4.57/2.09 YES 4.57/2.12 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 4.57/2.12 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.57/2.12 4.57/2.12 4.57/2.12 Quasi decreasingness of the given CTRS could be proven: 4.57/2.12 4.57/2.12 (0) CTRS 4.57/2.12 (1) CTRSToQTRSProof [SOUND, 0 ms] 4.57/2.12 (2) QTRS 4.57/2.12 (3) QTRSRRRProof [EQUIVALENT, 38 ms] 4.57/2.12 (4) QTRS 4.57/2.12 (5) DependencyPairsProof [EQUIVALENT, 0 ms] 4.57/2.12 (6) QDP 4.57/2.12 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (8) QDP 4.57/2.12 (9) UsableRulesReductionPairsProof [EQUIVALENT, 9 ms] 4.57/2.12 (10) QDP 4.57/2.12 (11) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (12) QDP 4.57/2.12 (13) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (14) QDP 4.57/2.12 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (16) QDP 4.57/2.12 (17) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (18) QDP 4.57/2.12 (19) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (20) QDP 4.57/2.12 (21) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (22) QDP 4.57/2.12 (23) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (24) QDP 4.57/2.12 (25) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (26) QDP 4.57/2.12 (27) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (28) QDP 4.57/2.12 (29) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (30) QDP 4.57/2.12 (31) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (32) QDP 4.57/2.12 (33) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (34) QDP 4.57/2.12 (35) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (36) QDP 4.57/2.12 (37) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (38) QDP 4.57/2.12 (39) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (40) QDP 4.57/2.12 (41) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (42) QDP 4.57/2.12 (43) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (44) QDP 4.57/2.12 (45) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (46) QDP 4.57/2.12 (47) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (48) QDP 4.57/2.12 (49) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (50) QDP 4.57/2.12 (51) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (52) QDP 4.57/2.12 (53) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (54) QDP 4.57/2.12 (55) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (56) QDP 4.57/2.12 (57) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (58) QDP 4.57/2.12 (59) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (60) QDP 4.57/2.12 (61) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (62) QDP 4.57/2.12 (63) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (64) QDP 4.57/2.12 (65) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (66) QDP 4.57/2.12 (67) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (68) QDP 4.57/2.12 (69) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (70) QDP 4.57/2.12 (71) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (72) QDP 4.57/2.12 (73) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (74) QDP 4.57/2.12 (75) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (76) QDP 4.57/2.12 (77) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (78) QDP 4.57/2.12 (79) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (80) QDP 4.57/2.12 (81) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (82) QDP 4.57/2.12 (83) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (84) QDP 4.57/2.12 (85) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (86) QDP 4.57/2.12 (87) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (88) QDP 4.57/2.12 (89) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (90) QDP 4.57/2.12 (91) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (92) QDP 4.57/2.12 (93) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (94) QDP 4.57/2.12 (95) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (96) QDP 4.57/2.12 (97) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (98) QDP 4.57/2.12 (99) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (100) QDP 4.57/2.12 (101) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (102) QDP 4.57/2.12 (103) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (104) QDP 4.57/2.12 (105) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (106) QDP 4.57/2.12 (107) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (108) QDP 4.57/2.12 (109) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (110) QDP 4.57/2.12 (111) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (112) QDP 4.57/2.12 (113) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (114) QDP 4.57/2.12 (115) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (116) QDP 4.57/2.12 (117) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (118) QDP 4.57/2.12 (119) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (120) QDP 4.57/2.12 (121) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (122) QDP 4.57/2.12 (123) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (124) QDP 4.57/2.12 (125) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (126) QDP 4.57/2.12 (127) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (128) QDP 4.57/2.12 (129) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (130) QDP 4.57/2.12 (131) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (132) QDP 4.57/2.12 (133) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (134) QDP 4.57/2.12 (135) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (136) QDP 4.57/2.12 (137) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (138) QDP 4.57/2.12 (139) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (140) QDP 4.57/2.12 (141) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (142) QDP 4.57/2.12 (143) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (144) QDP 4.57/2.12 (145) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (146) QDP 4.57/2.12 (147) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (148) QDP 4.57/2.12 (149) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (150) QDP 4.57/2.12 (151) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (152) QDP 4.57/2.12 (153) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (154) QDP 4.57/2.12 (155) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (156) QDP 4.57/2.12 (157) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (158) QDP 4.57/2.12 (159) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (160) QDP 4.57/2.12 (161) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (162) QDP 4.57/2.12 (163) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (164) QDP 4.57/2.12 (165) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (166) QDP 4.57/2.12 (167) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (168) QDP 4.57/2.12 (169) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (170) QDP 4.57/2.12 (171) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (172) QDP 4.57/2.12 (173) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (174) QDP 4.57/2.12 (175) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (176) QDP 4.57/2.12 (177) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (178) QDP 4.57/2.12 (179) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (180) QDP 4.57/2.12 (181) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (182) QDP 4.57/2.12 (183) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (184) QDP 4.57/2.12 (185) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (186) QDP 4.57/2.12 (187) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (188) QDP 4.57/2.12 (189) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (190) QDP 4.57/2.12 (191) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (192) QDP 4.57/2.12 (193) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (194) QDP 4.57/2.12 (195) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (196) QDP 4.57/2.12 (197) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (198) QDP 4.57/2.12 (199) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (200) QDP 4.57/2.12 (201) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (202) QDP 4.57/2.12 (203) TransformationProof [EQUIVALENT, 1 ms] 4.57/2.12 (204) QDP 4.57/2.12 (205) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (206) QDP 4.57/2.12 (207) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (208) QDP 4.57/2.12 (209) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (210) QDP 4.57/2.12 (211) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (212) QDP 4.57/2.12 (213) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (214) QDP 4.57/2.12 (215) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (216) QDP 4.57/2.12 (217) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (218) QDP 4.57/2.12 (219) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (220) QDP 4.57/2.12 (221) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (222) QDP 4.57/2.12 (223) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (224) QDP 4.57/2.12 (225) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (226) QDP 4.57/2.12 (227) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (228) QDP 4.57/2.12 (229) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (230) QDP 4.57/2.12 (231) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (232) QDP 4.57/2.12 (233) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (234) QDP 4.57/2.12 (235) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (236) QDP 4.57/2.12 (237) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (238) QDP 4.57/2.12 (239) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (240) QDP 4.57/2.12 (241) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (242) QDP 4.57/2.12 (243) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (244) QDP 4.57/2.12 (245) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (246) QDP 4.57/2.12 (247) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (248) QDP 4.57/2.12 (249) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (250) QDP 4.57/2.12 (251) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (252) QDP 4.57/2.12 (253) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (254) QDP 4.57/2.12 (255) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (256) QDP 4.57/2.12 (257) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (258) QDP 4.57/2.12 (259) UsableRulesProof [EQUIVALENT, 0 ms] 4.57/2.12 (260) QDP 4.57/2.12 (261) MNOCProof [EQUIVALENT, 0 ms] 4.57/2.12 (262) QDP 4.57/2.12 (263) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (264) QDP 4.57/2.12 (265) UsableRulesProof [EQUIVALENT, 0 ms] 4.57/2.12 (266) QDP 4.57/2.12 (267) QReductionProof [EQUIVALENT, 0 ms] 4.57/2.12 (268) QDP 4.57/2.12 (269) TransformationProof [EQUIVALENT, 0 ms] 4.57/2.12 (270) QDP 4.57/2.12 (271) DependencyGraphProof [EQUIVALENT, 0 ms] 4.57/2.12 (272) TRUE 4.57/2.12 4.57/2.12 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (0) 4.57/2.12 Obligation: 4.57/2.12 Conditional term rewrite system: 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 or(0, x) -> x 4.57/2.12 or(x, 0) -> x 4.57/2.12 or(1, x) -> 1 4.57/2.12 or(x, 1) -> 1 4.57/2.12 or(x, not(x)) -> 1 4.57/2.12 or(not(x), x) -> 1 4.57/2.12 and(0, x) -> 0 4.57/2.12 and(x, 0) -> 0 4.57/2.12 and(1, x) -> x 4.57/2.12 and(x, 1) -> x 4.57/2.12 and(x, not(x)) -> 0 4.57/2.12 and(not(x), x) -> 0 4.57/2.12 not(1) -> 0 4.57/2.12 not(0) -> 1 4.57/2.12 4.57/2.12 The conditional TRS C consists of the following conditional rules: 4.57/2.12 4.57/2.12 implies(x, y) -> 1 <= not(x) -> 1 4.57/2.12 implies(x, y) -> 1 <= y -> 1 4.57/2.12 implies(x, y) -> 0 <= x -> 1, y -> 0 4.57/2.12 f(x) -> f(0) <= implies(implies(x, implies(x, 0)), 0) -> 1 4.57/2.12 4.57/2.12 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (1) CTRSToQTRSProof (SOUND) 4.57/2.12 The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC]. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (2) 4.57/2.12 Obligation: 4.57/2.12 Q restricted rewrite system: 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 implies(x, y) -> U1(not(x)) 4.57/2.12 U1(1) -> 1 4.57/2.12 implies(x, y) -> U2(y) 4.57/2.12 U2(1) -> 1 4.57/2.12 implies(x, y) -> U3(x, y) 4.57/2.12 U3(1, y) -> U4(y) 4.57/2.12 U4(0) -> 0 4.57/2.12 f(x) -> U5(implies(implies(x, implies(x, 0)), 0)) 4.57/2.12 U5(1) -> f(0) 4.57/2.12 or(0, x) -> x 4.57/2.12 or(x, 0) -> x 4.57/2.12 or(1, x) -> 1 4.57/2.12 or(x, 1) -> 1 4.57/2.12 or(x, not(x)) -> 1 4.57/2.12 or(not(x), x) -> 1 4.57/2.12 and(0, x) -> 0 4.57/2.12 and(x, 0) -> 0 4.57/2.12 and(1, x) -> x 4.57/2.12 and(x, 1) -> x 4.57/2.12 and(x, not(x)) -> 0 4.57/2.12 and(not(x), x) -> 0 4.57/2.12 not(1) -> 0 4.57/2.12 not(0) -> 1 4.57/2.12 4.57/2.12 Q is empty. 4.57/2.12 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (3) QTRSRRRProof (EQUIVALENT) 4.57/2.12 Used ordering: 4.57/2.12 Polynomial interpretation [POLO]: 4.57/2.12 4.57/2.12 POL(0) = 0 4.57/2.12 POL(1) = 0 4.57/2.12 POL(U1(x_1)) = x_1 4.57/2.12 POL(U2(x_1)) = x_1 4.57/2.12 POL(U3(x_1, x_2)) = x_1 + x_2 4.57/2.12 POL(U4(x_1)) = x_1 4.57/2.12 POL(U5(x_1)) = x_1 4.57/2.12 POL(and(x_1, x_2)) = 2 + x_1 + 2*x_2 4.57/2.12 POL(f(x_1)) = 2*x_1 4.57/2.12 POL(implies(x_1, x_2)) = x_1 + x_2 4.57/2.12 POL(not(x_1)) = x_1 4.57/2.12 POL(or(x_1, x_2)) = 1 + x_1 + x_2 4.57/2.12 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.57/2.12 4.57/2.12 or(0, x) -> x 4.57/2.12 or(x, 0) -> x 4.57/2.12 or(1, x) -> 1 4.57/2.12 or(x, 1) -> 1 4.57/2.12 or(x, not(x)) -> 1 4.57/2.12 or(not(x), x) -> 1 4.57/2.12 and(0, x) -> 0 4.57/2.12 and(x, 0) -> 0 4.57/2.12 and(1, x) -> x 4.57/2.12 and(x, 1) -> x 4.57/2.12 and(x, not(x)) -> 0 4.57/2.12 and(not(x), x) -> 0 4.57/2.12 4.57/2.12 4.57/2.12 4.57/2.12 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (4) 4.57/2.12 Obligation: 4.57/2.12 Q restricted rewrite system: 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 implies(x, y) -> U1(not(x)) 4.57/2.12 U1(1) -> 1 4.57/2.12 implies(x, y) -> U2(y) 4.57/2.12 U2(1) -> 1 4.57/2.12 implies(x, y) -> U3(x, y) 4.57/2.12 U3(1, y) -> U4(y) 4.57/2.12 U4(0) -> 0 4.57/2.12 f(x) -> U5(implies(implies(x, implies(x, 0)), 0)) 4.57/2.12 U5(1) -> f(0) 4.57/2.12 not(1) -> 0 4.57/2.12 not(0) -> 1 4.57/2.12 4.57/2.12 Q is empty. 4.57/2.12 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (5) DependencyPairsProof (EQUIVALENT) 4.57/2.12 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (6) 4.57/2.12 Obligation: 4.57/2.12 Q DP problem: 4.57/2.12 The TRS P consists of the following rules: 4.57/2.12 4.57/2.12 IMPLIES(x, y) -> U1^1(not(x)) 4.57/2.12 IMPLIES(x, y) -> NOT(x) 4.57/2.12 IMPLIES(x, y) -> U2^1(y) 4.57/2.12 IMPLIES(x, y) -> U3^1(x, y) 4.57/2.12 U3^1(1, y) -> U4^1(y) 4.57/2.12 F(x) -> U5^1(implies(implies(x, implies(x, 0)), 0)) 4.57/2.12 F(x) -> IMPLIES(implies(x, implies(x, 0)), 0) 4.57/2.12 F(x) -> IMPLIES(x, implies(x, 0)) 4.57/2.12 F(x) -> IMPLIES(x, 0) 4.57/2.12 U5^1(1) -> F(0) 4.57/2.12 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 implies(x, y) -> U1(not(x)) 4.57/2.12 U1(1) -> 1 4.57/2.12 implies(x, y) -> U2(y) 4.57/2.12 U2(1) -> 1 4.57/2.12 implies(x, y) -> U3(x, y) 4.57/2.12 U3(1, y) -> U4(y) 4.57/2.12 U4(0) -> 0 4.57/2.12 f(x) -> U5(implies(implies(x, implies(x, 0)), 0)) 4.57/2.12 U5(1) -> f(0) 4.57/2.12 not(1) -> 0 4.57/2.12 not(0) -> 1 4.57/2.12 4.57/2.12 Q is empty. 4.57/2.12 We have to consider all minimal (P,Q,R)-chains. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (7) DependencyGraphProof (EQUIVALENT) 4.57/2.12 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (8) 4.57/2.12 Obligation: 4.57/2.12 Q DP problem: 4.57/2.12 The TRS P consists of the following rules: 4.57/2.12 4.57/2.12 F(x) -> U5^1(implies(implies(x, implies(x, 0)), 0)) 4.57/2.12 U5^1(1) -> F(0) 4.57/2.12 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 implies(x, y) -> U1(not(x)) 4.57/2.12 U1(1) -> 1 4.57/2.12 implies(x, y) -> U2(y) 4.57/2.12 U2(1) -> 1 4.57/2.12 implies(x, y) -> U3(x, y) 4.57/2.12 U3(1, y) -> U4(y) 4.57/2.12 U4(0) -> 0 4.57/2.12 f(x) -> U5(implies(implies(x, implies(x, 0)), 0)) 4.57/2.12 U5(1) -> f(0) 4.57/2.12 not(1) -> 0 4.57/2.12 not(0) -> 1 4.57/2.12 4.57/2.12 Q is empty. 4.57/2.12 We have to consider all minimal (P,Q,R)-chains. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (9) UsableRulesReductionPairsProof (EQUIVALENT) 4.57/2.12 By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. 4.57/2.12 4.57/2.12 No dependency pairs are removed. 4.57/2.12 4.57/2.12 The following rules are removed from R: 4.57/2.12 4.57/2.12 f(x) -> U5(implies(implies(x, implies(x, 0)), 0)) 4.57/2.12 U5(1) -> f(0) 4.57/2.12 Used ordering: POLO with Polynomial interpretation [POLO]: 4.57/2.12 4.57/2.12 POL(0) = 0 4.57/2.12 POL(1) = 0 4.57/2.12 POL(F(x_1)) = 2*x_1 4.57/2.12 POL(U1(x_1)) = x_1 4.57/2.12 POL(U2(x_1)) = x_1 4.57/2.12 POL(U3(x_1, x_2)) = x_1 + x_2 4.57/2.12 POL(U4(x_1)) = x_1 4.57/2.12 POL(U5^1(x_1)) = x_1 4.57/2.12 POL(implies(x_1, x_2)) = x_1 + x_2 4.57/2.12 POL(not(x_1)) = x_1 4.57/2.12 4.57/2.12 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (10) 4.57/2.12 Obligation: 4.57/2.12 Q DP problem: 4.57/2.12 The TRS P consists of the following rules: 4.57/2.12 4.57/2.12 F(x) -> U5^1(implies(implies(x, implies(x, 0)), 0)) 4.57/2.12 U5^1(1) -> F(0) 4.57/2.12 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 implies(x, y) -> U1(not(x)) 4.57/2.12 implies(x, y) -> U2(y) 4.57/2.12 implies(x, y) -> U3(x, y) 4.57/2.12 U3(1, y) -> U4(y) 4.57/2.12 U4(0) -> 0 4.57/2.12 U2(1) -> 1 4.57/2.12 not(1) -> 0 4.57/2.12 not(0) -> 1 4.57/2.12 U1(1) -> 1 4.57/2.12 4.57/2.12 Q is empty. 4.57/2.12 We have to consider all minimal (P,Q,R)-chains. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (11) TransformationProof (EQUIVALENT) 4.57/2.12 By instantiating [LPAR04] the rule F(x) -> U5^1(implies(implies(x, implies(x, 0)), 0)) we obtained the following new rules [LPAR04]: 4.57/2.12 4.57/2.12 (F(0) -> U5^1(implies(implies(0, implies(0, 0)), 0)),F(0) -> U5^1(implies(implies(0, implies(0, 0)), 0))) 4.57/2.12 4.57/2.12 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (12) 4.57/2.12 Obligation: 4.57/2.12 Q DP problem: 4.57/2.12 The TRS P consists of the following rules: 4.57/2.12 4.57/2.12 U5^1(1) -> F(0) 4.57/2.12 F(0) -> U5^1(implies(implies(0, implies(0, 0)), 0)) 4.57/2.12 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 implies(x, y) -> U1(not(x)) 4.57/2.12 implies(x, y) -> U2(y) 4.57/2.12 implies(x, y) -> U3(x, y) 4.57/2.12 U3(1, y) -> U4(y) 4.57/2.12 U4(0) -> 0 4.57/2.12 U2(1) -> 1 4.57/2.12 not(1) -> 0 4.57/2.12 not(0) -> 1 4.57/2.12 U1(1) -> 1 4.57/2.12 4.57/2.12 Q is empty. 4.57/2.12 We have to consider all minimal (P,Q,R)-chains. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (13) TransformationProof (EQUIVALENT) 4.57/2.12 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(implies(0, implies(0, 0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.12 4.57/2.12 (F(0) -> U5^1(U1(not(implies(0, implies(0, 0))))),F(0) -> U5^1(U1(not(implies(0, implies(0, 0)))))) 4.57/2.12 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.12 (F(0) -> U5^1(U3(implies(0, implies(0, 0)), 0)),F(0) -> U5^1(U3(implies(0, implies(0, 0)), 0))) 4.57/2.12 (F(0) -> U5^1(implies(U1(not(0)), 0)),F(0) -> U5^1(implies(U1(not(0)), 0))) 4.57/2.12 (F(0) -> U5^1(implies(U2(implies(0, 0)), 0)),F(0) -> U5^1(implies(U2(implies(0, 0)), 0))) 4.57/2.12 (F(0) -> U5^1(implies(U3(0, implies(0, 0)), 0)),F(0) -> U5^1(implies(U3(0, implies(0, 0)), 0))) 4.57/2.12 (F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)),F(0) -> U5^1(implies(implies(0, U1(not(0))), 0))) 4.57/2.12 (F(0) -> U5^1(implies(implies(0, U2(0)), 0)),F(0) -> U5^1(implies(implies(0, U2(0)), 0))) 4.57/2.12 (F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)),F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0))) 4.57/2.12 4.57/2.12 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (14) 4.57/2.12 Obligation: 4.57/2.12 Q DP problem: 4.57/2.12 The TRS P consists of the following rules: 4.57/2.12 4.57/2.12 U5^1(1) -> F(0) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, implies(0, 0))))) 4.57/2.12 F(0) -> U5^1(U2(0)) 4.57/2.12 F(0) -> U5^1(U3(implies(0, implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U2(implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U3(0, implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.12 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 implies(x, y) -> U1(not(x)) 4.57/2.12 implies(x, y) -> U2(y) 4.57/2.12 implies(x, y) -> U3(x, y) 4.57/2.12 U3(1, y) -> U4(y) 4.57/2.12 U4(0) -> 0 4.57/2.12 U2(1) -> 1 4.57/2.12 not(1) -> 0 4.57/2.12 not(0) -> 1 4.57/2.12 U1(1) -> 1 4.57/2.12 4.57/2.12 Q is empty. 4.57/2.12 We have to consider all minimal (P,Q,R)-chains. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (15) DependencyGraphProof (EQUIVALENT) 4.57/2.12 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (16) 4.57/2.12 Obligation: 4.57/2.12 Q DP problem: 4.57/2.12 The TRS P consists of the following rules: 4.57/2.12 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, implies(0, 0))))) 4.57/2.12 U5^1(1) -> F(0) 4.57/2.12 F(0) -> U5^1(U3(implies(0, implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U2(implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U3(0, implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.12 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 implies(x, y) -> U1(not(x)) 4.57/2.12 implies(x, y) -> U2(y) 4.57/2.12 implies(x, y) -> U3(x, y) 4.57/2.12 U3(1, y) -> U4(y) 4.57/2.12 U4(0) -> 0 4.57/2.12 U2(1) -> 1 4.57/2.12 not(1) -> 0 4.57/2.12 not(0) -> 1 4.57/2.12 U1(1) -> 1 4.57/2.12 4.57/2.12 Q is empty. 4.57/2.12 We have to consider all minimal (P,Q,R)-chains. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (17) TransformationProof (EQUIVALENT) 4.57/2.12 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(implies(0, implies(0, 0))))) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.12 4.57/2.12 (F(0) -> U5^1(U1(not(U1(not(0))))),F(0) -> U5^1(U1(not(U1(not(0)))))) 4.57/2.12 (F(0) -> U5^1(U1(not(U2(implies(0, 0))))),F(0) -> U5^1(U1(not(U2(implies(0, 0)))))) 4.57/2.12 (F(0) -> U5^1(U1(not(U3(0, implies(0, 0))))),F(0) -> U5^1(U1(not(U3(0, implies(0, 0)))))) 4.57/2.12 (F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))),F(0) -> U5^1(U1(not(implies(0, U1(not(0))))))) 4.57/2.12 (F(0) -> U5^1(U1(not(implies(0, U2(0))))),F(0) -> U5^1(U1(not(implies(0, U2(0)))))) 4.57/2.12 (F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))),F(0) -> U5^1(U1(not(implies(0, U3(0, 0)))))) 4.57/2.12 4.57/2.12 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (18) 4.57/2.12 Obligation: 4.57/2.12 Q DP problem: 4.57/2.12 The TRS P consists of the following rules: 4.57/2.12 4.57/2.12 U5^1(1) -> F(0) 4.57/2.12 F(0) -> U5^1(U3(implies(0, implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U2(implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U3(0, implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.12 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.12 F(0) -> U5^1(U1(not(U3(0, implies(0, 0))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.12 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 implies(x, y) -> U1(not(x)) 4.57/2.12 implies(x, y) -> U2(y) 4.57/2.12 implies(x, y) -> U3(x, y) 4.57/2.12 U3(1, y) -> U4(y) 4.57/2.12 U4(0) -> 0 4.57/2.12 U2(1) -> 1 4.57/2.12 not(1) -> 0 4.57/2.12 not(0) -> 1 4.57/2.12 U1(1) -> 1 4.57/2.12 4.57/2.12 Q is empty. 4.57/2.12 We have to consider all minimal (P,Q,R)-chains. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (19) DependencyGraphProof (EQUIVALENT) 4.57/2.12 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (20) 4.57/2.12 Obligation: 4.57/2.12 Q DP problem: 4.57/2.12 The TRS P consists of the following rules: 4.57/2.12 4.57/2.12 F(0) -> U5^1(U3(implies(0, implies(0, 0)), 0)) 4.57/2.12 U5^1(1) -> F(0) 4.57/2.12 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U2(implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U3(0, implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.12 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.12 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 implies(x, y) -> U1(not(x)) 4.57/2.12 implies(x, y) -> U2(y) 4.57/2.12 implies(x, y) -> U3(x, y) 4.57/2.12 U3(1, y) -> U4(y) 4.57/2.12 U4(0) -> 0 4.57/2.12 U2(1) -> 1 4.57/2.12 not(1) -> 0 4.57/2.12 not(0) -> 1 4.57/2.12 U1(1) -> 1 4.57/2.12 4.57/2.12 Q is empty. 4.57/2.12 We have to consider all minimal (P,Q,R)-chains. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (21) TransformationProof (EQUIVALENT) 4.57/2.12 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(implies(0, implies(0, 0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.12 4.57/2.12 (F(0) -> U5^1(U3(U1(not(0)), 0)),F(0) -> U5^1(U3(U1(not(0)), 0))) 4.57/2.12 (F(0) -> U5^1(U3(U2(implies(0, 0)), 0)),F(0) -> U5^1(U3(U2(implies(0, 0)), 0))) 4.57/2.12 (F(0) -> U5^1(U3(U3(0, implies(0, 0)), 0)),F(0) -> U5^1(U3(U3(0, implies(0, 0)), 0))) 4.57/2.12 (F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)),F(0) -> U5^1(U3(implies(0, U1(not(0))), 0))) 4.57/2.12 (F(0) -> U5^1(U3(implies(0, U2(0)), 0)),F(0) -> U5^1(U3(implies(0, U2(0)), 0))) 4.57/2.12 (F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)),F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0))) 4.57/2.12 4.57/2.12 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (22) 4.57/2.12 Obligation: 4.57/2.12 Q DP problem: 4.57/2.12 The TRS P consists of the following rules: 4.57/2.12 4.57/2.12 U5^1(1) -> F(0) 4.57/2.12 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U2(implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U3(0, implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.12 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.12 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.12 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(U3(U3(0, implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.12 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.12 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.12 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 implies(x, y) -> U1(not(x)) 4.57/2.12 implies(x, y) -> U2(y) 4.57/2.12 implies(x, y) -> U3(x, y) 4.57/2.12 U3(1, y) -> U4(y) 4.57/2.12 U4(0) -> 0 4.57/2.12 U2(1) -> 1 4.57/2.12 not(1) -> 0 4.57/2.12 not(0) -> 1 4.57/2.12 U1(1) -> 1 4.57/2.12 4.57/2.12 Q is empty. 4.57/2.12 We have to consider all minimal (P,Q,R)-chains. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (23) DependencyGraphProof (EQUIVALENT) 4.57/2.12 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (24) 4.57/2.12 Obligation: 4.57/2.12 Q DP problem: 4.57/2.12 The TRS P consists of the following rules: 4.57/2.12 4.57/2.12 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.12 U5^1(1) -> F(0) 4.57/2.12 F(0) -> U5^1(implies(U2(implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U3(0, implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.12 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.12 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.12 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.12 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.12 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.12 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 implies(x, y) -> U1(not(x)) 4.57/2.12 implies(x, y) -> U2(y) 4.57/2.12 implies(x, y) -> U3(x, y) 4.57/2.12 U3(1, y) -> U4(y) 4.57/2.12 U4(0) -> 0 4.57/2.12 U2(1) -> 1 4.57/2.12 not(1) -> 0 4.57/2.12 not(0) -> 1 4.57/2.12 U1(1) -> 1 4.57/2.12 4.57/2.12 Q is empty. 4.57/2.12 We have to consider all minimal (P,Q,R)-chains. 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (25) TransformationProof (EQUIVALENT) 4.57/2.12 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U1(not(0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.12 4.57/2.12 (F(0) -> U5^1(U1(not(U1(not(0))))),F(0) -> U5^1(U1(not(U1(not(0)))))) 4.57/2.12 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.12 (F(0) -> U5^1(U3(U1(not(0)), 0)),F(0) -> U5^1(U3(U1(not(0)), 0))) 4.57/2.12 (F(0) -> U5^1(implies(U1(1), 0)),F(0) -> U5^1(implies(U1(1), 0))) 4.57/2.12 4.57/2.12 4.57/2.12 ---------------------------------------- 4.57/2.12 4.57/2.12 (26) 4.57/2.12 Obligation: 4.57/2.12 Q DP problem: 4.57/2.12 The TRS P consists of the following rules: 4.57/2.12 4.57/2.12 U5^1(1) -> F(0) 4.57/2.12 F(0) -> U5^1(implies(U2(implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(U3(0, implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.12 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.12 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.12 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.12 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.12 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.12 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.12 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.12 F(0) -> U5^1(U2(0)) 4.57/2.12 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.12 4.57/2.12 The TRS R consists of the following rules: 4.57/2.12 4.57/2.12 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (27) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (28) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(implies(U2(implies(0, 