178.13/95.72 MAYBE 178.13/95.73 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 178.13/95.73 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 178.13/95.73 178.13/95.73 178.13/95.73 Outermost Termination of the given OTRS could not be shown: 178.13/95.73 178.13/95.73 (0) OTRS 178.13/95.73 (1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms] 178.13/95.73 (2) QTRS 178.13/95.73 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 178.13/95.73 (4) QDP 178.13/95.73 (5) DependencyGraphProof [EQUIVALENT, 1 ms] 178.13/95.73 (6) AND 178.13/95.73 (7) QDP 178.13/95.73 (8) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (9) QDP 178.13/95.73 (10) QReductionProof [EQUIVALENT, 0 ms] 178.13/95.73 (11) QDP 178.13/95.73 (12) MRRProof [EQUIVALENT, 0 ms] 178.13/95.73 (13) QDP 178.13/95.73 (14) PisEmptyProof [EQUIVALENT, 0 ms] 178.13/95.73 (15) YES 178.13/95.73 (16) QDP 178.13/95.73 (17) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (18) QDP 178.13/95.73 (19) QReductionProof [EQUIVALENT, 0 ms] 178.13/95.73 (20) QDP 178.13/95.73 (21) UsableRulesReductionPairsProof [EQUIVALENT, 5 ms] 178.13/95.73 (22) QDP 178.13/95.73 (23) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (24) TRUE 178.13/95.73 (25) QDP 178.13/95.73 (26) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (27) QDP 178.13/95.73 (28) QReductionProof [EQUIVALENT, 0 ms] 178.13/95.73 (29) QDP 178.13/95.73 (30) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (31) QDP 178.13/95.73 (32) QDPOrderProof [EQUIVALENT, 6 ms] 178.13/95.73 (33) QDP 178.13/95.73 (34) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (35) QDP 178.13/95.73 (36) QReductionProof [EQUIVALENT, 0 ms] 178.13/95.73 (37) QDP 178.13/95.73 (38) Trivial-Transformation [SOUND, 0 ms] 178.13/95.73 (39) QTRS 178.13/95.73 (40) DependencyPairsProof [EQUIVALENT, 0 ms] 178.13/95.73 (41) QDP 178.13/95.73 (42) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (43) AND 178.13/95.73 (44) QDP 178.13/95.73 (45) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (46) QDP 178.13/95.73 (47) MRRProof [EQUIVALENT, 0 ms] 178.13/95.73 (48) QDP 178.13/95.73 (49) PisEmptyProof [EQUIVALENT, 0 ms] 178.13/95.73 (50) YES 178.13/95.73 (51) QDP 178.13/95.73 (52) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (53) QDP 178.13/95.73 (54) NonTerminationLoopProof [COMPLETE, 0 ms] 178.13/95.73 (55) NO 178.13/95.73 (56) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] 178.13/95.73 (57) QTRS 178.13/95.73 (58) AAECC Innermost [EQUIVALENT, 0 ms] 178.13/95.73 (59) QTRS 178.13/95.73 (60) DependencyPairsProof [EQUIVALENT, 1 ms] 178.13/95.73 (61) QDP 178.13/95.73 (62) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (63) AND 178.13/95.73 (64) QDP 178.13/95.73 (65) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (66) QDP 178.13/95.73 (67) QReductionProof [EQUIVALENT, 0 ms] 178.13/95.73 (68) QDP 178.13/95.73 (69) QDPSizeChangeProof [EQUIVALENT, 0 ms] 178.13/95.73 (70) YES 178.13/95.73 (71) QDP 178.13/95.73 (72) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (73) QDP 178.13/95.73 (74) QReductionProof [EQUIVALENT, 0 ms] 178.13/95.73 (75) QDP 178.13/95.73 (76) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (77) QDP 178.13/95.73 (78) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (79) QDP 178.13/95.73 (80) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (81) QDP 178.13/95.73 (82) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (83) QDP 178.13/95.73 (84) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (85) QDP 178.13/95.73 (86) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (87) QDP 178.13/95.73 (88) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (89) QDP 178.13/95.73 (90) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (91) QDP 178.13/95.73 (92) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (93) QDP 178.13/95.73 (94) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (95) QDP 178.13/95.73 (96) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (97) QDP 178.13/95.73 (98) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (99) QDP 178.13/95.73 (100) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (101) QDP 178.13/95.73 (102) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (103) QDP 178.13/95.73 (104) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (105) QDP 178.13/95.73 (106) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (107) QDP 178.13/95.73 (108) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (109) QDP 178.13/95.73 (110) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (111) QDP 178.13/95.73 (112) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (113) QDP 178.13/95.73 (114) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (115) QDP 178.13/95.73 (116) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (117) QDP 178.13/95.73 (118) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (119) QDP 178.13/95.73 (120) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (121) QDP 178.13/95.73 (122) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (123) QDP 178.13/95.73 (124) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (125) QDP 178.13/95.73 (126) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (127) QDP 178.13/95.73 (128) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (129) QDP 178.13/95.73 (130) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (131) QDP 178.13/95.73 (132) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (133) QDP 178.13/95.73 (134) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (135) QDP 178.13/95.73 (136) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (137) QDP 178.13/95.73 (138) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (139) QDP 178.13/95.73 (140) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (141) QDP 178.13/95.73 (142) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (143) QDP 178.13/95.73 (144) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (145) QDP 178.13/95.73 (146) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (147) QDP 178.13/95.73 (148) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (149) QDP 178.13/95.73 (150) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (151) QDP 178.13/95.73 (152) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (153) QDP 178.13/95.73 (154) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (155) QDP 178.13/95.73 (156) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (157) QDP 178.13/95.73 (158) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (159) QDP 178.13/95.73 (160) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (161) QDP 178.13/95.73 (162) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (163) QDP 178.13/95.73 (164) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (165) QDP 178.13/95.73 (166) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (167) QDP 178.13/95.73 (168) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (169) QDP 178.13/95.73 (170) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (171) QDP 178.13/95.73 (172) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (173) QDP 178.13/95.73 (174) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (175) QDP 178.13/95.73 (176) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (177) QDP 178.13/95.73 (178) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (179) QDP 178.13/95.73 (180) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (181) QDP 178.13/95.73 (182) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (183) QDP 178.13/95.73 (184) UsableRulesProof [EQUIVALENT, 0 ms] 178.13/95.73 (185) QDP 178.13/95.73 (186) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (187) QDP 178.13/95.73 (188) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (189) QDP 178.13/95.73 (190) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (191) QDP 178.13/95.73 (192) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (193) QDP 178.13/95.73 (194) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (195) QDP 178.13/95.73 (196) TransformationProof [EQUIVALENT, 0 ms] 178.13/95.73 (197) QDP 178.13/95.73 (198) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (199) QDP 178.13/95.73 (200) QDPOrderProof [EQUIVALENT, 11 ms] 178.13/95.73 (201) QDP 178.13/95.73 (202) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (203) QDP 178.13/95.73 (204) QDPOrderProof [EQUIVALENT, 9 ms] 178.13/95.73 (205) QDP 178.13/95.73 (206) QDPOrderProof [EQUIVALENT, 388 ms] 178.13/95.73 (207) QDP 178.13/95.73 (208) SplitQDPProof [EQUIVALENT, 0 ms] 178.13/95.73 (209) AND 178.13/95.73 (210) QDP 178.13/95.73 (211) SemLabProof [SOUND, 0 ms] 178.13/95.73 (212) QDP 178.13/95.73 (213) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 178.13/95.73 (214) QDP 178.13/95.73 (215) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (216) QDP 178.13/95.73 (217) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 178.13/95.73 (218) QDP 178.13/95.73 (219) PisEmptyProof [SOUND, 0 ms] 178.13/95.73 (220) TRUE 178.13/95.73 (221) QDP 178.13/95.73 (222) QReductionProof [EQUIVALENT, 0 ms] 178.13/95.73 (223) QDP 178.13/95.73 (224) SplitQDPProof [EQUIVALENT, 0 ms] 178.13/95.73 (225) AND 178.13/95.73 (226) QDP 178.13/95.73 (227) SemLabProof [SOUND, 0 ms] 178.13/95.73 (228) QDP 178.13/95.73 (229) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 178.13/95.73 (230) QDP 178.13/95.73 (231) DependencyGraphProof [EQUIVALENT, 0 ms] 178.13/95.73 (232) QDP 178.13/95.73 (233) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 178.13/95.73 (234) QDP 178.13/95.73 (235) PisEmptyProof [SOUND, 0 ms] 178.13/95.73 (236) TRUE 178.13/95.73 (237) QDP 178.13/95.73 (238) QReductionProof [EQUIVALENT, 0 ms] 178.13/95.73 (239) QDP 178.13/95.73 178.13/95.73 178.13/95.73 ---------------------------------------- 178.13/95.73 178.13/95.73 (0) 178.13/95.73 Obligation: 178.13/95.73 Term rewrite system R: 178.13/95.73 The TRS R consists of the following rules: 178.13/95.73 178.13/95.73 a -> f(a) 178.13/95.73 g(f(x)) -> f(g(g(x))) 178.13/95.73 f(f(f(f(f(x))))) -> b 178.13/95.73 178.13/95.73 178.13/95.73 178.13/95.73 Outermost Strategy. 178.13/95.73 178.13/95.73 ---------------------------------------- 178.13/95.73 178.13/95.73 (1) Thiemann-SpecialC-Transformation (EQUIVALENT) 178.13/95.73 We applied the Thiemann-SpecialC transformation to transform the outermost TRS to an innermost TRS. 178.13/95.73 ---------------------------------------- 178.13/95.73 178.13/95.73 (2) 178.13/95.73 Obligation: 178.13/95.73 Q restricted rewrite system: 178.13/95.73 The TRS R consists of the following rules: 178.13/95.73 178.13/95.73 top(go_up(x)) -> top(reduce(x)) 178.13/95.73 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.13/95.73 reduce(f(x_1)) -> check_f(redex_f(x_1)) 178.13/95.73 reduce(a) -> go_up(f(a)) 178.13/95.73 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.73 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.73 check_g(result_g(x)) -> go_up(x) 178.13/95.73 check_f(result_f(x)) -> go_up(x) 178.13/95.73 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.13/95.73 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 178.13/95.73 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 178.13/95.73 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.13/95.73 178.13/95.73 The set Q consists of the following terms: 178.13/95.73 178.13/95.73 top(go_up(x0)) 178.13/95.73 reduce(g(x0)) 178.13/95.73 reduce(f(x0)) 178.13/95.73 reduce(a) 178.13/95.73 redex_g(f(x0)) 178.13/95.73 redex_f(f(f(f(f(x0))))) 178.13/95.73 check_g(result_g(x0)) 178.13/95.73 check_f(result_f(x0)) 178.13/95.73 check_g(redex_g(x0)) 178.13/95.73 check_f(redex_f(x0)) 178.13/95.73 in_f_1(go_up(x0)) 178.13/95.73 in_g_1(go_up(x0)) 178.13/95.73 178.13/95.73 178.13/95.73 ---------------------------------------- 178.13/95.73 178.13/95.73 (3) DependencyPairsProof (EQUIVALENT) 178.13/95.73 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 178.13/95.73 ---------------------------------------- 178.13/95.73 178.13/95.73 (4) 178.13/95.73 Obligation: 178.13/95.73 Q DP problem: 178.13/95.73 The TRS P consists of the following rules: 178.13/95.73 178.13/95.73 TOP(go_up(x)) -> TOP(reduce(x)) 178.13/95.73 TOP(go_up(x)) -> REDUCE(x) 178.13/95.73 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 178.13/95.73 REDUCE(g(x_1)) -> REDEX_G(x_1) 178.13/95.73 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 178.13/95.73 REDUCE(f(x_1)) -> REDEX_F(x_1) 178.13/95.73 CHECK_G(redex_g(x_1)) -> IN_G_1(reduce(x_1)) 178.13/95.73 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 178.13/95.73 CHECK_F(redex_f(x_1)) -> IN_F_1(reduce(x_1)) 178.13/95.73 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 178.13/95.73 178.13/95.73 The TRS R consists of the following rules: 178.13/95.73 178.13/95.73 top(go_up(x)) -> top(reduce(x)) 178.13/95.73 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.13/95.73 reduce(f(x_1)) -> check_f(redex_f(x_1)) 178.13/95.73 reduce(a) -> go_up(f(a)) 178.13/95.73 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.73 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.73 check_g(result_g(x)) -> go_up(x) 178.13/95.73 check_f(result_f(x)) -> go_up(x) 178.13/95.73 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.13/95.73 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 178.13/95.73 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 178.13/95.73 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.13/95.73 178.13/95.73 The set Q consists of the following terms: 178.13/95.73 178.13/95.73 top(go_up(x0)) 178.13/95.73 reduce(g(x0)) 178.13/95.73 reduce(f(x0)) 178.13/95.73 reduce(a) 178.13/95.73 redex_g(f(x0)) 178.13/95.73 redex_f(f(f(f(f(x0))))) 178.13/95.73 check_g(result_g(x0)) 178.13/95.73 check_f(result_f(x0)) 178.13/95.73 check_g(redex_g(x0)) 178.13/95.73 check_f(redex_f(x0)) 178.13/95.73 in_f_1(go_up(x0)) 178.13/95.73 in_g_1(go_up(x0)) 178.13/95.73 178.13/95.73 We have to consider all minimal (P,Q,R)-chains. 178.13/95.73 ---------------------------------------- 178.13/95.73 178.13/95.73 (5) DependencyGraphProof (EQUIVALENT) 178.13/95.73 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 5 less nodes. 178.13/95.73 ---------------------------------------- 178.13/95.73 178.13/95.73 (6) 178.13/95.73 Complex Obligation (AND) 178.13/95.73 178.13/95.73 ---------------------------------------- 178.13/95.73 178.13/95.73 (7) 178.13/95.73 Obligation: 178.13/95.73 Q DP problem: 178.13/95.73 The TRS P consists of the following rules: 178.13/95.73 178.13/95.73 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 178.13/95.73 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 178.13/95.73 178.13/95.73 The TRS R consists of the following rules: 178.13/95.73 178.13/95.73 top(go_up(x)) -> top(reduce(x)) 178.13/95.73 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.13/95.73 reduce(f(x_1)) -> check_f(redex_f(x_1)) 178.13/95.73 reduce(a) -> go_up(f(a)) 178.13/95.73 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.73 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.73 check_g(result_g(x)) -> go_up(x) 178.13/95.73 check_f(result_f(x)) -> go_up(x) 178.13/95.73 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.13/95.73 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 178.13/95.73 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 178.13/95.73 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.13/95.73 178.13/95.73 The set Q consists of the following terms: 178.13/95.73 178.13/95.73 top(go_up(x0)) 178.13/95.73 reduce(g(x0)) 178.13/95.73 reduce(f(x0)) 178.13/95.73 reduce(a) 178.13/95.73 redex_g(f(x0)) 178.13/95.73 redex_f(f(f(f(f(x0))))) 178.13/95.73 check_g(result_g(x0)) 178.13/95.73 check_f(result_f(x0)) 178.13/95.73 check_g(redex_g(x0)) 178.13/95.73 check_f(redex_f(x0)) 178.13/95.73 in_f_1(go_up(x0)) 178.13/95.73 in_g_1(go_up(x0)) 178.13/95.73 178.13/95.73 We have to consider all minimal (P,Q,R)-chains. 178.13/95.73 ---------------------------------------- 178.13/95.73 178.13/95.73 (8) UsableRulesProof (EQUIVALENT) 178.13/95.73 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. 178.13/95.73 ---------------------------------------- 178.13/95.73 178.13/95.73 (9) 178.13/95.73 Obligation: 178.13/95.73 Q DP problem: 178.13/95.73 The TRS P consists of the following rules: 178.13/95.73 178.13/95.73 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 178.13/95.73 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 178.13/95.73 178.13/95.73 The TRS R consists of the following rules: 178.13/95.73 178.13/95.73 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.73 178.13/95.73 The set Q consists of the following terms: 178.13/95.73 178.13/95.73 top(go_up(x0)) 178.13/95.73 reduce(g(x0)) 178.13/95.73 reduce(f(x0)) 178.13/95.73 reduce(a) 178.13/95.73 redex_g(f(x0)) 178.13/95.73 redex_f(f(f(f(f(x0))))) 178.13/95.73 check_g(result_g(x0)) 178.13/95.73 check_f(result_f(x0)) 178.13/95.73 check_g(redex_g(x0)) 178.13/95.73 check_f(redex_f(x0)) 178.13/95.73 in_f_1(go_up(x0)) 178.13/95.73 in_g_1(go_up(x0)) 178.13/95.73 178.13/95.73 We have to consider all minimal (P,Q,R)-chains. 178.13/95.73 ---------------------------------------- 178.13/95.73 178.13/95.73 (10) QReductionProof (EQUIVALENT) 178.13/95.73 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 178.13/95.73 178.13/95.73 top(go_up(x0)) 178.13/95.73 reduce(g(x0)) 178.13/95.73 reduce(f(x0)) 178.13/95.73 reduce(a) 178.13/95.73 redex_f(f(f(f(f(x0))))) 178.13/95.73 check_g(result_g(x0)) 178.13/95.73 check_f(result_f(x0)) 178.13/95.73 check_g(redex_g(x0)) 178.13/95.73 check_f(redex_f(x0)) 178.13/95.73 in_f_1(go_up(x0)) 178.13/95.73 in_g_1(go_up(x0)) 178.13/95.73 178.13/95.73 178.13/95.73 ---------------------------------------- 178.13/95.73 178.13/95.73 (11) 178.13/95.73 Obligation: 178.13/95.73 Q DP problem: 178.13/95.73 The TRS P consists of the following rules: 178.13/95.73 178.13/95.73 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 178.13/95.73 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 178.13/95.73 178.13/95.73 The TRS R consists of the following rules: 178.13/95.73 178.13/95.73 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.73 178.13/95.73 The set Q consists of the following terms: 178.13/95.73 178.13/95.73 redex_g(f(x0)) 178.13/95.73 178.13/95.73 We have to consider all minimal (P,Q,R)-chains. 178.13/95.73 ---------------------------------------- 178.13/95.73 178.13/95.73 (12) MRRProof (EQUIVALENT) 178.13/95.73 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 178.13/95.74 178.13/95.74 Strictly oriented dependency pairs: 178.13/95.74 178.13/95.74 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 178.13/95.74 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 178.13/95.74 178.13/95.74 178.13/95.74 Used ordering: Polynomial interpretation [POLO]: 178.13/95.74 178.13/95.74 POL(CHECK_G(x_1)) = 1 + x_1 178.13/95.74 POL(REDUCE(x_1)) = 2*x_1 178.13/95.74 POL(f(x_1)) = 2 + x_1 178.13/95.74 POL(g(x_1)) = 2 + x_1 178.13/95.74 POL(redex_g(x_1)) = 2 + 2*x_1 178.13/95.74 POL(result_g(x_1)) = x_1 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (13) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 P is empty. 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 redex_g(f(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (14) PisEmptyProof (EQUIVALENT) 178.13/95.74 The TRS P is empty. Hence, there is no (P,Q,R) chain. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (15) 178.13/95.74 YES 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (16) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 178.13/95.74 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 top(go_up(x)) -> top(reduce(x)) 178.13/95.74 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.13/95.74 reduce(f(x_1)) -> check_f(redex_f(x_1)) 178.13/95.74 reduce(a) -> go_up(f(a)) 178.13/95.74 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.74 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.74 check_g(result_g(x)) -> go_up(x) 178.13/95.74 check_f(result_f(x)) -> go_up(x) 178.13/95.74 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.13/95.74 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 178.13/95.74 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 178.13/95.74 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 top(go_up(x0)) 178.13/95.74 reduce(g(x0)) 178.13/95.74 reduce(f(x0)) 178.13/95.74 reduce(a) 178.13/95.74 redex_g(f(x0)) 178.13/95.74 redex_f(f(f(f(f(x0))))) 178.13/95.74 check_g(result_g(x0)) 178.13/95.74 check_f(result_f(x0)) 178.13/95.74 check_g(redex_g(x0)) 178.13/95.74 check_f(redex_f(x0)) 178.13/95.74 in_f_1(go_up(x0)) 178.13/95.74 in_g_1(go_up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (17) UsableRulesProof (EQUIVALENT) 178.13/95.74 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. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (18) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 178.13/95.74 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 top(go_up(x0)) 178.13/95.74 reduce(g(x0)) 178.13/95.74 reduce(f(x0)) 178.13/95.74 reduce(a) 178.13/95.74 redex_g(f(x0)) 178.13/95.74 redex_f(f(f(f(f(x0))))) 178.13/95.74 check_g(result_g(x0)) 178.13/95.74 check_f(result_f(x0)) 178.13/95.74 check_g(redex_g(x0)) 178.13/95.74 check_f(redex_f(x0)) 178.13/95.74 in_f_1(go_up(x0)) 178.13/95.74 in_g_1(go_up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (19) QReductionProof (EQUIVALENT) 178.13/95.74 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 178.13/95.74 178.13/95.74 top(go_up(x0)) 178.13/95.74 reduce(g(x0)) 178.13/95.74 reduce(f(x0)) 178.13/95.74 reduce(a) 178.13/95.74 redex_g(f(x0)) 178.13/95.74 check_g(result_g(x0)) 178.13/95.74 check_f(result_f(x0)) 178.13/95.74 check_g(redex_g(x0)) 178.13/95.74 check_f(redex_f(x0)) 178.13/95.74 in_f_1(go_up(x0)) 178.13/95.74 in_g_1(go_up(x0)) 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (20) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 178.13/95.74 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 redex_f(f(f(f(f(x0))))) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (21) UsableRulesReductionPairsProof (EQUIVALENT) 178.13/95.74 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. 178.13/95.74 178.13/95.74 The following dependency pairs can be deleted: 178.13/95.74 178.13/95.74 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 178.13/95.74 The following rules are removed from R: 178.13/95.74 178.13/95.74 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.74 Used ordering: POLO with Polynomial interpretation [POLO]: 178.13/95.74 178.13/95.74 POL(CHECK_F(x_1)) = x_1 178.13/95.74 POL(REDUCE(x_1)) = 2*x_1 178.13/95.74 POL(b) = 0 178.13/95.74 POL(f(x_1)) = 2*x_1 178.13/95.74 POL(redex_f(x_1)) = 2*x_1 178.13/95.74 POL(result_f(x_1)) = x_1 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (22) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 178.13/95.74 178.13/95.74 R is empty. 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 redex_f(f(f(f(f(x0))))) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (23) DependencyGraphProof (EQUIVALENT) 178.13/95.74 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (24) 178.13/95.74 TRUE 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (25) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 TOP(go_up(x)) -> TOP(reduce(x)) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 top(go_up(x)) -> top(reduce(x)) 178.13/95.74 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.13/95.74 reduce(f(x_1)) -> check_f(redex_f(x_1)) 178.13/95.74 reduce(a) -> go_up(f(a)) 178.13/95.74 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.74 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.74 check_g(result_g(x)) -> go_up(x) 178.13/95.74 check_f(result_f(x)) -> go_up(x) 178.13/95.74 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.13/95.74 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 178.13/95.74 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 178.13/95.74 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 top(go_up(x0)) 178.13/95.74 reduce(g(x0)) 178.13/95.74 reduce(f(x0)) 178.13/95.74 reduce(a) 178.13/95.74 redex_g(f(x0)) 178.13/95.74 redex_f(f(f(f(f(x0))))) 178.13/95.74 check_g(result_g(x0)) 178.13/95.74 check_f(result_f(x0)) 178.13/95.74 check_g(redex_g(x0)) 178.13/95.74 check_f(redex_f(x0)) 178.13/95.74 in_f_1(go_up(x0)) 178.13/95.74 in_g_1(go_up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (26) UsableRulesProof (EQUIVALENT) 178.13/95.74 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. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (27) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 TOP(go_up(x)) -> TOP(reduce(x)) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.13/95.74 reduce(f(x_1)) -> check_f(redex_f(x_1)) 178.13/95.74 reduce(a) -> go_up(f(a)) 178.13/95.74 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.74 check_f(result_f(x)) -> go_up(x) 178.13/95.74 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 178.13/95.74 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 178.13/95.74 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.74 check_g(result_g(x)) -> go_up(x) 178.13/95.74 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.13/95.74 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 top(go_up(x0)) 178.13/95.74 reduce(g(x0)) 178.13/95.74 reduce(f(x0)) 178.13/95.74 reduce(a) 178.13/95.74 redex_g(f(x0)) 178.13/95.74 redex_f(f(f(f(f(x0))))) 178.13/95.74 check_g(result_g(x0)) 178.13/95.74 check_f(result_f(x0)) 178.13/95.74 check_g(redex_g(x0)) 178.13/95.74 check_f(redex_f(x0)) 178.13/95.74 in_f_1(go_up(x0)) 178.13/95.74 in_g_1(go_up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (28) QReductionProof (EQUIVALENT) 178.13/95.74 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 178.13/95.74 178.13/95.74 top(go_up(x0)) 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (29) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 TOP(go_up(x)) -> TOP(reduce(x)) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.13/95.74 reduce(f(x_1)) -> check_f(redex_f(x_1)) 178.13/95.74 reduce(a) -> go_up(f(a)) 178.13/95.74 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.74 check_f(result_f(x)) -> go_up(x) 178.13/95.74 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 178.13/95.74 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 178.13/95.74 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.74 check_g(result_g(x)) -> go_up(x) 178.13/95.74 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.13/95.74 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 reduce(g(x0)) 178.13/95.74 reduce(f(x0)) 178.13/95.74 reduce(a) 178.13/95.74 redex_g(f(x0)) 178.13/95.74 redex_f(f(f(f(f(x0))))) 178.13/95.74 check_g(result_g(x0)) 178.13/95.74 check_f(result_f(x0)) 178.13/95.74 check_g(redex_g(x0)) 178.13/95.74 check_f(redex_f(x0)) 178.13/95.74 in_f_1(go_up(x0)) 178.13/95.74 in_g_1(go_up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (30) TransformationProof (EQUIVALENT) 178.13/95.74 By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]: 178.13/95.74 178.13/95.74 (TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))),TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))) 178.13/95.74 (TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))),TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))) 178.13/95.74 (TOP(go_up(a)) -> TOP(go_up(f(a))),TOP(go_up(a)) -> TOP(go_up(f(a)))) 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (31) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))) 178.13/95.74 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 178.13/95.74 TOP(go_up(a)) -> TOP(go_up(f(a))) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.13/95.74 reduce(f(x_1)) -> check_f(redex_f(x_1)) 178.13/95.74 reduce(a) -> go_up(f(a)) 178.13/95.74 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.74 check_f(result_f(x)) -> go_up(x) 178.13/95.74 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 178.13/95.74 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 178.13/95.74 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.74 check_g(result_g(x)) -> go_up(x) 178.13/95.74 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.13/95.74 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 reduce(g(x0)) 178.13/95.74 reduce(f(x0)) 178.13/95.74 reduce(a) 178.13/95.74 redex_g(f(x0)) 178.13/95.74 redex_f(f(f(f(f(x0))))) 178.13/95.74 check_g(result_g(x0)) 178.13/95.74 check_f(result_f(x0)) 178.13/95.74 check_g(redex_g(x0)) 178.13/95.74 check_f(redex_f(x0)) 178.13/95.74 in_f_1(go_up(x0)) 178.13/95.74 in_g_1(go_up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (32) QDPOrderProof (EQUIVALENT) 178.13/95.74 We use the reduction pair processor [LPAR04,JAR06]. 178.13/95.74 178.13/95.74 178.13/95.74 The following pairs can be oriented strictly and are deleted. 178.13/95.74 178.13/95.74 TOP(go_up(a)) -> TOP(go_up(f(a))) 178.13/95.74 The remaining pairs can at least be oriented weakly. 178.13/95.74 Used ordering: Polynomial interpretation [POLO]: 178.13/95.74 178.13/95.74 POL(TOP(x_1)) = x_1 178.13/95.74 POL(a) = 1 178.13/95.74 POL(b) = 0 178.13/95.74 POL(check_f(x_1)) = x_1 178.13/95.74 POL(check_g(x_1)) = x_1 178.13/95.74 POL(f(x_1)) = 0 178.13/95.74 POL(g(x_1)) = 0 178.13/95.74 POL(go_up(x_1)) = x_1 178.13/95.74 POL(in_f_1(x_1)) = 0 178.13/95.74 POL(in_g_1(x_1)) = 0 178.13/95.74 POL(redex_f(x_1)) = 0 178.13/95.74 POL(redex_g(x_1)) = 0 178.13/95.74 POL(reduce(x_1)) = 0 178.13/95.74 POL(result_f(x_1)) = x_1 178.13/95.74 POL(result_g(x_1)) = x_1 178.13/95.74 178.13/95.74 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 178.13/95.74 178.13/95.74 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.74 check_g(result_g(x)) -> go_up(x) 178.13/95.74 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.13/95.74 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.74 check_f(result_f(x)) -> go_up(x) 178.13/95.74 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 178.13/95.74 reduce(f(x_1)) -> check_f(redex_f(x_1)) 178.13/95.74 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.13/95.74 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 178.13/95.74 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (33) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))) 178.13/95.74 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.13/95.74 reduce(f(x_1)) -> check_f(redex_f(x_1)) 178.13/95.74 reduce(a) -> go_up(f(a)) 178.13/95.74 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.74 check_f(result_f(x)) -> go_up(x) 178.13/95.74 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 178.13/95.74 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 178.13/95.74 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.74 check_g(result_g(x)) -> go_up(x) 178.13/95.74 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.13/95.74 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 reduce(g(x0)) 178.13/95.74 reduce(f(x0)) 178.13/95.74 reduce(a) 178.13/95.74 redex_g(f(x0)) 178.13/95.74 redex_f(f(f(f(f(x0))))) 178.13/95.74 check_g(result_g(x0)) 178.13/95.74 check_f(result_f(x0)) 178.13/95.74 check_g(redex_g(x0)) 178.13/95.74 check_f(redex_f(x0)) 178.13/95.74 in_f_1(go_up(x0)) 178.13/95.74 in_g_1(go_up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (34) UsableRulesProof (EQUIVALENT) 178.13/95.74 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. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (35) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 TOP(go_up(x)) -> TOP(reduce(x)) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.13/95.74 reduce(f(x_1)) -> check_f(redex_f(x_1)) 178.13/95.74 reduce(a) -> go_up(f(a)) 178.13/95.74 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.74 check_f(result_f(x)) -> go_up(x) 178.13/95.74 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 178.13/95.74 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 178.13/95.74 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.74 check_g(result_g(x)) -> go_up(x) 178.13/95.74 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.13/95.74 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 top(go_up(x0)) 178.13/95.74 reduce(g(x0)) 178.13/95.74 reduce(f(x0)) 178.13/95.74 reduce(a) 178.13/95.74 redex_g(f(x0)) 178.13/95.74 redex_f(f(f(f(f(x0))))) 178.13/95.74 check_g(result_g(x0)) 178.13/95.74 check_f(result_f(x0)) 178.13/95.74 check_g(redex_g(x0)) 178.13/95.74 check_f(redex_f(x0)) 178.13/95.74 in_f_1(go_up(x0)) 178.13/95.74 in_g_1(go_up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (36) QReductionProof (EQUIVALENT) 178.13/95.74 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 178.13/95.74 178.13/95.74 top(go_up(x0)) 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (37) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 TOP(go_up(x)) -> TOP(reduce(x)) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 reduce(g(x_1)) -> check_g(redex_g(x_1)) 178.13/95.74 reduce(f(x_1)) -> check_f(redex_f(x_1)) 178.13/95.74 reduce(a) -> go_up(f(a)) 178.13/95.74 redex_f(f(f(f(f(x))))) -> result_f(b) 178.13/95.74 check_f(result_f(x)) -> go_up(x) 178.13/95.74 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 178.13/95.74 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 178.13/95.74 redex_g(f(x)) -> result_g(f(g(g(x)))) 178.13/95.74 check_g(result_g(x)) -> go_up(x) 178.13/95.74 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 178.13/95.74 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 reduce(g(x0)) 178.13/95.74 reduce(f(x0)) 178.13/95.74 reduce(a) 178.13/95.74 redex_g(f(x0)) 178.13/95.74 redex_f(f(f(f(f(x0))))) 178.13/95.74 check_g(result_g(x0)) 178.13/95.74 check_f(result_f(x0)) 178.13/95.74 check_g(redex_g(x0)) 178.13/95.74 check_f(redex_f(x0)) 178.13/95.74 in_f_1(go_up(x0)) 178.13/95.74 in_g_1(go_up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (38) Trivial-Transformation (SOUND) 178.13/95.74 We applied the Trivial transformation to transform the outermost TRS to a standard TRS. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (39) 178.13/95.74 Obligation: 178.13/95.74 Q restricted rewrite system: 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 a -> f(a) 178.13/95.74 g(f(x)) -> f(g(g(x))) 178.13/95.74 f(f(f(f(f(x))))) -> b 178.13/95.74 178.13/95.74 Q is empty. 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (40) DependencyPairsProof (EQUIVALENT) 178.13/95.74 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (41) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 A -> F(a) 178.13/95.74 A -> A 178.13/95.74 G(f(x)) -> F(g(g(x))) 178.13/95.74 G(f(x)) -> G(g(x)) 178.13/95.74 G(f(x)) -> G(x) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 a -> f(a) 178.13/95.74 g(f(x)) -> f(g(g(x))) 178.13/95.74 f(f(f(f(f(x))))) -> b 178.13/95.74 178.13/95.74 Q is empty. 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (42) DependencyGraphProof (EQUIVALENT) 178.13/95.74 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (43) 178.13/95.74 Complex Obligation (AND) 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (44) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 G(f(x)) -> G(x) 178.13/95.74 G(f(x)) -> G(g(x)) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 a -> f(a) 178.13/95.74 g(f(x)) -> f(g(g(x))) 178.13/95.74 f(f(f(f(f(x))))) -> b 178.13/95.74 178.13/95.74 Q is empty. 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (45) UsableRulesProof (EQUIVALENT) 178.13/95.74 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. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (46) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 G(f(x)) -> G(x) 178.13/95.74 G(f(x)) -> G(g(x)) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 g(f(x)) -> f(g(g(x))) 178.13/95.74 f(f(f(f(f(x))))) -> b 178.13/95.74 178.13/95.74 Q is empty. 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (47) MRRProof (EQUIVALENT) 178.13/95.74 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 178.13/95.74 178.13/95.74 Strictly oriented dependency pairs: 178.13/95.74 178.13/95.74 G(f(x)) -> G(x) 178.13/95.74 G(f(x)) -> G(g(x)) 178.13/95.74 178.13/95.74 Strictly oriented rules of the TRS R: 178.13/95.74 178.13/95.74 f(f(f(f(f(x))))) -> b 178.13/95.74 178.13/95.74 Used ordering: Polynomial interpretation [POLO]: 178.13/95.74 178.13/95.74 POL(G(x_1)) = x_1 178.13/95.74 POL(b) = 1 178.13/95.74 POL(f(x_1)) = 2 + 2*x_1 178.13/95.74 POL(g(x_1)) = x_1 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (48) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 P is empty. 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 g(f(x)) -> f(g(g(x))) 178.13/95.74 178.13/95.74 Q is empty. 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (49) PisEmptyProof (EQUIVALENT) 178.13/95.74 The TRS P is empty. Hence, there is no (P,Q,R) chain. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (50) 178.13/95.74 YES 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (51) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 A -> A 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 a -> f(a) 178.13/95.74 g(f(x)) -> f(g(g(x))) 178.13/95.74 f(f(f(f(f(x))))) -> b 178.13/95.74 178.13/95.74 Q is empty. 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (52) UsableRulesProof (EQUIVALENT) 178.13/95.74 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. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (53) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 A -> A 178.13/95.74 178.13/95.74 R is empty. 178.13/95.74 Q is empty. 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (54) NonTerminationLoopProof (COMPLETE) 178.13/95.74 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 178.13/95.74 Found a loop by semiunifying a rule from P directly. 178.13/95.74 178.13/95.74 s = A evaluates to t =A 178.13/95.74 178.13/95.74 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 178.13/95.74 * Matcher: [ ] 178.13/95.74 * Semiunifier: [ ] 178.13/95.74 178.13/95.74 -------------------------------------------------------------------------------- 178.13/95.74 Rewriting sequence 178.13/95.74 178.13/95.74 The DP semiunifies directly so there is only one rewrite step from A to A. 178.13/95.74 178.13/95.74 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (55) 178.13/95.74 NO 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (56) Raffelsieper-Zantema-Transformation (SOUND) 178.13/95.74 We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (57) 178.13/95.74 Obligation: 178.13/95.74 Q restricted rewrite system: 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 down(a) -> up(f(a)) 178.13/95.74 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.74 down(f(f(f(f(f(x)))))) -> up(b) 178.13/95.74 top(up(x)) -> top(down(x)) 178.13/95.74 down(f(a)) -> f_flat(down(a)) 178.13/95.74 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.74 down(f(b)) -> f_flat(down(b)) 178.13/95.74 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.74 down(g(a)) -> g_flat(down(a)) 178.13/95.74 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.74 down(g(b)) -> g_flat(down(b)) 178.13/95.74 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.74 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.74 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.74 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.74 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.74 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.74 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) 178.13/95.74 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.74 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.74 178.13/95.74 Q is empty. 