275.76/137.29 MAYBE 275.82/137.32 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 275.82/137.32 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 275.82/137.32 275.82/137.32 275.82/137.32 Outermost Termination of the given OTRS could not be shown: 275.82/137.32 275.82/137.32 (0) OTRS 275.82/137.32 (1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms] 275.82/137.32 (2) QTRS 275.82/137.32 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 275.82/137.32 (4) QDP 275.82/137.32 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 275.82/137.32 (6) AND 275.82/137.32 (7) QDP 275.82/137.32 (8) UsableRulesProof [EQUIVALENT, 0 ms] 275.82/137.32 (9) QDP 275.82/137.32 (10) QReductionProof [EQUIVALENT, 0 ms] 275.82/137.32 (11) QDP 275.82/137.32 (12) MRRProof [EQUIVALENT, 5 ms] 275.82/137.32 (13) QDP 275.82/137.32 (14) UsableRulesProof [EQUIVALENT, 0 ms] 275.82/137.32 (15) QDP 275.82/137.32 (16) QReductionProof [EQUIVALENT, 0 ms] 275.82/137.32 (17) QDP 275.82/137.32 (18) UsableRulesReductionPairsProof [EQUIVALENT, 5 ms] 275.82/137.32 (19) QDP 275.82/137.32 (20) DependencyGraphProof [EQUIVALENT, 0 ms] 275.82/137.32 (21) TRUE 275.82/137.32 (22) QDP 275.82/137.32 (23) UsableRulesProof [EQUIVALENT, 0 ms] 275.82/137.32 (24) QDP 275.82/137.32 (25) QReductionProof [EQUIVALENT, 0 ms] 275.82/137.32 (26) QDP 275.82/137.32 (27) TransformationProof [EQUIVALENT, 0 ms] 275.82/137.32 (28) QDP 275.82/137.32 (29) UsableRulesProof [EQUIVALENT, 0 ms] 275.82/137.32 (30) QDP 275.82/137.32 (31) QDPOrderProof [EQUIVALENT, 12 ms] 275.82/137.32 (32) QDP 275.82/137.32 (33) SplitQDPProof [EQUIVALENT, 0 ms] 275.82/137.32 (34) AND 275.82/137.32 (35) QDP 275.82/137.32 (36) SemLabProof [SOUND, 0 ms] 275.82/137.32 (37) QDP 275.82/137.32 (38) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 275.82/137.32 (39) QDP 275.82/137.32 (40) MRRProof [EQUIVALENT, 0 ms] 275.82/137.32 (41) QDP 275.82/137.32 (42) QDPOrderProof [EQUIVALENT, 0 ms] 275.82/137.32 (43) QDP 275.82/137.32 (44) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 275.82/137.32 (45) QDP 275.82/137.32 (46) QDPOrderProof [EQUIVALENT, 0 ms] 275.82/137.32 (47) QDP 275.82/137.32 (48) QDPOrderProof [EQUIVALENT, 4 ms] 275.82/137.32 (49) QDP 275.82/137.32 (50) PisEmptyProof [SOUND, 0 ms] 275.82/137.32 (51) TRUE 275.82/137.32 (52) QDP 275.82/137.32 (53) TransformationProof [SOUND, 0 ms] 275.82/137.32 (54) QDP 275.82/137.32 (55) UsableRulesProof [EQUIVALENT, 0 ms] 275.82/137.32 (56) QDP 275.82/137.32 (57) QReductionProof [EQUIVALENT, 0 ms] 275.82/137.32 (58) QDP 275.82/137.32 (59) UsableRulesProof [EQUIVALENT, 0 ms] 275.82/137.32 (60) QDP 275.82/137.32 (61) QReductionProof [EQUIVALENT, 0 ms] 275.82/137.32 (62) QDP 275.82/137.32 (63) Trivial-Transformation [SOUND, 0 ms] 275.82/137.32 (64) QTRS 275.82/137.32 (65) DependencyPairsProof [EQUIVALENT, 0 ms] 275.82/137.32 (66) QDP 275.82/137.32 (67) DependencyGraphProof [EQUIVALENT, 0 ms] 275.82/137.32 (68) AND 275.82/137.32 (69) QDP 275.82/137.32 (70) UsableRulesProof [EQUIVALENT, 0 ms] 275.82/137.32 (71) QDP 275.82/137.32 (72) QDPSizeChangeProof [EQUIVALENT, 0 ms] 275.82/137.32 (73) YES 275.82/137.32 (74) QDP 275.82/137.32 (75) UsableRulesProof [EQUIVALENT, 0 ms] 275.82/137.32 (76) QDP 275.82/137.32 (77) NonTerminationLoopProof [COMPLETE, 0 ms] 275.82/137.32 (78) NO 275.82/137.32 (79) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] 275.82/137.32 (80) QTRS 275.82/137.32 (81) AAECC Innermost [EQUIVALENT, 0 ms] 275.82/137.32 (82) QTRS 275.82/137.32 (83) DependencyPairsProof [EQUIVALENT, 0 ms] 275.82/137.32 (84) QDP 275.82/137.32 (85) DependencyGraphProof [EQUIVALENT, 0 ms] 275.82/137.32 (86) AND 275.82/137.32 (87) QDP 275.82/137.32 (88) UsableRulesProof [EQUIVALENT, 0 ms] 275.82/137.32 (89) QDP 275.82/137.32 (90) QReductionProof [EQUIVALENT, 0 ms] 275.82/137.32 (91) QDP 275.82/137.32 (92) QDPSizeChangeProof [EQUIVALENT, 0 ms] 275.82/137.32 (93) YES 275.82/137.32 (94) QDP 275.82/137.32 (95) UsableRulesProof [EQUIVALENT, 0 ms] 275.82/137.32 (96) QDP 275.82/137.32 (97) QReductionProof [EQUIVALENT, 0 ms] 275.82/137.32 (98) QDP 275.82/137.32 (99) TransformationProof [EQUIVALENT, 0 ms] 275.82/137.32 (100) QDP 275.82/137.32 (101) DependencyGraphProof [EQUIVALENT, 0 ms] 275.82/137.32 (102) QDP 275.82/137.32 (103) UsableRulesProof [EQUIVALENT, 0 ms] 275.82/137.32 (104) QDP 275.82/137.32 (105) TransformationProof [EQUIVALENT, 0 ms] 275.82/137.32 (106) QDP 275.82/137.32 (107) TransformationProof [EQUIVALENT, 0 ms] 275.82/137.32 (108) QDP 275.82/137.32 (109) TransformationProof [EQUIVALENT, 0 ms] 275.82/137.32 (110) QDP 275.82/137.32 (111) TransformationProof [EQUIVALENT, 0 ms] 275.82/137.32 (112) QDP 275.82/137.32 (113) DependencyGraphProof [EQUIVALENT, 0 ms] 275.82/137.32 (114) QDP 275.82/137.32 (115) TransformationProof [EQUIVALENT, 0 ms] 275.82/137.32 (116) QDP 275.82/137.32 (117) TransformationProof [EQUIVALENT, 0 ms] 275.82/137.32 (118) QDP 275.82/137.32 (119) TransformationProof [EQUIVALENT, 0 ms] 275.82/137.32 (120) QDP 275.82/137.32 (121) QDPOrderProof [EQUIVALENT, 14 ms] 275.82/137.32 (122) QDP 275.82/137.32 (123) DependencyGraphProof [EQUIVALENT, 0 ms] 275.82/137.32 (124) QDP 275.82/137.32 (125) QDPOrderProof [EQUIVALENT, 14 ms] 275.82/137.32 (126) QDP 275.82/137.32 (127) DependencyGraphProof [EQUIVALENT, 0 ms] 