0)), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U3(0, implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (29) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U2(implies(0, 0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U2(implies(0, 0))))),F(0) -> U5^1(U1(not(U2(implies(0, 0)))))) 4.57/2.13 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.13 (F(0) -> U5^1(U3(U2(implies(0, 0)), 0)),F(0) -> U5^1(U3(U2(implies(0, 0)), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U2(U1(not(0))), 0)),F(0) -> U5^1(implies(U2(U1(not(0))), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U2(U2(0)), 0)),F(0) -> U5^1(implies(U2(U2(0)), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U2(U3(0, 0)), 0)),F(0) -> U5^1(implies(U2(U3(0, 0)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (30) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U3(0, implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U2(0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (31) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (32) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(implies(U3(0, implies(0, 0)), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (33) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U3(0, implies(0, 0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U3(0, implies(0, 0))))),F(0) -> U5^1(U1(not(U3(0, implies(0, 0)))))) 4.57/2.13 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.13 (F(0) -> U5^1(U3(U3(0, implies(0, 0)), 0)),F(0) -> U5^1(U3(U3(0, implies(0, 0)), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)),F(0) -> U5^1(implies(U3(0, U1(not(0))), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U3(0, U2(0)), 0)),F(0) -> U5^1(implies(U3(0, U2(0)), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)),F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (34) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U3(0, implies(0, 0))))) 4.57/2.13 F(0) -> U5^1(U2(0)) 4.57/2.13 F(0) -> U5^1(U3(U3(0, implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (35) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (36) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (37) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(implies(0, U1(not(0))), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))),F(0) -> U5^1(U1(not(implies(0, U1(not(0))))))) 4.57/2.13 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.13 (F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)),F(0) -> U5^1(U3(implies(0, U1(not(0))), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U1(not(0)), 0)),F(0) -> U5^1(implies(U1(not(0)), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U2(U1(not(0))), 0)),F(0) -> U5^1(implies(U2(U1(not(0))), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)),F(0) -> U5^1(implies(U3(0, U1(not(0))), 0))) 4.57/2.13 (F(0) -> U5^1(implies(implies(0, U1(1)), 0)),F(0) -> U5^1(implies(implies(0, U1(1)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (38) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U2(0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (39) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (40) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(implies(implies(0, U2(0)), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (41) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(implies(0, U2(0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(implies(0, U2(0))))),F(0) -> U5^1(U1(not(implies(0, U2(0)))))) 4.57/2.13 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.13 (F(0) -> U5^1(U3(implies(0, U2(0)), 0)),F(0) -> U5^1(U3(implies(0, U2(0)), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U1(not(0)), 0)),F(0) -> U5^1(implies(U1(not(0)), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U2(U2(0)), 0)),F(0) -> U5^1(implies(U2(U2(0)), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U3(0, U2(0)), 0)),F(0) -> U5^1(implies(U3(0, U2(0)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (42) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U2(0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (43) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (44) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (45) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(implies(0, U3(0, 0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))),F(0) -> U5^1(U1(not(implies(0, U3(0, 0)))))) 4.57/2.13 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.13 (F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)),F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U1(not(0)), 0)),F(0) -> U5^1(implies(U1(not(0)), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U2(U3(0, 0)), 0)),F(0) -> U5^1(implies(U2(U3(0, 0)), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)),F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (46) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U2(0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (47) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (48) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (49) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(U1(not(0))))) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U1(1)))),F(0) -> U5^1(U1(not(U1(1))))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (50) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U1(not(U2(implies(0, 0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (51) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(U2(implies(0, 0))))) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U2(U1(not(0)))))),F(0) -> U5^1(U1(not(U2(U1(not(0))))))) 4.57/2.13 (F(0) -> U5^1(U1(not(U2(U2(0))))),F(0) -> U5^1(U1(not(U2(U2(0)))))) 4.57/2.13 (F(0) -> U5^1(U1(not(U2(U3(0, 0))))),F(0) -> U5^1(U1(not(U2(U3(0, 0)))))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (52) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U3(0, 0))))) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (53) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (54) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (55) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(implies(0, U1(not(0)))))) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U1(not(0))))),F(0) -> U5^1(U1(not(U1(not(0)))))) 4.57/2.13 (F(0) -> U5^1(U1(not(U2(U1(not(0)))))),F(0) -> U5^1(U1(not(U2(U1(not(0))))))) 4.57/2.13 (F(0) -> U5^1(U1(not(U3(0, U1(not(0)))))),F(0) -> U5^1(U1(not(U3(0, U1(not(0))))))) 4.57/2.13 (F(0) -> U5^1(U1(not(implies(0, U1(1))))),F(0) -> U5^1(U1(not(implies(0, U1(1)))))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (56) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U3(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (57) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (58) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U2(0))))) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (59) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(implies(0, U2(0))))) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U1(not(0))))),F(0) -> U5^1(U1(not(U1(not(0)))))) 4.57/2.13 (F(0) -> U5^1(U1(not(U2(U2(0))))),F(0) -> U5^1(U1(not(U2(U2(0)))))) 4.57/2.13 (F(0) -> U5^1(U1(not(U3(0, U2(0))))),F(0) -> U5^1(U1(not(U3(0, U2(0)))))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (60) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U2(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U3(0, U2(0))))) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (61) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (62) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (63) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(implies(0, U3(0, 0))))) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U1(not(0))))),F(0) -> U5^1(U1(not(U1(not(0)))))) 4.57/2.13 (F(0) -> U5^1(U1(not(U2(U3(0, 0))))),F(0) -> U5^1(U1(not(U2(U3(0, 0)))))) 4.57/2.13 (F(0) -> U5^1(U1(not(U3(0, U3(0, 0))))),F(0) -> U5^1(U1(not(U3(0, U3(0, 0)))))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (64) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U1(not(U3(0, U3(0, 0))))) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (65) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (66) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (67) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(U1(not(0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U3(U1(1), 0)),F(0) -> U5^1(U3(U1(1), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (68) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (69) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(U2(implies(0, 0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U3(U2(U1(not(0))), 0)),F(0) -> U5^1(U3(U2(U1(not(0))), 0))) 4.57/2.13 (F(0) -> U5^1(U3(U2(U2(0)), 0)),F(0) -> U5^1(U3(U2(U2(0)), 0))) 4.57/2.13 (F(0) -> U5^1(U3(U2(U3(0, 0)), 0)),F(0) -> U5^1(U3(U2(U3(0, 0)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (70) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U3(0, 0)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (71) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (72) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (73) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(implies(0, U1(not(0))), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U3(U1(not(0)), 0)),F(0) -> U5^1(U3(U1(not(0)), 0))) 4.57/2.13 (F(0) -> U5^1(U3(U2(U1(not(0))), 0)),F(0) -> U5^1(U3(U2(U1(not(0))), 0))) 4.57/2.13 (F(0) -> U5^1(U3(U3(0, U1(not(0))), 0)),F(0) -> U5^1(U3(U3(0, U1(not(0))), 0))) 4.57/2.13 (F(0) -> U5^1(U3(implies(0, U1(1)), 