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (58) AAECC Innermost (EQUIVALENT) 178.13/95.74 We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 178.13/95.74 down(f(a)) -> f_flat(down(a)) 178.13/95.74 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.74 down(f(b)) -> f_flat(down(b)) 178.13/95.74 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.74 down(g(a)) -> g_flat(down(a)) 178.13/95.74 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.74 down(g(b)) -> g_flat(down(b)) 178.13/95.74 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.74 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.74 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.74 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.74 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.74 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.74 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) 178.13/95.74 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.74 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.74 down(a) -> up(f(a)) 178.13/95.74 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.74 down(f(f(f(f(f(x)))))) -> up(b) 178.13/95.74 178.13/95.74 The TRS R 2 is 178.13/95.74 top(up(x)) -> top(down(x)) 178.13/95.74 178.13/95.74 The signature Sigma is {top_1} 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (59) 178.13/95.74 Obligation: 178.13/95.74 Q restricted rewrite system: 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 down(a) -> up(f(a)) 178.13/95.74 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.74 down(f(f(f(f(f(x)))))) -> up(b) 178.13/95.74 top(up(x)) -> top(down(x)) 178.13/95.74 down(f(a)) -> f_flat(down(a)) 178.13/95.74 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.74 down(f(b)) -> f_flat(down(b)) 178.13/95.74 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.74 down(g(a)) -> g_flat(down(a)) 178.13/95.74 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.74 down(g(b)) -> g_flat(down(b)) 178.13/95.74 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.74 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.74 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.74 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.74 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.74 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.74 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) 178.13/95.74 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.74 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 down(a) 178.13/95.74 down(g(f(x0))) 178.13/95.74 down(f(f(f(f(f(x0)))))) 178.13/95.74 top(up(x0)) 178.13/95.74 down(f(a)) 178.13/95.74 down(f(g(x0))) 178.13/95.74 down(f(b)) 178.13/95.74 down(f(fresh_constant)) 178.13/95.74 down(g(a)) 178.13/95.74 down(g(g(x0))) 178.13/95.74 down(g(b)) 178.13/95.74 down(g(fresh_constant)) 178.13/95.74 down(f(f(a))) 178.13/95.74 down(f(f(g(x0)))) 178.13/95.74 down(f(f(b))) 178.13/95.74 down(f(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) 178.13/95.74 down(f(f(f(g(x0))))) 178.13/95.74 down(f(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(x0)))))) 178.13/95.74 down(f(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x0)) 178.13/95.74 g_flat(up(x0)) 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (60) DependencyPairsProof (EQUIVALENT) 178.13/95.74 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (61) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 TOP(up(x)) -> TOP(down(x)) 178.13/95.74 TOP(up(x)) -> DOWN(x) 178.13/95.74 DOWN(f(a)) -> F_FLAT(down(a)) 178.13/95.74 DOWN(f(a)) -> DOWN(a) 178.13/95.74 DOWN(f(g(y4))) -> F_FLAT(down(g(y4))) 178.13/95.74 DOWN(f(g(y4))) -> DOWN(g(y4)) 178.13/95.74 DOWN(f(b)) -> F_FLAT(down(b)) 178.13/95.74 DOWN(f(b)) -> DOWN(b) 178.13/95.74 DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant)) 178.13/95.74 DOWN(f(fresh_constant)) -> DOWN(fresh_constant) 178.13/95.74 DOWN(g(a)) -> G_FLAT(down(a)) 178.13/95.74 DOWN(g(a)) -> DOWN(a) 178.13/95.74 DOWN(g(g(y7))) -> G_FLAT(down(g(y7))) 178.13/95.74 DOWN(g(g(y7))) -> DOWN(g(y7)) 178.13/95.74 DOWN(g(b)) -> G_FLAT(down(b)) 178.13/95.74 DOWN(g(b)) -> DOWN(b) 178.13/95.74 DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant)) 178.13/95.74 DOWN(g(fresh_constant)) -> DOWN(fresh_constant) 178.13/95.74 DOWN(f(f(a))) -> F_FLAT(down(f(a))) 178.13/95.74 DOWN(f(f(a))) -> DOWN(f(a)) 178.13/95.74 DOWN(f(f(g(y10)))) -> F_FLAT(down(f(g(y10)))) 178.13/95.74 DOWN(f(f(g(y10)))) -> DOWN(f(g(y10))) 178.13/95.74 DOWN(f(f(b))) -> F_FLAT(down(f(b))) 178.13/95.74 DOWN(f(f(b))) -> DOWN(f(b)) 178.13/95.74 DOWN(f(f(fresh_constant))) -> F_FLAT(down(f(fresh_constant))) 178.13/95.74 DOWN(f(f(fresh_constant))) -> DOWN(f(fresh_constant)) 178.13/95.74 DOWN(f(f(f(a)))) -> F_FLAT(down(f(f(a)))) 178.13/95.74 DOWN(f(f(f(a)))) -> DOWN(f(f(a))) 178.13/95.74 DOWN(f(f(f(g(y13))))) -> F_FLAT(down(f(f(g(y13))))) 178.13/95.74 DOWN(f(f(f(g(y13))))) -> DOWN(f(f(g(y13)))) 178.13/95.74 DOWN(f(f(f(b)))) -> F_FLAT(down(f(f(b)))) 178.13/95.74 DOWN(f(f(f(b)))) -> DOWN(f(f(b))) 178.13/95.74 DOWN(f(f(f(fresh_constant)))) -> F_FLAT(down(f(f(fresh_constant)))) 178.13/95.74 DOWN(f(f(f(fresh_constant)))) -> DOWN(f(f(fresh_constant))) 178.13/95.74 DOWN(f(f(f(f(a))))) -> F_FLAT(down(f(f(f(a))))) 178.13/95.74 DOWN(f(f(f(f(a))))) -> DOWN(f(f(f(a)))) 178.13/95.74 DOWN(f(f(f(f(g(y16)))))) -> F_FLAT(down(f(f(f(g(y16)))))) 178.13/95.74 DOWN(f(f(f(f(g(y16)))))) -> DOWN(f(f(f(g(y16))))) 178.13/95.74 DOWN(f(f(f(f(b))))) -> F_FLAT(down(f(f(f(b))))) 178.13/95.74 DOWN(f(f(f(f(b))))) -> DOWN(f(f(f(b)))) 178.13/95.74 DOWN(f(f(f(f(fresh_constant))))) -> F_FLAT(down(f(f(f(fresh_constant))))) 178.13/95.74 DOWN(f(f(f(f(fresh_constant))))) -> DOWN(f(f(f(fresh_constant)))) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 down(a) -> up(f(a)) 178.13/95.74 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.74 down(f(f(f(f(f(x)))))) -> up(b) 178.13/95.74 top(up(x)) -> top(down(x)) 178.13/95.74 down(f(a)) -> f_flat(down(a)) 178.13/95.74 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.74 down(f(b)) -> f_flat(down(b)) 178.13/95.74 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.74 down(g(a)) -> g_flat(down(a)) 178.13/95.74 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.74 down(g(b)) -> g_flat(down(b)) 178.13/95.74 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.74 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.74 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.74 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.74 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.74 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.74 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) 178.13/95.74 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.74 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 down(a) 178.13/95.74 down(g(f(x0))) 178.13/95.74 down(f(f(f(f(f(x0)))))) 178.13/95.74 top(up(x0)) 178.13/95.74 down(f(a)) 178.13/95.74 down(f(g(x0))) 178.13/95.74 down(f(b)) 178.13/95.74 down(f(fresh_constant)) 178.13/95.74 down(g(a)) 178.13/95.74 down(g(g(x0))) 178.13/95.74 down(g(b)) 178.13/95.74 down(g(fresh_constant)) 178.13/95.74 down(f(f(a))) 178.13/95.74 down(f(f(g(x0)))) 178.13/95.74 down(f(f(b))) 178.13/95.74 down(f(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) 178.13/95.74 down(f(f(f(g(x0))))) 178.13/95.74 down(f(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(x0)))))) 178.13/95.74 down(f(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x0)) 178.13/95.74 g_flat(up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (62) DependencyGraphProof (EQUIVALENT) 178.13/95.74 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 40 less nodes. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (63) 178.13/95.74 Complex Obligation (AND) 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (64) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 DOWN(g(g(y7))) -> DOWN(g(y7)) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 down(a) -> up(f(a)) 178.13/95.74 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.74 down(f(f(f(f(f(x)))))) -> up(b) 178.13/95.74 top(up(x)) -> top(down(x)) 178.13/95.74 down(f(a)) -> f_flat(down(a)) 178.13/95.74 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.74 down(f(b)) -> f_flat(down(b)) 178.13/95.74 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.74 down(g(a)) -> g_flat(down(a)) 178.13/95.74 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.74 down(g(b)) -> g_flat(down(b)) 178.13/95.74 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.74 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.74 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.74 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.74 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.74 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.74 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) 178.13/95.74 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.74 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 down(a) 178.13/95.74 down(g(f(x0))) 178.13/95.74 down(f(f(f(f(f(x0)))))) 178.13/95.74 top(up(x0)) 178.13/95.74 down(f(a)) 178.13/95.74 down(f(g(x0))) 178.13/95.74 down(f(b)) 178.13/95.74 down(f(fresh_constant)) 178.13/95.74 down(g(a)) 178.13/95.74 down(g(g(x0))) 178.13/95.74 down(g(b)) 178.13/95.74 down(g(fresh_constant)) 178.13/95.74 down(f(f(a))) 178.13/95.74 down(f(f(g(x0)))) 178.13/95.74 down(f(f(b))) 178.13/95.74 down(f(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) 178.13/95.74 down(f(f(f(g(x0))))) 178.13/95.74 down(f(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(x0)))))) 178.13/95.74 down(f(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x0)) 178.13/95.74 g_flat(up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (65) UsableRulesProof (EQUIVALENT) 178.13/95.74 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. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (66) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 DOWN(g(g(y7))) -> DOWN(g(y7)) 178.13/95.74 178.13/95.74 R is empty. 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 down(a) 178.13/95.74 down(g(f(x0))) 178.13/95.74 down(f(f(f(f(f(x0)))))) 178.13/95.74 top(up(x0)) 178.13/95.74 down(f(a)) 178.13/95.74 down(f(g(x0))) 178.13/95.74 down(f(b)) 178.13/95.74 down(f(fresh_constant)) 178.13/95.74 down(g(a)) 178.13/95.74 down(g(g(x0))) 178.13/95.74 down(g(b)) 178.13/95.74 down(g(fresh_constant)) 178.13/95.74 down(f(f(a))) 178.13/95.74 down(f(f(g(x0)))) 178.13/95.74 down(f(f(b))) 178.13/95.74 down(f(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) 178.13/95.74 down(f(f(f(g(x0))))) 178.13/95.74 down(f(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(x0)))))) 178.13/95.74 down(f(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x0)) 178.13/95.74 g_flat(up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (67) QReductionProof (EQUIVALENT) 178.13/95.74 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 178.13/95.74 178.13/95.74 down(a) 178.13/95.74 down(g(f(x0))) 178.13/95.74 down(f(f(f(f(f(x0)))))) 178.13/95.74 top(up(x0)) 178.13/95.74 down(f(a)) 178.13/95.74 down(f(g(x0))) 178.13/95.74 down(f(b)) 178.13/95.74 down(f(fresh_constant)) 178.13/95.74 down(g(a)) 178.13/95.74 down(g(g(x0))) 178.13/95.74 down(g(b)) 178.13/95.74 down(g(fresh_constant)) 178.13/95.74 down(f(f(a))) 178.13/95.74 down(f(f(g(x0)))) 178.13/95.74 down(f(f(b))) 178.13/95.74 down(f(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) 178.13/95.74 down(f(f(f(g(x0))))) 178.13/95.74 down(f(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(x0)))))) 178.13/95.74 down(f(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x0)) 178.13/95.74 g_flat(up(x0)) 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (68) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 DOWN(g(g(y7))) -> DOWN(g(y7)) 178.13/95.74 178.13/95.74 R is empty. 178.13/95.74 Q is empty. 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (69) QDPSizeChangeProof (EQUIVALENT) 178.13/95.74 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 178.13/95.74 178.13/95.74 From the DPs we obtained the following set of size-change graphs: 178.13/95.74 *DOWN(g(g(y7))) -> DOWN(g(y7)) 178.13/95.74 The graph contains the following edges 1 > 1 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (70) 178.13/95.74 YES 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (71) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 TOP(up(x)) -> TOP(down(x)) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 down(a) -> up(f(a)) 178.13/95.74 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.74 down(f(f(f(f(f(x)))))) -> up(b) 178.13/95.74 top(up(x)) -> top(down(x)) 178.13/95.74 down(f(a)) -> f_flat(down(a)) 178.13/95.74 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.74 down(f(b)) -> f_flat(down(b)) 178.13/95.74 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.74 down(g(a)) -> g_flat(down(a)) 178.13/95.74 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.74 down(g(b)) -> g_flat(down(b)) 178.13/95.74 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.74 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.74 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.74 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.74 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.74 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.74 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) 178.13/95.74 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.74 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 down(a) 178.13/95.74 down(g(f(x0))) 178.13/95.74 down(f(f(f(f(f(x0)))))) 178.13/95.74 top(up(x0)) 178.13/95.74 down(f(a)) 178.13/95.74 down(f(g(x0))) 178.13/95.74 down(f(b)) 178.13/95.74 down(f(fresh_constant)) 178.13/95.74 down(g(a)) 178.13/95.74 down(g(g(x0))) 178.13/95.74 down(g(b)) 178.13/95.74 down(g(fresh_constant)) 178.13/95.74 down(f(f(a))) 178.13/95.74 down(f(f(g(x0)))) 178.13/95.74 down(f(f(b))) 178.13/95.74 down(f(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) 178.13/95.74 down(f(f(f(g(x0))))) 178.13/95.74 down(f(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(x0)))))) 178.13/95.74 down(f(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x0)) 178.13/95.74 g_flat(up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (72) UsableRulesProof (EQUIVALENT) 178.13/95.74 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. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (73) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 TOP(up(x)) -> TOP(down(x)) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 down(a) -> up(f(a)) 178.13/95.74 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.74 down(f(f(f(f(f(x)))))) -> up(b) 178.13/95.74 down(f(a)) -> f_flat(down(a)) 178.13/95.74 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.74 down(f(b)) -> f_flat(down(b)) 178.13/95.74 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.74 down(g(a)) -> g_flat(down(a)) 178.13/95.74 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.74 down(g(b)) -> g_flat(down(b)) 178.13/95.74 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.74 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.74 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.74 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.74 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.74 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.74 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) 178.13/95.74 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.74 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 down(a) 178.13/95.74 down(g(f(x0))) 178.13/95.74 down(f(f(f(f(f(x0)))))) 178.13/95.74 top(up(x0)) 178.13/95.74 down(f(a)) 178.13/95.74 down(f(g(x0))) 178.13/95.74 down(f(b)) 178.13/95.74 down(f(fresh_constant)) 178.13/95.74 down(g(a)) 178.13/95.74 down(g(g(x0))) 178.13/95.74 down(g(b)) 178.13/95.74 down(g(fresh_constant)) 178.13/95.74 down(f(f(a))) 178.13/95.74 down(f(f(g(x0)))) 178.13/95.74 down(f(f(b))) 178.13/95.74 down(f(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) 178.13/95.74 down(f(f(f(g(x0))))) 178.13/95.74 down(f(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(x0)))))) 178.13/95.74 down(f(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x0)) 178.13/95.74 g_flat(up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (74) QReductionProof (EQUIVALENT) 178.13/95.74 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 178.13/95.74 178.13/95.74 top(up(x0)) 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (75) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 TOP(up(x)) -> TOP(down(x)) 178.13/95.74 178.13/95.74 The TRS R consists of the following rules: 178.13/95.74 178.13/95.74 down(a) -> up(f(a)) 178.13/95.74 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.74 down(f(f(f(f(f(x)))))) -> up(b) 178.13/95.74 down(f(a)) -> f_flat(down(a)) 178.13/95.74 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.74 down(f(b)) -> f_flat(down(b)) 178.13/95.74 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.74 down(g(a)) -> g_flat(down(a)) 178.13/95.74 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.74 down(g(b)) -> g_flat(down(b)) 178.13/95.74 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.74 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.74 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.74 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.74 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.74 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.74 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) 178.13/95.74 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.74 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.74 178.13/95.74 The set Q consists of the following terms: 178.13/95.74 178.13/95.74 down(a) 178.13/95.74 down(g(f(x0))) 178.13/95.74 down(f(f(f(f(f(x0)))))) 178.13/95.74 down(f(a)) 178.13/95.74 down(f(g(x0))) 178.13/95.74 down(f(b)) 178.13/95.74 down(f(fresh_constant)) 178.13/95.74 down(g(a)) 178.13/95.74 down(g(g(x0))) 178.13/95.74 down(g(b)) 178.13/95.74 down(g(fresh_constant)) 178.13/95.74 down(f(f(a))) 178.13/95.74 down(f(f(g(x0)))) 178.13/95.74 down(f(f(b))) 178.13/95.74 down(f(f(fresh_constant))) 178.13/95.74 down(f(f(f(a)))) 178.13/95.74 down(f(f(f(g(x0))))) 178.13/95.74 down(f(f(f(b)))) 178.13/95.74 down(f(f(f(fresh_constant)))) 178.13/95.74 down(f(f(f(f(a))))) 178.13/95.74 down(f(f(f(f(g(x0)))))) 178.13/95.74 down(f(f(f(f(b))))) 178.13/95.74 down(f(f(f(f(fresh_constant))))) 178.13/95.74 f_flat(up(x0)) 178.13/95.74 g_flat(up(x0)) 178.13/95.74 178.13/95.74 We have to consider all minimal (P,Q,R)-chains. 