[label="",style="solid", color="black", weight=3]; 82.06/55.94 3549[label="yx570",fontsize=16,color="green",shape="box"];3550 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3550[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3550 -> 3576[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3551[label="yx26",fontsize=16,color="green",shape="box"];3389 -> 3582[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3389[label="map toEnum (takeWhile (flip (<=) yx26) (Pos (Succ yx5700) : (numericEnumFrom $! Pos (Succ yx5700) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3389 -> 3584[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3389 -> 3585[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3389 -> 3586[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3364[label="map toEnum (takeWhile (flip (<=) yx18) (numericEnumFrom (Pos Zero)))",fontsize=16,color="black",shape="box"];3364 -> 3388[label="",style="solid", color="black", weight=3]; 82.06/55.94 3576[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3584 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3584[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3584 -> 3635[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3585[label="yx26",fontsize=16,color="green",shape="box"];3586[label="yx5700",fontsize=16,color="green",shape="box"];3388 -> 3587[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3388[label="map toEnum (takeWhile (flip (<=) yx18) (Pos Zero : (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3388 -> 3589[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3388 -> 3590[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3635[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3589[label="yx18",fontsize=16,color="green",shape="box"];3590 -> 576[label="",style="dashed", color="red", weight=0]; 82.06/55.94 3590[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3590 -> 3634[label="",style="dashed", color="magenta", weight=3]; 82.06/55.94 3634[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];} 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (14) 82.06/55.94 Complex Obligation (AND) 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (15) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_primDivNatS0(Succ(yx40000), Succ(yx41000)) -> new_primDivNatS00(yx40000, yx41000, yx40000, yx41000) 82.06/55.94 new_primDivNatS00(yx255, yx256, Succ(yx2570), Zero) -> new_primDivNatS(new_primMinusNatS2(Succ(yx255), Succ(yx256)), Succ(yx256)) 82.06/55.94 new_primDivNatS(Succ(yx4000), yx4100) -> new_primDivNatS0(yx4000, yx4100) 82.06/55.94 new_primDivNatS0(Succ(yx40000), Zero) -> new_primDivNatS(new_primMinusNatS0(yx40000), Zero) 82.06/55.94 new_primDivNatS01(yx255, yx256) -> new_primDivNatS(new_primMinusNatS2(Succ(yx255), Succ(yx256)), Succ(yx256)) 82.06/55.94 new_primDivNatS0(Zero, Zero) -> new_primDivNatS(new_primMinusNatS1, Zero) 82.06/55.94 new_primDivNatS00(yx255, yx256, Succ(yx2570), Succ(yx2580)) -> new_primDivNatS00(yx255, yx256, yx2570, yx2580) 82.06/55.94 new_primDivNatS00(yx255, yx256, Zero, Zero) -> new_primDivNatS01(yx255, yx256) 82.06/55.94 82.06/55.94 The TRS R consists of the following rules: 82.06/55.94 82.06/55.94 new_primMinusNatS1 -> Zero 82.06/55.94 new_primMinusNatS2(Succ(yx2000), Zero) -> Succ(yx2000) 82.06/55.94 new_primMinusNatS2(Zero, Zero) -> Zero 82.06/55.94 new_primMinusNatS2(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS2(yx2000, yx2010) 82.06/55.94 new_primMinusNatS2(Zero, Succ(yx2010)) -> Zero 82.06/55.94 new_primMinusNatS0(yx30000) -> Succ(yx30000) 82.06/55.94 82.06/55.94 The set Q consists of the following terms: 82.06/55.94 82.06/55.94 new_primMinusNatS0(x0) 82.06/55.94 new_primMinusNatS2(Succ(x0), Succ(x1)) 82.06/55.94 new_primMinusNatS2(Zero, Succ(x0)) 82.06/55.94 new_primMinusNatS2(Zero, Zero) 82.06/55.94 new_primMinusNatS1 82.06/55.94 new_primMinusNatS2(Succ(x0), Zero) 82.06/55.94 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (16) DependencyGraphProof (EQUIVALENT) 82.06/55.94 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (17) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_primDivNatS00(yx255, yx256, Succ(yx2570), Zero) -> new_primDivNatS(new_primMinusNatS2(Succ(yx255), Succ(yx256)), Succ(yx256)) 82.06/55.94 new_primDivNatS(Succ(yx4000), yx4100) -> new_primDivNatS0(yx4000, yx4100) 82.06/55.94 new_primDivNatS0(Succ(yx40000), Succ(yx41000)) -> new_primDivNatS00(yx40000, yx41000, yx40000, yx41000) 82.06/55.94 new_primDivNatS00(yx255, yx256, Succ(yx2570), Succ(yx2580)) -> new_primDivNatS00(yx255, yx256, yx2570, yx2580) 82.06/55.94 new_primDivNatS00(yx255, yx256, Zero, Zero) -> new_primDivNatS01(yx255, yx256) 82.06/55.94 new_primDivNatS01(yx255, yx256) -> new_primDivNatS(new_primMinusNatS2(Succ(yx255), Succ(yx256)), Succ(yx256)) 82.06/55.94 new_primDivNatS0(Succ(yx40000), Zero) -> new_primDivNatS(new_primMinusNatS0(yx40000), Zero) 82.06/55.94 82.06/55.94 The TRS R consists of the following rules: 82.06/55.94 82.06/55.94 new_primMinusNatS1 -> Zero 82.06/55.94 new_primMinusNatS2(Succ(yx2000), Zero) -> Succ(yx2000) 82.06/55.94 new_primMinusNatS2(Zero, Zero) -> Zero 82.06/55.94 new_primMinusNatS2(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS2(yx2000, yx2010) 82.06/55.94 new_primMinusNatS2(Zero, Succ(yx2010)) -> Zero 82.06/55.94 new_primMinusNatS0(yx30000) -> Succ(yx30000) 82.06/55.94 82.06/55.94 The set Q consists of the following terms: 82.06/55.94 82.06/55.94 new_primMinusNatS0(x0) 82.06/55.94 new_primMinusNatS2(Succ(x0), Succ(x1)) 82.06/55.94 new_primMinusNatS2(Zero, Succ(x0)) 82.06/55.94 new_primMinusNatS2(Zero, Zero) 82.06/55.94 new_primMinusNatS1 82.06/55.94 new_primMinusNatS2(Succ(x0), Zero) 82.06/55.94 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (18) QDPOrderProof (EQUIVALENT) 82.06/55.94 We use the reduction pair processor [LPAR04,JAR06]. 82.06/55.94 82.06/55.94 82.06/55.94 The following pairs can be oriented strictly and are deleted. 82.06/55.94 82.06/55.94 new_primDivNatS(Succ(yx4000), yx4100) -> new_primDivNatS0(yx4000, yx4100) 82.06/55.94 The remaining pairs can at least be oriented weakly. 82.06/55.94 Used ordering: Polynomial interpretation [POLO]: 82.06/55.94 82.06/55.94 POL(Succ(x_1)) = 1 + x_1 82.06/55.94 POL(Zero) = 0 82.06/55.94 POL(new_primDivNatS(x_1, x_2)) = x_1 82.06/55.94 POL(new_primDivNatS0(x_1, x_2)) = x_1 82.06/55.94 POL(new_primDivNatS00(x_1, x_2, x_3, x_4)) = 1 + x_1 82.06/55.94 POL(new_primDivNatS01(x_1, x_2)) = 1 + x_1 82.06/55.94 POL(new_primMinusNatS0(x_1)) = 1 + x_1 82.06/55.94 POL(new_primMinusNatS2(x_1, x_2)) = x_1 82.06/55.94 82.06/55.94 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 82.06/55.94 82.06/55.94 new_primMinusNatS2(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS2(yx2000, yx2010) 82.06/55.94 new_primMinusNatS0(yx30000) -> Succ(yx30000) 82.06/55.94 new_primMinusNatS2(Succ(yx2000), Zero) -> Succ(yx2000) 82.06/55.94 new_primMinusNatS2(Zero, Zero) -> Zero 82.06/55.94 new_primMinusNatS2(Zero, Succ(yx2010)) -> Zero 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (19) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_primDivNatS00(yx255, yx256, Succ(yx2570), Zero) -> new_primDivNatS(new_primMinusNatS2(Succ(yx255), Succ(yx256)), Succ(yx256)) 82.06/55.94 new_primDivNatS0(Succ(yx40000), Succ(yx41000)) -> new_primDivNatS00(yx40000, yx41000, yx40000, yx41000) 82.06/55.94 new_primDivNatS00(yx255, yx256, Succ(yx2570), Succ(yx2580)) -> new_primDivNatS00(yx255, yx256, yx2570, yx2580) 82.06/55.94 new_primDivNatS00(yx255, yx256, Zero, Zero) -> new_primDivNatS01(yx255, yx256) 82.06/55.94 new_primDivNatS01(yx255, yx256) -> new_primDivNatS(new_primMinusNatS2(Succ(yx255), Succ(yx256)), Succ(yx256)) 82.06/55.94 new_primDivNatS0(Succ(yx40000), Zero) -> new_primDivNatS(new_primMinusNatS0(yx40000), Zero) 82.06/55.94 82.06/55.94 The TRS R consists of the following rules: 82.06/55.94 82.06/55.94 new_primMinusNatS1 -> Zero 82.06/55.94 new_primMinusNatS2(Succ(yx2000), Zero) -> Succ(yx2000) 82.06/55.94 new_primMinusNatS2(Zero, Zero) -> Zero 82.06/55.94 new_primMinusNatS2(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS2(yx2000, yx2010) 82.06/55.94 new_primMinusNatS2(Zero, Succ(yx2010)) -> Zero 82.06/55.94 new_primMinusNatS0(yx30000) -> Succ(yx30000) 82.06/55.94 82.06/55.94 The set Q consists of the following terms: 82.06/55.94 82.06/55.94 new_primMinusNatS0(x0) 82.06/55.94 new_primMinusNatS2(Succ(x0), Succ(x1)) 82.06/55.94 new_primMinusNatS2(Zero, Succ(x0)) 82.06/55.94 new_primMinusNatS2(Zero, Zero) 82.06/55.94 new_primMinusNatS1 82.06/55.94 new_primMinusNatS2(Succ(x0), Zero) 82.06/55.94 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (20) DependencyGraphProof (EQUIVALENT) 82.06/55.94 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (21) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_primDivNatS00(yx255, yx256, Succ(yx2570), Succ(yx2580)) -> new_primDivNatS00(yx255, yx256, yx2570, yx2580) 82.06/55.94 82.06/55.94 The TRS R consists of the following rules: 82.06/55.94 82.06/55.94 new_primMinusNatS1 -> Zero 82.06/55.94 new_primMinusNatS2(Succ(yx2000), Zero) -> Succ(yx2000) 82.06/55.94 new_primMinusNatS2(Zero, Zero) -> Zero 82.06/55.94 new_primMinusNatS2(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS2(yx2000, yx2010) 82.06/55.94 new_primMinusNatS2(Zero, Succ(yx2010)) -> Zero 82.06/55.94 new_primMinusNatS0(yx30000) -> Succ(yx30000) 82.06/55.94 82.06/55.94 The set Q consists of the following terms: 82.06/55.94 82.06/55.94 new_primMinusNatS0(x0) 82.06/55.94 new_primMinusNatS2(Succ(x0), Succ(x1)) 82.06/55.94 new_primMinusNatS2(Zero, Succ(x0)) 82.06/55.94 new_primMinusNatS2(Zero, Zero) 82.06/55.94 new_primMinusNatS1 82.06/55.94 new_primMinusNatS2(Succ(x0), Zero) 82.06/55.94 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (22) QDPSizeChangeProof (EQUIVALENT) 82.06/55.94 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. 82.06/55.94 82.06/55.94 From the DPs we obtained the following set of size-change graphs: 82.06/55.94 *new_primDivNatS00(yx255, yx256, Succ(yx2570), Succ(yx2580)) -> new_primDivNatS00(yx255, yx256, yx2570, yx2580) 82.06/55.94 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (23) 82.06/55.94 YES 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (24) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map21(yx207, Zero, yx234, h) -> new_map20(yx207, yx234, yx207, h) 82.06/55.94 new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, new_fromInt(Pos(Succ(Zero))), h) 82.06/55.94 new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, new_fromInt(Pos(Succ(Zero))), h) 82.06/55.94 new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) 82.06/55.94 82.06/55.94 The TRS R consists of the following rules: 82.06/55.94 82.06/55.94 new_fromInt(yx8) -> yx8 82.06/55.94 82.06/55.94 The set Q consists of the following terms: 82.06/55.94 82.06/55.94 new_fromInt(x0) 82.06/55.94 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (25) TransformationProof (EQUIVALENT) 82.06/55.94 By rewriting [LPAR04] the rule new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, new_fromInt(Pos(Succ(Zero))), h) at position [2] we obtained the following new rules [LPAR04]: 82.06/55.94 82.06/55.94 (new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h),new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h)) 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (26) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map21(yx207, Zero, yx234, h) -> new_map20(yx207, yx234, yx207, h) 82.06/55.94 new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, new_fromInt(Pos(Succ(Zero))), h) 82.06/55.94 new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) 82.06/55.94 new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 82.06/55.94 The TRS R consists of the following rules: 82.06/55.94 82.06/55.94 new_fromInt(yx8) -> yx8 82.06/55.94 82.06/55.94 The set Q consists of the following terms: 82.06/55.94 82.06/55.94 new_fromInt(x0) 82.06/55.94 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (27) TransformationProof (EQUIVALENT) 82.06/55.94 By rewriting [LPAR04] the rule new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, new_fromInt(Pos(Succ(Zero))), h) at position [2] we obtained the following new rules [LPAR04]: 82.06/55.94 82.06/55.94 (new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h),new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h)) 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (28) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map21(yx207, Zero, yx234, h) -> new_map20(yx207, yx234, yx207, h) 82.06/55.94 new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) 82.06/55.94 new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 82.06/55.94 The TRS R consists of the following rules: 82.06/55.94 82.06/55.94 new_fromInt(yx8) -> yx8 82.06/55.94 82.06/55.94 The set Q consists of the following terms: 82.06/55.94 82.06/55.94 new_fromInt(x0) 82.06/55.94 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (29) UsableRulesProof (EQUIVALENT) 82.06/55.94 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. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (30) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map21(yx207, Zero, yx234, h) -> new_map20(yx207, yx234, yx207, h) 82.06/55.94 new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) 82.06/55.94 new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 82.06/55.94 R is empty. 82.06/55.94 The set Q consists of the following terms: 82.06/55.94 82.06/55.94 new_fromInt(x0) 82.06/55.94 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (31) QReductionProof (EQUIVALENT) 82.06/55.94 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 82.06/55.94 82.06/55.94 new_fromInt(x0) 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (32) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map21(yx207, Zero, yx234, h) -> new_map20(yx207, yx234, yx207, h) 82.06/55.94 new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) 82.06/55.94 new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 82.06/55.94 R is empty. 82.06/55.94 Q is empty. 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (33) TransformationProof (EQUIVALENT) 82.06/55.94 By instantiating [LPAR04] the rule new_map21(yx207, Zero, yx234, h) -> new_map20(yx207, yx234, yx207, h) we obtained the following new rules [LPAR04]: 82.06/55.94 82.06/55.94 (new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2),new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2)) 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (34) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) 82.06/55.94 new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2) 82.06/55.94 82.06/55.94 R is empty. 82.06/55.94 Q is empty. 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (35) DependencyGraphProof (EQUIVALENT) 82.06/55.94 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (36) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2) 82.06/55.94 new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) 82.06/55.94 new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) 82.06/55.94 82.06/55.94 R is empty. 82.06/55.94 Q is empty. 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (37) TransformationProof (EQUIVALENT) 82.06/55.94 By instantiating [LPAR04] the rule new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) we obtained the following new rules [LPAR04]: 82.06/55.94 82.06/55.94 (new_map20(Pos(Zero), Pos(Succ(Zero)), Pos(Zero), z1) -> new_map22(Pos(Zero), Pos(Succ(Zero)), z1),new_map20(Pos(Zero), Pos(Succ(Zero)), Pos(Zero), z1) -> new_map22(Pos(Zero), Pos(Succ(Zero)), z1)) 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (38) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2) 82.06/55.94 new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) 82.06/55.94 new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map20(Pos(Zero), Pos(Succ(Zero)), Pos(Zero), z1) -> new_map22(Pos(Zero), Pos(Succ(Zero)), z1) 82.06/55.94 82.06/55.94 R is empty. 82.06/55.94 Q is empty. 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (39) DependencyGraphProof (EQUIVALENT) 82.06/55.94 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (40) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2) 82.06/55.94 new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) 82.06/55.94 new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 82.06/55.94 R is empty. 82.06/55.94 Q is empty. 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (41) TransformationProof (EQUIVALENT) 82.06/55.94 By instantiating [LPAR04] the rule new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) we obtained the following new rules [LPAR04]: 82.06/55.94 82.06/55.94 (new_map20(Neg(Zero), Pos(Succ(Zero)), Neg(Zero), z1) -> new_map23(Neg(Zero), Pos(Succ(Zero)), z1),new_map20(Neg(Zero), Pos(Succ(Zero)), Neg(Zero), z1) -> new_map23(Neg(Zero), Pos(Succ(Zero)), z1)) 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (42) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2) 82.06/55.94 new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) 82.06/55.94 new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 new_map20(Neg(Zero), Pos(Succ(Zero)), Neg(Zero), z1) -> new_map23(Neg(Zero), Pos(Succ(Zero)), z1) 82.06/55.94 82.06/55.94 R is empty. 82.06/55.94 Q is empty. 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (43) TransformationProof (EQUIVALENT) 82.06/55.94 By instantiating [LPAR04] the rule new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) we obtained the following new rules [LPAR04]: 82.06/55.94 82.06/55.94 (new_map23(Neg(Zero), Pos(Succ(Zero)), z0) -> new_map22(Neg(Zero), Pos(Succ(Zero)), z0),new_map23(Neg(Zero), Pos(Succ(Zero)), z0) -> new_map22(Neg(Zero), Pos(Succ(Zero)), z0)) 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (44) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2) 82.06/55.94 new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) 82.06/55.94 new_map20(Neg(Zero), Pos(Succ(Zero)), Neg(Zero), z1) -> new_map23(Neg(Zero), Pos(Succ(Zero)), z1) 82.06/55.94 new_map23(Neg(Zero), Pos(Succ(Zero)), z0) -> new_map22(Neg(Zero), Pos(Succ(Zero)), z0) 82.06/55.94 82.06/55.94 R is empty. 82.06/55.94 Q is empty. 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (45) DependencyGraphProof (EQUIVALENT) 82.06/55.94 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (46) 82.06/55.94 TRUE 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (47) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map24(yx207, yx208, yx209, Succ(yx2100), Succ(yx2110), yx212, yx213, h) -> new_map24(yx207, yx208, yx209, yx2100, yx2110, yx212, yx213, h) 82.06/55.94 82.06/55.94 R is empty. 82.06/55.94 Q is empty. 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (48) QDPSizeChangeProof (EQUIVALENT) 82.06/55.94 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. 82.06/55.94 82.06/55.94 From the DPs we obtained the following set of size-change graphs: 82.06/55.94 *new_map24(yx207, yx208, yx209, Succ(yx2100), Succ(yx2110), yx212, yx213, h) -> new_map24(yx207, yx208, yx209, yx2100, yx2110, yx212, yx213, h) 82.06/55.94 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5, 6 >= 6, 7 >= 7, 8 >= 8 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (49) 82.06/55.94 YES 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (50) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.94 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.94 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.94 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.94 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map11(yx207, yx212000, bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.94 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.94 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, new_fromInt(Pos(Succ(Zero))), h) 82.06/55.94 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.94 new_map5(yx260, yx259, ba) -> new_map7(yx259, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.94 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.94 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.94 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.94 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.94 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map18(yx207, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map6(yx262, yx261, ba) -> new_map19(yx261, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.94 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.94 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.94 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.94 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.94 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.94 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.94 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.94 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.94 82.06/55.94 The TRS R consists of the following rules: 82.06/55.94 82.06/55.94 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.94 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.94 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.94 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.94 new_fromInt(yx8) -> yx8 82.06/55.94 82.06/55.94 The set Q consists of the following terms: 82.06/55.94 82.06/55.94 new_primPlusNat0(Succ(x0), Zero) 82.06/55.94 new_fromInt(x0) 82.06/55.94 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.94 new_primPlusNat0(Zero, Zero) 82.06/55.94 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.94 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (51) DependencyGraphProof (EQUIVALENT) 82.06/55.94 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (52) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.94 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.94 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.94 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.94 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.94 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.94 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, new_fromInt(Pos(Succ(Zero))), h) 82.06/55.94 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.94 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.94 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.94 