0)),F(0) -> U5^1(U3(implies(0, U1(1)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (74) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (75) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (76) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(U3(implies(0, U2(0)), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (77) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(implies(0, U2(0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U3(U1(not(0)), 0)),F(0) -> U5^1(U3(U1(not(0)), 0))) 4.57/2.13 (F(0) -> U5^1(U3(U2(U2(0)), 0)),F(0) -> U5^1(U3(U2(U2(0)), 0))) 4.57/2.13 (F(0) -> U5^1(U3(U3(0, U2(0)), 0)),F(0) -> U5^1(U3(U3(0, U2(0)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (78) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U3(0, U2(0)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (79) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (80) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (81) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(implies(0, U3(0, 0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U3(U1(not(0)), 0)),F(0) -> U5^1(U3(U1(not(0)), 0))) 4.57/2.13 (F(0) -> U5^1(U3(U2(U3(0, 0)), 0)),F(0) -> U5^1(U3(U2(U3(0, 0)), 0))) 4.57/2.13 (F(0) -> U5^1(U3(U3(0, U3(0, 0)), 0)),F(0) -> U5^1(U3(U3(0, U3(0, 0)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (82) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(U3(0, U3(0, 0)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (83) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (84) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (85) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U1(1), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U1(1)))),F(0) -> U5^1(U1(not(U1(1))))) 4.57/2.13 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.13 (F(0) -> U5^1(U3(U1(1), 0)),F(0) -> U5^1(U3(U1(1), 0))) 4.57/2.13 (F(0) -> U5^1(implies(1, 0)),F(0) -> U5^1(implies(1, 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (86) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U2(0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (87) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (88) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(implies(U2(U1(not(0))), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (89) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U2(U1(not(0))), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U2(U1(not(0)))))),F(0) -> U5^1(U1(not(U2(U1(not(0))))))) 4.57/2.13 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.13 (F(0) -> U5^1(U3(U2(U1(not(0))), 0)),F(0) -> U5^1(U3(U2(U1(not(0))), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U2(U1(1)), 0)),F(0) -> U5^1(implies(U2(U1(1)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (90) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 F(0) -> U5^1(U2(0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (91) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (92) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(implies(U2(U2(0)), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (93) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U2(U2(0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U2(U2(0))))),F(0) -> U5^1(U1(not(U2(U2(0)))))) 4.57/2.13 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.13 (F(0) -> U5^1(U3(U2(U2(0)), 0)),F(0) -> U5^1(U3(U2(U2(0)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (94) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U2(0))))) 4.57/2.13 F(0) -> U5^1(U2(0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U2(0)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (95) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (96) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (97) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U2(U3(0, 0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U2(U3(0, 0))))),F(0) -> U5^1(U1(not(U2(U3(0, 0)))))) 4.57/2.13 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.13 (F(0) -> U5^1(U3(U2(U3(0, 0)), 0)),F(0) -> U5^1(U3(U2(U3(0, 0)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (98) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U2(0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U3(0, 0)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (99) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (100) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (101) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U3(0, U1(not(0))), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U3(0, U1(not(0)))))),F(0) -> U5^1(U1(not(U3(0, U1(not(0))))))) 4.57/2.13 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.13 (F(0) -> U5^1(U3(U3(0, U1(not(0))), 0)),F(0) -> U5^1(U3(U3(0, U1(not(0))), 0))) 4.57/2.13 (F(0) -> U5^1(implies(U3(0, U1(1)), 0)),F(0) -> U5^1(implies(U3(0, U1(1)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (102) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U3(0, U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U2(0)) 4.57/2.13 F(0) -> U5^1(U3(U3(0, U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (103) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (104) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(implies(U3(0, U2(0)), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (105) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U3(0, U2(0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U3(0, U2(0))))),F(0) -> U5^1(U1(not(U3(0, U2(0)))))) 4.57/2.13 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.13 (F(0) -> U5^1(U3(U3(0, U2(0)), 0)),F(0) -> U5^1(U3(U3(0, U2(0)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (106) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U3(0, U2(0))))) 4.57/2.13 F(0) -> U5^1(U2(0)) 4.57/2.13 F(0) -> U5^1(U3(U3(0, U2(0)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (107) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (108) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (109) TransformationProof (EQUIVALENT) 4.57/2.13 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U3(0, U3(0, 0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.13 4.57/2.13 (F(0) -> U5^1(U1(not(U3(0, U3(0, 0))))),F(0) -> U5^1(U1(not(U3(0, U3(0, 0)))))) 4.57/2.13 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.13 (F(0) -> U5^1(U3(U3(0, U3(0, 0)), 0)),F(0) -> U5^1(U3(U3(0, U3(0, 0)), 0))) 4.57/2.13 4.57/2.13 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (110) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U3(0, U3(0, 0))))) 4.57/2.13 F(0) -> U5^1(U2(0)) 4.57/2.13 F(0) -> U5^1(U3(U3(0, U3(0, 0)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.13 implies(x, y) -> U2(y) 4.57/2.13 implies(x, y) -> U3(x, y) 4.57/2.13 U3(1, y) -> U4(y) 4.57/2.13 U4(0) -> 0 4.57/2.13 U2(1) -> 1 4.57/2.13 not(1) -> 0 4.57/2.13 not(0) -> 1 4.57/2.13 U1(1) -> 1 4.57/2.13 4.57/2.13 Q is empty. 4.57/2.13 We have to consider all minimal (P,Q,R)-chains. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (111) DependencyGraphProof (EQUIVALENT) 4.57/2.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 4.57/2.13 ---------------------------------------- 4.57/2.13 4.57/2.13 (112) 4.57/2.13 Obligation: 4.57/2.13 Q DP problem: 4.57/2.13 The TRS P consists of the following rules: 4.57/2.13 4.57/2.13 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.57/2.13 U5^1(1) -> F(0) 4.57/2.13 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.13 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.13 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.13 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.13 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.13 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.13 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.13 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(1, 0)) 4.57/2.13 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.57/2.13 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.57/2.13 4.57/2.13 The TRS R consists of the following rules: 4.57/2.13 4.57/2.13 implies(x, y) -> U1(not(x)) 4.57/2.14 implies(x, y) -> U2(y) 4.57/2.14 implies(x, y) -> U3(x, y) 4.57/2.14 U3(1, y) -> U4(y) 4.57/2.14 U4(0) -> 0 4.57/2.14 U2(1) -> 1 4.57/2.14 not(1) -> 0 4.57/2.14 not(0) -> 1 4.57/2.14 U1(1) -> 1 4.57/2.14 4.57/2.14 Q is empty. 4.57/2.14 We have to consider all minimal (P,Q,R)-chains. 4.57/2.14 ---------------------------------------- 4.57/2.14 4.57/2.14 (113) TransformationProof (EQUIVALENT) 4.57/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U1(not(0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.57/2.14 4.57/2.14 (F(0) -> U5^1(U1(not(U1(not(0))))),F(0) -> U5^1(U1(not(U1(not(0)))))) 4.57/2.14 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.57/2.14 (F(0) -> U5^1(U3(U1(not(0)), 0)),F(0) -> U5^1(U3(U1(not(0)), 0))) 4.57/2.14 (F(0) -> U5^1(implies(U1(1), 0)),F(0) -> U5^1(implies(U1(1), 0))) 4.57/2.14 4.57/2.14 4.57/2.14 ---------------------------------------- 4.57/2.14 4.57/2.14 (114) 4.57/2.14 Obligation: 4.57/2.14 Q DP problem: 4.57/2.14 The TRS P consists of the following rules: 4.57/2.14 4.57/2.14 U5^1(1) -> F(0) 4.57/2.14 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.14 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.57/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.57/2.14 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.57/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.57/2.14 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.57/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.57/2.14 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.57/2.14 F(0) -> U5^1(implies(1, 0)) 4.57/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.57/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.57/2.14 F(0) -> U5^1(U2(0)) 4.57/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.57/2.14 4.57/2.14 The TRS R consists of the following rules: 4.57/2.14 4.57/2.14 implies(x, y) -> U1(not(x)) 4.57/2.14 implies(x, y) -> U2(y) 4.57/2.14 implies(x, y) -> U3(x, y) 4.57/2.14 U3(1, y) -> U4(y) 4.57/2.14 U4(0) -> 0 4.57/2.14 U2(1) -> 1 4.57/2.14 not(1) -> 0 4.57/2.14 not(0) -> 1 4.57/2.14 U1(1) -> 1 4.57/2.14 4.57/2.14 Q is empty. 