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (76) TransformationProof (EQUIVALENT) 178.13/95.74 By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]: 178.13/95.74 178.13/95.74 (TOP(up(a)) -> TOP(up(f(a))),TOP(up(a)) -> TOP(up(f(a)))) 178.13/95.74 (TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))),TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0)))))) 178.13/95.74 (TOP(up(f(f(f(f(f(x0))))))) -> TOP(up(b)),TOP(up(f(f(f(f(f(x0))))))) -> TOP(up(b))) 178.13/95.74 (TOP(up(f(a))) -> TOP(f_flat(down(a))),TOP(up(f(a))) -> TOP(f_flat(down(a)))) 178.13/95.74 (TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))),TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))) 178.13/95.74 (TOP(up(f(b))) -> TOP(f_flat(down(b))),TOP(up(f(b))) -> TOP(f_flat(down(b)))) 178.13/95.74 (TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))),TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant)))) 178.13/95.74 (TOP(up(g(a))) -> TOP(g_flat(down(a))),TOP(up(g(a))) -> TOP(g_flat(down(a)))) 178.13/95.74 (TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))),TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))) 178.13/95.74 (TOP(up(g(b))) -> TOP(g_flat(down(b))),TOP(up(g(b))) -> TOP(g_flat(down(b)))) 178.13/95.74 (TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))),TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant)))) 178.13/95.74 (TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))),TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))) 178.13/95.74 (TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))),TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))) 178.13/95.74 (TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))),TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))) 178.13/95.74 (TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))),TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))) 178.13/95.74 (TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))) 178.13/95.74 (TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))),TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))) 178.13/95.74 (TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))) 178.13/95.74 (TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))),TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))) 178.13/95.74 (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))) 178.13/95.74 (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))) 178.13/95.74 (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))) 178.13/95.74 (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))) 178.13/95.74 178.13/95.74 178.13/95.74 ---------------------------------------- 178.13/95.74 178.13/95.74 (77) 178.13/95.74 Obligation: 178.13/95.74 Q DP problem: 178.13/95.74 The TRS P consists of the following rules: 178.13/95.74 178.13/95.74 TOP(up(a)) -> TOP(up(f(a))) 178.13/95.74 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.74 TOP(up(f(f(f(f(f(x0))))))) -> TOP(up(b)) 178.13/95.75 TOP(up(f(a))) -> TOP(f_flat(down(a))) 178.13/95.75 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.75 TOP(up(f(b))) -> TOP(f_flat(down(b))) 178.13/95.75 TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))) 178.13/95.75 TOP(up(g(a))) -> TOP(g_flat(down(a))) 178.13/95.75 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.75 TOP(up(g(b))) -> TOP(g_flat(down(b))) 178.13/95.75 TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))) 178.13/95.75 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 178.13/95.75 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 178.13/95.75 TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) 178.13/95.75 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 178.13/95.75 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.75 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.75 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.75 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.75 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.75 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.75 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.75 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.75 178.13/95.75 The TRS R consists of the following rules: 178.13/95.75 178.13/95.75 down(a) -> up(f(a)) 178.13/95.75 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.75 down(f(f(f(f(f(x)))))) -> up(b) 178.13/95.75 down(f(a)) -> f_flat(down(a)) 178.13/95.75 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.75 down(f(b)) -> f_flat(down(b)) 178.13/95.75 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.75 down(g(a)) -> g_flat(down(a)) 178.13/95.75 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.75 down(g(b)) -> g_flat(down(b)) 178.13/95.75 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.75 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.75 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.75 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.75 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.75 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.75 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.75 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.75 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.75 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 178.13/95.75 down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) 178.13/95.75 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 178.13/95.75 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 178.13/95.75 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.75 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.75 178.13/95.75 The set Q consists of the following terms: 178.13/95.75 178.13/95.75 down(a) 178.13/95.75 down(g(f(x0))) 178.13/95.75 down(f(f(f(f(f(x0)))))) 178.13/95.75 down(f(a)) 178.13/95.75 down(f(g(x0))) 178.13/95.75 down(f(b)) 178.13/95.75 down(f(fresh_constant)) 178.13/95.75 down(g(a)) 178.13/95.75 down(g(g(x0))) 178.13/95.75 down(g(b)) 178.13/95.75 down(g(fresh_constant)) 178.13/95.75 down(f(f(a))) 178.13/95.75 down(f(f(g(x0)))) 178.13/95.75 down(f(f(b))) 178.13/95.75 down(f(f(fresh_constant))) 178.13/95.75 down(f(f(f(a)))) 178.13/95.75 down(f(f(f(g(x0))))) 178.13/95.75 down(f(f(f(b)))) 178.13/95.75 down(f(f(f(fresh_constant)))) 178.13/95.75 down(f(f(f(f(a))))) 178.13/95.75 down(f(f(f(f(g(x0)))))) 178.13/95.75 down(f(f(f(f(b))))) 178.13/95.75 down(f(f(f(f(fresh_constant))))) 178.13/95.75 f_flat(up(x0)) 178.13/95.75 g_flat(up(x0)) 178.13/95.75 178.13/95.75 We have to consider all minimal (P,Q,R)-chains. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (78) DependencyGraphProof (EQUIVALENT) 178.13/95.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (79) 178.13/95.75 Obligation: 178.13/95.75 Q DP problem: 178.13/95.75 The TRS P consists of the following rules: 178.13/95.75 178.13/95.75 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.75 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.75 TOP(up(f(a))) -> TOP(f_flat(down(a))) 178.13/95.75 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 178.13/95.75 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 178.13/95.75 TOP(up(g(a))) -> TOP(g_flat(down(a))) 178.13/95.75 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.75 TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) 178.13/95.75 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 178.13/95.75 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.75 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.75 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.75 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.75 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.75 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.75 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.75 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.75 178.13/95.75 The TRS R consists of the following rules: 178.13/95.75 178.13/95.75 down(a) -> up(f(a)) 178.13/95.75 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.75 down(f(f(f(f(f(x)))))) -> up(b) 178.13/95.75 down(f(a)) -> f_flat(down(a)) 178.13/95.75 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.75 down(f(b)) -> f_flat(down(b)) 178.13/95.75 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.75 down(g(a)) -> g_flat(down(a)) 178.13/95.75 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.75 down(g(b)) -> g_flat(down(b)) 178.13/95.75 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.75 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.75 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.75 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.75 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.75 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.75 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.75 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.75 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.75 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 178.13/95.75 down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) 178.13/95.75 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 178.13/95.75 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 178.13/95.75 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.75 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.75 178.13/95.75 The set Q consists of the following terms: 178.13/95.75 178.13/95.75 down(a) 178.13/95.75 down(g(f(x0))) 178.13/95.75 down(f(f(f(f(f(x0)))))) 178.13/95.75 down(f(a)) 178.13/95.75 down(f(g(x0))) 178.13/95.75 down(f(b)) 178.13/95.75 down(f(fresh_constant)) 178.13/95.75 down(g(a)) 178.13/95.75 down(g(g(x0))) 178.13/95.75 down(g(b)) 178.13/95.75 down(g(fresh_constant)) 178.13/95.75 down(f(f(a))) 178.13/95.75 down(f(f(g(x0)))) 178.13/95.75 down(f(f(b))) 178.13/95.75 down(f(f(fresh_constant))) 178.13/95.75 down(f(f(f(a)))) 178.13/95.75 down(f(f(f(g(x0))))) 178.13/95.75 down(f(f(f(b)))) 178.13/95.75 down(f(f(f(fresh_constant)))) 178.13/95.75 down(f(f(f(f(a))))) 178.13/95.75 down(f(f(f(f(g(x0)))))) 178.13/95.75 down(f(f(f(f(b))))) 178.13/95.75 down(f(f(f(f(fresh_constant))))) 178.13/95.75 f_flat(up(x0)) 178.13/95.75 g_flat(up(x0)) 178.13/95.75 178.13/95.75 We have to consider all minimal (P,Q,R)-chains. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (80) UsableRulesProof (EQUIVALENT) 178.13/95.75 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. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (81) 178.13/95.75 Obligation: 178.13/95.75 Q DP problem: 178.13/95.75 The TRS P consists of the following rules: 178.13/95.75 178.13/95.75 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.75 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.75 TOP(up(f(a))) -> TOP(f_flat(down(a))) 178.13/95.75 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 178.13/95.75 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 178.13/95.75 TOP(up(g(a))) -> TOP(g_flat(down(a))) 178.13/95.75 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.75 TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) 178.13/95.75 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 178.13/95.75 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.75 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.75 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.75 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.75 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.75 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.75 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.75 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.75 178.13/95.75 The TRS R consists of the following rules: 178.13/95.75 178.13/95.75 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.75 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.75 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.75 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.75 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.75 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.75 down(f(b)) -> f_flat(down(b)) 178.13/95.75 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.75 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.75 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.75 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.75 down(g(a)) -> g_flat(down(a)) 178.13/95.75 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.75 down(g(b)) -> g_flat(down(b)) 178.13/95.75 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.75 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.75 down(a) -> up(f(a)) 178.13/95.75 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.75 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.75 down(f(a)) -> f_flat(down(a)) 178.13/95.75 178.13/95.75 The set Q consists of the following terms: 178.13/95.75 178.13/95.75 down(a) 178.13/95.75 down(g(f(x0))) 178.13/95.75 down(f(f(f(f(f(x0)))))) 178.13/95.75 down(f(a)) 178.13/95.75 down(f(g(x0))) 178.13/95.75 down(f(b)) 178.13/95.75 down(f(fresh_constant)) 178.13/95.75 down(g(a)) 178.13/95.75 down(g(g(x0))) 178.13/95.75 down(g(b)) 178.13/95.75 down(g(fresh_constant)) 178.13/95.75 down(f(f(a))) 178.13/95.75 down(f(f(g(x0)))) 178.13/95.75 down(f(f(b))) 178.13/95.75 down(f(f(fresh_constant))) 178.13/95.75 down(f(f(f(a)))) 178.13/95.75 down(f(f(f(g(x0))))) 178.13/95.75 down(f(f(f(b)))) 178.13/95.75 down(f(f(f(fresh_constant)))) 178.13/95.75 down(f(f(f(f(a))))) 178.13/95.75 down(f(f(f(f(g(x0)))))) 178.13/95.75 down(f(f(f(f(b))))) 178.13/95.75 down(f(f(f(f(fresh_constant))))) 178.13/95.75 f_flat(up(x0)) 178.13/95.75 g_flat(up(x0)) 178.13/95.75 178.13/95.75 We have to consider all minimal (P,Q,R)-chains. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (82) TransformationProof (EQUIVALENT) 178.13/95.75 By rewriting [LPAR04] the rule TOP(up(f(a))) -> TOP(f_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.75 178.13/95.75 (TOP(up(f(a))) -> TOP(f_flat(up(f(a)))),TOP(up(f(a))) -> TOP(f_flat(up(f(a))))) 178.13/95.75 178.13/95.75 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (83) 178.13/95.75 Obligation: 178.13/95.75 Q DP problem: 178.13/95.75 The TRS P consists of the following rules: 178.13/95.75 178.13/95.75 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.75 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.75 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 178.13/95.75 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 178.13/95.75 TOP(up(g(a))) -> TOP(g_flat(down(a))) 178.13/95.75 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.75 TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) 178.13/95.75 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 178.13/95.75 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.75 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.75 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.75 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.75 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.75 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.75 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.75 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.75 TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) 178.13/95.75 178.13/95.75 The TRS R consists of the following rules: 178.13/95.75 178.13/95.75 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.75 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.75 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.75 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.75 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.75 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.75 down(f(b)) -> f_flat(down(b)) 178.13/95.75 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.75 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.75 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.75 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.75 down(g(a)) -> g_flat(down(a)) 178.13/95.75 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.75 down(g(b)) -> g_flat(down(b)) 178.13/95.75 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.75 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.75 down(a) -> up(f(a)) 178.13/95.75 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.75 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.75 down(f(a)) -> f_flat(down(a)) 178.13/95.75 178.13/95.75 The set Q consists of the following terms: 178.13/95.75 178.13/95.75 down(a) 178.13/95.75 down(g(f(x0))) 178.13/95.75 down(f(f(f(f(f(x0)))))) 178.13/95.75 down(f(a)) 178.13/95.75 down(f(g(x0))) 178.13/95.75 down(f(b)) 178.13/95.75 down(f(fresh_constant)) 178.13/95.75 down(g(a)) 178.13/95.75 down(g(g(x0))) 178.13/95.75 down(g(b)) 178.13/95.75 down(g(fresh_constant)) 178.13/95.75 down(f(f(a))) 178.13/95.75 down(f(f(g(x0)))) 178.13/95.75 down(f(f(b))) 178.13/95.75 down(f(f(fresh_constant))) 178.13/95.75 down(f(f(f(a)))) 178.13/95.75 down(f(f(f(g(x0))))) 178.13/95.75 down(f(f(f(b)))) 178.13/95.75 down(f(f(f(fresh_constant)))) 178.13/95.75 down(f(f(f(f(a))))) 178.13/95.75 down(f(f(f(f(g(x0)))))) 178.13/95.75 down(f(f(f(f(b))))) 178.13/95.75 down(f(f(f(f(fresh_constant))))) 178.13/95.75 f_flat(up(x0)) 178.13/95.75 g_flat(up(x0)) 178.13/95.75 178.13/95.75 We have to consider all minimal (P,Q,R)-chains. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (84) TransformationProof (EQUIVALENT) 178.13/95.75 By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.75 178.13/95.75 (TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))),TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a))))) 178.13/95.75 178.13/95.75 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (85) 178.13/95.75 Obligation: 178.13/95.75 Q DP problem: 178.13/95.75 The TRS P consists of the following rules: 178.13/95.75 178.13/95.75 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.75 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.75 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 178.13/95.75 TOP(up(g(a))) -> TOP(g_flat(down(a))) 178.13/95.75 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.75 TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) 178.13/95.75 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 178.13/95.75 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.75 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.75 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.75 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.75 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.75 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.75 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.75 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.75 TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) 178.13/95.75 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) 178.13/95.75 178.13/95.75 The TRS R consists of the following rules: 178.13/95.75 178.13/95.75 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.75 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.75 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.75 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.75 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.75 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.75 down(f(b)) -> f_flat(down(b)) 178.13/95.75 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.75 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.75 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.75 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.75 down(g(a)) -> g_flat(down(a)) 178.13/95.75 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.75 down(g(b)) -> g_flat(down(b)) 178.13/95.75 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.75 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.75 down(a) -> up(f(a)) 178.13/95.75 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.75 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.75 down(f(a)) -> f_flat(down(a)) 178.13/95.75 178.13/95.75 The set Q consists of the following terms: 178.13/95.75 178.13/95.75 down(a) 178.13/95.75 down(g(f(x0))) 178.13/95.75 down(f(f(f(f(f(x0)))))) 178.13/95.75 down(f(a)) 178.13/95.75 down(f(g(x0))) 178.13/95.75 down(f(b)) 178.13/95.75 down(f(fresh_constant)) 178.13/95.75 down(g(a)) 178.13/95.75 down(g(g(x0))) 178.13/95.75 down(g(b)) 178.13/95.75 down(g(fresh_constant)) 178.13/95.75 down(f(f(a))) 178.13/95.75 down(f(f(g(x0)))) 178.13/95.75 down(f(f(b))) 178.13/95.75 down(f(f(fresh_constant))) 178.13/95.75 down(f(f(f(a)))) 178.13/95.75 down(f(f(f(g(x0))))) 178.13/95.75 down(f(f(f(b)))) 178.13/95.75 down(f(f(f(fresh_constant)))) 178.13/95.75 down(f(f(f(f(a))))) 178.13/95.75 down(f(f(f(f(g(x0)))))) 178.13/95.75 down(f(f(f(f(b))))) 178.13/95.75 down(f(f(f(f(fresh_constant))))) 178.13/95.75 f_flat(up(x0)) 178.13/95.75 g_flat(up(x0)) 178.13/95.75 178.13/95.75 We have to consider all minimal (P,Q,R)-chains. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (86) TransformationProof (EQUIVALENT) 178.13/95.75 By rewriting [LPAR04] the rule TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.75 178.13/95.75 (TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))),TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))) 178.13/95.75 178.13/95.75 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (87) 178.13/95.75 Obligation: 178.13/95.75 Q DP problem: 178.13/95.75 The TRS P consists of the following rules: 178.13/95.75 178.13/95.75 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.75 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.75 TOP(up(g(a))) -> TOP(g_flat(down(a))) 178.13/95.75 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.75 TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) 178.13/95.75 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 178.13/95.75 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.75 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.75 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.75 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.75 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.75 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.75 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.75 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.75 TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) 178.13/95.75 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) 178.13/95.75 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.75 178.13/95.75 The TRS R consists of the following rules: 178.13/95.75 178.13/95.75 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.75 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.75 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.75 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.75 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.75 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.75 down(f(b)) -> f_flat(down(b)) 178.13/95.75 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.75 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.75 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.75 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.75 down(g(a)) -> g_flat(down(a)) 178.13/95.75 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.75 down(g(b)) -> g_flat(down(b)) 178.13/95.75 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.75 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.75 down(a) -> up(f(a)) 178.13/95.75 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.75 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.75 down(f(a)) -> f_flat(down(a)) 178.13/95.75 178.13/95.75 The set Q consists of the following terms: 178.13/95.75 178.13/95.75 down(a) 178.13/95.75 down(g(f(x0))) 178.13/95.75 down(f(f(f(f(f(x0)))))) 178.13/95.75 down(f(a)) 178.13/95.75 down(f(g(x0))) 178.13/95.75 down(f(b)) 178.13/95.75 down(f(fresh_constant)) 178.13/95.75 down(g(a)) 178.13/95.75 down(g(g(x0))) 178.13/95.75 down(g(b)) 178.13/95.75 down(g(fresh_constant)) 178.13/95.75 down(f(f(a))) 178.13/95.75 down(f(f(g(x0)))) 178.13/95.75 down(f(f(b))) 178.13/95.75 down(f(f(fresh_constant))) 178.13/95.75 down(f(f(f(a)))) 178.13/95.75 down(f(f(f(g(x0))))) 178.13/95.75 down(f(f(f(b)))) 178.13/95.75 down(f(f(f(fresh_constant)))) 178.13/95.75 down(f(f(f(f(a))))) 178.13/95.75 down(f(f(f(f(g(x0)))))) 178.13/95.75 down(f(f(f(f(b))))) 178.13/95.75 down(f(f(f(f(fresh_constant))))) 178.13/95.75 f_flat(up(x0)) 178.13/95.75 g_flat(up(x0)) 178.13/95.75 178.13/95.75 We have to consider all minimal (P,Q,R)-chains. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (88) TransformationProof (EQUIVALENT) 178.13/95.75 By rewriting [LPAR04] the rule TOP(up(g(a))) -> TOP(g_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.75 178.13/95.75 (TOP(up(g(a))) -> TOP(g_flat(up(f(a)))),TOP(up(g(a))) -> TOP(g_flat(up(f(a))))) 178.13/95.75 178.13/95.75 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (89) 178.13/95.75 Obligation: 178.13/95.75 Q DP problem: 178.13/95.75 The TRS P consists of the following rules: 178.13/95.75 178.13/95.75 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.75 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.75 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.75 TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) 178.13/95.75 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 178.13/95.75 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.75 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.75 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.75 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.75 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.75 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.75 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.75 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.75 TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) 178.13/95.75 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) 178.13/95.75 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.75 TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) 178.13/95.75 178.13/95.75 The TRS R consists of the following rules: 178.13/95.75 178.13/95.75 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.75 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.75 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.75 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.75 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.75 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.75 down(f(b)) -> f_flat(down(b)) 178.13/95.75 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.75 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.75 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.75 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.75 down(g(a)) -> g_flat(down(a)) 178.13/95.75 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.75 down(g(b)) -> g_flat(down(b)) 178.13/95.75 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.75 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.75 down(a) -> up(f(a)) 178.13/95.75 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.75 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.75 down(f(a)) -> f_flat(down(a)) 178.13/95.75 178.13/95.75 The set Q consists of the following terms: 178.13/95.75 178.13/95.75 down(a) 178.13/95.75 down(g(f(x0))) 178.13/95.75 down(f(f(f(f(f(x0)))))) 178.13/95.75 down(f(a)) 178.13/95.75 down(f(g(x0))) 178.13/95.75 down(f(b)) 178.13/95.75 down(f(fresh_constant)) 178.13/95.75 down(g(a)) 178.13/95.75 down(g(g(x0))) 178.13/95.75 down(g(b)) 178.13/95.75 down(g(fresh_constant)) 178.13/95.75 down(f(f(a))) 178.13/95.75 down(f(f(g(x0)))) 178.13/95.75 down(f(f(b))) 178.13/95.75 down(f(f(fresh_constant))) 178.13/95.75 down(f(f(f(a)))) 178.13/95.75 down(f(f(f(g(x0))))) 178.13/95.75 down(f(f(f(b)))) 178.13/95.75 down(f(f(f(fresh_constant)))) 178.13/95.75 down(f(f(f(f(a))))) 178.13/95.75 down(f(f(f(f(g(x0)))))) 178.13/95.75 down(f(f(f(f(b))))) 178.13/95.75 down(f(f(f(f(fresh_constant))))) 178.13/95.75 f_flat(up(x0)) 178.13/95.75 g_flat(up(x0)) 178.13/95.75 178.13/95.75 We have to consider all minimal (P,Q,R)-chains. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (90) TransformationProof (EQUIVALENT) 178.13/95.75 By rewriting [LPAR04] the rule TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.75 178.13/95.75 (TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))),TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b))))) 178.13/95.75 178.13/95.75 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (91) 178.13/95.75 Obligation: 178.13/95.75 Q DP problem: 178.13/95.75 The TRS P consists of the following rules: 178.13/95.75 178.13/95.75 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.75 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.75 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.75 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 178.13/95.75 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.75 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.75 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.75 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.75 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.75 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.75 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.75 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.75 TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) 178.13/95.75 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) 178.13/95.75 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.75 TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) 178.13/95.75 TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) 178.13/95.75 178.13/95.75 The TRS R consists of the following rules: 178.13/95.75 178.13/95.75 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.75 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.75 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.75 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.75 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.75 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.75 down(f(b)) -> f_flat(down(b)) 178.13/95.75 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.75 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.75 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.75 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.75 down(g(a)) -> g_flat(down(a)) 178.13/95.75 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.75 down(g(b)) -> g_flat(down(b)) 178.13/95.75 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.75 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.75 down(a) -> up(f(a)) 178.13/95.75 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.75 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.75 down(f(a)) -> f_flat(down(a)) 178.13/95.75 178.13/95.75 The set Q consists of the following terms: 178.13/95.75 178.13/95.75 down(a) 178.13/95.75 down(g(f(x0))) 178.13/95.75 down(f(f(f(f(f(x0)))))) 178.13/95.75 down(f(a)) 178.13/95.75 down(f(g(x0))) 178.13/95.75 down(f(b)) 178.13/95.75 down(f(fresh_constant)) 178.13/95.75 down(g(a)) 178.13/95.75 down(g(g(x0))) 178.13/95.75 down(g(b)) 178.13/95.75 down(g(fresh_constant)) 178.13/95.75 down(f(f(a))) 178.13/95.75 down(f(f(g(x0)))) 178.13/95.75 down(f(f(b))) 178.13/95.75 down(f(f(fresh_constant))) 178.13/95.75 down(f(f(f(a)))) 178.13/95.75 down(f(f(f(g(x0))))) 178.13/95.75 down(f(f(f(b)))) 178.13/95.75 down(f(f(f(fresh_constant)))) 178.13/95.75 down(f(f(f(f(a))))) 178.13/95.75 down(f(f(f(f(g(x0)))))) 178.13/95.75 down(f(f(f(f(b))))) 178.13/95.75 down(f(f(f(f(fresh_constant))))) 178.13/95.75 f_flat(up(x0)) 178.13/95.75 g_flat(up(x0)) 178.13/95.75 178.13/95.75 We have to consider all minimal (P,Q,R)-chains. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (92) DependencyGraphProof (EQUIVALENT) 178.13/95.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (93) 178.13/95.75 Obligation: 178.13/95.75 Q DP problem: 178.13/95.75 The TRS P consists of the following rules: 178.13/95.75 178.13/95.75 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.75 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.75 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.75 TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) 178.13/95.75 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 178.13/95.75 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.75 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.75 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.75 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.75 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.75 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.75 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.75 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.75 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) 178.13/95.75 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.75 TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) 178.13/95.75 178.13/95.75 The TRS R consists of the following rules: 178.13/95.75 178.13/95.75 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.75 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.75 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.75 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.75 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.75 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.75 down(f(b)) -> f_flat(down(b)) 178.13/95.75 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.75 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.75 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.75 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.75 down(g(a)) -> g_flat(down(a)) 178.13/95.75 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.75 down(g(b)) -> g_flat(down(b)) 178.13/95.75 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.75 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.75 down(a) -> up(f(a)) 178.13/95.75 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.75 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.75 down(f(a)) -> f_flat(down(a)) 178.13/95.75 178.13/95.75 The set Q consists of the following terms: 178.13/95.75 178.13/95.75 down(a) 178.13/95.75 down(g(f(x0))) 178.13/95.75 down(f(f(f(f(f(x0)))))) 178.13/95.75 down(f(a)) 178.13/95.75 down(f(g(x0))) 178.13/95.75 down(f(b)) 178.13/95.75 down(f(fresh_constant)) 178.13/95.75 down(g(a)) 178.13/95.75 down(g(g(x0))) 178.13/95.75 down(g(b)) 178.13/95.75 down(g(fresh_constant)) 178.13/95.75 down(f(f(a))) 178.13/95.75 down(f(f(g(x0)))) 178.13/95.75 down(f(f(b))) 178.13/95.75 down(f(f(fresh_constant))) 178.13/95.75 down(f(f(f(a)))) 178.13/95.75 down(f(f(f(g(x0))))) 178.13/95.75 down(f(f(f(b)))) 178.13/95.75 down(f(f(f(fresh_constant)))) 178.13/95.75 down(f(f(f(f(a))))) 178.13/95.75 down(f(f(f(f(g(x0)))))) 178.13/95.75 down(f(f(f(f(b))))) 178.13/95.75 down(f(f(f(f(fresh_constant))))) 178.13/95.75 f_flat(up(x0)) 178.13/95.75 g_flat(up(x0)) 178.13/95.75 178.13/95.75 We have to consider all minimal (P,Q,R)-chains. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (94) TransformationProof (EQUIVALENT) 178.13/95.75 By rewriting [LPAR04] the rule TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]: 178.13/95.75 178.13/95.75 (TOP(up(f(a))) -> TOP(up(f(f(a)))),TOP(up(f(a))) -> TOP(up(f(f(a))))) 178.13/95.75 178.13/95.75 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (95) 178.13/95.75 Obligation: 178.13/95.75 Q DP problem: 178.13/95.75 The TRS P consists of the following rules: 178.13/95.75 178.13/95.75 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.75 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.75 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.75 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 178.13/95.75 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.75 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.75 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.75 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.75 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.75 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.75 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.75 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.75 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) 178.13/95.75 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.75 TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) 178.13/95.75 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.75 178.13/95.75 The TRS R consists of the following rules: 178.13/95.75 178.13/95.75 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.75 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.75 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.75 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.75 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.75 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.75 down(f(b)) -> f_flat(down(b)) 178.13/95.75 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.75 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.75 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.75 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.75 down(g(a)) -> g_flat(down(a)) 178.13/95.75 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.75 down(g(b)) -> g_flat(down(b)) 178.13/95.75 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.75 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.75 down(a) -> up(f(a)) 178.13/95.75 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.75 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.75 down(f(a)) -> f_flat(down(a)) 178.13/95.75 178.13/95.75 The set Q consists of the following terms: 178.13/95.75 178.13/95.75 down(a) 178.13/95.75 down(g(f(x0))) 178.13/95.75 down(f(f(f(f(f(x0)))))) 178.13/95.75 down(f(a)) 178.13/95.75 down(f(g(x0))) 178.13/95.75 down(f(b)) 178.13/95.75 down(f(fresh_constant)) 178.13/95.75 down(g(a)) 178.13/95.75 down(g(g(x0))) 178.13/95.75 down(g(b)) 178.13/95.75 down(g(fresh_constant)) 178.13/95.75 down(f(f(a))) 178.13/95.75 down(f(f(g(x0)))) 178.13/95.75 down(f(f(b))) 178.13/95.75 down(f(f(fresh_constant))) 178.13/95.75 down(f(f(f(a)))) 178.13/95.75 down(f(f(f(g(x0))))) 178.13/95.75 down(f(f(f(b)))) 178.13/95.75 down(f(f(f(fresh_constant)))) 178.13/95.75 down(f(f(f(f(a))))) 178.13/95.75 down(f(f(f(f(g(x0)))))) 178.13/95.75 down(f(f(f(f(b))))) 178.13/95.75 down(f(f(f(f(fresh_constant))))) 178.13/95.75 f_flat(up(x0)) 178.13/95.75 g_flat(up(x0)) 178.13/95.75 178.13/95.75 We have to consider all minimal (P,Q,R)-chains. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (96) TransformationProof (EQUIVALENT) 178.13/95.75 By rewriting [LPAR04] the rule TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.75 178.13/95.75 (TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant)))),TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant))))) 178.13/95.75 178.13/95.75 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (97) 178.13/95.75 Obligation: 178.13/95.75 Q DP problem: 178.13/95.75 The TRS P consists of the following rules: 178.13/95.75 178.13/95.75 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.75 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.75 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.75 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.75 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.75 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.75 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.75 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.75 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.75 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.75 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.75 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) 178.13/95.75 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.75 TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) 178.13/95.75 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.75 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant)))) 178.13/95.75 178.13/95.75 The TRS R consists of the following rules: 178.13/95.75 178.13/95.75 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.75 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.75 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.75 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.75 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.75 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.75 down(f(b)) -> f_flat(down(b)) 178.13/95.75 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.75 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.75 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.75 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.75 down(g(a)) -> g_flat(down(a)) 178.13/95.75 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.75 down(g(b)) -> g_flat(down(b)) 178.13/95.75 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.75 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.75 down(a) -> up(f(a)) 178.13/95.75 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.75 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.75 down(f(a)) -> f_flat(down(a)) 178.13/95.75 178.13/95.75 The set Q consists of the following terms: 178.13/95.75 178.13/95.75 down(a) 178.13/95.75 down(g(f(x0))) 178.13/95.75 down(f(f(f(f(f(x0)))))) 178.13/95.75 down(f(a)) 178.13/95.75 down(f(g(x0))) 178.13/95.75 down(f(b)) 178.13/95.75 down(f(fresh_constant)) 178.13/95.75 down(g(a)) 178.13/95.75 down(g(g(x0))) 178.13/95.75 down(g(b)) 178.13/95.75 down(g(fresh_constant)) 178.13/95.75 down(f(f(a))) 178.13/95.75 down(f(f(g(x0)))) 178.13/95.75 down(f(f(b))) 178.13/95.75 down(f(f(fresh_constant))) 178.13/95.75 down(f(f(f(a)))) 178.13/95.75 down(f(f(f(g(x0))))) 178.13/95.75 down(f(f(f(b)))) 178.13/95.75 down(f(f(f(fresh_constant)))) 178.13/95.75 down(f(f(f(f(a))))) 178.13/95.75 down(f(f(f(f(g(x0)))))) 178.13/95.75 down(f(f(f(f(b))))) 178.13/95.75 down(f(f(f(f(fresh_constant))))) 178.13/95.75 f_flat(up(x0)) 178.13/95.75 g_flat(up(x0)) 178.13/95.75 178.13/95.75 We have to consider all minimal (P,Q,R)-chains. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (98) DependencyGraphProof (EQUIVALENT) 178.13/95.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 178.13/95.75 ---------------------------------------- 178.13/95.75 178.13/95.75 (99) 178.13/95.75 Obligation: 178.13/95.75 Q DP problem: 178.13/95.75 The TRS P consists of the following rules: 178.13/95.75 178.13/95.75 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.75 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.75 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.75 TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) 178.13/95.75 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.75 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) 178.13/95.75 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.75 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.75 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.75 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.75 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.75 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.75 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.75 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.75 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.75 178.13/95.75 The TRS R consists of the following rules: 178.13/95.75 178.13/95.75 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.75 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.75 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.75 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.75 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.75 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.75 down(f(b)) -> f_flat(down(b)) 178.13/95.75 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (100) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(g(a))) -> TOP(up(g(f(a)))),TOP(up(g(a))) -> TOP(up(g(f(a))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (101) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (102) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) at position [0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))),TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (103) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (104) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (105) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (106) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))),TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (107) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (108) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (109) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (110) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))),TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (111) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (112) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (113) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (114) UsableRulesProof (EQUIVALENT) 178.13/95.76 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. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (115) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (116) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (117) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (118) UsableRulesProof (EQUIVALENT) 178.13/95.76 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. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (119) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (120) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (121) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (122) UsableRulesProof (EQUIVALENT) 178.13/95.76 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. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (123) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (124) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (125) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (126) UsableRulesProof (EQUIVALENT) 178.13/95.76 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. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (127) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (128) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))),TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a)))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (129) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (130) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) at position [0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a)))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (131) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (132) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))),TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (133) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (134) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) at position [0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b)))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (135) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (136) DependencyGraphProof (EQUIVALENT) 178.13/95.76 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (137) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (138) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) at position [0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))),TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (139) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (140) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) at position [0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(f_flat(down(fresh_constant))))),TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(f_flat(down(fresh_constant)))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (141) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.76 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(f_flat(down(fresh_constant))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (142) DependencyGraphProof (EQUIVALENT) 178.13/95.76 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (143) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (144) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a))))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (145) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.76 178.13/95.76 down(a) 178.13/95.76 down(g(f(x0))) 178.13/95.76 down(f(f(f(f(f(x0)))))) 178.13/95.76 down(f(a)) 178.13/95.76 down(f(g(x0))) 178.13/95.76 down(f(b)) 178.13/95.76 down(f(fresh_constant)) 178.13/95.76 down(g(a)) 178.13/95.76 down(g(g(x0))) 178.13/95.76 down(g(b)) 178.13/95.76 down(g(fresh_constant)) 178.13/95.76 down(f(f(a))) 178.13/95.76 down(f(f(g(x0)))) 178.13/95.76 down(f(f(b))) 178.13/95.76 down(f(f(fresh_constant))) 178.13/95.76 down(f(f(f(a)))) 178.13/95.76 down(f(f(f(g(x0))))) 178.13/95.76 down(f(f(f(b)))) 178.13/95.76 down(f(f(f(fresh_constant)))) 178.13/95.76 down(f(f(f(f(a))))) 178.13/95.76 down(f(f(f(f(g(x0)))))) 178.13/95.76 down(f(f(f(f(b))))) 178.13/95.76 down(f(f(f(f(fresh_constant))))) 178.13/95.76 f_flat(up(x0)) 178.13/95.76 g_flat(up(x0)) 178.13/95.76 178.13/95.76 We have to consider all minimal (P,Q,R)-chains. 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (146) TransformationProof (EQUIVALENT) 178.13/95.76 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.76 178.13/95.76 (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a))))))) 178.13/95.76 178.13/95.76 178.13/95.76 ---------------------------------------- 178.13/95.76 178.13/95.76 (147) 178.13/95.76 Obligation: 178.13/95.76 Q DP problem: 178.13/95.76 The TRS P consists of the following rules: 178.13/95.76 178.13/95.76 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.76 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.76 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.76 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.76 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.76 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.76 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.76 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.76 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.76 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.76 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.76 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) 178.13/95.76 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) 178.13/95.76 178.13/95.76 The TRS R consists of the following rules: 178.13/95.76 178.13/95.76 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.76 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.76 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.76 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.76 down(f(b)) -> f_flat(down(b)) 178.13/95.76 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.76 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.76 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.76 down(g(a)) -> g_flat(down(a)) 178.13/95.76 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.76 down(g(b)) -> g_flat(down(b)) 178.13/95.76 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.76 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.76 down(a) -> up(f(a)) 178.13/95.76 down(f(f(a))) -> f_flat(down(f(a))) 178.13/95.76 down(f(a)) -> f_flat(down(a)) 178.13/95.76 178.13/95.76 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (148) UsableRulesProof (EQUIVALENT) 178.13/95.77 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. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (149) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(f(a)) -> f_flat(down(a)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.77 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (150) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0)))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (151) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(f(a)) -> f_flat(down(a)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 178.13/95.77 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (152) UsableRulesProof (EQUIVALENT) 178.13/95.77 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. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (153) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(a)) -> f_flat(down(a)) 178.13/95.77 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (154) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (155) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(a)) -> f_flat(down(a)) 178.13/95.77 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 down(f(f(b))) -> f_flat(down(f(b))) 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (156) UsableRulesProof (EQUIVALENT) 178.13/95.77 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. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (157) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(a)) -> f_flat(down(a)) 178.13/95.77 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (158) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (159) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(a)) -> f_flat(down(a)) 178.13/95.77 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (160) UsableRulesProof (EQUIVALENT) 178.13/95.77 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. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (161) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(a)) -> f_flat(down(a)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (162) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (163) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(a)) -> f_flat(down(a)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (164) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (165) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(a)) -> f_flat(down(a)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (166) UsableRulesProof (EQUIVALENT) 178.13/95.77 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. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (167) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (168) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (169) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 down(f(g(y4))) -> f_flat(down(g(y4))) 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (170) UsableRulesProof (EQUIVALENT) 178.13/95.77 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. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (171) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (172) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (173) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (174) DependencyGraphProof (EQUIVALENT) 178.13/95.77 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (175) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 down(f(b)) -> f_flat(down(b)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (176) UsableRulesProof (EQUIVALENT) 178.13/95.77 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. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (177) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (178) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (179) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (180) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(fresh_constant))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (181) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (182) DependencyGraphProof (EQUIVALENT) 178.13/95.77 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (183) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (184) UsableRulesProof (EQUIVALENT) 178.13/95.77 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. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (185) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (186) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) at position [0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (187) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (188) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (189) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (190) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a)))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (191) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (192) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a)))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (193) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (194) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) at position [0,0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a)))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (195) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (196) TransformationProof (EQUIVALENT) 178.13/95.77 By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))) at position [0] we obtained the following new rules [LPAR04]: 178.13/95.77 178.13/95.77 (TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a)))))))) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (197) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(a))) -> TOP(up(f(f(a)))) 178.13/95.77 TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) 178.13/95.77 TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (198) DependencyGraphProof (EQUIVALENT) 178.13/95.77 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (199) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (200) QDPOrderProof (EQUIVALENT) 178.13/95.77 We use the reduction pair processor [LPAR04,JAR06]. 178.13/95.77 178.13/95.77 178.13/95.77 The following pairs can be oriented strictly and are deleted. 178.13/95.77 178.13/95.77 TOP(up(g(f(x0)))) -> TOP(up(f(g(g(x0))))) 178.13/95.77 The remaining pairs can at least be oriented weakly. 