275.82/137.32 (128) QDP 275.82/137.32 (129) SplitQDPProof [EQUIVALENT, 0 ms] 275.82/137.32 (130) AND 275.82/137.32 (131) QDP 275.82/137.32 (132) SemLabProof [SOUND, 0 ms] 275.82/137.32 (133) QDP 275.82/137.32 (134) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 275.82/137.32 (135) QDP 275.82/137.32 (136) MRRProof [EQUIVALENT, 0 ms] 275.82/137.32 (137) QDP 275.82/137.32 (138) DependencyGraphProof [EQUIVALENT, 0 ms] 275.82/137.32 (139) QDP 275.82/137.32 (140) PisEmptyProof [SOUND, 0 ms] 275.82/137.32 (141) TRUE 275.82/137.32 (142) QDP 275.82/137.32 (143) QReductionProof [EQUIVALENT, 0 ms] 275.82/137.32 (144) QDP 275.82/137.32 (145) SplitQDPProof [EQUIVALENT, 0 ms] 275.82/137.32 (146) AND 275.82/137.32 (147) QDP 275.82/137.32 (148) SemLabProof [SOUND, 0 ms] 275.82/137.32 (149) QDP 275.82/137.32 (150) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 275.82/137.32 (151) QDP 275.82/137.32 (152) MRRProof [EQUIVALENT, 0 ms] 275.82/137.32 (153) QDP 275.82/137.32 (154) DependencyGraphProof [EQUIVALENT, 0 ms] 275.82/137.32 (155) QDP 275.82/137.32 (156) PisEmptyProof [SOUND, 0 ms] 275.82/137.32 (157) TRUE 275.82/137.32 (158) QDP 275.82/137.32 (159) QReductionProof [EQUIVALENT, 0 ms] 275.82/137.32 (160) QDP 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (0) 275.82/137.32 Obligation: 275.82/137.32 Term rewrite system R: 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 a -> f(a) 275.82/137.32 f(f(x)) -> b 275.82/137.32 0(f(x)) -> 1(x) 275.82/137.32 1(f(x)) -> f(0(x)) 275.82/137.32 275.82/137.32 275.82/137.32 275.82/137.32 Outermost Strategy. 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (1) Thiemann-SpecialC-Transformation (EQUIVALENT) 275.82/137.32 We applied the Thiemann-SpecialC transformation to transform the outermost TRS to an innermost TRS. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (2) 275.82/137.32 Obligation: 275.82/137.32 Q restricted rewrite system: 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 top(go_up(x)) -> top(reduce(x)) 275.82/137.32 reduce(f(x_1)) -> check_f(redex_f(x_1)) 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 reduce(a) -> go_up(f(a)) 275.82/137.32 redex_f(f(x)) -> result_f(b) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_f(result_f(x)) -> go_up(x) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 top(go_up(x0)) 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (3) DependencyPairsProof (EQUIVALENT) 275.82/137.32 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (4) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP(go_up(x)) -> TOP(reduce(x)) 275.82/137.32 TOP(go_up(x)) -> REDUCE(x) 275.82/137.32 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 275.82/137.32 REDUCE(f(x_1)) -> REDEX_F(x_1) 275.82/137.32 REDUCE(0(x_1)) -> CHECK_0(redex_0(x_1)) 275.82/137.32 REDUCE(0(x_1)) -> REDEX_0(x_1) 275.82/137.32 REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) 275.82/137.32 REDUCE(1(x_1)) -> REDEX_1(x_1) 275.82/137.32 CHECK_F(redex_f(x_1)) -> IN_F_1(reduce(x_1)) 275.82/137.32 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 275.82/137.32 CHECK_0(redex_0(x_1)) -> IN_0_1(reduce(x_1)) 275.82/137.32 CHECK_0(redex_0(x_1)) -> REDUCE(x_1) 275.82/137.32 CHECK_1(redex_1(x_1)) -> IN_1_1(reduce(x_1)) 275.82/137.32 CHECK_1(redex_1(x_1)) -> REDUCE(x_1) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 top(go_up(x)) -> top(reduce(x)) 275.82/137.32 reduce(f(x_1)) -> check_f(redex_f(x_1)) 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 reduce(a) -> go_up(f(a)) 275.82/137.32 redex_f(f(x)) -> result_f(b) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_f(result_f(x)) -> go_up(x) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 top(go_up(x0)) 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (5) DependencyGraphProof (EQUIVALENT) 275.82/137.32 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 9 less nodes. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (6) 275.82/137.32 Complex Obligation (AND) 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (7) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 REDUCE(0(x_1)) -> CHECK_0(redex_0(x_1)) 275.82/137.32 CHECK_0(redex_0(x_1)) -> REDUCE(x_1) 275.82/137.32 REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) 275.82/137.32 CHECK_1(redex_1(x_1)) -> REDUCE(x_1) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 top(go_up(x)) -> top(reduce(x)) 275.82/137.32 reduce(f(x_1)) -> check_f(redex_f(x_1)) 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 reduce(a) -> go_up(f(a)) 275.82/137.32 redex_f(f(x)) -> result_f(b) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_f(result_f(x)) -> go_up(x) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 top(go_up(x0)) 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (8) UsableRulesProof (EQUIVALENT) 275.82/137.32 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. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (9) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 REDUCE(0(x_1)) -> CHECK_0(redex_0(x_1)) 275.82/137.32 CHECK_0(redex_0(x_1)) -> REDUCE(x_1) 275.82/137.32 REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) 275.82/137.32 CHECK_1(redex_1(x_1)) -> REDUCE(x_1) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 top(go_up(x0)) 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (10) QReductionProof (EQUIVALENT) 275.82/137.32 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 275.82/137.32 275.82/137.32 top(go_up(x0)) 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (11) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 REDUCE(0(x_1)) -> CHECK_0(redex_0(x_1)) 275.82/137.32 CHECK_0(redex_0(x_1)) -> REDUCE(x_1) 275.82/137.32 REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) 275.82/137.32 CHECK_1(redex_1(x_1)) -> REDUCE(x_1) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (12) MRRProof (EQUIVALENT) 275.82/137.32 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. 275.82/137.32 275.82/137.32 Strictly oriented dependency pairs: 275.82/137.32 275.82/137.32 REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) 275.82/137.32 CHECK_1(redex_1(x_1)) -> REDUCE(x_1) 275.82/137.32 275.82/137.32 275.82/137.32 Used ordering: Polynomial interpretation [POLO]: 275.82/137.32 275.82/137.32 POL(0(x_1)) = x_1 275.82/137.32 POL(1(x_1)) = 2 + 2*x_1 275.82/137.32 POL(CHECK_0(x_1)) = 1 + x_1 275.82/137.32 POL(CHECK_1(x_1)) = 2 + 2*x_1 275.82/137.32 POL(REDUCE(x_1)) = 1 + 2*x_1 275.82/137.32 POL(f(x_1)) = 2 + 2*x_1 275.82/137.32 POL(redex_0(x_1)) = 2*x_1 275.82/137.32 POL(redex_1(x_1)) = 2*x_1 275.82/137.32 POL(result_0(x_1)) = 2*x_1 275.82/137.32 POL(result_1(x_1)) = 2 + x_1 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (13) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 REDUCE(0(x_1)) -> CHECK_0(redex_0(x_1)) 275.82/137.32 CHECK_0(redex_0(x_1)) -> REDUCE(x_1) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (14) UsableRulesProof (EQUIVALENT) 275.82/137.32 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. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (15) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 REDUCE(0(x_1)) -> CHECK_0(redex_0(x_1)) 275.82/137.32 CHECK_0(redex_0(x_1)) -> REDUCE(x_1) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (16) QReductionProof (EQUIVALENT) 275.82/137.32 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 275.82/137.32 275.82/137.32 redex_1(f(x0)) 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (17) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 REDUCE(0(x_1)) -> CHECK_0(redex_0(x_1)) 275.82/137.32 CHECK_0(redex_0(x_1)) -> REDUCE(x_1) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 redex_0(f(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (18) UsableRulesReductionPairsProof (EQUIVALENT) 275.82/137.32 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. 275.82/137.32 275.82/137.32 The following dependency pairs can be deleted: 275.82/137.32 275.82/137.32 REDUCE(0(x_1)) -> CHECK_0(redex_0(x_1)) 275.82/137.32 The following rules are removed from R: 275.82/137.32 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 Used ordering: POLO with Polynomial interpretation [POLO]: 275.82/137.32 275.82/137.32 POL(0(x_1)) = 2*x_1 275.82/137.32 POL(1(x_1)) = x_1 275.82/137.32 POL(CHECK_0(x_1)) = x_1 275.82/137.32 POL(REDUCE(x_1)) = 2*x_1 275.82/137.32 POL(f(x_1)) = 2*x_1 275.82/137.32 POL(redex_0(x_1)) = 2*x_1 275.82/137.32 POL(result_0(x_1)) = 2*x_1 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (19) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 CHECK_0(redex_0(x_1)) -> REDUCE(x_1) 275.82/137.32 275.82/137.32 R is empty. 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 redex_0(f(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (20) DependencyGraphProof (EQUIVALENT) 275.82/137.32 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (21) 275.82/137.32 TRUE 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (22) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP(go_up(x)) -> TOP(reduce(x)) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 top(go_up(x)) -> top(reduce(x)) 275.82/137.32 reduce(f(x_1)) -> check_f(redex_f(x_1)) 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 reduce(a) -> go_up(f(a)) 275.82/137.32 redex_f(f(x)) -> result_f(b) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_f(result_f(x)) -> go_up(x) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 top(go_up(x0)) 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (23) UsableRulesProof (EQUIVALENT) 275.82/137.32 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. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (24) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP(go_up(x)) -> TOP(reduce(x)) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 reduce(f(x_1)) -> check_f(redex_f(x_1)) 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 reduce(a) -> go_up(f(a)) 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 redex_f(f(x)) -> result_f(b) 275.82/137.32 check_f(result_f(x)) -> go_up(x) 275.82/137.32 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 top(go_up(x0)) 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (25) QReductionProof (EQUIVALENT) 275.82/137.32 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 275.82/137.32 275.82/137.32 top(go_up(x0)) 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (26) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP(go_up(x)) -> TOP(reduce(x)) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 reduce(f(x_1)) -> check_f(redex_f(x_1)) 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 reduce(a) -> go_up(f(a)) 