new_map6(yx262, yx261, ba) -> new_map19(yx261, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.94 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.94 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.94 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.94 new_map5(yx260, yx259, ba) -> new_map7(yx259, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.94 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.94 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.94 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.94 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.94 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.94 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.94 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 82.06/55.94 The TRS R consists of the following rules: 82.06/55.94 82.06/55.94 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.94 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.94 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.94 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.94 new_fromInt(yx8) -> yx8 82.06/55.94 82.06/55.94 The set Q consists of the following terms: 82.06/55.94 82.06/55.94 new_primPlusNat0(Succ(x0), Zero) 82.06/55.94 new_fromInt(x0) 82.06/55.94 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.94 new_primPlusNat0(Zero, Zero) 82.06/55.94 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.94 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (53) TransformationProof (EQUIVALENT) 82.06/55.94 By rewriting [LPAR04] the rule new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: 82.06/55.94 82.06/55.94 (new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb),new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb)) 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (54) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.94 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.94 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.94 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.94 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.94 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.94 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, new_fromInt(Pos(Succ(Zero))), h) 82.06/55.94 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.94 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.94 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.94 new_map6(yx262, yx261, ba) -> new_map19(yx261, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.94 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.94 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.94 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.94 new_map5(yx260, yx259, ba) -> new_map7(yx259, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.94 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.94 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.94 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.94 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.94 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.94 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.94 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.94 82.06/55.94 The TRS R consists of the following rules: 82.06/55.94 82.06/55.94 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.94 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.94 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.94 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.94 new_fromInt(yx8) -> yx8 82.06/55.94 82.06/55.94 The set Q consists of the following terms: 82.06/55.94 82.06/55.94 new_primPlusNat0(Succ(x0), Zero) 82.06/55.94 new_fromInt(x0) 82.06/55.94 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.94 new_primPlusNat0(Zero, Zero) 82.06/55.94 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.94 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (55) TransformationProof (EQUIVALENT) 82.06/55.94 By rewriting [LPAR04] the rule new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: 82.06/55.94 82.06/55.94 (new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb),new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb)) 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (56) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.94 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.94 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.94 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.94 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.94 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.94 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, new_fromInt(Pos(Succ(Zero))), h) 82.06/55.94 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.94 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.94 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.94 new_map6(yx262, yx261, ba) -> new_map19(yx261, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.94 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.94 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.94 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.94 new_map5(yx260, yx259, ba) -> new_map7(yx259, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.94 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.94 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.94 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.94 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.94 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.94 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.94 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.94 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.94 82.06/55.94 The TRS R consists of the following rules: 82.06/55.94 82.06/55.94 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.94 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.94 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.94 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.94 new_fromInt(yx8) -> yx8 82.06/55.94 82.06/55.94 The set Q consists of the following terms: 82.06/55.94 82.06/55.94 new_primPlusNat0(Succ(x0), Zero) 82.06/55.94 new_fromInt(x0) 82.06/55.94 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.94 new_primPlusNat0(Zero, Zero) 82.06/55.94 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.94 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (57) TransformationProof (EQUIVALENT) 82.06/55.94 By rewriting [LPAR04] the rule new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: 82.06/55.94 82.06/55.94 (new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb),new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb)) 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (58) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.94 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.94 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.94 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.94 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.94 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.94 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, new_fromInt(Pos(Succ(Zero))), h) 82.06/55.94 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.94 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.94 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.94 new_map6(yx262, yx261, ba) -> new_map19(yx261, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.94 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.94 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.94 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.94 new_map5(yx260, yx259, ba) -> new_map7(yx259, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.94 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.94 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.94 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.94 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.94 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.94 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.94 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.94 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.94 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.94 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.94 82.06/55.94 The TRS R consists of the following rules: 82.06/55.94 82.06/55.94 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.94 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.94 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.94 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.94 new_fromInt(yx8) -> yx8 82.06/55.94 82.06/55.94 The set Q consists of the following terms: 82.06/55.94 82.06/55.94 new_primPlusNat0(Succ(x0), Zero) 82.06/55.94 new_fromInt(x0) 82.06/55.94 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.94 new_primPlusNat0(Zero, Zero) 82.06/55.94 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.94 82.06/55.94 We have to consider all minimal (P,Q,R)-chains. 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (59) TransformationProof (EQUIVALENT) 82.06/55.94 By rewriting [LPAR04] the rule new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) at position [1] we obtained the following new rules [LPAR04]: 82.06/55.94 82.06/55.94 (new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb),new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb)) 82.06/55.94 82.06/55.94 82.06/55.94 ---------------------------------------- 82.06/55.94 82.06/55.94 (60) 82.06/55.94 Obligation: 82.06/55.94 Q DP problem: 82.06/55.94 The TRS P consists of the following rules: 82.06/55.94 82.06/55.94 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.94 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.94 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.94 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.94 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.94 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.94 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.94 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.94 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.94 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, new_fromInt(Pos(Succ(Zero))), h) 82.06/55.94 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.94 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.94 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.94 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.94 new_map6(yx262, yx261, ba) -> new_map19(yx261, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.94 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.94 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.94 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.94 new_map5(yx260, yx259, ba) -> new_map7(yx259, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.94 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.94 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.94 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.94 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.94 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.94 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (61) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, new_fromInt(Pos(Succ(Zero))), h) at position [2] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h),new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (62) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (63) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map6(yx262, yx261, ba) -> new_map19(yx261, new_fromInt(Pos(Succ(Zero))), ba) at position [1] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba),new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (64) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, new_fromInt(Pos(Succ(Zero))), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (65) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map5(yx260, yx259, ba) -> new_map7(yx259, new_fromInt(Pos(Succ(Zero))), ba) at position [1] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba),new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (66) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (67) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) at position [3] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb),new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (68) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (69) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb),new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (70) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (71) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) at position [1] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb),new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (72) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (73) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb),new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (74) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (75) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb),new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (76) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (77) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) at position [3] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb),new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (78) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (79) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) at position [1] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb),new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (80) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (81) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) at position [3] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb),new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (82) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (83) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb),new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (84) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (85) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) at position [1] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb),new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (86) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (87) TransformationProof (EQUIVALENT) 82.06/55.95 By rewriting [LPAR04] the rule new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb),new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (88) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 new_fromInt(yx8) -> yx8 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (89) UsableRulesProof (EQUIVALENT) 82.06/55.95 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. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (90) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_fromInt(x0) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (91) QReductionProof (EQUIVALENT) 82.06/55.95 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 82.06/55.95 82.06/55.95 new_fromInt(x0) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (92) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (93) TransformationProof (EQUIVALENT) 82.06/55.95 By instantiating [LPAR04] the rule new_map12(yx207, yx218, yx233, bb) -> new_map13(yx207, yx218, yx218, yx233, yx218, yx207, bb) we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2),new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (94) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (95) TransformationProof (EQUIVALENT) 82.06/55.95 By instantiating [LPAR04] the rule new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4),new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4)) 82.06/55.95 (new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3),new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3)) 82.06/55.95 (new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2),new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (96) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.06/55.95 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.06/55.95 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.06/55.95 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (97) TransformationProof (EQUIVALENT) 82.06/55.95 By instantiating [LPAR04] the rule new_map8(yx207, yx212000, yx237, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx237, Succ(yx212000), bb) we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3),new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (98) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.06/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.06/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.06/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.06/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.06/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.06/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.06/55.95 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.06/55.95 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.06/55.95 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.06/55.95 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.06/55.95 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.06/55.95 82.06/55.95 The TRS R consists of the following rules: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.06/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.06/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.06/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.06/55.95 82.06/55.95 The set Q consists of the following terms: 82.06/55.95 82.06/55.95 new_primPlusNat0(Succ(x0), Zero) 82.06/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.06/55.95 new_primPlusNat0(Zero, Zero) 82.06/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.06/55.95 82.06/55.95 We have to consider all minimal (P,Q,R)-chains. 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (99) TransformationProof (EQUIVALENT) 82.06/55.95 By instantiating [LPAR04] the rule new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx236, yx2510, yx19900, ba) we obtained the following new rules [LPAR04]: 82.06/55.95 82.06/55.95 (new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3),new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3)) 82.06/55.95 (new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2),new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2)) 82.06/55.95 82.06/55.95 82.06/55.95 ---------------------------------------- 82.06/55.95 82.06/55.95 (100) 82.06/55.95 Obligation: 82.06/55.95 Q DP problem: 82.06/55.95 The TRS P consists of the following rules: 82.06/55.95 82.06/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) 82.06/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.06/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.06/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.06/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.06/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.06/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.06/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.06/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.06/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.06/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.06/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.06/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.06/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.95 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.95 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.95 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.95 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.95 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.95 new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) 82.15/55.95 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.95 82.15/55.95 The TRS R consists of the following rules: 82.15/55.95 82.15/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.95 82.15/55.95 The set Q consists of the following terms: 82.15/55.95 82.15/55.95 new_primPlusNat0(Succ(x0), Zero) 82.15/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.95 new_primPlusNat0(Zero, Zero) 82.15/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.95 82.15/55.95 We have to consider all minimal (P,Q,R)-chains. 82.15/55.95 ---------------------------------------- 82.15/55.95 82.15/55.95 (101) TransformationProof (EQUIVALENT) 82.15/55.95 By instantiating [LPAR04] the rule new_map10(yx207, yx238, bb) -> new_map4(yx207, Zero, Zero, yx238, Zero, bb) we obtained the following new rules [LPAR04]: 82.15/55.95 82.15/55.95 (new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1),new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1)) 82.15/55.95 82.15/55.95 82.15/55.95 ---------------------------------------- 82.15/55.95 82.15/55.95 (102) 82.15/55.95 Obligation: 82.15/55.95 Q DP problem: 82.15/55.95 The TRS P consists of the following rules: 82.15/55.95 82.15/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.95 new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) 82.15/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.15/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.15/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.15/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.15/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.95 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.95 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.95 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.95 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.95 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.95 new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) 82.15/55.95 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.95 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.95 82.15/55.95 The TRS R consists of the following rules: 82.15/55.95 82.15/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.95 82.15/55.95 The set Q consists of the following terms: 82.15/55.95 82.15/55.95 new_primPlusNat0(Succ(x0), Zero) 82.15/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.95 new_primPlusNat0(Zero, Zero) 82.15/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.95 82.15/55.95 We have to consider all minimal (P,Q,R)-chains. 