4.57/2.14 We have to consider all minimal (P,Q,R)-chains. 4.57/2.14 ---------------------------------------- 4.57/2.14 4.57/2.14 (115) DependencyGraphProof (EQUIVALENT) 4.57/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.57/2.14 ---------------------------------------- 4.57/2.14 4.57/2.14 (116) 4.57/2.14 Obligation: 4.57/2.14 Q DP problem: 4.57/2.14 The TRS P consists of the following rules: 4.57/2.14 4.57/2.14 F(0) -> U5^1(implies(implies(0, U1(1)), 0)) 4.57/2.14 U5^1(1) -> F(0) 4.57/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.57/2.14 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (117) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(implies(0, U1(1)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(implies(0, U1(1))))),F(0) -> U5^1(U1(not(implies(0, U1(1)))))) 4.88/2.14 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.14 (F(0) -> U5^1(U3(implies(0, U1(1)), 0)),F(0) -> U5^1(U3(implies(0, U1(1)), 0))) 4.88/2.14 (F(0) -> U5^1(implies(U1(not(0)), 0)),F(0) -> U5^1(implies(U1(not(0)), 0))) 4.88/2.14 (F(0) -> U5^1(implies(U2(U1(1)), 0)),F(0) -> U5^1(implies(U2(U1(1)), 0))) 4.88/2.14 (F(0) -> U5^1(implies(U3(0, U1(1)), 0)),F(0) -> U5^1(implies(U3(0, U1(1)), 0))) 4.88/2.14 (F(0) -> U5^1(implies(implies(0, 1), 0)),F(0) -> U5^1(implies(implies(0, 1), 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (118) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U2(0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (119) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (120) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (121) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(U1(1)))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(1))),F(0) -> U5^1(U1(not(1)))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (122) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(not(0)))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (123) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(U2(U1(not(0)))))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(U2(U1(1))))),F(0) -> U5^1(U1(not(U2(U1(1)))))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (124) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (125) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(U1(not(0))))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(U1(1)))),F(0) -> U5^1(U1(not(U1(1))))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (126) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, U1(1))))) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (127) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(implies(0, U1(1))))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(U1(not(0))))),F(0) -> U5^1(U1(not(U1(not(0)))))) 4.88/2.14 (F(0) -> U5^1(U1(not(U2(U1(1))))),F(0) -> U5^1(U1(not(U2(U1(1)))))) 4.88/2.14 (F(0) -> U5^1(U1(not(U3(0, U1(1))))),F(0) -> U5^1(U1(not(U3(0, U1(1)))))) 4.88/2.14 (F(0) -> U5^1(U1(not(implies(0, 1)))),F(0) -> U5^1(U1(not(implies(0, 1))))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (128) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(U3(0, U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (129) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (130) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (131) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(U1(1), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U3(1, 0)),F(0) -> U5^1(U3(1, 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (132) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(not(0))), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (133) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(U2(U1(not(0))), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U3(U2(U1(1)), 0)),F(0) -> U5^1(U3(U2(U1(1)), 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (134) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (135) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(U1(not(0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U3(U1(1), 0)),F(0) -> U5^1(U3(U1(1), 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (136) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U3(implies(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (137) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(implies(0, U1(1)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U3(U1(not(0)), 0)),F(0) -> U5^1(U3(U1(not(0)), 0))) 4.88/2.14 (F(0) -> U5^1(U3(U2(U1(1)), 0)),F(0) -> U5^1(U3(U2(U1(1)), 0))) 4.88/2.14 (F(0) -> U5^1(U3(U3(0, U1(1)), 0)),F(0) -> U5^1(U3(U3(0, U1(1)), 0))) 4.88/2.14 (F(0) -> U5^1(U3(implies(0, 1), 0)),F(0) -> U5^1(U3(implies(0, 1), 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (138) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (139) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (140) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (141) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(1, 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(1))),F(0) -> U5^1(U1(not(1)))) 4.88/2.14 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.14 (F(0) -> U5^1(U3(1, 0)),F(0) -> U5^1(U3(1, 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (142) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U2(0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (143) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (144) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(implies(U2(U1(1)), 0)) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (145) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U2(U1(1)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(U2(U1(1))))),F(0) -> U5^1(U1(not(U2(U1(1)))))) 4.88/2.14 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.14 (F(0) -> U5^1(U3(U2(U1(1)), 0)),F(0) -> U5^1(U3(U2(U1(1)), 0))) 4.88/2.14 (F(0) -> U5^1(implies(U2(1), 0)),F(0) -> U5^1(implies(U2(1), 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (146) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U2(0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (147) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (148) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(implies(U3(0, U1(1)), 0)) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (149) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U3(0, U1(1)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(U3(0, U1(1))))),F(0) -> U5^1(U1(not(U3(0, U1(1)))))) 4.88/2.14 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.14 (F(0) -> U5^1(U3(U3(0, U1(1)), 0)),F(0) -> U5^1(U3(U3(0, U1(1)), 0))) 4.88/2.14 (F(0) -> U5^1(implies(U3(0, 1), 0)),F(0) -> U5^1(implies(U3(0, 1), 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (150) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U3(0, U1(1))))) 4.88/2.14 F(0) -> U5^1(U2(0)) 4.88/2.14 F(0) -> U5^1(U3(U3(0, U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (151) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (152) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (153) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U1(1), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(U1(1)))),F(0) -> U5^1(U1(not(U1(1))))) 4.88/2.14 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.14 (F(0) -> U5^1(U3(U1(1), 0)),F(0) -> U5^1(U3(U1(1), 0))) 4.88/2.14 (F(0) -> U5^1(implies(1, 0)),F(0) -> U5^1(implies(1, 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (154) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U2(0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (155) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (156) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (157) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U1(not(0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(U1(not(0))))),F(0) -> U5^1(U1(not(U1(not(0)))))) 4.88/2.14 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.14 (F(0) -> U5^1(U3(U1(not(0)), 0)),F(0) -> U5^1(U3(U1(not(0)), 0))) 4.88/2.14 (F(0) -> U5^1(implies(U1(1), 0)),F(0) -> U5^1(implies(U1(1), 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (158) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(U2(0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (159) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (160) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(implies(implies(0, 1), 0)) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (161) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(implies(0, 1), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(implies(0, 1)))),F(0) -> U5^1(U1(not(implies(0, 1))))) 4.88/2.14 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.14 (F(0) -> U5^1(U3(implies(0, 1), 0)),F(0) -> U5^1(U3(implies(0, 1), 0))) 4.88/2.14 (F(0) -> U5^1(implies(U1(not(0)), 0)),F(0) -> U5^1(implies(U1(not(0)), 0))) 4.88/2.14 (F(0) -> U5^1(implies(U2(1), 0)),F(0) -> U5^1(implies(U2(1), 0))) 4.88/2.14 (F(0) -> U5^1(implies(U3(0, 1), 0)),F(0) -> U5^1(implies(U3(0, 1), 