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 redex_f(f(x)) -> result_f(b) 275.82/137.32 check_f(result_f(x)) -> go_up(x) 275.82/137.32 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (27) TransformationProof (EQUIVALENT) 275.82/137.32 By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]: 275.82/137.32 275.82/137.32 (TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))),TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))) 275.82/137.32 (TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))),TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0)))) 275.82/137.32 (TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))),TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0)))) 275.82/137.32 (TOP(go_up(a)) -> TOP(go_up(f(a))),TOP(go_up(a)) -> TOP(go_up(f(a)))) 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (28) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 275.82/137.32 TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) 275.82/137.32 TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) 275.82/137.32 TOP(go_up(a)) -> TOP(go_up(f(a))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 reduce(f(x_1)) -> check_f(redex_f(x_1)) 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 reduce(a) -> go_up(f(a)) 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 redex_f(f(x)) -> result_f(b) 275.82/137.32 check_f(result_f(x)) -> go_up(x) 275.82/137.32 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (29) UsableRulesProof (EQUIVALENT) 275.82/137.32 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. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (30) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 275.82/137.32 TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) 275.82/137.32 TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) 275.82/137.32 TOP(go_up(a)) -> TOP(go_up(f(a))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 reduce(a) -> go_up(f(a)) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 redex_f(f(x)) -> result_f(b) 275.82/137.32 check_f(result_f(x)) -> go_up(x) 275.82/137.32 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (31) QDPOrderProof (EQUIVALENT) 275.82/137.32 We use the reduction pair processor [LPAR04,JAR06]. 275.82/137.32 275.82/137.32 275.82/137.32 The following pairs can be oriented strictly and are deleted. 275.82/137.32 275.82/137.32 TOP(go_up(a)) -> TOP(go_up(f(a))) 275.82/137.32 The remaining pairs can at least be oriented weakly. 275.82/137.32 Used ordering: Polynomial interpretation [POLO]: 275.82/137.32 275.82/137.32 POL(0(x_1)) = 0 275.82/137.32 POL(1(x_1)) = 0 275.82/137.32 POL(TOP(x_1)) = x_1 275.82/137.32 POL(a) = 1 275.82/137.32 POL(b) = 0 275.82/137.32 POL(check_0(x_1)) = x_1 275.82/137.32 POL(check_1(x_1)) = x_1 275.82/137.32 POL(check_f(x_1)) = x_1 275.82/137.32 POL(f(x_1)) = 0 275.82/137.32 POL(go_up(x_1)) = x_1 275.82/137.32 POL(in_0_1(x_1)) = 0 275.82/137.32 POL(in_1_1(x_1)) = 0 275.82/137.32 POL(in_f_1(x_1)) = 0 275.82/137.32 POL(redex_0(x_1)) = 0 275.82/137.32 POL(redex_1(x_1)) = 0 275.82/137.32 POL(redex_f(x_1)) = 0 275.82/137.32 POL(reduce(x_1)) = 0 275.82/137.32 POL(result_0(x_1)) = x_1 275.82/137.32 POL(result_1(x_1)) = x_1 275.82/137.32 POL(result_f(x_1)) = x_1 275.82/137.32 275.82/137.32 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 275.82/137.32 275.82/137.32 redex_f(f(x)) -> result_f(b) 275.82/137.32 check_f(result_f(x)) -> go_up(x) 275.82/137.32 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (32) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 275.82/137.32 TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) 275.82/137.32 TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 reduce(a) -> go_up(f(a)) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 redex_f(f(x)) -> result_f(b) 275.82/137.32 check_f(result_f(x)) -> go_up(x) 275.82/137.32 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (33) SplitQDPProof (EQUIVALENT) 275.82/137.32 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (34) 275.82/137.32 Complex Obligation (AND) 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (35) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 275.82/137.32 TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) 275.82/137.32 TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 reduce(a) -> go_up(f(a)) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 redex_f(f(x)) -> result_f(b) 275.82/137.32 check_f(result_f(x)) -> go_up(x) 275.82/137.32 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (36) SemLabProof (SOUND) 275.82/137.32 We found the following model for the rules of the TRSs R and P. 275.82/137.32 Interpretation over the domain with elements from 0 to 1. 