82.15/55.95 ---------------------------------------- 82.15/55.95 82.15/55.95 (103) TransformationProof (EQUIVALENT) 82.15/55.95 By instantiating [LPAR04] the rule new_map4(Pos(Succ(yx19900)), yx249, yx250, yx236, Zero, ba) -> new_map(yx19900, yx249, yx250, yx236, Zero, Succ(yx19900), ba) we obtained the following new rules [LPAR04]: 82.15/55.95 82.15/55.95 (new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3),new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3)) 82.15/55.95 (new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1),new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1)) 82.15/55.95 82.15/55.95 82.15/55.95 ---------------------------------------- 82.15/55.95 82.15/55.95 (104) 82.15/55.95 Obligation: 82.15/55.95 Q DP problem: 82.15/55.95 The TRS P consists of the following rules: 82.15/55.95 82.15/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.95 new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) 82.15/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.15/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.95 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.95 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.15/55.95 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.15/55.95 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.95 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.95 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.95 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.95 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.95 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.95 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.95 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.95 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.95 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.95 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.95 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.95 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.95 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.95 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.95 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.95 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.95 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.95 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.95 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.95 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.95 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.95 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.95 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.95 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.95 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.95 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.95 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.95 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.95 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.95 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.95 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.95 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.95 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.95 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.95 new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) 82.15/55.95 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.95 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.95 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.95 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.95 82.15/55.95 The TRS R consists of the following rules: 82.15/55.95 82.15/55.95 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.95 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.95 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.95 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.95 82.15/55.95 The set Q consists of the following terms: 82.15/55.95 82.15/55.95 new_primPlusNat0(Succ(x0), Zero) 82.15/55.95 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.95 new_primPlusNat0(Zero, Zero) 82.15/55.95 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.95 82.15/55.95 We have to consider all minimal (P,Q,R)-chains. 82.15/55.95 ---------------------------------------- 82.15/55.95 82.15/55.95 (105) TransformationProof (EQUIVALENT) 82.15/55.95 By instantiating [LPAR04] the rule new_map3(yx279, yx285, yx287, h) -> new_map4(Pos(Succ(yx279)), yx285, yx285, yx287, yx285, h) we obtained the following new rules [LPAR04]: 82.15/55.95 82.15/55.95 (new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3),new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3)) 82.15/55.95 82.15/55.95 82.15/55.95 ---------------------------------------- 82.15/55.95 82.15/55.95 (106) 82.15/55.95 Obligation: 82.15/55.95 Q DP problem: 82.15/55.95 The TRS P consists of the following rules: 82.15/55.95 82.15/55.95 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.95 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.95 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.95 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.95 new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) 82.15/55.95 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.96 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.15/55.96 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.15/55.96 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.96 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (107) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map4(Pos(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2360, ba) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2),new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (108) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.96 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.15/55.96 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.15/55.96 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.96 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (109) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map4(Neg(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2),new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2)) 82.15/55.96 (new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (110) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.15/55.96 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.15/55.96 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.96 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (111) TransformationProof (EQUIVALENT) 82.15/55.96 By rewriting [LPAR04] the rule new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [0] we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (112) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.15/55.96 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.15/55.96 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.96 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (113) TransformationProof (EQUIVALENT) 82.15/55.96 By rewriting [LPAR04] the rule new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [1] we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1),new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (114) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) 82.15/55.96 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.15/55.96 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.96 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (115) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map19(yx261, yx266, ba) -> new_map4(Neg(Zero), yx261, yx261, yx266, yx261, ba) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2),new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (116) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map4(Neg(Zero), yx249, yx250, Neg(yx2360), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2360, ba) 82.15/55.96 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.96 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) 82.15/55.96 new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (117) DependencyGraphProof (EQUIVALENT) 82.15/55.96 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (118) 82.15/55.96 Complex Obligation (AND) 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (119) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (120) UsableRulesProof (EQUIVALENT) 82.15/55.96 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. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (121) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (122) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (123) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (124) UsableRulesProof (EQUIVALENT) 82.15/55.96 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. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (125) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (126) TransformationProof (EQUIVALENT) 82.15/55.96 By rewriting [LPAR04] the rule new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [0] we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (127) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (128) TransformationProof (EQUIVALENT) 82.15/55.96 By rewriting [LPAR04] the rule new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [1] we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1),new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (129) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (130) UsableRulesProof (EQUIVALENT) 82.15/55.96 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. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (131) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) 82.15/55.96 82.15/55.96 R is empty. 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (132) QReductionProof (EQUIVALENT) 82.15/55.96 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (133) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) 82.15/55.96 82.15/55.96 R is empty. 82.15/55.96 Q is empty. 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (134) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map6(yx262, yx261, ba) -> new_map19(yx261, Pos(Succ(Zero)), ba) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map6(Succ(Zero), Succ(Zero), z0) -> new_map19(Succ(Zero), Pos(Succ(Zero)), z0),new_map6(Succ(Zero), Succ(Zero), z0) -> new_map19(Succ(Zero), Pos(Succ(Zero)), z0)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (135) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) 82.15/55.96 new_map6(Succ(Zero), Succ(Zero), z0) -> new_map19(Succ(Zero), Pos(Succ(Zero)), z0) 82.15/55.96 82.15/55.96 R is empty. 82.15/55.96 Q is empty. 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (136) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map19(Succ(Zero), Pos(Succ(Zero)), z0) -> new_map4(Neg(Zero), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), z0),new_map19(Succ(Zero), Pos(Succ(Zero)), z0) -> new_map4(Neg(Zero), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), z0)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (137) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) 82.15/55.96 new_map6(Succ(Zero), Succ(Zero), z0) -> new_map19(Succ(Zero), Pos(Succ(Zero)), z0) 82.15/55.96 new_map19(Succ(Zero), Pos(Succ(Zero)), z0) -> new_map4(Neg(Zero), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), z0) 82.15/55.96 82.15/55.96 R is empty. 82.15/55.96 Q is empty. 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (138) DependencyGraphProof (EQUIVALENT) 82.15/55.96 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (139) 82.15/55.96 TRUE 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (140) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (141) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2),new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2)) 82.15/55.96 (new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4),new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (142) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.96 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (143) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2),new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2)) 82.15/55.96 (new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4),new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4)) 82.15/55.96 (new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3),new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (144) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.96 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (145) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2),new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (146) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.96 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (147) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map4(Pos(Zero), yx249, yx250, Pos(yx2360), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2360), new_primPlusNat0(yx250, yx2360), ba) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1)) 82.15/55.96 (new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2),new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (148) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.96 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (149) TransformationProof (EQUIVALENT) 82.15/55.96 By rewriting [LPAR04] the rule new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [0] we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (150) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.96 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (151) TransformationProof (EQUIVALENT) 82.15/55.96 By rewriting [LPAR04] the rule new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [1] we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1),new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (152) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.96 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (153) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map7(yx259, yx265, ba) -> new_map4(Pos(Zero), yx259, yx259, yx265, yx259, ba) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2),new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (154) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.96 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) 82.15/55.96 new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (155) DependencyGraphProof (EQUIVALENT) 82.15/55.96 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (156) 82.15/55.96 Complex Obligation (AND) 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (157) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (158) UsableRulesProof (EQUIVALENT) 82.15/55.96 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. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (159) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (160) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (161) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (162) UsableRulesProof (EQUIVALENT) 82.15/55.96 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. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (163) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (164) TransformationProof (EQUIVALENT) 82.15/55.96 By rewriting [LPAR04] the rule new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [0] we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (165) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (166) TransformationProof (EQUIVALENT) 82.15/55.96 By rewriting [LPAR04] the rule new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [1] we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1),new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (167) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (168) UsableRulesProof (EQUIVALENT) 82.15/55.96 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. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (169) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) 82.15/55.96 82.15/55.96 R is empty. 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (170) QReductionProof (EQUIVALENT) 82.15/55.96 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (171) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) 82.15/55.96 82.15/55.96 R is empty. 82.15/55.96 Q is empty. 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (172) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map5(yx260, yx259, ba) -> new_map7(yx259, Pos(Succ(Zero)), ba) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map5(Succ(Zero), Succ(Zero), z0) -> new_map7(Succ(Zero), Pos(Succ(Zero)), z0),new_map5(Succ(Zero), Succ(Zero), z0) -> new_map7(Succ(Zero), Pos(Succ(Zero)), z0)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (173) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) 82.15/55.96 new_map5(Succ(Zero), Succ(Zero), z0) -> new_map7(Succ(Zero), Pos(Succ(Zero)), z0) 82.15/55.96 82.15/55.96 R is empty. 82.15/55.96 Q is empty. 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (174) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map7(Succ(Zero), Pos(Succ(Zero)), z0) -> new_map4(Pos(Zero), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), z0),new_map7(Succ(Zero), Pos(Succ(Zero)), z0) -> new_map4(Pos(Zero), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), z0)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (175) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) 82.15/55.96 new_map5(Succ(Zero), Succ(Zero), z0) -> new_map7(Succ(Zero), Pos(Succ(Zero)), z0) 82.15/55.96 new_map7(Succ(Zero), Pos(Succ(Zero)), z0) -> new_map4(Pos(Zero), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), z0) 82.15/55.96 82.15/55.96 R is empty. 82.15/55.96 Q is empty. 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (176) DependencyGraphProof (EQUIVALENT) 82.15/55.96 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (177) 82.15/55.96 TRUE 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (178) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.96 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (179) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2),new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2)) 82.15/55.96 (new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4),new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4)) 82.15/55.96 (new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3),new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (180) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.96 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.96 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.96 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.96 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.96 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.96 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.96 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.96 82.15/55.96 The TRS R consists of the following rules: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.96 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.96 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.96 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.96 82.15/55.96 The set Q consists of the following terms: 82.15/55.96 82.15/55.96 new_primPlusNat0(Succ(x0), Zero) 82.15/55.96 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.96 new_primPlusNat0(Zero, Zero) 82.15/55.96 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.96 82.15/55.96 We have to consider all minimal (P,Q,R)-chains. 