178.13/95.77 Used ordering: Polynomial interpretation [POLO]: 178.13/95.77 178.13/95.77 POL(TOP(x_1)) = x_1 178.13/95.77 POL(a) = 0 178.13/95.77 POL(b) = 0 178.13/95.77 POL(down(x_1)) = 0 178.13/95.77 POL(f(x_1)) = 0 178.13/95.77 POL(f_flat(x_1)) = 0 178.13/95.77 POL(fresh_constant) = 0 178.13/95.77 POL(g(x_1)) = 1 178.13/95.77 POL(g_flat(x_1)) = 1 178.13/95.77 POL(up(x_1)) = x_1 178.13/95.77 178.13/95.77 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 178.13/95.77 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 178.13/95.77 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (201) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(a))) -> TOP(up(g(f(a)))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.77 178.13/95.77 The TRS R consists of the following rules: 178.13/95.77 178.13/95.77 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.77 down(g(a)) -> g_flat(down(a)) 178.13/95.77 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.77 down(g(b)) -> g_flat(down(b)) 178.13/95.77 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.77 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.77 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.77 down(a) -> up(f(a)) 178.13/95.77 178.13/95.77 The set Q consists of the following terms: 178.13/95.77 178.13/95.77 down(a) 178.13/95.77 down(g(f(x0))) 178.13/95.77 down(f(f(f(f(f(x0)))))) 178.13/95.77 down(f(a)) 178.13/95.77 down(f(g(x0))) 178.13/95.77 down(f(b)) 178.13/95.77 down(f(fresh_constant)) 178.13/95.77 down(g(a)) 178.13/95.77 down(g(g(x0))) 178.13/95.77 down(g(b)) 178.13/95.77 down(g(fresh_constant)) 178.13/95.77 down(f(f(a))) 178.13/95.77 down(f(f(g(x0)))) 178.13/95.77 down(f(f(b))) 178.13/95.77 down(f(f(fresh_constant))) 178.13/95.77 down(f(f(f(a)))) 178.13/95.77 down(f(f(f(g(x0))))) 178.13/95.77 down(f(f(f(b)))) 178.13/95.77 down(f(f(f(fresh_constant)))) 178.13/95.77 down(f(f(f(f(a))))) 178.13/95.77 down(f(f(f(f(g(x0)))))) 178.13/95.77 down(f(f(f(f(b))))) 178.13/95.77 down(f(f(f(f(fresh_constant))))) 178.13/95.77 f_flat(up(x0)) 178.13/95.77 g_flat(up(x0)) 178.13/95.77 178.13/95.77 We have to consider all minimal (P,Q,R)-chains. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (202) DependencyGraphProof (EQUIVALENT) 178.13/95.77 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 178.13/95.77 ---------------------------------------- 178.13/95.77 178.13/95.77 (203) 178.13/95.77 Obligation: 178.13/95.77 Q DP problem: 178.13/95.77 The TRS P consists of the following rules: 178.13/95.77 178.13/95.77 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.77 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.77 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.77 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.77 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.78 down(g(a)) -> g_flat(down(a)) 178.13/95.78 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.78 down(g(b)) -> g_flat(down(b)) 178.13/95.78 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.78 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.78 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.78 down(a) -> up(f(a)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down(a) 178.13/95.78 down(g(f(x0))) 178.13/95.78 down(f(f(f(f(f(x0)))))) 178.13/95.78 down(f(a)) 178.13/95.78 down(f(g(x0))) 178.13/95.78 down(f(b)) 178.13/95.78 down(f(fresh_constant)) 178.13/95.78 down(g(a)) 178.13/95.78 down(g(g(x0))) 178.13/95.78 down(g(b)) 178.13/95.78 down(g(fresh_constant)) 178.13/95.78 down(f(f(a))) 178.13/95.78 down(f(f(g(x0)))) 178.13/95.78 down(f(f(b))) 178.13/95.78 down(f(f(fresh_constant))) 178.13/95.78 down(f(f(f(a)))) 178.13/95.78 down(f(f(f(g(x0))))) 178.13/95.78 down(f(f(f(b)))) 178.13/95.78 down(f(f(f(fresh_constant)))) 178.13/95.78 down(f(f(f(f(a))))) 178.13/95.78 down(f(f(f(f(g(x0)))))) 178.13/95.78 down(f(f(f(f(b))))) 178.13/95.78 down(f(f(f(f(fresh_constant))))) 178.13/95.78 f_flat(up(x0)) 178.13/95.78 g_flat(up(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (204) QDPOrderProof (EQUIVALENT) 178.13/95.78 We use the reduction pair processor [LPAR04,JAR06]. 178.13/95.78 178.13/95.78 178.13/95.78 The following pairs can be oriented strictly and are deleted. 178.13/95.78 178.13/95.78 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 178.13/95.78 The remaining pairs can at least be oriented weakly. 178.13/95.78 Used ordering: Polynomial interpretation [POLO]: 178.13/95.78 178.13/95.78 POL(TOP(x_1)) = x_1 178.13/95.78 POL(a) = 1 178.13/95.78 POL(b) = 0 178.13/95.78 POL(down(x_1)) = x_1 178.13/95.78 POL(f(x_1)) = 0 178.13/95.78 POL(f_flat(x_1)) = 1 178.13/95.78 POL(fresh_constant) = 1 178.13/95.78 POL(g(x_1)) = 1 + x_1 178.13/95.78 POL(g_flat(x_1)) = 1 + x_1 178.13/95.78 POL(up(x_1)) = 1 + x_1 178.13/95.78 178.13/95.78 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 178.13/95.78 178.13/95.78 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.78 down(g(a)) -> g_flat(down(a)) 178.13/95.78 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.78 down(g(b)) -> g_flat(down(b)) 178.13/95.78 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.78 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.78 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.78 down(a) -> up(f(a)) 178.13/95.78 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (205) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.78 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.78 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.78 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.78 down(g(a)) -> g_flat(down(a)) 178.13/95.78 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.78 down(g(b)) -> g_flat(down(b)) 178.13/95.78 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.78 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.78 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.78 down(a) -> up(f(a)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down(a) 178.13/95.78 down(g(f(x0))) 178.13/95.78 down(f(f(f(f(f(x0)))))) 178.13/95.78 down(f(a)) 178.13/95.78 down(f(g(x0))) 178.13/95.78 down(f(b)) 178.13/95.78 down(f(fresh_constant)) 178.13/95.78 down(g(a)) 178.13/95.78 down(g(g(x0))) 178.13/95.78 down(g(b)) 178.13/95.78 down(g(fresh_constant)) 178.13/95.78 down(f(f(a))) 178.13/95.78 down(f(f(g(x0)))) 178.13/95.78 down(f(f(b))) 178.13/95.78 down(f(f(fresh_constant))) 178.13/95.78 down(f(f(f(a)))) 178.13/95.78 down(f(f(f(g(x0))))) 178.13/95.78 down(f(f(f(b)))) 178.13/95.78 down(f(f(f(fresh_constant)))) 178.13/95.78 down(f(f(f(f(a))))) 178.13/95.78 down(f(f(f(f(g(x0)))))) 178.13/95.78 down(f(f(f(f(b))))) 178.13/95.78 down(f(f(f(f(fresh_constant))))) 178.13/95.78 f_flat(up(x0)) 178.13/95.78 g_flat(up(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (206) QDPOrderProof (EQUIVALENT) 178.13/95.78 We use the reduction pair processor [LPAR04,JAR06]. 178.13/95.78 178.13/95.78 178.13/95.78 The following pairs can be oriented strictly and are deleted. 178.13/95.78 178.13/95.78 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 178.13/95.78 The remaining pairs can at least be oriented weakly. 178.13/95.78 Used ordering: Matrix interpretation [MATRO]: 178.13/95.78 178.13/95.78 Non-tuple symbols: 178.13/95.78 <<< 178.13/95.78 M( a ) = [[1], [1]] 178.13/95.78 >>> 178.13/95.78 178.13/95.78 <<< 178.13/95.78 M( b ) = [[0], [0]] 178.13/95.78 >>> 178.13/95.78 178.13/95.78 <<< 178.13/95.78 M( down_1(x_1) ) = [[1], [0]] + [[1, 1], [0, 1]] * x_1 178.13/95.78 >>> 178.13/95.78 178.13/95.78 <<< 178.13/95.78 M( f_1(x_1) ) = [[0], [0]] + [[0, 1], [0, 0]] * x_1 178.13/95.78 >>> 178.13/95.78 178.13/95.78 <<< 178.13/95.78 M( fresh_constant ) = [[0], [0]] 178.13/95.78 >>> 178.13/95.78 178.13/95.78 <<< 178.13/95.78 M( up_1(x_1) ) = [[1], [1]] + [[1, 0], [0, 1]] * x_1 178.13/95.78 >>> 178.13/95.78 178.13/95.78 <<< 178.13/95.78 M( f_flat_1(x_1) ) = [[0], [1]] + [[0, 1], [0, 0]] * x_1 178.13/95.78 >>> 178.13/95.78 178.13/95.78 <<< 178.13/95.78 M( g_1(x_1) ) = [[1], [1]] + [[1, 0], [0, 1]] * x_1 178.13/95.78 >>> 178.13/95.78 178.13/95.78 <<< 178.13/95.78 M( g_flat_1(x_1) ) = [[1], [1]] + [[1, 0], [0, 1]] * x_1 178.13/95.78 >>> 178.13/95.78 178.13/95.78 Tuple symbols: 178.13/95.78 <<< 178.13/95.78 M( TOP_1(x_1) ) = [[0]] + [[1, 0]] * x_1 178.13/95.78 >>> 178.13/95.78 178.13/95.78 178.13/95.78 178.13/95.78 Matrix type: 178.13/95.78 178.13/95.78 We used a basic matrix type which is not further parametrizeable. 178.13/95.78 178.13/95.78 178.13/95.78 178.13/95.78 178.13/95.78 178.13/95.78 As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. 178.13/95.78 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 178.13/95.78 178.13/95.78 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.78 down(g(a)) -> g_flat(down(a)) 178.13/95.78 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.78 down(g(b)) -> g_flat(down(b)) 178.13/95.78 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.78 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.78 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.78 down(a) -> up(f(a)) 178.13/95.78 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (207) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.78 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.78 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.78 down(g(a)) -> g_flat(down(a)) 178.13/95.78 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.78 down(g(b)) -> g_flat(down(b)) 178.13/95.78 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.78 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.78 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.78 down(a) -> up(f(a)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down(a) 178.13/95.78 down(g(f(x0))) 178.13/95.78 down(f(f(f(f(f(x0)))))) 178.13/95.78 down(f(a)) 178.13/95.78 down(f(g(x0))) 178.13/95.78 down(f(b)) 178.13/95.78 down(f(fresh_constant)) 178.13/95.78 down(g(a)) 178.13/95.78 down(g(g(x0))) 178.13/95.78 down(g(b)) 178.13/95.78 down(g(fresh_constant)) 178.13/95.78 down(f(f(a))) 178.13/95.78 down(f(f(g(x0)))) 178.13/95.78 down(f(f(b))) 178.13/95.78 down(f(f(fresh_constant))) 178.13/95.78 down(f(f(f(a)))) 178.13/95.78 down(f(f(f(g(x0))))) 178.13/95.78 down(f(f(f(b)))) 178.13/95.78 down(f(f(f(fresh_constant)))) 178.13/95.78 down(f(f(f(f(a))))) 178.13/95.78 down(f(f(f(f(g(x0)))))) 178.13/95.78 down(f(f(f(f(b))))) 178.13/95.78 down(f(f(f(f(fresh_constant))))) 178.13/95.78 f_flat(up(x0)) 178.13/95.78 g_flat(up(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (208) SplitQDPProof (EQUIVALENT) 178.13/95.78 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (209) 178.13/95.78 Complex Obligation (AND) 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (210) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.78 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.78 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.78 down(g(a)) -> g_flat(down(a)) 178.13/95.78 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.78 down(g(b)) -> g_flat(down(b)) 178.13/95.78 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 178.13/95.78 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.78 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.78 down(a) -> up(f(a)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down(a) 178.13/95.78 down(g(f(x0))) 178.13/95.78 down(f(f(f(f(f(x0)))))) 178.13/95.78 down(f(a)) 178.13/95.78 down(f(g(x0))) 178.13/95.78 down(f(b)) 178.13/95.78 down(f(fresh_constant)) 178.13/95.78 down(g(a)) 178.13/95.78 down(g(g(x0))) 178.13/95.78 down(g(b)) 178.13/95.78 down(g(fresh_constant)) 178.13/95.78 down(f(f(a))) 178.13/95.78 down(f(f(g(x0)))) 178.13/95.78 down(f(f(b))) 178.13/95.78 down(f(f(fresh_constant))) 178.13/95.78 down(f(f(f(a)))) 178.13/95.78 down(f(f(f(g(x0))))) 178.13/95.78 down(f(f(f(b)))) 178.13/95.78 down(f(f(f(fresh_constant)))) 178.13/95.78 down(f(f(f(f(a))))) 178.13/95.78 down(f(f(f(f(g(x0)))))) 178.13/95.78 down(f(f(f(f(b))))) 178.13/95.78 down(f(f(f(f(fresh_constant))))) 178.13/95.78 f_flat(up(x0)) 178.13/95.78 g_flat(up(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (211) SemLabProof (SOUND) 178.13/95.78 We found the following model for the rules of the TRSs R and P. 178.13/95.78 Interpretation over the domain with elements from 0 to 1. 178.13/95.78 a: 0 178.13/95.78 b: 0 178.13/95.78 down: 0 178.13/95.78 f: 0 178.13/95.78 fresh_constant: 1 178.13/95.78 up: 0 178.13/95.78 f_flat: 0 178.13/95.78 TOP: 0 178.13/95.78 g_flat: 0 178.13/95.78 g: 0 178.13/95.78 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (212) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 down.0(g.0(f.0(x))) -> up.0(f.0(g.0(g.0(x)))) 178.13/95.78 down.0(g.0(f.1(x))) -> up.0(f.0(g.0(g.1(x)))) 178.13/95.78 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 178.13/95.78 down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) 178.13/95.78 down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) 178.13/95.78 down.0(g.0(b.)) -> g_flat.0(down.0(b.)) 178.13/95.78 down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.)) 178.13/95.78 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 178.13/95.78 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 178.13/95.78 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 178.13/95.78 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 178.13/95.78 down.0(a.) -> up.0(f.0(a.)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down.0(a.) 178.13/95.78 down.0(g.0(f.0(x0))) 178.13/95.78 down.0(g.0(f.1(x0))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) 178.13/95.78 down.0(f.0(a.)) 178.13/95.78 down.0(f.0(g.0(x0))) 178.13/95.78 down.0(f.0(g.1(x0))) 178.13/95.78 down.0(f.0(b.)) 178.13/95.78 down.0(f.1(fresh_constant.)) 178.13/95.78 down.0(g.0(a.)) 178.13/95.78 down.0(g.0(g.0(x0))) 178.13/95.78 down.0(g.0(g.1(x0))) 178.13/95.78 down.0(g.0(b.)) 178.13/95.78 down.0(g.1(fresh_constant.)) 178.13/95.78 down.0(f.0(f.0(a.))) 178.13/95.78 down.0(f.0(f.0(g.0(x0)))) 178.13/95.78 down.0(f.0(f.0(g.1(x0)))) 178.13/95.78 down.0(f.0(f.0(b.))) 178.13/95.78 down.0(f.0(f.1(fresh_constant.))) 178.13/95.78 down.0(f.0(f.0(f.0(a.)))) 178.13/95.78 down.0(f.0(f.0(f.0(g.0(x0))))) 178.13/95.78 down.0(f.0(f.0(f.0(g.1(x0))))) 178.13/95.78 down.0(f.0(f.0(f.0(b.)))) 178.13/95.78 down.0(f.0(f.0(f.1(fresh_constant.)))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(a.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(b.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) 178.13/95.78 f_flat.0(up.0(x0)) 178.13/95.78 f_flat.0(up.1(x0)) 178.13/95.78 g_flat.0(up.0(x0)) 178.13/95.78 g_flat.0(up.1(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (213) UsableRulesReductionPairsProof (EQUIVALENT) 178.13/95.78 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. 178.13/95.78 178.13/95.78 No dependency pairs are removed. 178.13/95.78 178.13/95.78 The following rules are removed from R: 178.13/95.78 178.13/95.78 down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.)) 178.13/95.78 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 178.13/95.78 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 178.13/95.78 Used ordering: POLO with Polynomial interpretation [POLO]: 178.13/95.78 178.13/95.78 POL(TOP.0(x_1)) = x_1 178.13/95.78 POL(a.) = 0 178.13/95.78 POL(b.) = 0 178.13/95.78 POL(down.0(x_1)) = 1 + x_1 178.13/95.78 POL(down.1(x_1)) = x_1 178.13/95.78 POL(f.0(x_1)) = x_1 178.13/95.78 POL(f.1(x_1)) = x_1 178.13/95.78 POL(f_flat.0(x_1)) = x_1 178.13/95.78 POL(fresh_constant.) = 0 178.13/95.78 POL(g.0(x_1)) = x_1 178.13/95.78 POL(g.1(x_1)) = x_1 178.13/95.78 POL(g_flat.0(x_1)) = x_1 178.13/95.78 POL(up.0(x_1)) = 1 + x_1 178.13/95.78 POL(up.1(x_1)) = 1 + x_1 178.13/95.78 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (214) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 178.13/95.78 down.0(g.0(f.0(x))) -> up.0(f.0(g.0(g.0(x)))) 178.13/95.78 down.0(g.0(f.1(x))) -> up.0(f.0(g.0(g.1(x)))) 178.13/95.78 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 178.13/95.78 down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) 178.13/95.78 down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) 178.13/95.78 down.0(g.0(b.)) -> g_flat.0(down.0(b.)) 178.13/95.78 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 178.13/95.78 down.0(a.) -> up.0(f.0(a.)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down.0(a.) 178.13/95.78 down.0(g.0(f.0(x0))) 178.13/95.78 down.0(g.0(f.1(x0))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) 178.13/95.78 down.0(f.0(a.)) 178.13/95.78 down.0(f.0(g.0(x0))) 178.13/95.78 down.0(f.0(g.1(x0))) 178.13/95.78 down.0(f.0(b.)) 178.13/95.78 down.0(f.1(fresh_constant.)) 178.13/95.78 down.0(g.0(a.)) 178.13/95.78 down.0(g.0(g.0(x0))) 178.13/95.78 down.0(g.0(g.1(x0))) 178.13/95.78 down.0(g.0(b.)) 178.13/95.78 down.0(g.1(fresh_constant.)) 178.13/95.78 down.0(f.0(f.0(a.))) 178.13/95.78 down.0(f.0(f.0(g.0(x0)))) 178.13/95.78 down.0(f.0(f.0(g.1(x0)))) 178.13/95.78 down.0(f.0(f.0(b.))) 178.13/95.78 down.0(f.0(f.1(fresh_constant.))) 178.13/95.78 down.0(f.0(f.0(f.0(a.)))) 178.13/95.78 down.0(f.0(f.0(f.0(g.0(x0))))) 178.13/95.78 down.0(f.0(f.0(f.0(g.1(x0))))) 178.13/95.78 down.0(f.0(f.0(f.0(b.)))) 178.13/95.78 down.0(f.0(f.0(f.1(fresh_constant.)))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(a.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(b.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) 178.13/95.78 f_flat.0(up.0(x0)) 178.13/95.78 f_flat.0(up.1(x0)) 178.13/95.78 g_flat.0(up.0(x0)) 178.13/95.78 g_flat.0(up.1(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (215) DependencyGraphProof (EQUIVALENT) 178.13/95.78 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (216) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 178.13/95.78 down.0(g.0(f.0(x))) -> up.0(f.0(g.0(g.0(x)))) 178.13/95.78 down.0(g.0(f.1(x))) -> up.0(f.0(g.0(g.1(x)))) 178.13/95.78 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 178.13/95.78 down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) 178.13/95.78 down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) 178.13/95.78 down.0(g.0(b.)) -> g_flat.0(down.0(b.)) 178.13/95.78 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 178.13/95.78 down.0(a.) -> up.0(f.0(a.)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down.0(a.) 178.13/95.78 down.0(g.0(f.0(x0))) 178.13/95.78 down.0(g.0(f.1(x0))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) 178.13/95.78 down.0(f.0(a.)) 178.13/95.78 down.0(f.0(g.0(x0))) 178.13/95.78 down.0(f.0(g.1(x0))) 178.13/95.78 down.0(f.0(b.)) 178.13/95.78 down.0(f.1(fresh_constant.)) 178.13/95.78 down.0(g.0(a.)) 178.13/95.78 down.0(g.0(g.0(x0))) 178.13/95.78 down.0(g.0(g.1(x0))) 178.13/95.78 down.0(g.0(b.)) 178.13/95.78 down.0(g.1(fresh_constant.)) 178.13/95.78 down.0(f.0(f.0(a.))) 178.13/95.78 down.0(f.0(f.0(g.0(x0)))) 178.13/95.78 down.0(f.0(f.0(g.1(x0)))) 178.13/95.78 down.0(f.0(f.0(b.))) 178.13/95.78 down.0(f.0(f.1(fresh_constant.))) 178.13/95.78 down.0(f.0(f.0(f.0(a.)))) 178.13/95.78 down.0(f.0(f.0(f.0(g.0(x0))))) 178.13/95.78 down.0(f.0(f.0(f.0(g.1(x0))))) 178.13/95.78 down.0(f.0(f.0(f.0(b.)))) 178.13/95.78 down.0(f.0(f.0(f.1(fresh_constant.)))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(a.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(b.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) 178.13/95.78 f_flat.0(up.0(x0)) 178.13/95.78 f_flat.0(up.1(x0)) 178.13/95.78 g_flat.0(up.0(x0)) 178.13/95.78 g_flat.0(up.1(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (217) UsableRulesReductionPairsProof (EQUIVALENT) 178.13/95.78 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. 178.13/95.78 178.13/95.78 No dependency pairs are removed. 178.13/95.78 178.13/95.78 The following rules are removed from R: 178.13/95.78 178.13/95.78 down.0(g.0(f.1(x))) -> up.0(f.0(g.0(g.1(x)))) 178.13/95.78 Used ordering: POLO with Polynomial interpretation [POLO]: 178.13/95.78 178.13/95.78 POL(TOP.0(x_1)) = x_1 178.13/95.78 POL(a.) = 0 178.13/95.78 POL(b.) = 0 178.13/95.78 POL(down.0(x_1)) = 1 + x_1 178.13/95.78 POL(f.0(x_1)) = x_1 178.13/95.78 POL(f.1(x_1)) = 1 + x_1 178.13/95.78 POL(f_flat.0(x_1)) = x_1 178.13/95.78 POL(g.0(x_1)) = x_1 178.13/95.78 POL(g.1(x_1)) = 1 + x_1 178.13/95.78 POL(g_flat.0(x_1)) = x_1 178.13/95.78 POL(up.0(x_1)) = 1 + x_1 178.13/95.78 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (218) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 down.0(g.0(f.0(x))) -> up.0(f.0(g.0(g.0(x)))) 178.13/95.78 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 178.13/95.78 down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) 178.13/95.78 down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) 178.13/95.78 down.0(g.0(b.)) -> g_flat.0(down.0(b.)) 178.13/95.78 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 178.13/95.78 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 178.13/95.78 down.0(a.) -> up.0(f.0(a.)