275.82/137.32 result_f: 0 275.82/137.32 a: 1 275.82/137.32 reduce: 0 275.82/137.32 redex_0: 0 275.82/137.32 check_1: 0 275.82/137.32 redex_f: 0 275.82/137.32 check_f: 0 275.82/137.32 in_1_1: 0 275.82/137.32 TOP: 0 275.82/137.32 redex_1: 0 275.82/137.32 go_up: 0 275.82/137.32 in_f_1: 0 275.82/137.32 b: 0 275.82/137.32 in_0_1: 0 275.82/137.32 result_0: 0 275.82/137.32 f: 0 275.82/137.32 0: 1 275.82/137.32 1: 1 275.82/137.32 check_0: 0 275.82/137.32 result_1: 0 275.82/137.32 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (37) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP.0(go_up.0(f.0(x0))) -> TOP.0(check_f.0(redex_f.0(x0))) 275.82/137.32 TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) 275.82/137.32 TOP.0(go_up.1(0.0(x0))) -> TOP.0(check_0.0(redex_0.0(x0))) 275.82/137.32 TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) 275.82/137.32 TOP.0(go_up.1(1.0(x0))) -> TOP.0(check_1.0(redex_1.0(x0))) 275.82/137.32 TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 redex_1.0(f.0(x)) -> result_1.0(f.1(0.0(x))) 275.82/137.32 redex_1.0(f.1(x)) -> result_1.0(f.1(0.1(x))) 275.82/137.32 check_1.0(result_1.0(x)) -> go_up.0(x) 275.82/137.32 check_1.0(result_1.1(x)) -> go_up.1(x) 275.82/137.32 check_1.0(redex_1.0(x_1)) -> in_1_1.0(reduce.0(x_1)) 275.82/137.32 check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) 275.82/137.32 reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) 275.82/137.32 reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) 275.82/137.32 reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) 275.82/137.32 reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) 275.82/137.32 reduce.1(a.) -> go_up.0(f.1(a.)) 275.82/137.32 in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) 275.82/137.32 in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) 275.82/137.32 redex_0.0(f.0(x)) -> result_0.1(1.0(x)) 275.82/137.32 redex_0.0(f.1(x)) -> result_0.1(1.1(x)) 275.82/137.32 check_0.0(result_0.0(x)) -> go_up.0(x) 275.82/137.32 check_0.0(result_0.1(x)) -> go_up.1(x) 275.82/137.32 check_0.0(redex_0.0(x_1)) -> in_0_1.0(reduce.0(x_1)) 275.82/137.32 check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) 275.82/137.32 in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) 275.82/137.32 in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) 275.82/137.32 redex_f.0(f.0(x)) -> result_f.0(b.) 275.82/137.32 redex_f.0(f.1(x)) -> result_f.0(b.) 275.82/137.32 check_f.0(result_f.0(x)) -> go_up.0(x) 275.82/137.32 check_f.0(result_f.1(x)) -> go_up.1(x) 275.82/137.32 check_f.0(redex_f.0(x_1)) -> in_f_1.0(reduce.0(x_1)) 275.82/137.32 check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) 275.82/137.32 in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) 275.82/137.32 in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce.0(f.0(x0)) 275.82/137.32 reduce.0(f.1(x0)) 275.82/137.32 reduce.1(0.0(x0)) 275.82/137.32 reduce.1(0.1(x0)) 275.82/137.32 reduce.1(1.0(x0)) 275.82/137.32 reduce.1(1.1(x0)) 275.82/137.32 reduce.1(a.) 275.82/137.32 redex_f.0(f.0(x0)) 275.82/137.32 redex_f.0(f.1(x0)) 275.82/137.32 redex_0.0(f.0(x0)) 275.82/137.32 redex_0.0(f.1(x0)) 275.82/137.32 redex_1.0(f.0(x0)) 275.82/137.32 redex_1.0(f.1(x0)) 275.82/137.32 check_f.0(result_f.0(x0)) 275.82/137.32 check_f.0(result_f.1(x0)) 275.82/137.32 check_0.0(result_0.0(x0)) 275.82/137.32 check_0.0(result_0.1(x0)) 275.82/137.32 check_1.0(result_1.0(x0)) 275.82/137.32 check_1.0(result_1.1(x0)) 275.82/137.32 check_f.0(redex_f.0(x0)) 275.82/137.32 check_f.0(redex_f.1(x0)) 275.82/137.32 check_0.0(redex_0.0(x0)) 275.82/137.32 check_0.0(redex_0.1(x0)) 275.82/137.32 check_1.0(redex_1.0(x0)) 275.82/137.32 check_1.0(redex_1.1(x0)) 275.82/137.32 in_f_1.0(go_up.0(x0)) 275.82/137.32 in_f_1.0(go_up.1(x0)) 275.82/137.32 in_0_1.0(go_up.0(x0)) 275.82/137.32 in_0_1.0(go_up.1(x0)) 275.82/137.32 in_1_1.0(go_up.0(x0)) 275.82/137.32 in_1_1.0(go_up.1(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (38) UsableRulesReductionPairsProof (EQUIVALENT) 275.82/137.32 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. 275.82/137.32 275.82/137.32 No dependency pairs are removed. 275.82/137.32 275.82/137.32 The following rules are removed from R: 275.82/137.32 275.82/137.32 check_1.0(result_1.1(x)) -> go_up.1(x) 275.82/137.32 check_0.0(result_0.0(x)) -> go_up.0(x) 275.82/137.32 check_f.0(result_f.1(x)) -> go_up.1(x) 275.82/137.32 Used ordering: POLO with Polynomial interpretation [POLO]: 275.82/137.32 275.82/137.32 POL(0.0(x_1)) = x_1 275.82/137.32 POL(0.1(x_1)) = x_1 275.82/137.32 POL(1.0(x_1)) = x_1 275.82/137.32 POL(1.1(x_1)) = x_1 275.82/137.32 POL(TOP.0(x_1)) = x_1 275.82/137.32 POL(a.) = 0 275.82/137.32 POL(b.) = 0 275.82/137.32 POL(check_0.0(x_1)) = x_1 275.82/137.32 POL(check_1.0(x_1)) = x_1 275.82/137.32 POL(check_f.0(x_1)) = x_1 275.82/137.32 POL(f.0(x_1)) = x_1 275.82/137.32 POL(f.1(x_1)) = x_1 275.82/137.32 POL(go_up.0(x_1)) = x_1 275.82/137.32 POL(go_up.1(x_1)) = x_1 275.82/137.32 POL(in_0_1.0(x_1)) = x_1 275.82/137.32 POL(in_1_1.0(x_1)) = x_1 275.82/137.32 POL(in_f_1.0(x_1)) = x_1 275.82/137.32 POL(redex_0.0(x_1)) = x_1 275.82/137.32 POL(redex_0.1(x_1)) = x_1 275.82/137.32 POL(redex_1.0(x_1)) = x_1 275.82/137.32 POL(redex_1.1(x_1)) = x_1 275.82/137.32 POL(redex_f.0(x_1)) = x_1 275.82/137.32 POL(redex_f.1(x_1)) = x_1 275.82/137.32 POL(reduce.0(x_1)) = x_1 275.82/137.32 POL(reduce.1(x_1)) = x_1 275.82/137.32 POL(result_0.0(x_1)) = x_1 275.82/137.32 POL(result_0.1(x_1)) = x_1 275.82/137.32 POL(result_1.0(x_1)) = x_1 275.82/137.32 POL(result_1.1(x_1)) = x_1 275.82/137.32 POL(result_f.0(x_1)) = x_1 275.82/137.32 POL(result_f.1(x_1)) = x_1 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (39) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP.0(go_up.0(f.0(x0))) -> TOP.0(check_f.0(redex_f.0(x0))) 275.82/137.32 TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) 275.82/137.32 