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (181) TransformationProof (EQUIVALENT) 82.15/55.96 By instantiating [LPAR04] the rule new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) we obtained the following new rules [LPAR04]: 82.15/55.96 82.15/55.96 (new_map4(Pos(Succ(z0)), Zero, Zero, Pos(Succ(Zero)), Zero, z2) -> new_map(z0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(z0), z2),new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1)) 82.15/55.96 82.15/55.96 82.15/55.96 ---------------------------------------- 82.15/55.96 82.15/55.96 (182) 82.15/55.96 Obligation: 82.15/55.96 Q DP problem: 82.15/55.96 The TRS P consists of the following rules: 82.15/55.96 82.15/55.96 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.96 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.96 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.96 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.96 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.96 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.96 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.96 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.96 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.96 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.96 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) 82.15/55.96 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.96 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.96 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.96 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (183) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map14(yx207, yx218, Succ(yx21200), yx231, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx231, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4),new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (184) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.97 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (185) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map13(Pos(x5), Succ(x2), Succ(x2), Pos(Succ(Succ(x3))), Succ(x2), Pos(x5), z3) -> new_map1(Pos(x5), x3, x2, z3),new_map13(Pos(x5), Succ(x2), Succ(x2), Pos(Succ(Succ(x3))), Succ(x2), Pos(x5), z3) -> new_map1(Pos(x5), x3, x2, z3)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (186) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.97 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map13(Pos(x5), Succ(x2), Succ(x2), Pos(Succ(Succ(x3))), Succ(x2), Pos(x5), z3) -> new_map1(Pos(x5), x3, x2, z3) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (187) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map13(Pos(x4), z1, z1, Pos(Zero), z1, Pos(x4), z3) -> new_map12(Pos(x4), z1, Pos(Succ(Zero)), z3),new_map13(Pos(x4), z1, z1, Pos(Zero), z1, Pos(x4), z3) -> new_map12(Pos(x4), z1, Pos(Succ(Zero)), z3)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (188) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.97 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map13(Pos(x5), Succ(x2), Succ(x2), Pos(Succ(Succ(x3))), Succ(x2), Pos(x5), z3) -> new_map1(Pos(x5), x3, x2, z3) 82.15/55.97 new_map13(Pos(x4), z1, z1, Pos(Zero), z1, Pos(x4), z3) -> new_map12(Pos(x4), z1, Pos(Succ(Zero)), z3) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (189) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map13(Pos(x5), z1, z1, Neg(x3), z1, Pos(x5), z3) -> new_map14(Pos(x5), z1, x3, Pos(Succ(Zero)), z3),new_map13(Pos(x5), z1, z1, Neg(x3), z1, Pos(x5), z3) -> new_map14(Pos(x5), z1, x3, Pos(Succ(Zero)), z3)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (190) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) 82.15/55.97 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map13(Pos(x5), Succ(x2), Succ(x2), Pos(Succ(Succ(x3))), Succ(x2), Pos(x5), z3) -> new_map1(Pos(x5), x3, x2, z3) 82.15/55.97 new_map13(Pos(x4), z1, z1, Pos(Zero), z1, Pos(x4), z3) -> new_map12(Pos(x4), z1, Pos(Succ(Zero)), z3) 82.15/55.97 new_map13(Pos(x5), z1, z1, Neg(x3), z1, Pos(x5), z3) -> new_map14(Pos(x5), z1, x3, Pos(Succ(Zero)), z3) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (191) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map14(yx207, yx218, Zero, yx231, bb) -> new_map13(yx207, yx218, yx218, yx231, yx218, yx207, bb) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4),new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4)) 82.15/55.97 (new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3),new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (192) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map13(Pos(x5), Succ(x2), Succ(x2), Pos(Succ(Succ(x3))), Succ(x2), Pos(x5), z3) -> new_map1(Pos(x5), x3, x2, z3) 82.15/55.97 new_map13(Pos(x4), z1, z1, Pos(Zero), z1, Pos(x4), z3) -> new_map12(Pos(x4), z1, Pos(Succ(Zero)), z3) 82.15/55.97 new_map13(Pos(x5), z1, z1, Neg(x3), z1, Pos(x5), z3) -> new_map14(Pos(x5), z1, x3, Pos(Succ(Zero)), z3) 82.15/55.97 new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) 82.15/55.97 new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (193) DependencyGraphProof (EQUIVALENT) 82.15/55.97 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (194) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (195) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2),new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (196) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (197) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2),new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (198) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (199) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map13(Neg(Succ(Zero)), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2),new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2)) 82.15/55.97 (new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3),new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (200) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (201) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map13(Neg(Succ(Zero)), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2),new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (202) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (203) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map13(Neg(Succ(Succ(x0))), Succ(x1), Succ(x1), Pos(Succ(Zero)), Succ(x1), Neg(Succ(Succ(x0))), z2) -> new_map15(Neg(Succ(Succ(x0))), Succ(x1), Succ(x1), Pos(Succ(Zero)), x0, x1, z2),new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2)) 82.15/55.97 (new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3),new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (204) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.97 new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (205) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map13(Neg(Succ(Succ(x0))), Succ(x1), Succ(x1), Pos(Succ(Zero)), Succ(x1), Neg(Succ(Succ(x0))), z2) -> new_map15(Neg(Succ(Succ(x0))), Succ(x1), Succ(x1), Pos(Succ(Zero)), x0, x1, z2),new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (206) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.97 new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (207) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2),new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2)) 82.15/55.97 (new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3),new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (208) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) 82.15/55.97 new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.97 new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) 82.15/55.97 new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (209) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2),new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2)) 82.15/55.97 (new_map13(Neg(Zero), Succ(y_0), Succ(y_0), Pos(Succ(Zero)), Succ(y_0), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_0), Pos(Succ(Zero)), z3),new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (210) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.97 new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) 82.15/55.97 new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (211) QDPOrderProof (EQUIVALENT) 82.15/55.97 We use the reduction pair processor [LPAR04,JAR06]. 82.15/55.97 82.15/55.97 82.15/55.97 The following pairs can be oriented strictly and are deleted. 82.15/55.97 82.15/55.97 new_map(yx279, yx280, yx281, Neg(yx2820), Zero, Succ(yx2840), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 The remaining pairs can at least be oriented weakly. 82.15/55.97 Used ordering: Polynomial interpretation [POLO]: 82.15/55.97 82.15/55.97 POL(Neg(x_1)) = 1 82.15/55.97 POL(Pos(x_1)) = 0 82.15/55.97 POL(Succ(x_1)) = 0 82.15/55.97 POL(Zero) = 0 82.15/55.97 POL(new_map(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_4 82.15/55.97 POL(new_map0(x_1, x_2, x_3, x_4)) = 0 82.15/55.97 POL(new_map1(x_1, x_2, x_3, x_4)) = 0 82.15/55.97 POL(new_map10(x_1, x_2, x_3)) = 0 82.15/55.97 POL(new_map12(x_1, x_2, x_3, x_4)) = x_3 82.15/55.97 POL(new_map13(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 82.15/55.97 POL(new_map14(x_1, x_2, x_3, x_4, x_5)) = 0 82.15/55.97 POL(new_map15(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_4 82.15/55.97 POL(new_map16(x_1, x_2, x_3, x_4, x_5)) = 0 82.15/55.97 POL(new_map17(x_1, x_2, x_3, x_4, x_5)) = x_4 82.15/55.97 POL(new_map2(x_1, x_2, x_3, x_4, x_5)) = 0 82.15/55.97 POL(new_map3(x_1, x_2, x_3, x_4)) = x_3 82.15/55.97 POL(new_map4(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 82.15/55.97 POL(new_map8(x_1, x_2, x_3, x_4)) = 0 82.15/55.97 POL(new_map9(x_1, x_2, x_3)) = 0 82.15/55.97 POL(new_primPlusNat0(x_1, x_2)) = 0 82.15/55.97 82.15/55.97 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 82.15/55.97 none 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (212) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.97 new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) 82.15/55.97 new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (213) QDPOrderProof (EQUIVALENT) 82.15/55.97 We use the reduction pair processor [LPAR04,JAR06]. 82.15/55.97 82.15/55.97 82.15/55.97 The following pairs can be oriented strictly and are deleted. 82.15/55.97 82.15/55.97 new_map2(yx279, yx280, yx281, Neg(yx2820), h) -> new_map1(Pos(Succ(yx279)), yx281, yx2820, h) 82.15/55.97 new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) 82.15/55.97 The remaining pairs can at least be oriented weakly. 82.15/55.97 Used ordering: Polynomial interpretation [POLO]: 82.15/55.97 82.15/55.97 POL(Neg(x_1)) = 1 82.15/55.97 POL(Pos(x_1)) = 0 82.15/55.97 POL(Succ(x_1)) = 0 82.15/55.97 POL(Zero) = 0 82.15/55.97 POL(new_map(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_4 82.15/55.97 POL(new_map0(x_1, x_2, x_3, x_4)) = 0 82.15/55.97 POL(new_map1(x_1, x_2, x_3, x_4)) = 0 82.15/55.97 POL(new_map10(x_1, x_2, x_3)) = 0 82.15/55.97 POL(new_map12(x_1, x_2, x_3, x_4)) = x_3 82.15/55.97 POL(new_map13(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 82.15/55.97 POL(new_map14(x_1, x_2, x_3, x_4, x_5)) = 0 82.15/55.97 POL(new_map15(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_4 82.15/55.97 POL(new_map16(x_1, x_2, x_3, x_4, x_5)) = x_4 82.15/55.97 POL(new_map17(x_1, x_2, x_3, x_4, x_5)) = 0 82.15/55.97 POL(new_map2(x_1, x_2, x_3, x_4, x_5)) = x_4 82.15/55.97 POL(new_map3(x_1, x_2, x_3, x_4)) = x_3 82.15/55.97 POL(new_map4(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 82.15/55.97 POL(new_map8(x_1, x_2, x_3, x_4)) = 0 82.15/55.97 POL(new_map9(x_1, x_2, x_3)) = 0 82.15/55.97 POL(new_primPlusNat0(x_1, x_2)) = 0 82.15/55.97 82.15/55.97 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 82.15/55.97 none 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (214) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.97 new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) 82.15/55.97 new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.97 new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) 82.15/55.97 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.97 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.97 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.97 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) 82.15/55.97 new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) 82.15/55.97 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.97 new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) 82.15/55.97 new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) 82.15/55.97 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.97 new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.97 new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) 82.15/55.97 new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) 82.15/55.97 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (215) DependencyGraphProof (EQUIVALENT) 82.15/55.97 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 13 less nodes. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (216) 82.15/55.97 Complex Obligation (AND) 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (217) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (218) QDPOrderProof (EQUIVALENT) 82.15/55.97 We use the reduction pair processor [LPAR04,JAR06]. 82.15/55.97 82.15/55.97 82.15/55.97 The following pairs can be oriented strictly and are deleted. 82.15/55.97 82.15/55.97 new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) 82.15/55.97 The remaining pairs can at least be oriented weakly. 82.15/55.97 Used ordering: Polynomial interpretation [POLO]: 82.15/55.97 82.15/55.97 POL(Pos(x_1)) = x_1 82.15/55.97 POL(Succ(x_1)) = 0 82.15/55.97 POL(Zero) = 1 82.15/55.97 POL(new_map(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_4 82.15/55.97 POL(new_map0(x_1, x_2, x_3, x_4)) = x_3 82.15/55.97 POL(new_map2(x_1, x_2, x_3, x_4, x_5)) = x_4 82.15/55.97 POL(new_map3(x_1, x_2, x_3, x_4)) = x_2 + x_3 82.15/55.97 POL(new_map4(x_1, x_2, x_3, x_4, x_5, x_6)) = x_5 82.15/55.97 POL(new_primPlusNat0(x_1, x_2)) = x_2 82.15/55.97 82.15/55.97 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (219) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (220) QDPPairToRuleProof (EQUIVALENT) 82.15/55.97 The dependency pair new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) was transformed to the following new rules: 82.15/55.97 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.97 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 the following new pairs maintain the fan-in: 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.97 82.15/55.97 the following new pairs maintain the fan-out: 82.15/55.97 H(yx279, yx280, yx281, Pos(yx2820), h, cons_new_map(Zero, Succ(yx2840))) -> new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) 82.15/55.97 H(yx279, yx280, yx281, yx282, h, cons_new_map(Zero, Zero)) -> new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (221) 82.15/55.97 Complex Obligation (AND) 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (222) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.97 H(yx279, yx280, yx281, Pos(yx2820), h, cons_new_map(Zero, Succ(yx2840))) -> new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) 82.15/55.97 H(yx279, yx280, yx281, yx282, h, cons_new_map(Zero, Zero)) -> new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.97 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Succ(x0), Succ(x1)) 82.15/55.97 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Zero, Succ(x0)) 82.15/55.97 new_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (223) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule H(yx279, yx280, yx281, Pos(yx2820), h, cons_new_map(Zero, Succ(yx2840))) -> new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2),H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (224) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.97 H(yx279, yx280, yx281, yx282, h, cons_new_map(Zero, Zero)) -> new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) 82.15/55.97 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.97 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Succ(x0), Succ(x1)) 82.15/55.97 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Zero, Succ(x0)) 82.15/55.97 new_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (225) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule new_map(yx279, yx280, yx281, Pos(yx2820), Zero, Succ(yx2840), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2),new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2)) 82.15/55.97 (new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2),new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (226) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.97 H(yx279, yx280, yx281, yx282, h, cons_new_map(Zero, Zero)) -> new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) 82.15/55.97 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.97 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) 82.15/55.97 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.97 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Succ(x0), Succ(x1)) 82.15/55.97 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Zero, Succ(x0)) 82.15/55.97 new_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (227) TransformationProof (EQUIVALENT) 82.15/55.97 By rewriting [LPAR04] the rule new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) at position [1] we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2),new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (228) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.97 H(yx279, yx280, yx281, yx282, h, cons_new_map(Zero, Zero)) -> new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) 82.15/55.97 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.97 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) 82.15/55.97 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.97 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Succ(x0), Succ(x1)) 82.15/55.97 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Zero, Succ(x0)) 82.15/55.97 new_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (229) TransformationProof (EQUIVALENT) 82.15/55.97 By rewriting [LPAR04] the rule new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) at position [1] we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2),new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (230) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.97 H(yx279, yx280, yx281, yx282, h, cons_new_map(Zero, Zero)) -> new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) 82.15/55.97 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.97 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) 82.15/55.97 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.97 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Succ(x0), Succ(x1)) 82.15/55.97 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Zero, Succ(x0)) 82.15/55.97 new_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (231) TransformationProof (EQUIVALENT) 82.15/55.97 By rewriting [LPAR04] the rule new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) at position [1,0,0] we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2),new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (232) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.97 H(yx279, yx280, yx281, yx282, h, cons_new_map(Zero, Zero)) -> new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) 82.15/55.97 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.97 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) 82.15/55.97 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.97 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Succ(x0), Succ(x1)) 82.15/55.97 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Zero, Succ(x0)) 82.15/55.97 new_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (233) TransformationProof (EQUIVALENT) 82.15/55.97 By rewriting [LPAR04] the rule new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) at position [2] we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2),new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (234) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.97 H(yx279, yx280, yx281, yx282, h, cons_new_map(Zero, Zero)) -> new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) 82.15/55.97 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.97 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) 82.15/55.97 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.97 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Succ(x0), Succ(x1)) 82.15/55.97 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Zero, Succ(x0)) 82.15/55.97 new_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (235) TransformationProof (EQUIVALENT) 82.15/55.97 By rewriting [LPAR04] the rule new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) at position [2] we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z2),new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (236) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.97 H(yx279, yx280, yx281, yx282, h, cons_new_map(Zero, Zero)) -> new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) 82.15/55.97 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.97 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.97 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z2) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.97 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Succ(x0), Succ(x1)) 82.15/55.97 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Zero, Succ(x0)) 82.15/55.97 new_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (237) TransformationProof (EQUIVALENT) 82.15/55.97 By rewriting [LPAR04] the rule new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z2) at position [2,0,0] we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2),new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (238) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.97 H(yx279, yx280, yx281, yx282, h, cons_new_map(Zero, Zero)) -> new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) 82.15/55.97 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.97 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.97 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.97 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Succ(x0), Succ(x1)) 82.15/55.97 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.97 new_new_map(Zero, Succ(x0)) 82.15/55.97 new_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 We have to consider all minimal (P,Q,R)-chains. 