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (162) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U2(0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (163) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (164) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (165) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(1))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(0)),F(0) -> U5^1(U1(0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (166) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (167) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (168) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(U1(not(U2(U1(1))))) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (169) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(U2(U1(1))))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(U2(1)))),F(0) -> U5^1(U1(not(U2(1))))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (170) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (171) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(U1(1)))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(1))),F(0) -> U5^1(U1(not(1)))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (172) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (173) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(U1(not(0))))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(U1(1)))),F(0) -> U5^1(U1(not(U1(1))))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (174) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(implies(0, 1)))) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (175) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(implies(0, 1)))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(U1(not(0))))),F(0) -> U5^1(U1(not(U1(not(0)))))) 4.88/2.14 (F(0) -> U5^1(U1(not(U2(1)))),F(0) -> U5^1(U1(not(U2(1))))) 4.88/2.14 (F(0) -> U5^1(U1(not(U3(0, 1)))),F(0) -> U5^1(U1(not(U3(0, 1))))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (176) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U1(not(U3(0, 1)))) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (177) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (178) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (179) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(1, 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U4(0)),F(0) -> U5^1(U4(0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (180) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U3(U2(U1(1)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U4(0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (181) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(U2(U1(1)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U3(U2(1), 0)),F(0) -> U5^1(U3(U2(1), 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (182) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U4(0)) 4.88/2.14 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (183) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(U1(1), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U3(1, 0)),F(0) -> U5^1(U3(1, 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (184) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U4(0)) 4.88/2.14 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (185) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(U1(not(0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U3(U1(1), 0)),F(0) -> U5^1(U3(U1(1), 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (186) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U3(implies(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U4(0)) 4.88/2.14 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (187) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(implies(0, 1), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U3(U1(not(0)), 0)),F(0) -> U5^1(U3(U1(not(0)), 0))) 4.88/2.14 (F(0) -> U5^1(U3(U2(1), 0)),F(0) -> U5^1(U3(U2(1), 0))) 4.88/2.14 (F(0) -> U5^1(U3(U3(0, 1), 0)),F(0) -> U5^1(U3(U3(0, 1), 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (188) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U4(0)) 4.88/2.14 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U3(U3(0, 1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (189) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (190) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(implies(U2(1), 0)) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U4(0)) 4.88/2.14 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (191) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U2(1), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(U2(1)))),F(0) -> U5^1(U1(not(U2(1))))) 4.88/2.14 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.14 (F(0) -> U5^1(U3(U2(1), 0)),F(0) -> U5^1(U3(U2(1), 0))) 4.88/2.14 (F(0) -> U5^1(implies(1, 0)),F(0) -> U5^1(implies(1, 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (192) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U4(0)) 4.88/2.14 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U2(0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (193) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (194) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(implies(U3(0, 1), 0)) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U4(0)) 4.88/2.14 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (195) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U3(0, 1), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(U3(0, 1)))),F(0) -> U5^1(U1(not(U3(0, 1))))) 4.88/2.14 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.14 (F(0) -> U5^1(U3(U3(0, 1), 0)),F(0) -> U5^1(U3(U3(0, 1), 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (196) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U4(0)) 4.88/2.14 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U3(0, 1)))) 4.88/2.14 F(0) -> U5^1(U2(0)) 4.88/2.14 F(0) -> U5^1(U3(U3(0, 1), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (197) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (198) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U4(0)) 4.88/2.14 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (199) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(1, 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(1))),F(0) -> U5^1(U1(not(1)))) 4.88/2.14 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.14 (F(0) -> U5^1(U3(1, 0)),F(0) -> U5^1(U3(1, 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (200) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U4(0)) 4.88/2.14 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U2(0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (201) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (202) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U4(0)) 4.88/2.14 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (203) TransformationProof (EQUIVALENT) 4.88/2.14 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U1(1), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.14 4.88/2.14 (F(0) -> U5^1(U1(not(U1(1)))),F(0) -> U5^1(U1(not(U1(1))))) 4.88/2.14 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.14 (F(0) -> U5^1(U3(U1(1), 0)),F(0) -> U5^1(U3(U1(1), 0))) 4.88/2.14 (F(0) -> U5^1(implies(1, 0)),F(0) -> U5^1(implies(1, 0))) 4.88/2.14 4.88/2.14 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (204) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.14 F(0) -> U5^1(U4(0)) 4.88/2.14 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(1, 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.14 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.14 F(0) -> U5^1(U2(0)) 4.88/2.14 F(0) -> U5^1(implies(1, 0)) 4.88/2.14 4.88/2.14 The TRS R consists of the following rules: 4.88/2.14 4.88/2.14 implies(x, y) -> U1(not(x)) 4.88/2.14 implies(x, y) -> U2(y) 4.88/2.14 implies(x, y) -> U3(x, y) 4.88/2.14 U3(1, y) -> U4(y) 4.88/2.14 U4(0) -> 0 4.88/2.14 U2(1) -> 1 4.88/2.14 not(1) -> 0 4.88/2.14 not(0) -> 1 4.88/2.14 U1(1) -> 1 4.88/2.14 4.88/2.14 Q is empty. 4.88/2.14 We have to consider all minimal (P,Q,R)-chains. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (205) DependencyGraphProof (EQUIVALENT) 4.88/2.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.14 ---------------------------------------- 4.88/2.14 4.88/2.14 (206) 4.88/2.14 Obligation: 4.88/2.14 Q DP problem: 4.88/2.14 The TRS P consists of the following rules: 4.88/2.14 4.88/2.14 F(0) -> U5^1(implies(U1(not(0)), 0)) 4.88/2.14 U5^1(1) -> F(0) 4.88/2.14 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.14 F(0) -> U5^1(U1(not(1))) 4.88/2.14 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (207) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U1(not(0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U1(not(U1(not(0))))),F(0) -> U5^1(U1(not(U1(not(0)))))) 4.88/2.15 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.15 (F(0) -> U5^1(U3(U1(not(0)), 0)),F(0) -> U5^1(U3(U1(not(0)), 0))) 4.88/2.15 (F(0) -> U5^1(implies(U1(1), 0)),F(0) -> U5^1(implies(U1(1), 0))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (208) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(U2(0)) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (209) DependencyGraphProof (EQUIVALENT) 4.88/2.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (210) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 F(0) -> U5^1(U1(not(U2(1)))) 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (211) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(U2(1)))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U1(not(1))),F(0) -> U5^1(U1(not(1)))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (212) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (213) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(1))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U1(0)),F(0) -> U5^1(U1(0))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (214) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U1(0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (215) DependencyGraphProof (EQUIVALENT) 4.88/2.