TOP.0(go_up.1(0.0(x0))) -> TOP.0(check_0.0(redex_0.0(x0))) 275.82/137.32 TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) 275.82/137.32 TOP.0(go_up.1(1.0(x0))) -> TOP.0(check_1.0(redex_1.0(x0))) 275.82/137.32 TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) 275.82/137.32 reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) 275.82/137.32 reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) 275.82/137.32 reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) 275.82/137.32 reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) 275.82/137.32 reduce.1(a.) -> go_up.0(f.1(a.)) 275.82/137.32 in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) 275.82/137.32 in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) 275.82/137.32 redex_1.0(f.0(x)) -> result_1.0(f.1(0.0(x))) 275.82/137.32 redex_1.0(f.1(x)) -> result_1.0(f.1(0.1(x))) 275.82/137.32 check_1.0(result_1.0(x)) -> go_up.0(x) 275.82/137.32 check_1.0(redex_1.0(x_1)) -> in_1_1.0(reduce.0(x_1)) 275.82/137.32 check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) 275.82/137.32 in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) 275.82/137.32 in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) 275.82/137.32 redex_0.0(f.0(x)) -> result_0.1(1.0(x)) 275.82/137.32 redex_0.0(f.1(x)) -> result_0.1(1.1(x)) 275.82/137.32 check_0.0(result_0.1(x)) -> go_up.1(x) 275.82/137.32 check_0.0(redex_0.0(x_1)) -> in_0_1.0(reduce.0(x_1)) 275.82/137.32 check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) 275.82/137.32 in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) 275.82/137.32 in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) 275.82/137.32 redex_f.0(f.0(x)) -> result_f.0(b.) 275.82/137.32 redex_f.0(f.1(x)) -> result_f.0(b.) 275.82/137.32 check_f.0(result_f.0(x)) -> go_up.0(x) 275.82/137.32 check_f.0(redex_f.0(x_1)) -> in_f_1.0(reduce.0(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce.0(f.0(x0)) 275.82/137.32 reduce.0(f.1(x0)) 275.82/137.32 reduce.1(0.0(x0)) 275.82/137.32 reduce.1(0.1(x0)) 275.82/137.32 reduce.1(1.0(x0)) 275.82/137.32 reduce.1(1.1(x0)) 275.82/137.32 reduce.1(a.) 275.82/137.32 redex_f.0(f.0(x0)) 275.82/137.32 redex_f.0(f.1(x0)) 275.82/137.32 redex_0.0(f.0(x0)) 275.82/137.32 redex_0.0(f.1(x0)) 275.82/137.32 redex_1.0(f.0(x0)) 275.82/137.32 redex_1.0(f.1(x0)) 275.82/137.32 check_f.0(result_f.0(x0)) 275.82/137.32 check_f.0(result_f.1(x0)) 275.82/137.32 check_0.0(result_0.0(x0)) 275.82/137.32 check_0.0(result_0.1(x0)) 275.82/137.32 check_1.0(result_1.0(x0)) 275.82/137.32 check_1.0(result_1.1(x0)) 275.82/137.32 check_f.0(redex_f.0(x0)) 275.82/137.32 check_f.0(redex_f.1(x0)) 275.82/137.32 check_0.0(redex_0.0(x0)) 275.82/137.32 check_0.0(redex_0.1(x0)) 275.82/137.32 check_1.0(redex_1.0(x0)) 275.82/137.32 check_1.0(redex_1.1(x0)) 275.82/137.32 in_f_1.0(go_up.0(x0)) 275.82/137.32 in_f_1.0(go_up.1(x0)) 275.82/137.32 in_0_1.0(go_up.0(x0)) 275.82/137.32 in_0_1.0(go_up.1(x0)) 275.82/137.32 in_1_1.0(go_up.0(x0)) 275.82/137.32 in_1_1.0(go_up.1(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (40) MRRProof (EQUIVALENT) 275.82/137.32 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. 275.82/137.32 275.82/137.32 275.82/137.32 Strictly oriented rules of the TRS R: 275.82/137.32 275.82/137.32 check_1.0(redex_1.0(x_1)) -> in_1_1.0(reduce.0(x_1)) 275.82/137.32 check_0.0(redex_0.0(x_1)) -> in_0_1.0(reduce.0(x_1)) 275.82/137.32 check_f.0(redex_f.0(x_1)) -> in_f_1.0(reduce.0(x_1)) 275.82/137.32 275.82/137.32 Used ordering: Polynomial interpretation [POLO]: 275.82/137.32 275.82/137.32 POL(0.0(x_1)) = x_1 275.82/137.32 POL(0.1(x_1)) = x_1 275.82/137.32 POL(1.0(x_1)) = x_1 275.82/137.32 POL(1.1(x_1)) = x_1 275.82/137.32 POL(TOP.0(x_1)) = x_1 275.82/137.32 POL(a.) = 0 275.82/137.32 POL(b.) = 0 275.82/137.32 POL(check_0.0(x_1)) = 1 + x_1 275.82/137.32 POL(check_1.0(x_1)) = x_1 275.82/137.32 POL(check_f.0(x_1)) = 1 + x_1 275.82/137.32 POL(f.0(x_1)) = x_1 275.82/137.32 POL(f.1(x_1)) = x_1 275.82/137.32 POL(go_up.0(x_1)) = 1 + x_1 275.82/137.32 POL(go_up.1(x_1)) = 1 + x_1 275.82/137.32 POL(in_0_1.0(x_1)) = x_1 275.82/137.32 POL(in_1_1.0(x_1)) = x_1 275.82/137.32 POL(in_f_1.0(x_1)) = x_1 275.82/137.32 POL(redex_0.0(x_1)) = x_1 275.82/137.32 POL(redex_0.1(x_1)) = x_1 275.82/137.32 POL(redex_1.0(x_1)) = 1 + x_1 275.82/137.32 POL(redex_1.1(x_1)) = 1 + x_1 275.82/137.32 POL(redex_f.0(x_1)) = x_1 275.82/137.32 POL(redex_f.1(x_1)) = x_1 275.82/137.32 POL(reduce.0(x_1)) = x_1 275.82/137.32 POL(reduce.1(x_1)) = 1 + x_1 275.82/137.32 POL(result_0.1(x_1)) = x_1 275.82/137.32 POL(result_1.0(x_1)) = 1 + x_1 275.82/137.32 POL(result_f.0(x_1)) = x_1 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (41) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP.0(go_up.0(f.0(x0))) -> TOP.0(check_f.0(redex_f.0(x0))) 275.82/137.32 TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) 275.82/137.32 TOP.0(go_up.1(0.0(x0))) -> TOP.0(check_0.0(redex_0.0(x0))) 275.82/137.32 TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) 275.82/137.32 TOP.0(go_up.1(1.0(x0))) -> TOP.0(check_1.0(redex_1.0(x0))) 275.82/137.32 TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) 275.82/137.32 reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) 275.82/137.32 reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) 275.82/137.32 reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) 275.82/137.32 reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) 275.82/137.32 reduce.1(a.) -> go_up.0(f.1(a.)) 275.82/137.32 in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) 275.82/137.32 in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) 275.82/137.32 redex_1.0(f.0(x)) -> result_1.0(f.1(0.0(x))) 275.82/137.32 redex_1.0(f.1(x)) -> result_1.0(f.1(0.1(x))) 275.82/137.32 check_1.0(result_1.0(x)) -> go_up.0(x) 275.82/137.32 check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) 275.82/137.32 in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) 275.82/137.32 in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) 275.82/137.32 redex_0.0(f.0(x)) -> result_0.1(1.0(x)) 275.82/137.32 redex_0.0(f.1(x)) -> result_0.1(1.1(x)) 275.82/137.32 check_0.0(result_0.1(x)) -> go_up.1(x) 275.82/137.32 check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) 275.82/137.32 in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) 275.82/137.32 in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) 275.82/137.32 redex_f.0(f.0(x)) -> result_f.0(b.) 275.82/137.32 redex_f.0(f.1(x)) -> result_f.0(b.) 275.82/137.32 check_f.0(result_f.0(x)) -> go_up.0(x) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce.0(f.0(x0)) 275.82/137.32 reduce.0(f.1(x0)) 275.82/137.32 reduce.1(0.0(x0)) 275.82/137.32 reduce.1(0.1(x0)) 275.82/137.32 reduce.1(1.0(x0)) 275.82/137.32 reduce.1(1.1(x0)) 275.82/137.32 reduce.1(a.) 275.82/137.32 redex_f.0(f.0(x0)) 275.82/137.32 redex_f.0(f.1(x0)) 275.82/137.32 redex_0.0(f.0(x0)) 275.82/137.32 redex_0.0(f.1(x0)) 275.82/137.32 redex_1.0(f.0(x0)) 275.82/137.32 redex_1.0(f.1(x0)) 275.82/137.32 check_f.0(result_f.0(x0)) 275.82/137.32 check_f.0(result_f.1(x0)) 275.82/137.32 check_0.0(result_0.0(x0)) 275.82/137.32 check_0.0(result_0.1(x0)) 275.82/137.32 check_1.0(result_1.0(x0)) 275.82/137.32 check_1.0(result_1.1(x0)) 275.82/137.32 check_f.0(redex_f.0(x0)) 275.82/137.32 check_f.0(redex_f.1(x0)) 275.82/137.32 check_0.0(redex_0.0(x0)) 275.82/137.32 check_0.0(redex_0.1(x0)) 275.82/137.32 check_1.0(redex_1.0(x0)) 275.82/137.32 check_1.0(redex_1.1(x0)) 275.82/137.32 in_f_1.0(go_up.0(x0)) 275.82/137.32 in_f_1.0(go_up.1(x0)) 275.82/137.32 in_0_1.0(go_up.0(x0)) 275.82/137.32 in_0_1.0(go_up.1(x0)) 275.82/137.32 in_1_1.0(go_up.0(x0)) 275.82/137.32 in_1_1.0(go_up.1(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (42) QDPOrderProof (EQUIVALENT) 275.82/137.32 We use the reduction pair processor [LPAR04,JAR06]. 275.82/137.32 275.82/137.32 275.82/137.32 The following pairs can be oriented strictly and are deleted. 275.82/137.32 275.82/137.32 TOP.0(go_up.0(f.0(x0))) -> TOP.0(check_f.0(redex_f.0(x0))) 275.82/137.32 The remaining pairs can at least be oriented weakly. 275.82/137.32 Used ordering: Polynomial interpretation [POLO]: 275.82/137.32 275.82/137.32 POL(0.0(x_1)) = x_1 275.82/137.32 POL(0.1(x_1)) = x_1 275.82/137.32 POL(1.0(x_1)) = 0 275.82/137.32 POL(1.1(x_1)) = x_1 275.82/137.32 POL(TOP.0(x_1)) = x_1 275.82/137.32 POL(a.) = 0 275.82/137.32 POL(b.) = 0 275.82/137.32 POL(check_0.0(x_1)) = 1 275.82/137.32 POL(check_1.0(x_1)) = x_1 275.82/137.32 POL(check_f.0(x_1)) = x_1 275.82/137.32 POL(f.0(x_1)) = 1 275.82/137.32 POL(f.1(x_1)) = 1 275.82/137.32 POL(go_up.0(x_1)) = x_1 275.82/137.32 POL(go_up.1(x_1)) = 1 275.82/137.32 POL(in_0_1.0(x_1)) = 1 275.82/137.32 POL(in_1_1.0(x_1)) = 1 275.82/137.32 POL(in_f_1.0(x_1)) = 1 275.82/137.32 POL(redex_0.0(x_1)) = 0 275.82/137.32 POL(redex_0.1(x_1)) = x_1 275.82/137.32 POL(redex_1.0(x_1)) = 1 275.82/137.32 POL(redex_1.1(x_1)) = 1 275.82/137.32 POL(redex_f.0(x_1)) = 0 275.82/137.32 POL(redex_f.1(x_1)) = 1 275.82/137.32 POL(reduce.1(x_1)) = 1 275.82/137.32 POL(result_0.1(x_1)) = 0 275.82/137.32 POL(result_1.0(x_1)) = x_1 275.82/137.32 POL(result_f.0(x_1)) = x_1 275.82/137.32 275.82/137.32 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 275.82/137.32 275.82/137.32 redex_f.0(f.0(x)) -> result_f.0(b.) 275.82/137.32 redex_f.0(f.1(x)) -> result_f.0(b.) 275.82/137.32 check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) 275.82/137.32 check_f.0(result_f.0(x)) -> go_up.0(x) 275.82/137.32 check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) 275.82/137.32 check_0.0(result_0.1(x)) -> go_up.1(x) 275.82/137.32 redex_1.0(f.0(x)) -> result_1.0(f.1(0.0(x))) 275.82/137.32 redex_1.0(f.1(x)) -> result_1.0(f.1(0.1(x))) 275.82/137.32 check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) 275.82/137.32 check_1.0(result_1.0(x)) -> go_up.0(x) 275.82/137.32 reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) 275.82/137.32 reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) 275.82/137.32 reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) 275.82/137.32 reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) 275.82/137.32 in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) 275.82/137.32 in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) 275.82/137.32 in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) 275.82/137.32 in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) 275.82/137.32 in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) 275.82/137.32 in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (43) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) 