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (239) TransformationProof (EQUIVALENT) 82.15/55.97 By instantiating [LPAR04] the rule H(yx279, yx280, yx281, yx282, h, cons_new_map(Zero, Zero)) -> new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) we obtained the following new rules [LPAR04]: 82.15/55.97 82.15/55.97 (H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2),H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2)) 82.15/55.97 82.15/55.97 82.15/55.97 ---------------------------------------- 82.15/55.97 82.15/55.97 (240) 82.15/55.97 Obligation: 82.15/55.97 Q DP problem: 82.15/55.97 The TRS P consists of the following rules: 82.15/55.97 82.15/55.97 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.97 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.97 new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) 82.15/55.97 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.97 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.97 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.97 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.97 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.97 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.97 82.15/55.97 The TRS R consists of the following rules: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.97 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.97 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.97 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.97 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.97 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.97 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.97 82.15/55.97 The set Q consists of the following terms: 82.15/55.97 82.15/55.97 new_primPlusNat0(Succ(x0), Zero) 82.15/55.97 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.97 new_primPlusNat0(Zero, Zero) 82.15/55.97 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (241) TransformationProof (EQUIVALENT) 82.15/55.98 By instantiating [LPAR04] the rule new_map(yx279, yx280, yx281, yx282, Zero, Zero, h) -> new_map2(yx279, yx280, yx281, yx282, h) we obtained the following new rules [LPAR04]: 82.15/55.98 82.15/55.98 (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2),new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2)) 82.15/55.98 (new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2),new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2)) 82.15/55.98 82.15/55.98 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (242) 82.15/55.98 Obligation: 82.15/55.98 Q DP problem: 82.15/55.98 The TRS P consists of the following rules: 82.15/55.98 82.15/55.98 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 82.15/55.98 The TRS R consists of the following rules: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.98 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.98 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.98 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.98 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.98 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 The set Q consists of the following terms: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(x0), Zero) 82.15/55.98 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.98 new_primPlusNat0(Zero, Zero) 82.15/55.98 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (243) TransformationProof (EQUIVALENT) 82.15/55.98 By instantiating [LPAR04] the rule new_map2(yx279, yx280, yx281, Pos(yx2820), h) -> new_map0(yx279, new_primPlusNat0(yx281, yx2820), new_primPlusNat0(yx281, yx2820), h) we obtained the following new rules [LPAR04]: 82.15/55.98 82.15/55.98 (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0),new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0)) 82.15/55.98 (new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2),new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2)) 82.15/55.98 82.15/55.98 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (244) 82.15/55.98 Obligation: 82.15/55.98 Q DP problem: 82.15/55.98 The TRS P consists of the following rules: 82.15/55.98 82.15/55.98 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) 82.15/55.98 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) 82.15/55.98 82.15/55.98 The TRS R consists of the following rules: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.98 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.98 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.98 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.98 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.98 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 The set Q consists of the following terms: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(x0), Zero) 82.15/55.98 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.98 new_primPlusNat0(Zero, Zero) 82.15/55.98 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (245) TransformationProof (EQUIVALENT) 82.15/55.98 By rewriting [LPAR04] the rule new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) at position [1] we obtained the following new rules [LPAR04]: 82.15/55.98 82.15/55.98 (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0),new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0)) 82.15/55.98 82.15/55.98 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (246) 82.15/55.98 Obligation: 82.15/55.98 Q DP problem: 82.15/55.98 The TRS P consists of the following rules: 82.15/55.98 82.15/55.98 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) 82.15/55.98 new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) 82.15/55.98 82.15/55.98 The TRS R consists of the following rules: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.98 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.98 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.98 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.98 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.98 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 The set Q consists of the following terms: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(x0), Zero) 82.15/55.98 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.98 new_primPlusNat0(Zero, Zero) 82.15/55.98 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (247) TransformationProof (EQUIVALENT) 82.15/55.98 By rewriting [LPAR04] the rule new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) at position [1] we obtained the following new rules [LPAR04]: 82.15/55.98 82.15/55.98 (new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2),new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2)) 82.15/55.98 82.15/55.98 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (248) 82.15/55.98 Obligation: 82.15/55.98 Q DP problem: 82.15/55.98 The TRS P consists of the following rules: 82.15/55.98 82.15/55.98 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) 82.15/55.98 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) 82.15/55.98 82.15/55.98 The TRS R consists of the following rules: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.98 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.98 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.98 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.98 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.98 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 The set Q consists of the following terms: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(x0), Zero) 82.15/55.98 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.98 new_primPlusNat0(Zero, Zero) 82.15/55.98 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (249) TransformationProof (EQUIVALENT) 82.15/55.98 By rewriting [LPAR04] the rule new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) at position [1,0,0] we obtained the following new rules [LPAR04]: 82.15/55.98 82.15/55.98 (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0),new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0)) 82.15/55.98 82.15/55.98 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (250) 82.15/55.98 Obligation: 82.15/55.98 Q DP problem: 82.15/55.98 The TRS P consists of the following rules: 82.15/55.98 82.15/55.98 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) 82.15/55.98 new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) 82.15/55.98 82.15/55.98 The TRS R consists of the following rules: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.98 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.98 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.98 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.98 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.98 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 The set Q consists of the following terms: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(x0), Zero) 82.15/55.98 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.98 new_primPlusNat0(Zero, Zero) 82.15/55.98 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (251) TransformationProof (EQUIVALENT) 82.15/55.98 By rewriting [LPAR04] the rule new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) at position [2] we obtained the following new rules [LPAR04]: 82.15/55.98 82.15/55.98 (new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2),new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2)) 82.15/55.98 82.15/55.98 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (252) 82.15/55.98 Obligation: 82.15/55.98 Q DP problem: 82.15/55.98 The TRS P consists of the following rules: 82.15/55.98 82.15/55.98 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) 82.15/55.98 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 82.15/55.98 The TRS R consists of the following rules: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.98 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.98 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.98 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.98 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.98 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 The set Q consists of the following terms: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(x0), Zero) 82.15/55.98 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.98 new_primPlusNat0(Zero, Zero) 82.15/55.98 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (253) TransformationProof (EQUIVALENT) 82.15/55.98 By rewriting [LPAR04] the rule new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) at position [2] we obtained the following new rules [LPAR04]: 82.15/55.98 82.15/55.98 (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z0),new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z0)) 82.15/55.98 82.15/55.98 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (254) 82.15/55.98 Obligation: 82.15/55.98 Q DP problem: 82.15/55.98 The TRS P consists of the following rules: 82.15/55.98 82.15/55.98 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z0) 82.15/55.98 82.15/55.98 The TRS R consists of the following rules: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.98 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.98 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.98 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.98 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.98 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 The set Q consists of the following terms: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(x0), Zero) 82.15/55.98 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.98 new_primPlusNat0(Zero, Zero) 82.15/55.98 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (255) UsableRulesProof (EQUIVALENT) 82.15/55.98 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. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (256) 82.15/55.98 Obligation: 82.15/55.98 Q DP problem: 82.15/55.98 The TRS P consists of the following rules: 82.15/55.98 82.15/55.98 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z0) 82.15/55.98 82.15/55.98 The TRS R consists of the following rules: 82.15/55.98 82.15/55.98 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.98 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.98 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.98 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 The set Q consists of the following terms: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(x0), Zero) 82.15/55.98 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.98 new_primPlusNat0(Zero, Zero) 82.15/55.98 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (257) TransformationProof (EQUIVALENT) 82.15/55.98 By rewriting [LPAR04] the rule new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z0) at position [2,0,0] we obtained the following new rules [LPAR04]: 82.15/55.98 82.15/55.98 (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0),new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0)) 82.15/55.98 82.15/55.98 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (258) 82.15/55.98 Obligation: 82.15/55.98 Q DP problem: 82.15/55.98 The TRS P consists of the following rules: 82.15/55.98 82.15/55.98 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) 82.15/55.98 82.15/55.98 The TRS R consists of the following rules: 82.15/55.98 82.15/55.98 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.98 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.98 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.98 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 The set Q consists of the following terms: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(x0), Zero) 82.15/55.98 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.98 new_primPlusNat0(Zero, Zero) 82.15/55.98 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (259) InductionCalculusProof (EQUIVALENT) 82.15/55.98 Note that final constraints are written in bold face. 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) the following chains were created: 82.15/55.98 *We consider the chain new_map0(x4, x5, x6, x7) -> new_map3(x4, x6, Pos(Succ(Zero)), x7), new_map3(x8, x9, Pos(Succ(Zero)), x10) -> new_map4(Pos(Succ(x8)), x9, x9, Pos(Succ(Zero)), x9, x10) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map3(x4, x6, Pos(Succ(Zero)), x7)=new_map3(x8, x9, Pos(Succ(Zero)), x10) ==> new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) the following chains were created: 82.15/55.98 *We consider the chain new_map3(x57, x58, Pos(Succ(Zero)), x59) -> new_map4(Pos(Succ(x57)), x58, x58, Pos(Succ(Zero)), x58, x59), new_map4(Pos(Succ(x60)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x62) -> new_map(x60, Succ(x61), Succ(x61), Pos(Succ(Zero)), x61, x60, x62) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map4(Pos(Succ(x57)), x58, x58, Pos(Succ(Zero)), x58, x59)=new_map4(Pos(Succ(x60)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x62) ==> new_map3(x57, x58, Pos(Succ(Zero)), x59)_>=_new_map4(Pos(Succ(x57)), x58, x58, Pos(Succ(Zero)), x58, x59)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map3(x57, Succ(x61), Pos(Succ(Zero)), x59)_>=_new_map4(Pos(Succ(x57)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x59)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain new_map3(x63, x64, Pos(Succ(Zero)), x65) -> new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65), new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) -> H(x66, Succ(x67), Succ(x67), Pos(Succ(Zero)), x68, anew_new_map(x67, x66)) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)=new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) ==> new_map3(x63, x64, Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) the following chains were created: 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x108)), Succ(x109), Succ(x109), Pos(Succ(Zero)), Succ(x109), x110) -> new_map(x108, Succ(x109), Succ(x109), Pos(Succ(Zero)), x109, x108, x110), new_map(x111, Succ(x112), Succ(x112), Pos(Succ(Zero)), Zero, Succ(x113), x114) -> new_map0(x111, Succ(Succ(new_primPlusNat0(x112, Zero))), Succ(Succ(new_primPlusNat0(x112, Zero))), x114) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x108, Succ(x109), Succ(x109), Pos(Succ(Zero)), x109, x108, x110)=new_map(x111, Succ(x112), Succ(x112), Pos(Succ(Zero)), Zero, Succ(x113), x114) ==> new_map4(Pos(Succ(x108)), Succ(x109), Succ(x109), Pos(Succ(Zero)), Succ(x109), x110)_>=_new_map(x108, Succ(x109), Succ(x109), Pos(Succ(Zero)), x109, x108, x110)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map4(Pos(Succ(Succ(x113))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x110)_>=_new_map(Succ(x113), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x113), x110)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x115)), Succ(x116), Succ(x116), Pos(Succ(Zero)), Succ(x116), x117) -> new_map(x115, Succ(x116), Succ(x116), Pos(Succ(Zero)), x116, x115, x117), new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x119) -> new_map0(Succ(x118), Succ(Succ(Zero)), Succ(Succ(Zero)), x119) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x115, Succ(x116), Succ(x116), Pos(Succ(Zero)), x116, x115, x117)=new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x119) ==> new_map4(Pos(Succ(x115)), Succ(x116), Succ(x116), Pos(Succ(Zero)), Succ(x116), x117)_>=_new_map(x115, Succ(x116), Succ(x116), Pos(Succ(Zero)), x116, x115, x117)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map4(Pos(Succ(Succ(x118))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x117)_>=_new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x117)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x123)), Succ(x124), Succ(x124), Pos(Succ(Zero)), Succ(x124), x125) -> new_map(x123, Succ(x124), Succ(x124), Pos(Succ(Zero)), x124, x123, x125), new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x126) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x126) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x123, Succ(x124), Succ(x124), Pos(Succ(Zero)), x124, x123, x125)=new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x126) ==> new_map4(Pos(Succ(x123)), Succ(x124), Succ(x124), Pos(Succ(Zero)), Succ(x124), x125)_>=_new_map(x123, Succ(x124), Succ(x124), Pos(Succ(Zero)), x124, x123, x125)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x125)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x125)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x127)), Succ(x128), Succ(x128), Pos(Succ(Zero)), Succ(x128), x129) -> new_map(x127, Succ(x128), Succ(x128), Pos(Succ(Zero)), x128, x127, x129), new_map(x130, Succ(x131), Succ(x131), Pos(Succ(Zero)), Zero, Zero, x132) -> new_map2(x130, Succ(x131), Succ(x131), Pos(Succ(Zero)), x132) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x127, Succ(x128), Succ(x128), Pos(Succ(Zero)), x128, x127, x129)=new_map(x130, Succ(x131), Succ(x131), Pos(Succ(Zero)), Zero, Zero, x132) ==> new_map4(Pos(Succ(x127)), Succ(x128), Succ(x128), Pos(Succ(Zero)), Succ(x128), x129)_>=_new_map(x127, Succ(x128), Succ(x128), Pos(Succ(Zero)), x128, x127, x129)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x129)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x129)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) the following chains were created: 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153) -> H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151)), H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) -> new_map(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), Zero, Succ(x157), x156) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))=H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) which results in the following new constraint: 82.15/55.98 82.15/55.98 (3) (new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(x468))), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), Succ(Succ(x469)), x153)_>=_H(Succ(x468), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x469), Succ(x468)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) which results in the following new constraints: 82.15/55.98 82.15/55.98 (4) (new_new_map(x471, x470)=cons_new_map(Zero, Succ(x157)) & (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) 82.15/55.98 82.15/55.98 (5) (cons_new_map(Zero, Succ(x474))=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) 82.15/55.98 82.15/55.98 (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) with sigma = [x472 / x157, x473 / x153] which results in the following new constraint: 82.15/55.98 82.15/55.98 (7) (new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (8) (new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We solved constraint (6) using rules (I), (II). 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166) -> H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164)), H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) -> new_map(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), Zero, Zero, x169) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))=H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (anew_new_map(x165, x164)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x165, x164)=cons_new_map(Zero, Zero) which results in the following new constraint: 82.15/55.98 82.15/55.98 (3) (new_new_map(x476, x475)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x475))), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), Succ(Succ(x476)), x166)_>=_H(Succ(x475), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x476), Succ(x475)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x476, x475)=cons_new_map(Zero, Zero) which results in the following new constraints: 82.15/55.98 82.15/55.98 (4) (new_new_map(x478, x477)=cons_new_map(Zero, Zero) & (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) 82.15/55.98 82.15/55.98 (5) (cons_new_map(Zero, Succ(x480))=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Succ(x480)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Succ(x480)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Succ(x480))))) 82.15/55.98 82.15/55.98 (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) with sigma = [x479 / x166] which results in the following new constraint: 82.15/55.98 82.15/55.98 (7) (new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: 82.15/55.98 82.15/55.98 (8) (new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) the following chains were created: 82.15/55.98 *We consider the chain H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205))) -> new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204), new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) -> new_map0(x206, Succ(Succ(new_primPlusNat0(x207, Zero))), Succ(Succ(new_primPlusNat0(x207, Zero))), x209) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)=new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) ==> H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain H(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213))) -> new_map(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), Zero, Succ(x213), x212), new_map(Succ(x214), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x214), x215) -> new_map0(Succ(x214), Succ(Succ(Zero)), Succ(Succ(Zero)), x215) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), Zero, Succ(x213), x212)=new_map(Succ(x214), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x214), x215) ==> H(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213)))_>=_new_map(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), Zero, Succ(x213), x212)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (H(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213)))_>=_new_map(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x213), x212)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: 82.15/55.98 *We consider the chain new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239) -> new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239), new_map0(x240, x241, x242, x243) -> new_map3(x240, x242, Pos(Succ(Zero)), x243) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)=new_map0(x240, x241, x242, x243) ==> new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) the following chains were created: 82.15/55.98 *We consider the chain new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289) -> new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289), new_map0(x290, x291, x292, x293) -> new_map3(x290, x292, Pos(Succ(Zero)), x293) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)=new_map0(x290, x291, x292, x293) ==> new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289)_>=_new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289)_>=_new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) the following chains were created: 82.15/55.98 *We consider the chain