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down.0(a.) 178.13/95.78 down.0(g.0(f.0(x0))) 178.13/95.78 down.0(g.0(f.1(x0))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) 178.13/95.78 down.0(f.0(a.)) 178.13/95.78 down.0(f.0(g.0(x0))) 178.13/95.78 down.0(f.0(g.1(x0))) 178.13/95.78 down.0(f.0(b.)) 178.13/95.78 down.0(f.1(fresh_constant.)) 178.13/95.78 down.0(g.0(a.)) 178.13/95.78 down.0(g.0(g.0(x0))) 178.13/95.78 down.0(g.0(g.1(x0))) 178.13/95.78 down.0(g.0(b.)) 178.13/95.78 down.0(g.1(fresh_constant.)) 178.13/95.78 down.0(f.0(f.0(a.))) 178.13/95.78 down.0(f.0(f.0(g.0(x0)))) 178.13/95.78 down.0(f.0(f.0(g.1(x0)))) 178.13/95.78 down.0(f.0(f.0(b.))) 178.13/95.78 down.0(f.0(f.1(fresh_constant.))) 178.13/95.78 down.0(f.0(f.0(f.0(a.)))) 178.13/95.78 down.0(f.0(f.0(f.0(g.0(x0))))) 178.13/95.78 down.0(f.0(f.0(f.0(g.1(x0))))) 178.13/95.78 down.0(f.0(f.0(f.0(b.)))) 178.13/95.78 down.0(f.0(f.0(f.1(fresh_constant.)))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(a.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(b.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) 178.13/95.78 f_flat.0(up.0(x0)) 178.13/95.78 f_flat.0(up.1(x0)) 178.13/95.78 g_flat.0(up.0(x0)) 178.13/95.78 g_flat.0(up.1(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (219) PisEmptyProof (SOUND) 178.13/95.78 The TRS P is empty. Hence, there is no (P,Q,R) chain. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (220) 178.13/95.78 TRUE 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (221) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.78 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.78 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.78 down(g(a)) -> g_flat(down(a)) 178.13/95.78 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.78 down(g(b)) -> g_flat(down(b)) 178.13/95.78 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.78 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.78 down(a) -> up(f(a)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down(a) 178.13/95.78 down(g(f(x0))) 178.13/95.78 down(f(f(f(f(f(x0)))))) 178.13/95.78 down(f(a)) 178.13/95.78 down(f(g(x0))) 178.13/95.78 down(f(b)) 178.13/95.78 down(f(fresh_constant)) 178.13/95.78 down(g(a)) 178.13/95.78 down(g(g(x0))) 178.13/95.78 down(g(b)) 178.13/95.78 down(g(fresh_constant)) 178.13/95.78 down(f(f(a))) 178.13/95.78 down(f(f(g(x0)))) 178.13/95.78 down(f(f(b))) 178.13/95.78 down(f(f(fresh_constant))) 178.13/95.78 down(f(f(f(a)))) 178.13/95.78 down(f(f(f(g(x0))))) 178.13/95.78 down(f(f(f(b)))) 178.13/95.78 down(f(f(f(fresh_constant)))) 178.13/95.78 down(f(f(f(f(a))))) 178.13/95.78 down(f(f(f(f(g(x0)))))) 178.13/95.78 down(f(f(f(f(b))))) 178.13/95.78 down(f(f(f(f(fresh_constant))))) 178.13/95.78 f_flat(up(x0)) 178.13/95.78 g_flat(up(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (222) QReductionProof (EQUIVALENT) 178.13/95.78 We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN]. 178.13/95.78 178.13/95.78 down(f(fresh_constant)) 178.13/95.78 down(g(fresh_constant)) 178.13/95.78 down(f(f(fresh_constant))) 178.13/95.78 down(f(f(f(fresh_constant)))) 178.13/95.78 down(f(f(f(f(fresh_constant))))) 178.13/95.78 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (223) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.78 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.78 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.78 down(g(a)) -> g_flat(down(a)) 178.13/95.78 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.78 down(g(b)) -> g_flat(down(b)) 178.13/95.78 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.78 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.78 down(a) -> up(f(a)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down(a) 178.13/95.78 down(g(f(x0))) 178.13/95.78 down(f(f(f(f(f(x0)))))) 178.13/95.78 down(f(a)) 178.13/95.78 down(f(g(x0))) 178.13/95.78 down(f(b)) 178.13/95.78 down(g(a)) 178.13/95.78 down(g(g(x0))) 178.13/95.78 down(g(b)) 178.13/95.78 down(f(f(a))) 178.13/95.78 down(f(f(g(x0)))) 178.13/95.78 down(f(f(b))) 178.13/95.78 down(f(f(f(a)))) 178.13/95.78 down(f(f(f(g(x0))))) 178.13/95.78 down(f(f(f(b)))) 178.13/95.78 down(f(f(f(f(a))))) 178.13/95.78 down(f(f(f(f(g(x0)))))) 178.13/95.78 down(f(f(f(f(b))))) 178.13/95.78 f_flat(up(x0)) 178.13/95.78 g_flat(up(x0)) 178.13/95.78 178.13/95.78 We have to consider all (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (224) SplitQDPProof (EQUIVALENT) 178.13/95.78 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (225) 178.13/95.78 Complex Obligation (AND) 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (226) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.78 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.78 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.78 down(g(a)) -> g_flat(down(a)) 178.13/95.78 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.78 down(g(b)) -> g_flat(down(b)) 178.13/95.78 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.78 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.78 down(a) -> up(f(a)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down(a) 178.13/95.78 down(g(f(x0))) 178.13/95.78 down(f(f(f(f(f(x0)))))) 178.13/95.78 down(f(a)) 178.13/95.78 down(f(g(x0))) 178.13/95.78 down(f(b)) 178.13/95.78 down(f(fresh_constant)) 178.13/95.78 down(g(a)) 178.13/95.78 down(g(g(x0))) 178.13/95.78 down(g(b)) 178.13/95.78 down(g(fresh_constant)) 178.13/95.78 down(f(f(a))) 178.13/95.78 down(f(f(g(x0)))) 178.13/95.78 down(f(f(b))) 178.13/95.78 down(f(f(fresh_constant))) 178.13/95.78 down(f(f(f(a)))) 178.13/95.78 down(f(f(f(g(x0))))) 178.13/95.78 down(f(f(f(b)))) 178.13/95.78 down(f(f(f(fresh_constant)))) 178.13/95.78 down(f(f(f(f(a))))) 178.13/95.78 down(f(f(f(f(g(x0)))))) 178.13/95.78 down(f(f(f(f(b))))) 178.13/95.78 down(f(f(f(f(fresh_constant))))) 178.13/95.78 f_flat(up(x0)) 178.13/95.78 g_flat(up(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (227) SemLabProof (SOUND) 178.13/95.78 We found the following model for the rules of the TRSs R and P. 178.13/95.78 Interpretation over the domain with elements from 0 to 1. 178.13/95.78 a: 0 178.13/95.78 b: 1 178.13/95.78 down: 0 178.13/95.78 f: 0 178.13/95.78 fresh_constant: 0 178.13/95.78 up: 0 178.13/95.78 f_flat: 0 178.13/95.78 TOP: 0 178.13/95.78 g_flat: 0 178.13/95.78 g: 0 178.13/95.78 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (228) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 down.0(g.0(f.0(x))) -> up.0(f.0(g.0(g.0(x)))) 178.13/95.78 down.0(g.0(f.1(x))) -> up.0(f.0(g.0(g.1(x)))) 178.13/95.78 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 178.13/95.78 down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) 178.13/95.78 down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) 178.13/95.78 down.0(g.1(b.)) -> g_flat.0(down.1(b.)) 178.13/95.78 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 178.13/95.78 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 178.13/95.78 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 178.13/95.78 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 178.13/95.78 down.0(a.) -> up.0(f.0(a.)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down.0(a.) 178.13/95.78 down.0(g.0(f.0(x0))) 178.13/95.78 down.0(g.0(f.1(x0))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) 178.13/95.78 down.0(f.0(a.)) 178.13/95.78 down.0(f.0(g.0(x0))) 178.13/95.78 down.0(f.0(g.1(x0))) 178.13/95.78 down.0(f.1(b.)) 178.13/95.78 down.0(f.0(fresh_constant.)) 178.13/95.78 down.0(g.0(a.)) 178.13/95.78 down.0(g.0(g.0(x0))) 178.13/95.78 down.0(g.0(g.1(x0))) 178.13/95.78 down.0(g.1(b.)) 178.13/95.78 down.0(g.0(fresh_constant.)) 178.13/95.78 down.0(f.0(f.0(a.))) 178.13/95.78 down.0(f.0(f.0(g.0(x0)))) 178.13/95.78 down.0(f.0(f.0(g.1(x0)))) 178.13/95.78 down.0(f.0(f.1(b.))) 178.13/95.78 down.0(f.0(f.0(fresh_constant.))) 178.13/95.78 down.0(f.0(f.0(f.0(a.)))) 178.13/95.78 down.0(f.0(f.0(f.0(g.0(x0))))) 178.13/95.78 down.0(f.0(f.0(f.0(g.1(x0))))) 178.13/95.78 down.0(f.0(f.0(f.1(b.)))) 178.13/95.78 down.0(f.0(f.0(f.0(fresh_constant.)))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(a.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.1(b.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) 178.13/95.78 f_flat.0(up.0(x0)) 178.13/95.78 f_flat.0(up.1(x0)) 178.13/95.78 g_flat.0(up.0(x0)) 178.13/95.78 g_flat.0(up.1(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (229) UsableRulesReductionPairsProof (EQUIVALENT) 178.13/95.78 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. 178.13/95.78 178.13/95.78 No dependency pairs are removed. 178.13/95.78 178.13/95.78 The following rules are removed from R: 178.13/95.78 178.13/95.78 down.0(g.1(b.)) -> g_flat.0(down.1(b.)) 178.13/95.78 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 178.13/95.78 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 178.13/95.78 Used ordering: POLO with Polynomial interpretation [POLO]: 178.13/95.78 178.13/95.78 POL(TOP.0(x_1)) = x_1 178.13/95.78 POL(a.) = 0 178.13/95.78 POL(b.) = 0 178.13/95.78 POL(down.0(x_1)) = 1 + x_1 178.13/95.78 POL(down.1(x_1)) = x_1 178.13/95.78 POL(f.0(x_1)) = x_1 178.13/95.78 POL(f.1(x_1)) = x_1 178.13/95.78 POL(f_flat.0(x_1)) = x_1 178.13/95.78 POL(g.0(x_1)) = x_1 178.13/95.78 POL(g.1(x_1)) = x_1 178.13/95.78 POL(g_flat.0(x_1)) = x_1 178.13/95.78 POL(up.0(x_1)) = 1 + x_1 178.13/95.78 POL(up.1(x_1)) = 1 + x_1 178.13/95.78 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (230) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 178.13/95.78 down.0(g.0(f.0(x))) -> up.0(f.0(g.0(g.0(x)))) 178.13/95.78 down.0(g.0(f.1(x))) -> up.0(f.0(g.0(g.1(x)))) 178.13/95.78 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 178.13/95.78 down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) 178.13/95.78 down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) 178.13/95.78 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 178.13/95.78 down.0(a.) -> up.0(f.0(a.)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down.0(a.) 178.13/95.78 down.0(g.0(f.0(x0))) 178.13/95.78 down.0(g.0(f.1(x0))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) 178.13/95.78 down.0(f.0(a.)) 178.13/95.78 down.0(f.0(g.0(x0))) 178.13/95.78 down.0(f.0(g.1(x0))) 178.13/95.78 down.0(f.1(b.)) 178.13/95.78 down.0(f.0(fresh_constant.)) 178.13/95.78 down.0(g.0(a.)) 178.13/95.78 down.0(g.0(g.0(x0))) 178.13/95.78 down.0(g.0(g.1(x0))) 178.13/95.78 down.0(g.1(b.)) 178.13/95.78 down.0(g.0(fresh_constant.)) 178.13/95.78 down.0(f.0(f.0(a.))) 178.13/95.78 down.0(f.0(f.0(g.0(x0)))) 178.13/95.78 down.0(f.0(f.0(g.1(x0)))) 178.13/95.78 down.0(f.0(f.1(b.))) 178.13/95.78 down.0(f.0(f.0(fresh_constant.))) 178.13/95.78 down.0(f.0(f.0(f.0(a.)))) 178.13/95.78 down.0(f.0(f.0(f.0(g.0(x0))))) 178.13/95.78 down.0(f.0(f.0(f.0(g.1(x0))))) 178.13/95.78 down.0(f.0(f.0(f.1(b.)))) 178.13/95.78 down.0(f.0(f.0(f.0(fresh_constant.)))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(a.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.1(b.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) 178.13/95.78 f_flat.0(up.0(x0)) 178.13/95.78 f_flat.0(up.1(x0)) 178.13/95.78 g_flat.0(up.0(x0)) 178.13/95.78 g_flat.0(up.1(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (231) DependencyGraphProof (EQUIVALENT) 178.13/95.78 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (232) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 178.13/95.78 down.0(g.0(f.0(x))) -> up.0(f.0(g.0(g.0(x)))) 178.13/95.78 down.0(g.0(f.1(x))) -> up.0(f.0(g.0(g.1(x)))) 178.13/95.78 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 178.13/95.78 down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) 178.13/95.78 down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) 178.13/95.78 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 178.13/95.78 down.0(a.) -> up.0(f.0(a.)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down.0(a.) 178.13/95.78 down.0(g.0(f.0(x0))) 178.13/95.78 down.0(g.0(f.1(x0))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) 178.13/95.78 down.0(f.0(a.)) 178.13/95.78 down.0(f.0(g.0(x0))) 178.13/95.78 down.0(f.0(g.1(x0))) 178.13/95.78 down.0(f.1(b.)) 178.13/95.78 down.0(f.0(fresh_constant.)) 178.13/95.78 down.0(g.0(a.)) 178.13/95.78 down.0(g.0(g.0(x0))) 178.13/95.78 down.0(g.0(g.1(x0))) 178.13/95.78 down.0(g.1(b.)) 178.13/95.78 down.0(g.0(fresh_constant.)) 178.13/95.78 down.0(f.0(f.0(a.))) 178.13/95.78 down.0(f.0(f.0(g.0(x0)))) 178.13/95.78 down.0(f.0(f.0(g.1(x0)))) 178.13/95.78 down.0(f.0(f.1(b.))) 178.13/95.78 down.0(f.0(f.0(fresh_constant.))) 178.13/95.78 down.0(f.0(f.0(f.0(a.)))) 178.13/95.78 down.0(f.0(f.0(f.0(g.0(x0))))) 178.13/95.78 down.0(f.0(f.0(f.0(g.1(x0))))) 178.13/95.78 down.0(f.0(f.0(f.1(b.)))) 178.13/95.78 down.0(f.0(f.0(f.0(fresh_constant.)))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(a.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.1(b.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) 178.13/95.78 f_flat.0(up.0(x0)) 178.13/95.78 f_flat.0(up.1(x0)) 178.13/95.78 g_flat.0(up.0(x0)) 178.13/95.78 g_flat.0(up.1(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (233) UsableRulesReductionPairsProof (EQUIVALENT) 178.13/95.78 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. 178.13/95.78 178.13/95.78 No dependency pairs are removed. 178.13/95.78 178.13/95.78 The following rules are removed from R: 178.13/95.78 178.13/95.78 down.0(g.0(f.1(x))) -> up.0(f.0(g.0(g.1(x)))) 178.13/95.78 Used ordering: POLO with Polynomial interpretation [POLO]: 178.13/95.78 178.13/95.78 POL(TOP.0(x_1)) = x_1 178.13/95.78 POL(a.) = 0 178.13/95.78 POL(down.0(x_1)) = 1 + x_1 178.13/95.78 POL(f.0(x_1)) = x_1 178.13/95.78 POL(f.1(x_1)) = 1 + x_1 178.13/95.78 POL(f_flat.0(x_1)) = x_1 178.13/95.78 POL(g.0(x_1)) = x_1 178.13/95.78 POL(g.1(x_1)) = 1 + x_1 178.13/95.78 POL(g_flat.0(x_1)) = x_1 178.13/95.78 POL(up.0(x_1)) = 1 + x_1 178.13/95.78 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (234) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) 178.13/95.78 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 down.0(g.0(f.0(x))) -> up.0(f.0(g.0(g.0(x)))) 178.13/95.78 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 178.13/95.78 down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) 178.13/95.78 down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) 178.13/95.78 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 178.13/95.78 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 178.13/95.78 down.0(a.) -> up.0(f.0(a.)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down.0(a.) 178.13/95.78 down.0(g.0(f.0(x0))) 178.13/95.78 down.0(g.0(f.1(x0))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) 178.13/95.78 down.0(f.0(a.)) 178.13/95.78 down.0(f.0(g.0(x0))) 178.13/95.78 down.0(f.0(g.1(x0))) 178.13/95.78 down.0(f.1(b.)) 178.13/95.78 down.0(f.0(fresh_constant.)) 178.13/95.78 down.0(g.0(a.)) 178.13/95.78 down.0(g.0(g.0(x0))) 178.13/95.78 down.0(g.0(g.1(x0))) 178.13/95.78 down.0(g.1(b.)) 178.13/95.78 down.0(g.0(fresh_constant.)) 178.13/95.78 down.0(f.0(f.0(a.))) 178.13/95.78 down.0(f.0(f.0(g.0(x0)))) 178.13/95.78 down.0(f.0(f.0(g.1(x0)))) 178.13/95.78 down.0(f.0(f.1(b.))) 178.13/95.78 down.0(f.0(f.0(fresh_constant.))) 178.13/95.78 down.0(f.0(f.0(f.0(a.)))) 178.13/95.78 down.0(f.0(f.0(f.0(g.0(x0))))) 178.13/95.78 down.0(f.0(f.0(f.0(g.1(x0))))) 178.13/95.78 down.0(f.0(f.0(f.1(b.)))) 178.13/95.78 down.0(f.0(f.0(f.0(fresh_constant.)))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(a.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.1(b.))))) 178.13/95.78 down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) 178.13/95.78 f_flat.0(up.0(x0)) 178.13/95.78 f_flat.0(up.1(x0)) 178.13/95.78 g_flat.0(up.0(x0)) 178.13/95.78 g_flat.0(up.1(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (235) PisEmptyProof (SOUND) 178.13/95.78 The TRS P is empty. Hence, there is no (P,Q,R) chain. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (236) 178.13/95.78 TRUE 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (237) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.78 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.78 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.78 down(g(a)) -> g_flat(down(a)) 178.13/95.78 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.78 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.78 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.78 down(a) -> up(f(a)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down(a) 178.13/95.78 down(g(f(x0))) 178.13/95.78 down(f(f(f(f(f(x0)))))) 178.13/95.78 down(f(a)) 178.13/95.78 down(f(g(x0))) 178.13/95.78 down(f(b)) 178.13/95.78 down(f(fresh_constant)) 178.13/95.78 down(g(a)) 178.13/95.78 down(g(g(x0))) 178.13/95.78 down(g(b)) 178.13/95.78 down(g(fresh_constant)) 178.13/95.78 down(f(f(a))) 178.13/95.78 down(f(f(g(x0)))) 178.13/95.78 down(f(f(b))) 178.13/95.78 down(f(f(fresh_constant))) 178.13/95.78 down(f(f(f(a)))) 178.13/95.78 down(f(f(f(g(x0))))) 178.13/95.78 down(f(f(f(b)))) 178.13/95.78 down(f(f(f(fresh_constant)))) 178.13/95.78 down(f(f(f(f(a))))) 178.13/95.78 down(f(f(f(f(g(x0)))))) 178.13/95.78 down(f(f(f(f(b))))) 178.13/95.78 down(f(f(f(f(fresh_constant))))) 178.13/95.78 f_flat(up(x0)) 178.13/95.78 g_flat(up(x0)) 178.13/95.78 178.13/95.78 We have to consider all minimal (P,Q,R)-chains. 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (238) QReductionProof (EQUIVALENT) 178.13/95.78 We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN]. 178.13/95.78 178.13/95.78 down(f(b)) 178.13/95.78 down(f(fresh_constant)) 178.13/95.78 down(g(b)) 178.13/95.78 down(g(fresh_constant)) 178.13/95.78 down(f(f(b))) 178.13/95.78 down(f(f(fresh_constant))) 178.13/95.78 down(f(f(f(b)))) 178.13/95.78 down(f(f(f(fresh_constant)))) 178.13/95.78 down(f(f(f(f(b))))) 178.13/95.78 down(f(f(f(f(fresh_constant))))) 178.13/95.78 178.13/95.78 178.13/95.78 ---------------------------------------- 178.13/95.78 178.13/95.78 (239) 178.13/95.78 Obligation: 178.13/95.78 Q DP problem: 178.13/95.78 The TRS P consists of the following rules: 178.13/95.78 178.13/95.78 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 178.13/95.78 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) 178.13/95.78 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) 178.13/95.78 178.13/95.78 The TRS R consists of the following rules: 178.13/95.78 178.13/95.78 down(g(f(x))) -> up(f(g(g(x)))) 178.13/95.78 down(g(a)) -> g_flat(down(a)) 178.13/95.78 down(g(g(y7))) -> g_flat(down(g(y7))) 178.13/95.78 f_flat(up(x_1)) -> up(f(x_1)) 178.13/95.78 g_flat(up(x_1)) -> up(g(x_1)) 178.13/95.78 down(a) -> up(f(a)) 178.13/95.78 178.13/95.78 The set Q consists of the following terms: 178.13/95.78 178.13/95.78 down(a) 178.13/95.78 down(g(f(x0))) 178.13/95.78 down(f(f(f(f(f(x0)))))) 178.13/95.78 down(f(a)) 178.13/95.78 down(f(g(x0))) 178.13/95.78 down(g(a)) 178.13/95.78 down(g(g(x0))) 178.13/95.78 down(f(f(a))) 178.13/95.78 down(f(f(g(x0)))) 178.13/95.78 down(f(f(f(a)))) 178.13/95.78 down(f(f(f(g(x0))))) 178.13/95.78 down(f(f(f(f(a))))) 178.13/95.78 down(f(f(f(f(g(x0)))))) 178.13/95.78 f_flat(up(x0)) 178.13/95.78 g_flat(up(x0)) 178.13/95.78 178.13/95.78 We have to consider all (P,Q,R)-chains. 179.97/97.29 EOF