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (216) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (217) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(U1(1)))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U1(not(1))),F(0) -> U5^1(U1(not(1)))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (218) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(U1(not(0))))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (219) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(U1(not(0))))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U1(not(U1(1)))),F(0) -> U5^1(U1(not(U1(1))))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (220) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (221) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U4(0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(0),F(0) -> U5^1(0)) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (222) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(0) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (223) DependencyGraphProof (EQUIVALENT) 4.88/2.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (224) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 F(0) -> U5^1(U3(U2(1), 0)) 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (225) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(U2(1), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U3(1, 0)),F(0) -> U5^1(U3(1, 0))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (226) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (227) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(1, 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U4(0)),F(0) -> U5^1(U4(0))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (228) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (229) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(U1(1), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U3(1, 0)),F(0) -> U5^1(U3(1, 0))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (230) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U3(U1(not(0)), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (231) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(U1(not(0)), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U3(U1(1), 0)),F(0) -> U5^1(U3(U1(1), 0))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (232) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (233) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(1, 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U1(not(1))),F(0) -> U5^1(U1(not(1)))) 4.88/2.15 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.15 (F(0) -> U5^1(U3(1, 0)),F(0) -> U5^1(U3(1, 0))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (234) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U2(0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (235) DependencyGraphProof (EQUIVALENT) 4.88/2.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (236) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 F(0) -> U5^1(implies(U1(1), 0)) 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (237) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(U1(1), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U1(not(U1(1)))),F(0) -> U5^1(U1(not(U1(1))))) 4.88/2.15 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.15 (F(0) -> U5^1(U3(U1(1), 0)),F(0) -> U5^1(U3(U1(1), 0))) 4.88/2.15 (F(0) -> U5^1(implies(1, 0)),F(0) -> U5^1(implies(1, 0))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (238) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(U2(0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (239) DependencyGraphProof (EQUIVALENT) 4.88/2.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (240) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (241) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(1))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U1(0)),F(0) -> U5^1(U1(0))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (242) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(U1(0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (243) DependencyGraphProof (EQUIVALENT) 4.88/2.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (244) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 F(0) -> U5^1(U1(not(U1(1)))) 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (245) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U1(not(U1(1)))) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U1(not(1))),F(0) -> U5^1(U1(not(1)))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (246) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (247) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U4(0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(0),F(0) -> U5^1(0)) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (248) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(0) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (249) DependencyGraphProof (EQUIVALENT) 4.88/2.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (250) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (251) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(1, 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U4(0)),F(0) -> U5^1(U4(0))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (252) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U3(U1(1), 0)) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (253) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(U3(U1(1), 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U3(1, 0)),F(0) -> U5^1(U3(1, 0))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (254) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(implies(1, 0)) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (255) TransformationProof (EQUIVALENT) 4.88/2.15 By narrowing [LPAR04] the rule F(0) -> U5^1(implies(1, 0)) at position [0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U1(not(1))),F(0) -> U5^1(U1(not(1)))) 4.88/2.15 (F(0) -> U5^1(U2(0)),F(0) -> U5^1(U2(0))) 4.88/2.15 (F(0) -> U5^1(U3(1, 0)),F(0) -> U5^1(U3(1, 0))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (256) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 F(0) -> U5^1(U2(0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (257) DependencyGraphProof (EQUIVALENT) 4.88/2.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (258) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 implies(x, y) -> U1(not(x)) 4.88/2.15 implies(x, y) -> U2(y) 4.88/2.15 implies(x, y) -> U3(x, y) 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 U2(1) -> 1 4.88/2.15 not(1) -> 0 4.88/2.15 not(0) -> 1 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (259) UsableRulesProof (EQUIVALENT) 4.88/2.15 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (260) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 not(1) -> 0 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 Q is empty. 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (261) MNOCProof (EQUIVALENT) 4.88/2.15 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (262) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U4(0)) 4.88/2.15 F(0) -> U5^1(U3(1, 0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 not(1) -> 0 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 The set Q consists of the following terms: 4.88/2.15 4.88/2.15 U3(1, x0) 4.88/2.15 U4(0) 4.88/2.15 not(1) 4.88/2.15 U1(1) 4.88/2.15 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (263) DependencyGraphProof (EQUIVALENT) 4.88/2.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (264) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 U3(1, y) -> U4(y) 4.88/2.15 U4(0) -> 0 4.88/2.15 not(1) -> 0 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 The set Q consists of the following terms: 4.88/2.15 4.88/2.15 U3(1, x0) 4.88/2.15 U4(0) 4.88/2.15 not(1) 4.88/2.15 U1(1) 4.88/2.15 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (265) UsableRulesProof (EQUIVALENT) 4.88/2.15 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. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (266) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 not(1) -> 0 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 The set Q consists of the following terms: 4.88/2.15 4.88/2.15 U3(1, x0) 4.88/2.15 U4(0) 4.88/2.15 not(1) 4.88/2.15 U1(1) 4.88/2.15 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (267) QReductionProof (EQUIVALENT) 4.88/2.15 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 4.88/2.15 4.88/2.15 U3(1, x0) 4.88/2.15 U4(0) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (268) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(not(1))) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 not(1) -> 0 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 The set Q consists of the following terms: 4.88/2.15 4.88/2.15 not(1) 4.88/2.15 U1(1) 4.88/2.15 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (269) TransformationProof (EQUIVALENT) 4.88/2.15 By rewriting [LPAR04] the rule F(0) -> U5^1(U1(not(1))) at position [0,0] we obtained the following new rules [LPAR04]: 4.88/2.15 4.88/2.15 (F(0) -> U5^1(U1(0)),F(0) -> U5^1(U1(0))) 4.88/2.15 4.88/2.15 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (270) 4.88/2.15 Obligation: 4.88/2.15 Q DP problem: 4.88/2.15 The TRS P consists of the following rules: 4.88/2.15 4.88/2.15 U5^1(1) -> F(0) 4.88/2.15 F(0) -> U5^1(U1(0)) 4.88/2.15 4.88/2.15 The TRS R consists of the following rules: 4.88/2.15 4.88/2.15 not(1) -> 0 4.88/2.15 U1(1) -> 1 4.88/2.15 4.88/2.15 The set Q consists of the following terms: 4.88/2.15 4.88/2.15 not(1) 4.88/2.15 U1(1) 4.88/2.15 4.88/2.15 We have to consider all minimal (P,Q,R)-chains. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (271) DependencyGraphProof (EQUIVALENT) 4.88/2.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 4.88/2.15 ---------------------------------------- 4.88/2.15 4.88/2.15 (272) 4.88/2.15 TRUE 4.88/2.19 EOF