H(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero)) -> new_map(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), Zero, Zero, x342), new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x343) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x343) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), Zero, Zero, x342)=new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x343) ==> H(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero))_>=_new_map(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), Zero, Zero, x342)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (H(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero))_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x342)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero)) -> new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346), new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) -> new_map2(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), x349) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)=new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) ==> H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) the following chains were created: 82.15/55.98 *We consider the chain new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366), new_map2(x367, Succ(x368), Succ(x368), Pos(Succ(Zero)), x369) -> new_map0(x367, Succ(Succ(new_primPlusNat0(x368, Zero))), Succ(Succ(new_primPlusNat0(x368, Zero))), x369) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)=new_map2(x367, Succ(x368), Succ(x368), Pos(Succ(Zero)), x369) ==> new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370), new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x371) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x371) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)=new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x371) ==> new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) the following chains were created: 82.15/55.98 *We consider the chain new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404) -> new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404), new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) -> new_map0(x405, Succ(Succ(new_primPlusNat0(x406, Zero))), Succ(Succ(new_primPlusNat0(x406, Zero))), x407) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)=new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) ==> new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain new_map(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), Zero, Zero, x410) -> new_map2(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), x410), new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x411) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x411) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map2(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), x410)=new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x411) ==> new_map(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), Zero, Zero, x410)_>=_new_map2(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), x410)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x410)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x410)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: 82.15/55.98 *We consider the chain new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414) -> new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414), new_map0(x415, x416, x417, x418) -> new_map3(x415, x417, Pos(Succ(Zero)), x418) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)=new_map0(x415, x416, x417, x418) ==> new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) the following chains were created: 82.15/55.98 *We consider the chain new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452), new_map0(x453, x454, x455, x456) -> new_map3(x453, x455, Pos(Succ(Zero)), x456) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)=new_map0(x453, x454, x455, x456) ==> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452)_>=_new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452)_>=_new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 To summarize, we get the following constraints P__>=_ for the following pairs. 82.15/55.98 82.15/55.98 *new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 82.15/55.98 *(new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 82.15/55.98 *(new_map3(x57, Succ(x61), Pos(Succ(Zero)), x59)_>=_new_map4(Pos(Succ(x57)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x59)) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(x113))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x110)_>=_new_map(Succ(x113), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x113), x110)) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(x118))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x117)_>=_new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x117)) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x125)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x125)) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x129)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x129)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 82.15/55.98 *(H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) 82.15/55.98 82.15/55.98 82.15/55.98 *(H(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213)))_>=_new_map(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x213), x212)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 82.15/55.98 *(new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 82.15/55.98 *(new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289)_>=_new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 82.15/55.98 *(H(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero))_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x342)) 82.15/55.98 82.15/55.98 82.15/55.98 *(H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 82.15/55.98 *(new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 82.15/55.98 *(new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x410)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x410)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 82.15/55.98 *(new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) 82.15/55.98 82.15/55.98 *(new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452)_>=_new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 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. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (260) 82.15/55.98 Obligation: 82.15/55.98 Q DP problem: 82.15/55.98 The TRS P consists of the following rules: 82.15/55.98 82.15/55.98 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) 82.15/55.98 82.15/55.98 The TRS R consists of the following rules: 82.15/55.98 82.15/55.98 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.98 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.98 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.98 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 The set Q consists of the following terms: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(x0), Zero) 82.15/55.98 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.98 new_primPlusNat0(Zero, Zero) 82.15/55.98 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (261) NonInfProof (EQUIVALENT) 82.15/55.98 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 82.15/55.98 82.15/55.98 Note that final constraints are written in bold face. 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) the following chains were created: 82.15/55.98 *We consider the chain new_map0(x4, x5, x6, x7) -> new_map3(x4, x6, Pos(Succ(Zero)), x7), new_map3(x8, x9, Pos(Succ(Zero)), x10) -> new_map4(Pos(Succ(x8)), x9, x9, Pos(Succ(Zero)), x9, x10) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map3(x4, x6, Pos(Succ(Zero)), x7)=new_map3(x8, x9, Pos(Succ(Zero)), x10) ==> new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) the following chains were created: 82.15/55.98 *We consider the chain new_map3(x57, x58, Pos(Succ(Zero)), x59) -> new_map4(Pos(Succ(x57)), x58, x58, Pos(Succ(Zero)), x58, x59), new_map4(Pos(Succ(x60)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x62) -> new_map(x60, Succ(x61), Succ(x61), Pos(Succ(Zero)), x61, x60, x62) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map4(Pos(Succ(x57)), x58, x58, Pos(Succ(Zero)), x58, x59)=new_map4(Pos(Succ(x60)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x62) ==> new_map3(x57, x58, Pos(Succ(Zero)), x59)_>=_new_map4(Pos(Succ(x57)), x58, x58, Pos(Succ(Zero)), x58, x59)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map3(x57, Succ(x61), Pos(Succ(Zero)), x59)_>=_new_map4(Pos(Succ(x57)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x59)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain new_map3(x63, x64, Pos(Succ(Zero)), x65) -> new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65), new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) -> H(x66, Succ(x67), Succ(x67), Pos(Succ(Zero)), x68, anew_new_map(x67, x66)) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)=new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) ==> new_map3(x63, x64, Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) the following chains were created: 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x108)), Succ(x109), Succ(x109), Pos(Succ(Zero)), Succ(x109), x110) -> new_map(x108, Succ(x109), Succ(x109), Pos(Succ(Zero)), x109, x108, x110), new_map(x111, Succ(x112), Succ(x112), Pos(Succ(Zero)), Zero, Succ(x113), x114) -> new_map0(x111, Succ(Succ(new_primPlusNat0(x112, Zero))), Succ(Succ(new_primPlusNat0(x112, Zero))), x114) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x108, Succ(x109), Succ(x109), Pos(Succ(Zero)), x109, x108, x110)=new_map(x111, Succ(x112), Succ(x112), Pos(Succ(Zero)), Zero, Succ(x113), x114) ==> new_map4(Pos(Succ(x108)), Succ(x109), Succ(x109), Pos(Succ(Zero)), Succ(x109), x110)_>=_new_map(x108, Succ(x109), Succ(x109), Pos(Succ(Zero)), x109, x108, x110)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map4(Pos(Succ(Succ(x113))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x110)_>=_new_map(Succ(x113), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x113), x110)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x115)), Succ(x116), Succ(x116), Pos(Succ(Zero)), Succ(x116), x117) -> new_map(x115, Succ(x116), Succ(x116), Pos(Succ(Zero)), x116, x115, x117), new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x119) -> new_map0(Succ(x118), Succ(Succ(Zero)), Succ(Succ(Zero)), x119) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x115, Succ(x116), Succ(x116), Pos(Succ(Zero)), x116, x115, x117)=new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x119) ==> new_map4(Pos(Succ(x115)), Succ(x116), Succ(x116), Pos(Succ(Zero)), Succ(x116), x117)_>=_new_map(x115, Succ(x116), Succ(x116), Pos(Succ(Zero)), x116, x115, x117)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map4(Pos(Succ(Succ(x118))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x117)_>=_new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x117)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x123)), Succ(x124), Succ(x124), Pos(Succ(Zero)), Succ(x124), x125) -> new_map(x123, Succ(x124), Succ(x124), Pos(Succ(Zero)), x124, x123, x125), new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x126) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x126) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x123, Succ(x124), Succ(x124), Pos(Succ(Zero)), x124, x123, x125)=new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x126) ==> new_map4(Pos(Succ(x123)), Succ(x124), Succ(x124), Pos(Succ(Zero)), Succ(x124), x125)_>=_new_map(x123, Succ(x124), Succ(x124), Pos(Succ(Zero)), x124, x123, x125)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x125)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x125)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x127)), Succ(x128), Succ(x128), Pos(Succ(Zero)), Succ(x128), x129) -> new_map(x127, Succ(x128), Succ(x128), Pos(Succ(Zero)), x128, x127, x129), new_map(x130, Succ(x131), Succ(x131), Pos(Succ(Zero)), Zero, Zero, x132) -> new_map2(x130, Succ(x131), Succ(x131), Pos(Succ(Zero)), x132) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x127, Succ(x128), Succ(x128), Pos(Succ(Zero)), x128, x127, x129)=new_map(x130, Succ(x131), Succ(x131), Pos(Succ(Zero)), Zero, Zero, x132) ==> new_map4(Pos(Succ(x127)), Succ(x128), Succ(x128), Pos(Succ(Zero)), Succ(x128), x129)_>=_new_map(x127, Succ(x128), Succ(x128), Pos(Succ(Zero)), x128, x127, x129)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x129)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x129)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) the following chains were created: 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153) -> H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151)), H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) -> new_map(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), Zero, Succ(x157), x156) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))=H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) which results in the following new constraint: 82.15/55.98 82.15/55.98 (3) (new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(x468))), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), Succ(Succ(x469)), x153)_>=_H(Succ(x468), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x469), Succ(x468)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) which results in the following new constraints: 82.15/55.98 82.15/55.98 (4) (new_new_map(x471, x470)=cons_new_map(Zero, Succ(x157)) & (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) 82.15/55.98 82.15/55.98 (5) (cons_new_map(Zero, Succ(x474))=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) 82.15/55.98 82.15/55.98 (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) with sigma = [x472 / x157, x473 / x153] which results in the following new constraint: 82.15/55.98 82.15/55.98 (7) (new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (8) (new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We solved constraint (6) using rules (I), (II). 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166) -> H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164)), H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) -> new_map(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), Zero, Zero, x169) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))=H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (anew_new_map(x165, x164)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x165, x164)=cons_new_map(Zero, Zero) which results in the following new constraint: 82.15/55.98 82.15/55.98 (3) (new_new_map(x476, x475)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x475))), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), Succ(Succ(x476)), x166)_>=_H(Succ(x475), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x476), Succ(x475)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x476, x475)=cons_new_map(Zero, Zero) which results in the following new constraints: 82.15/55.98 82.15/55.98 (4) (new_new_map(x478, x477)=cons_new_map(Zero, Zero) & (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) 82.15/55.98 82.15/55.98 (5) (cons_new_map(Zero, Succ(x480))=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Succ(x480)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Succ(x480)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Succ(x480))))) 82.15/55.98 82.15/55.98 (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) with sigma = [x479 / x166] which results in the following new constraint: 82.15/55.98 82.15/55.98 (7) (new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: 82.15/55.98 82.15/55.98 (8) (new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) the following chains were created: 82.15/55.98 *We consider the chain H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205))) -> new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204), new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) -> new_map0(x206, Succ(Succ(new_primPlusNat0(x207, Zero))), Succ(Succ(new_primPlusNat0(x207, Zero))), x209) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)=new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) ==> H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain H(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213))) -> new_map(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), Zero, Succ(x213), x212), new_map(Succ(x214), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x214), x215) -> new_map0(Succ(x214), Succ(Succ(Zero)), Succ(Succ(Zero)), x215) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), Zero, Succ(x213), x212)=new_map(Succ(x214), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x214), x215) ==> H(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213)))_>=_new_map(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), Zero, Succ(x213), x212)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (H(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213)))_>=_new_map(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x213), x212)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: 82.15/55.98 *We consider the chain new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239) -> new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239), new_map0(x240, x241, x242, x243) -> new_map3(x240, x242, Pos(Succ(Zero)), x243) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)=new_map0(x240, x241, x242, x243) ==> new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) the following chains were created: 82.15/55.98 *We consider the chain new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289) -> new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289), new_map0(x290, x291, x292, x293) -> new_map3(x290, x292, Pos(Succ(Zero)), x293) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)=new_map0(x290, x291, x292, x293) ==> new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289)_>=_new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289)_>=_new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) the following chains were created: 82.15/55.98 *We consider the chain H(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero)) -> new_map(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), Zero, Zero, x342), new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x343) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x343) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), Zero, Zero, x342)=new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x343) ==> H(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero))_>=_new_map(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), Zero, Zero, x342)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (H(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero))_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x342)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero)) -> new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346), new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) -> new_map2(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), x349) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)=new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) ==> H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) the following chains were created: 82.15/55.98 *We consider the chain new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366), new_map2(x367, Succ(x368), Succ(x368), Pos(Succ(Zero)), x369) -> new_map0(x367, Succ(Succ(new_primPlusNat0(x368, Zero))), Succ(Succ(new_primPlusNat0(x368, Zero))), x369) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)=new_map2(x367, Succ(x368), Succ(x368), Pos(Succ(Zero)), x369) ==> new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370), new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x371) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x371) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)=new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x371) ==> new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) the following chains were created: 82.15/55.98 *We consider the chain new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404) -> new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404), new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) -> new_map0(x405, Succ(Succ(new_primPlusNat0(x406, Zero))), Succ(Succ(new_primPlusNat0(x406, Zero))), x407) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)=new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) ==> new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *We consider the chain new_map(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), Zero, Zero, x410) -> new_map2(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), x410), new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x411) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x411) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map2(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), x410)=new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x411) ==> new_map(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), Zero, Zero, x410)_>=_new_map2(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), x410)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x410)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x410)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: 82.15/55.98 *We consider the chain new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414) -> new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414), new_map0(x415, x416, x417, x418) -> new_map3(x415, x417, Pos(Succ(Zero)), x418) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)=new_map0(x415, x416, x417, x418) ==> new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) the following chains were created: 82.15/55.98 *We consider the chain new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452), new_map0(x453, x454, x455, x456) -> new_map3(x453, x455, Pos(Succ(Zero)), x456) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)=new_map0(x453, x454, x455, x456) ==> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452)_>=_new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452)_>=_new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 To summarize, we get the following constraints P__>=_ for the following pairs. 82.15/55.98 82.15/55.98 *new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 82.15/55.98 *(new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 82.15/55.98 *(new_map3(x57, Succ(x61), Pos(Succ(Zero)), x59)_>=_new_map4(Pos(Succ(x57)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x59)) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(x113))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x110)_>=_new_map(Succ(x113), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x113), x110)) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(x118))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x117)_>=_new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x117)) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x125)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x125)) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x129)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x129)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 82.15/55.98 *(H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) 82.15/55.98 82.15/55.98 82.15/55.98 *(H(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213)))_>=_new_map(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x213), x212)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 82.15/55.98 *(new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 82.15/55.98 *(new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289)_>=_new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 82.15/55.98 *(H(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero))_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x342)) 82.15/55.98 82.15/55.98 82.15/55.98 *(H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 82.15/55.98 *(new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 82.15/55.98 *(new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x410)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x410)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 82.15/55.98 *(new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) 82.15/55.98 82.15/55.98 *(new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452)_>=_new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 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. 82.15/55.98 82.15/55.98 Using the following integer polynomial ordering the resulting constraints can be solved 82.15/55.98 82.15/55.98 Polynomial interpretation [NONINF]: 82.15/55.98 82.15/55.98 POL(H(x_1, x_2, x_3, x_4, x_5, x_6)) = -1 - x_2 - x_4 - x_6 82.15/55.98 POL(Pos(x_1)) = 0 82.15/55.98 POL(Succ(x_1)) = 1 + x_1 82.15/55.98 POL(Zero) = 0 82.15/55.98 POL(anew_new_map(x_1, x_2)) = 0 82.15/55.98 POL(c) = -3 82.15/55.98 POL(cons_new_map(x_1, x_2)) = 0 82.15/55.98 POL(new_map(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 - x_2 - x_4 + x_5 82.15/55.98 POL(new_map0(x_1, x_2, x_3, x_4)) = -1 - x_3 82.15/55.98 POL(new_map2(x_1, x_2, x_3, x_4, x_5)) = -1 - x_3 - x_4 82.15/55.98 POL(new_map3(x_1, x_2, x_3, x_4)) = -1 - x_2 - x_3 82.15/55.98 POL(new_map4(x_1, x_2, x_3, x_4, x_5, x_6)) = -1 - x_1 - x_2 + x_3 - x_4 - x_5 82.15/55.98 POL(new_new_map(x_1, x_2)) = 0 82.15/55.98 POL(new_primPlusNat0(x_1, x_2)) = x_1 82.15/55.98 82.15/55.98 82.15/55.98 The following pairs are in P_>: 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) 82.15/55.98 The following pairs are in P_bound: 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) 82.15/55.98 new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) 82.15/55.98 The following rules are usable: 82.15/55.98 new_new_map(yx2830, yx2840) -> anew_new_map(Succ(yx2830), Succ(yx2840)) 82.15/55.98 Succ(yx2180) -> new_primPlusNat0(Succ(yx2180), Zero) 82.15/55.98 Zero -> new_primPlusNat0(Zero, Zero) 82.15/55.98 new_new_map(yx2830, yx2840) -> new_new_map(Succ(yx2830), Succ(yx2840)) 82.15/55.98 cons_new_map(Zero, Succ(yx2840)) -> new_new_map(Zero, Succ(yx2840)) 82.15/55.98 cons_new_map(Zero, Zero) -> new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (262) 82.15/55.98 Complex Obligation (AND) 82.15/55.98 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (263) 82.15/55.98 Obligation: 82.15/55.98 Q DP problem: 82.15/55.98 The TRS P consists of the following rules: 82.15/55.98 82.15/55.98 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 82.15/55.98 The TRS R consists of the following rules: 82.15/55.98 82.15/55.98 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.98 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.98 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.98 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 The set Q consists of the following terms: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(x0), Zero) 82.15/55.98 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.98 new_primPlusNat0(Zero, Zero) 82.15/55.98 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (264) DependencyGraphProof (EQUIVALENT) 82.15/55.98 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 8 less nodes. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (265) 82.15/55.98 TRUE 82.15/55.98 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (266) 82.15/55.98 Obligation: 82.15/55.98 Q DP problem: 82.15/55.98 The TRS P consists of the following rules: 82.15/55.98 82.15/55.98 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 82.15/55.98 The TRS R consists of the following rules: 82.15/55.98 82.15/55.98 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.98 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.98 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.98 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 The set Q consists of the following terms: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(x0), Zero) 82.15/55.98 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.98 new_primPlusNat0(Zero, Zero) 82.15/55.98 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (267) InductionCalculusProof (EQUIVALENT) 82.15/55.98 Note that final constraints are written in bold face. 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) the following chains were created: 82.15/55.98 *We consider the chain new_map0(x4, x5, x6, x7) -> new_map3(x4, x6, Pos(Succ(Zero)), x7), new_map3(x8, x9, Pos(Succ(Zero)), x10) -> new_map4(Pos(Succ(x8)), x9, x9, Pos(Succ(Zero)), x9, x10) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map3(x4, x6, Pos(Succ(Zero)), x7)=new_map3(x8, x9, Pos(Succ(Zero)), x10) ==> new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) the following chains were created: 82.15/55.98 *We consider the chain new_map3(x63, x64, Pos(Succ(Zero)), x65) -> new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65), new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) -> H(x66, Succ(x67), Succ(x67), Pos(Succ(Zero)), x68, anew_new_map(x67, x66)) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)=new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) ==> new_map3(x63, x64, Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) the following chains were created: 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153) -> H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151)), H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) -> new_map(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), Zero, Succ(x157), x156) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))=H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) which results in the following new constraint: 82.15/55.98 82.15/55.98 (3) (new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(x468))), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), Succ(Succ(x469)), x153)_>=_H(Succ(x468), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x469), Succ(x468)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) which results in the following new constraints: 82.15/55.98 82.15/55.98 (4) (new_new_map(x471, x470)=cons_new_map(Zero, Succ(x157)) & (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) 82.15/55.98 82.15/55.98 (5) (cons_new_map(Zero, Succ(x474))=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) 82.15/55.98 82.15/55.98 (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) with sigma = [x472 / x157, x473 / x153] which results in the following new constraint: 82.15/55.98 82.15/55.98 (7) (new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (8) (new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We solved constraint (6) using rules (I), (II). 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166) -> H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164)), H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) -> new_map(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), Zero, Zero, x169) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))=H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (anew_new_map(x165, x164)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x165, x164)=cons_new_map(Zero, Zero) which results in the following new constraint: 82.15/55.98 82.15/55.98 (3) (new_new_map(x476, x475)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x475))), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), Succ(Succ(x476)), x166)_>=_H(Succ(x475), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x476), Succ(x475)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x476, x475)=cons_new_map(Zero, Zero) which results in the following new constraints: 82.15/55.98 82.15/55.98 (4) (new_new_map(x478, x477)=cons_new_map(Zero, Zero) & (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) 82.15/55.98 82.15/55.98 (5) (cons_new_map(Zero, Succ(x480))=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Succ(x480)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Succ(x480)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Succ(x480))))) 82.15/55.98 82.15/55.98 (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) with sigma = [x479 / x166] which results in the following new constraint: 82.15/55.98 82.15/55.98 (7) (new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: 82.15/55.98 82.15/55.98 (8) (new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) the following chains were created: 82.15/55.98 *We consider the chain H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205))) -> new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204), new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) -> new_map0(x206, Succ(Succ(new_primPlusNat0(x207, Zero))), Succ(Succ(new_primPlusNat0(x207, Zero))), x209) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)=new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) ==> H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: 82.15/55.98 *We consider the chain new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239) -> new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239), new_map0(x240, x241, x242, x243) -> new_map3(x240, x242, Pos(Succ(Zero)), x243) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)=new_map0(x240, x241, x242, x243) ==> new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) the following chains were created: 82.15/55.98 *We consider the chain H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero)) -> new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346), new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) -> new_map2(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), x349) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)=new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) ==> H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) the following chains were created: 82.15/55.98 *We consider the chain new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404) -> new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404), new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) -> new_map0(x405, Succ(Succ(new_primPlusNat0(x406, Zero))), Succ(Succ(new_primPlusNat0(x406, Zero))), x407) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)=new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) ==> new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: 82.15/55.98 *We consider the chain new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414) -> new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414), new_map0(x415, x416, x417, x418) -> new_map3(x415, x417, Pos(Succ(Zero)), x418) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)=new_map0(x415, x416, x417, x418) ==> new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 To summarize, we get the following constraints P__>=_ for the following pairs. 82.15/55.98 82.15/55.98 *new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 82.15/55.98 *(new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 82.15/55.98 *(new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) 82.15/55.98 82.15/55.98 82.15/55.98 *(new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 82.15/55.98 *(H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 82.15/55.98 *(new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 82.15/55.98 *(H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 82.15/55.98 *(new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 *new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 82.15/55.98 *(new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 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. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (268) 82.15/55.98 Obligation: 82.15/55.98 Q DP problem: 82.15/55.98 The TRS P consists of the following rules: 82.15/55.98 82.15/55.98 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.98 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.98 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.98 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.98 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.98 82.15/55.98 The TRS R consists of the following rules: 82.15/55.98 82.15/55.98 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.98 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.98 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.98 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.98 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 The set Q consists of the following terms: 82.15/55.98 82.15/55.98 new_primPlusNat0(Succ(x0), Zero) 82.15/55.98 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.98 new_primPlusNat0(Zero, Zero) 82.15/55.98 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Succ(x0), Succ(x1)) 82.15/55.98 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.98 new_new_map(Zero, Succ(x0)) 82.15/55.98 new_new_map(Zero, Zero) 82.15/55.98 82.15/55.98 We have to consider all minimal (P,Q,R)-chains. 82.15/55.98 ---------------------------------------- 82.15/55.98 82.15/55.98 (269) NonInfProof (EQUIVALENT) 82.15/55.98 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 82.15/55.98 82.15/55.98 Note that final constraints are written in bold face. 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) the following chains were created: 82.15/55.98 *We consider the chain new_map0(x4, x5, x6, x7) -> new_map3(x4, x6, Pos(Succ(Zero)), x7), new_map3(x8, x9, Pos(Succ(Zero)), x10) -> new_map4(Pos(Succ(x8)), x9, x9, Pos(Succ(Zero)), x9, x10) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map3(x4, x6, Pos(Succ(Zero)), x7)=new_map3(x8, x9, Pos(Succ(Zero)), x10) ==> new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) the following chains were created: 82.15/55.98 *We consider the chain new_map3(x63, x64, Pos(Succ(Zero)), x65) -> new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65), new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) -> H(x66, Succ(x67), Succ(x67), Pos(Succ(Zero)), x68, anew_new_map(x67, x66)) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)=new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) ==> new_map3(x63, x64, Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 82.15/55.98 82.15/55.98 (2) (new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 82.15/55.98 For Pair new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) the following chains were created: 82.15/55.98 *We consider the chain new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153) -> H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151)), H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) -> new_map(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), Zero, Succ(x157), x156) which results in the following constraint: 82.15/55.98 82.15/55.98 (1) (H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))=H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.99 82.15/55.99 (2) (anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) which results in the following new constraint: 82.15/55.99 82.15/55.99 (3) (new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(x468))), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), Succ(Succ(x469)), x153)_>=_H(Succ(x468), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x469), Succ(x468)))) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) which results in the following new constraints: 82.15/55.99 82.15/55.99 (4) (new_new_map(x471, x470)=cons_new_map(Zero, Succ(x157)) & (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) 82.15/55.99 82.15/55.99 (5) (cons_new_map(Zero, Succ(x474))=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) 82.15/55.99 82.15/55.99 (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) with sigma = [x472 / x157, x473 / x153] which results in the following new constraint: 82.15/55.99 82.15/55.99 (7) (new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.99 82.15/55.99 (8) (new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We solved constraint (6) using rules (I), (II). 82.15/55.99 *We consider the chain new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166) -> H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164)), H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) -> new_map(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), Zero, Zero, x169) which results in the following constraint: 82.15/55.99 82.15/55.99 (1) (H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))=H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.99 82.15/55.99 (2) (anew_new_map(x165, x164)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x165, x164)=cons_new_map(Zero, Zero) which results in the following new constraint: 82.15/55.99 82.15/55.99 (3) (new_new_map(x476, x475)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x475))), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), Succ(Succ(x476)), x166)_>=_H(Succ(x475), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x476), Succ(x475)))) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x476, x475)=cons_new_map(Zero, Zero) which results in the following new constraints: 82.15/55.99 82.15/55.99 (4) (new_new_map(x478, x477)=cons_new_map(Zero, Zero) & (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) 82.15/55.99 82.15/55.99 (5) (cons_new_map(Zero, Succ(x480))=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Succ(x480)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Succ(x480)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Succ(x480))))) 82.15/55.99 82.15/55.99 (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) with sigma = [x479 / x166] which results in the following new constraint: 82.15/55.99 82.15/55.99 (7) (new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: 82.15/55.99 82.15/55.99 (8) (new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) the following chains were created: 82.15/55.99 *We consider the chain H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205))) -> new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204), new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) -> new_map0(x206, Succ(Succ(new_primPlusNat0(x207, Zero))), Succ(Succ(new_primPlusNat0(x207, Zero))), x209) which results in the following constraint: 82.15/55.99 82.15/55.99 (1) (new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)=new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) ==> H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.99 82.15/55.99 (2) (H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: 82.15/55.99 *We consider the chain new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239) -> new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239), new_map0(x240, x241, x242, x243) -> new_map3(x240, x242, Pos(Succ(Zero)), x243) which results in the following constraint: 82.15/55.99 82.15/55.99 (1) (new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)=new_map0(x240, x241, x242, x243) ==> new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.99 82.15/55.99 (2) (new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) the following chains were created: 82.15/55.99 *We consider the chain H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero)) -> new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346), new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) -> new_map2(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), x349) which results in the following constraint: 82.15/55.99 82.15/55.99 (1) (new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)=new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) ==> H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.99 82.15/55.99 (2) (H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) the following chains were created: 82.15/55.99 *We consider the chain new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404) -> new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404), new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) -> new_map0(x405, Succ(Succ(new_primPlusNat0(x406, Zero))), Succ(Succ(new_primPlusNat0(x406, Zero))), x407) which results in the following constraint: 82.15/55.99 82.15/55.99 (1) (new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)=new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) ==> new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.99 82.15/55.99 (2) (new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 For Pair new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: 82.15/55.99 *We consider the chain new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414) -> new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414), new_map0(x415, x416, x417, x418) -> new_map3(x415, x417, Pos(Succ(Zero)), x418) which results in the following constraint: 82.15/55.99 82.15/55.99 (1) (new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)=new_map0(x415, x416, x417, x418) ==> new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 82.15/55.99 82.15/55.99 (2) (new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 To summarize, we get the following constraints P__>=_ for the following pairs. 82.15/55.99 82.15/55.99 *new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.99 82.15/55.99 *(new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 *new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.99 82.15/55.99 *(new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 *new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.99 82.15/55.99 *(new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) 82.15/55.99 82.15/55.99 82.15/55.99 *(new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) 82.15/55.99 82.15/55.99 82.15/55.99 *(new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) 82.15/55.99 82.15/55.99 82.15/55.99 *(new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.99 82.15/55.99 *(H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.99 82.15/55.99 *(new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.99 82.15/55.99 *(H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.99 82.15/55.99 *(new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 *new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.99 82.15/55.99 *(new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 82.15/55.99 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. 82.15/55.99 82.15/55.99 Using the following integer polynomial ordering the resulting constraints can be solved 82.15/55.99 82.15/55.99 Polynomial interpretation [NONINF]: 82.15/55.99 82.15/55.99 POL(H(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 - x_3 - x_4 82.15/55.99 POL(Pos(x_1)) = x_1 82.15/55.99 POL(Succ(x_1)) = 1 + x_1 82.15/55.99 POL(Zero) = 0 82.15/55.99 POL(anew_new_map(x_1, x_2)) = 0 82.15/55.99 POL(c) = -3 82.15/55.99 POL(cons_new_map(x_1, x_2)) = x_2 82.15/55.99 POL(new_map(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 + x_1 - x_2 - x_4 + x_5 82.15/55.99 POL(new_map0(x_1, x_2, x_3, x_4)) = -1 + x_1 - x_3 82.15/55.99 POL(new_map2(x_1, x_2, x_3, x_4, x_5)) = -1 + x_1 - x_3 - x_4 82.15/55.99 POL(new_map3(x_1, x_2, x_3, x_4)) = x_1 - x_2 - x_3 82.15/55.99 POL(new_map4(x_1, x_2, x_3, x_4, x_5, x_6)) = -1 + x_1 + x_2 - x_3 - x_4 - x_5 82.15/55.99 POL(new_new_map(x_1, x_2)) = 0 82.15/55.99 POL(new_primPlusNat0(x_1, x_2)) = x_1 82.15/55.99 82.15/55.99 82.15/55.99 The following pairs are in P_>: 82.15/55.99 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.99 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.99 The following pairs are in P_bound: 82.15/55.99 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.99 The following rules are usable: 82.15/55.99 Succ(yx2180) -> new_primPlusNat0(Succ(yx2180), Zero) 82.15/55.99 Zero -> new_primPlusNat0(Zero, Zero) 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (270) 82.15/55.99 Complex Obligation (AND) 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (271) 82.15/55.99 Obligation: 82.15/55.99 Q DP problem: 82.15/55.99 The TRS P consists of the following rules: 82.15/55.99 82.15/55.99 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.99 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.99 new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) 82.15/55.99 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.99 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.99 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.99 82.15/55.99 The TRS R consists of the following rules: 82.15/55.99 82.15/55.99 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.99 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.99 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.99 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.99 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.99 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.99 82.15/55.99 The set Q consists of the following terms: 82.15/55.99 82.15/55.99 new_primPlusNat0(Succ(x0), Zero) 82.15/55.99 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.99 new_primPlusNat0(Zero, Zero) 82.15/55.99 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.99 new_new_map(Succ(x0), Succ(x1)) 82.15/55.99 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.99 new_new_map(Zero, Succ(x0)) 82.15/55.99 new_new_map(Zero, Zero) 82.15/55.99 82.15/55.99 We have to consider all minimal (P,Q,R)-chains. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (272) DependencyGraphProof (EQUIVALENT) 82.15/55.99 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 6 less nodes. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (273) 82.15/55.99 TRUE 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (274) 82.15/55.99 Obligation: 82.15/55.99 Q DP problem: 82.15/55.99 The TRS P consists of the following rules: 82.15/55.99 82.15/55.99 new_map0(yx279, yx286, yx285, h) -> new_map3(yx279, yx285, Pos(Succ(Zero)), h) 82.15/55.99 new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) 82.15/55.99 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) 82.15/55.99 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.99 H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) 82.15/55.99 new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) 82.15/55.99 new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) 82.15/55.99 82.15/55.99 The TRS R consists of the following rules: 82.15/55.99 82.15/55.99 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.99 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.99 anew_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.99 new_new_map(Succ(yx2830), Succ(yx2840)) -> new_new_map(yx2830, yx2840) 82.15/55.99 new_new_map(Zero, Succ(yx2840)) -> cons_new_map(Zero, Succ(yx2840)) 82.15/55.99 new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) 82.15/55.99 82.15/55.99 The set Q consists of the following terms: 82.15/55.99 82.15/55.99 new_primPlusNat0(Succ(x0), Zero) 82.15/55.99 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.99 new_primPlusNat0(Zero, Zero) 82.15/55.99 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.99 new_new_map(Succ(x0), Succ(x1)) 82.15/55.99 anew_new_map(Succ(x0), Succ(x1)) 82.15/55.99 new_new_map(Zero, Succ(x0)) 82.15/55.99 new_new_map(Zero, Zero) 82.15/55.99 82.15/55.99 We have to consider all minimal (P,Q,R)-chains. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (275) DependencyGraphProof (EQUIVALENT) 82.15/55.99 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 7 less nodes. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (276) 82.15/55.99 TRUE 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (277) 82.15/55.99 Obligation: 82.15/55.99 Q DP problem: 82.15/55.99 The TRS P consists of the following rules: 82.15/55.99 82.15/55.99 new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.99 82.15/55.99 R is empty. 82.15/55.99 Q is empty. 82.15/55.99 We have to consider all minimal (P,Q,R)-chains. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (278) QDPSizeChangeProof (EQUIVALENT) 82.15/55.99 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. 82.15/55.99 82.15/55.99 From the DPs we obtained the following set of size-change graphs: 82.15/55.99 *new_map(yx279, yx280, yx281, yx282, Succ(yx2830), Succ(yx2840), h) -> new_map(yx279, yx280, yx281, yx282, yx2830, yx2840, h) 82.15/55.99 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 > 5, 6 > 6, 7 >= 7 82.15/55.99 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (279) 82.15/55.99 YES 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (280) 82.15/55.99 Obligation: 82.15/55.99 Q DP problem: 82.15/55.99 The TRS P consists of the following rules: 82.15/55.99 82.15/55.99 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.99 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.99 new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) 82.15/55.99 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.99 new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) 82.15/55.99 new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.99 82.15/55.99 The TRS R consists of the following rules: 82.15/55.99 82.15/55.99 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.99 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.99 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.99 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.99 82.15/55.99 The set Q consists of the following terms: 82.15/55.99 82.15/55.99 new_primPlusNat0(Succ(x0), Zero) 82.15/55.99 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.99 new_primPlusNat0(Zero, Zero) 82.15/55.99 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.99 82.15/55.99 We have to consider all minimal (P,Q,R)-chains. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (281) TransformationProof (EQUIVALENT) 82.15/55.99 By instantiating [LPAR04] the rule new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) we obtained the following new rules [LPAR04]: 82.15/55.99 82.15/55.99 (new_map13(Pos(x0), Succ(x1), Succ(x1), Pos(Succ(Zero)), Succ(x1), Pos(x0), z2) -> new_map9(Pos(x0), x1, z2),new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2)) 82.15/55.99 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (282) 82.15/55.99 Obligation: 82.15/55.99 Q DP problem: 82.15/55.99 The TRS P consists of the following rules: 82.15/55.99 82.15/55.99 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.99 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.99 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.99 new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) 82.15/55.99 new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.99 82.15/55.99 The TRS R consists of the following rules: 82.15/55.99 82.15/55.99 new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) 82.15/55.99 new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) 82.15/55.99 new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) 82.15/55.99 new_primPlusNat0(Zero, Zero) -> Zero 82.15/55.99 82.15/55.99 The set Q consists of the following terms: 82.15/55.99 82.15/55.99 new_primPlusNat0(Succ(x0), Zero) 82.15/55.99 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.99 new_primPlusNat0(Zero, Zero) 82.15/55.99 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.99 82.15/55.99 We have to consider all minimal (P,Q,R)-chains. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (283) UsableRulesProof (EQUIVALENT) 82.15/55.99 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. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (284) 82.15/55.99 Obligation: 82.15/55.99 Q DP problem: 82.15/55.99 The TRS P consists of the following rules: 82.15/55.99 82.15/55.99 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.99 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.99 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.99 new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) 82.15/55.99 new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.99 82.15/55.99 R is empty. 82.15/55.99 The set Q consists of the following terms: 82.15/55.99 82.15/55.99 new_primPlusNat0(Succ(x0), Zero) 82.15/55.99 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.99 new_primPlusNat0(Zero, Zero) 82.15/55.99 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.99 82.15/55.99 We have to consider all minimal (P,Q,R)-chains. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (285) QReductionProof (EQUIVALENT) 82.15/55.99 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 82.15/55.99 82.15/55.99 new_primPlusNat0(Succ(x0), Zero) 82.15/55.99 new_primPlusNat0(Zero, Succ(x0)) 82.15/55.99 new_primPlusNat0(Zero, Zero) 82.15/55.99 new_primPlusNat0(Succ(x0), Succ(x1)) 82.15/55.99 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (286) 82.15/55.99 Obligation: 82.15/55.99 Q DP problem: 82.15/55.99 The TRS P consists of the following rules: 82.15/55.99 82.15/55.99 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.99 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.99 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.99 new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) 82.15/55.99 new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.99 82.15/55.99 R is empty. 82.15/55.99 Q is empty. 82.15/55.99 We have to consider all minimal (P,Q,R)-chains. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (287) TransformationProof (EQUIVALENT) 82.15/55.99 By instantiating [LPAR04] the rule new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) we obtained the following new rules [LPAR04]: 82.15/55.99 82.15/55.99 (new_map13(Neg(Succ(Zero)), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2),new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2)) 82.15/55.99 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (288) 82.15/55.99 Obligation: 82.15/55.99 Q DP problem: 82.15/55.99 The TRS P consists of the following rules: 82.15/55.99 82.15/55.99 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.99 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.99 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) 82.15/55.99 new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.99 82.15/55.99 R is empty. 82.15/55.99 Q is empty. 82.15/55.99 We have to consider all minimal (P,Q,R)-chains. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (289) TransformationProof (EQUIVALENT) 82.15/55.99 By instantiating [LPAR04] the rule new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) we obtained the following new rules [LPAR04]: 82.15/55.99 82.15/55.99 (new_map13(Neg(Succ(Succ(x0))), Succ(x1), Succ(x1), Pos(Succ(Zero)), Succ(x1), Neg(Succ(Succ(x0))), z2) -> new_map15(Neg(Succ(Succ(x0))), Succ(x1), Succ(x1), Pos(Succ(Zero)), x0, x1, z2),new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2)) 82.15/55.99 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (290) 82.15/55.99 Obligation: 82.15/55.99 Q DP problem: 82.15/55.99 The TRS P consists of the following rules: 82.15/55.99 82.15/55.99 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.99 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.99 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) 82.15/55.99 82.15/55.99 R is empty. 82.15/55.99 Q is empty. 82.15/55.99 We have to consider all minimal (P,Q,R)-chains. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (291) TransformationProof (EQUIVALENT) 82.15/55.99 By instantiating [LPAR04] the rule new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) we obtained the following new rules [LPAR04]: 82.15/55.99 82.15/55.99 (new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2),new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2)) 82.15/55.99 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (292) 82.15/55.99 Obligation: 82.15/55.99 Q DP problem: 82.15/55.99 The TRS P consists of the following rules: 82.15/55.99 82.15/55.99 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.99 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.99 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) 82.15/55.99 82.15/55.99 R is empty. 82.15/55.99 Q is empty. 82.15/55.99 We have to consider all minimal (P,Q,R)-chains. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (293) QDPOrderProof (EQUIVALENT) 82.15/55.99 We use the reduction pair processor [LPAR04,JAR06]. 82.15/55.99 82.15/55.99 82.15/55.99 The following pairs can be oriented strictly and are deleted. 82.15/55.99 82.15/55.99 new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 The remaining pairs can at least be oriented weakly. 82.15/55.99 Used ordering: Polynomial interpretation [POLO]: 82.15/55.99 82.15/55.99 POL(Neg(x_1)) = 0 82.15/55.99 POL(Pos(x_1)) = x_1 82.15/55.99 POL(Succ(x_1)) = 0 82.15/55.99 POL(Zero) = 1 82.15/55.99 POL(new_map1(x_1, x_2, x_3, x_4)) = 0 82.15/55.99 POL(new_map12(x_1, x_2, x_3, x_4)) = x_3 82.15/55.99 POL(new_map13(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 82.15/55.99 POL(new_map15(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_4 82.15/55.99 POL(new_map16(x_1, x_2, x_3, x_4, x_5)) = x_4 82.15/55.99 POL(new_map17(x_1, x_2, x_3, x_4, x_5)) = x_4 82.15/55.99 POL(new_map9(x_1, x_2, x_3)) = 0 82.15/55.99 82.15/55.99 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 82.15/55.99 none 82.15/55.99 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (294) 82.15/55.99 Obligation: 82.15/55.99 Q DP problem: 82.15/55.99 The TRS P consists of the following rules: 82.15/55.99 82.15/55.99 new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.99 new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.99 new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.99 new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) 82.15/55.99 82.15/55.99 R is empty. 82.15/55.99 Q is empty. 82.15/55.99 We have to consider all minimal (P,Q,R)-chains. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (295) QDPSizeChangeProof (EQUIVALENT) 82.15/55.99 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. 82.15/55.99 82.15/55.99 From the DPs we obtained the following set of size-change graphs: 82.15/55.99 *new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) 82.15/55.99 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 4 82.15/55.99 82.15/55.99 82.15/55.99 *new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) 82.15/55.99 The graph contains the following edges 1 >= 1, 2 >= 2, 2 >= 3, 3 >= 4, 2 >= 5, 1 >= 6, 4 >= 7 82.15/55.99 82.15/55.99 82.15/55.99 *new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) 82.15/55.99 The graph contains the following edges 1 >= 1, 6 >= 1, 2 > 2, 3 > 2, 5 > 2, 7 >= 3 82.15/55.99 82.15/55.99 82.15/55.99 *new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 The graph contains the following edges 1 >= 1, 3 > 2, 5 >= 3 82.15/55.99 82.15/55.99 82.15/55.99 *new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) 82.15/55.99 The graph contains the following edges 1 >= 1, 6 >= 1, 2 >= 2, 3 >= 2, 5 >= 2, 2 >= 3, 3 >= 3, 5 >= 3, 4 >= 4, 1 > 5, 6 > 5, 2 > 6, 3 > 6, 5 > 6, 7 >= 7 82.15/55.99 82.15/55.99 82.15/55.99 *new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) 82.15/55.99 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 7 >= 5 82.15/55.99 82.15/55.99 82.15/55.99 *new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) 82.15/55.99 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 > 5, 6 > 6, 7 >= 7 82.15/55.99 82.15/55.99 82.15/55.99 *new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) 82.15/55.99 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 7 >= 5 82.15/55.99 82.15/55.99 82.15/55.99 *new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 The graph contains the following edges 1 >= 1, 4 > 2, 3 > 3, 5 >= 4 82.15/55.99 82.15/55.99 82.15/55.99 *new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4 82.15/55.99 82.15/55.99 82.15/55.99 *new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) 82.15/55.99 The graph contains the following edges 1 >= 1, 4 > 2, 3 > 3, 5 >= 4 82.15/55.99 82.15/55.99 82.15/55.99 *new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 The graph contains the following edges 1 >= 1, 3 > 2, 4 >= 3 82.15/55.99 82.15/55.99 82.15/55.99 *new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) 82.15/55.99 The graph contains the following edges 1 >= 1, 3 > 2, 5 >= 3 82.15/55.99 82.15/55.99 82.15/55.99 *new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) 82.15/55.99 The graph contains the following edges 1 >= 1, 6 >= 1, 2 >= 2, 3 >= 2, 5 >= 2, 2 >= 3, 3 >= 3, 5 >= 3, 4 >= 4, 7 >= 5 82.15/55.99 82.15/55.99 82.15/55.99 *new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) 82.15/55.99 The graph contains the following edges 1 >= 1, 6 >= 1, 2 >= 2, 3 >= 2, 5 >= 2, 2 >= 3, 3 >= 3, 5 >= 3, 4 >= 4, 7 >= 5 82.15/55.99 82.15/55.99 82.15/55.99 *new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) 82.15/55.99 The graph contains the following edges 1 >= 1, 6 >= 1, 2 >= 2, 3 >= 2, 5 >= 2, 2 >= 3, 3 >= 3, 5 >= 3, 4 >= 4, 7 >= 5 82.15/55.99 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (296) 82.15/55.99 YES 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (297) 82.15/55.99 Obligation: 82.15/55.99 Q DP problem: 82.15/55.99 The TRS P consists of the following rules: 82.15/55.99 82.15/55.99 new_primMinusNatS(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS(yx2000, yx2010) 82.15/55.99 82.15/55.99 R is empty. 82.15/55.99 Q is empty. 82.15/55.99 We have to consider all minimal (P,Q,R)-chains. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (298) QDPSizeChangeProof (EQUIVALENT) 82.15/55.99 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. 82.15/55.99 82.15/55.99 From the DPs we obtained the following set of size-change graphs: 82.15/55.99 *new_primMinusNatS(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS(yx2000, yx2010) 82.15/55.99 The graph contains the following edges 1 > 1, 2 > 2 82.15/55.99 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (299) 82.15/55.99 YES 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (300) 82.15/55.99 Obligation: 82.15/55.99 Q DP problem: 82.15/55.99 The TRS P consists of the following rules: 82.15/55.99 82.15/55.99 new_primPlusNat(Succ(yx2180), Succ(yx212000)) -> new_primPlusNat(yx2180, yx212000) 82.15/55.99 82.15/55.99 R is empty. 82.15/55.99 Q is empty. 82.15/55.99 We have to consider all minimal (P,Q,R)-chains. 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (301) QDPSizeChangeProof (EQUIVALENT) 82.15/55.99 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. 82.15/55.99 82.15/55.99 From the DPs we obtained the following set of size-change graphs: 82.15/55.99 *new_primPlusNat(Succ(yx2180), Succ(yx212000)) -> new_primPlusNat(yx2180, yx212000) 82.15/55.99 The graph contains the following edges 1 > 1, 2 > 2 82.15/55.99 82.15/55.99 82.15/55.99 ---------------------------------------- 82.15/55.99 82.15/55.99 (302) 82.15/55.99 YES 82.15/56.02 EOF