275.82/137.32 TOP.0(go_up.1(0.0(x0))) -> TOP.0(check_0.0(redex_0.0(x0))) 275.82/137.32 TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) 275.82/137.32 TOP.0(go_up.1(1.0(x0))) -> TOP.0(check_1.0(redex_1.0(x0))) 275.82/137.32 TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) 275.82/137.32 reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) 275.82/137.32 reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) 275.82/137.32 reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) 275.82/137.32 reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) 275.82/137.32 reduce.1(a.) -> go_up.0(f.1(a.)) 275.82/137.32 in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) 275.82/137.32 in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) 275.82/137.32 redex_1.0(f.0(x)) -> result_1.0(f.1(0.0(x))) 275.82/137.32 redex_1.0(f.1(x)) -> result_1.0(f.1(0.1(x))) 275.82/137.32 check_1.0(result_1.0(x)) -> go_up.0(x) 275.82/137.32 check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) 275.82/137.32 in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) 275.82/137.32 in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) 275.82/137.32 redex_0.0(f.0(x)) -> result_0.1(1.0(x)) 275.82/137.32 redex_0.0(f.1(x)) -> result_0.1(1.1(x)) 275.82/137.32 check_0.0(result_0.1(x)) -> go_up.1(x) 275.82/137.32 check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) 275.82/137.32 in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) 275.82/137.32 in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) 275.82/137.32 redex_f.0(f.0(x)) -> result_f.0(b.) 275.82/137.32 redex_f.0(f.1(x)) -> result_f.0(b.) 275.82/137.32 check_f.0(result_f.0(x)) -> go_up.0(x) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce.0(f.0(x0)) 275.82/137.32 reduce.0(f.1(x0)) 275.82/137.32 reduce.1(0.0(x0)) 275.82/137.32 reduce.1(0.1(x0)) 275.82/137.32 reduce.1(1.0(x0)) 275.82/137.32 reduce.1(1.1(x0)) 275.82/137.32 reduce.1(a.) 275.82/137.32 redex_f.0(f.0(x0)) 275.82/137.32 redex_f.0(f.1(x0)) 275.82/137.32 redex_0.0(f.0(x0)) 275.82/137.32 redex_0.0(f.1(x0)) 275.82/137.32 redex_1.0(f.0(x0)) 275.82/137.32 redex_1.0(f.1(x0)) 275.82/137.32 check_f.0(result_f.0(x0)) 275.82/137.32 check_f.0(result_f.1(x0)) 275.82/137.32 check_0.0(result_0.0(x0)) 275.82/137.32 check_0.0(result_0.1(x0)) 275.82/137.32 check_1.0(result_1.0(x0)) 275.82/137.32 check_1.0(result_1.1(x0)) 275.82/137.32 check_f.0(redex_f.0(x0)) 275.82/137.32 check_f.0(redex_f.1(x0)) 275.82/137.32 check_0.0(redex_0.0(x0)) 275.82/137.32 check_0.0(redex_0.1(x0)) 275.82/137.32 check_1.0(redex_1.0(x0)) 275.82/137.32 check_1.0(redex_1.1(x0)) 275.82/137.32 in_f_1.0(go_up.0(x0)) 275.82/137.32 in_f_1.0(go_up.1(x0)) 275.82/137.32 in_0_1.0(go_up.0(x0)) 275.82/137.32 in_0_1.0(go_up.1(x0)) 275.82/137.32 in_1_1.0(go_up.0(x0)) 275.82/137.32 in_1_1.0(go_up.1(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (44) UsableRulesReductionPairsProof (EQUIVALENT) 275.82/137.32 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. 275.82/137.32 275.82/137.32 No dependency pairs are removed. 275.82/137.32 275.82/137.32 The following rules are removed from R: 275.82/137.32 275.82/137.32 redex_f.0(f.0(x)) -> result_f.0(b.) 275.82/137.32 redex_f.0(f.1(x)) -> result_f.0(b.) 275.82/137.32 check_f.0(result_f.0(x)) -> go_up.0(x) 275.82/137.32 Used ordering: POLO with Polynomial interpretation [POLO]: 275.82/137.32 275.82/137.32 POL(0.0(x_1)) = x_1 275.82/137.32 POL(0.1(x_1)) = x_1 275.82/137.32 POL(1.0(x_1)) = x_1 275.82/137.32 POL(1.1(x_1)) = x_1 275.82/137.32 POL(TOP.0(x_1)) = x_1 275.82/137.32 POL(a.) = 0 275.82/137.32 POL(check_0.0(x_1)) = x_1 275.82/137.32 POL(check_1.0(x_1)) = x_1 275.82/137.32 POL(check_f.0(x_1)) = x_1 275.82/137.32 POL(f.0(x_1)) = x_1 275.82/137.32 POL(f.1(x_1)) = x_1 275.82/137.32 POL(go_up.0(x_1)) = x_1 275.82/137.32 POL(go_up.1(x_1)) = x_1 275.82/137.32 POL(in_0_1.0(x_1)) = x_1 275.82/137.32 POL(in_1_1.0(x_1)) = x_1 275.82/137.32 POL(in_f_1.0(x_1)) = x_1 275.82/137.32 POL(redex_0.0(x_1)) = x_1 275.82/137.32 POL(redex_0.1(x_1)) = x_1 275.82/137.32 POL(redex_1.0(x_1)) = x_1 275.82/137.32 POL(redex_1.1(x_1)) = x_1 275.82/137.32 POL(redex_f.1(x_1)) = x_1 275.82/137.32 POL(reduce.1(x_1)) = x_1 275.82/137.32 POL(result_0.1(x_1)) = x_1 275.82/137.32 POL(result_1.0(x_1)) = x_1 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (45) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) 275.82/137.32 TOP.0(go_up.1(0.0(x0))) -> TOP.0(check_0.0(redex_0.0(x0))) 275.82/137.32 TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) 275.82/137.32 TOP.0(go_up.1(1.0(x0))) -> TOP.0(check_1.0(redex_1.0(x0))) 275.82/137.32 TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) 275.82/137.32 reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) 275.82/137.32 reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) 275.82/137.32 reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) 275.82/137.32 reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) 275.82/137.32 reduce.1(a.) -> go_up.0(f.1(a.)) 275.82/137.32 in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) 275.82/137.32 in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) 275.82/137.32 redex_1.0(f.0(x)) -> result_1.0(f.1(0.0(x))) 275.82/137.32 redex_1.0(f.1(x)) -> result_1.0(f.1(0.1(x))) 275.82/137.32 check_1.0(result_1.0(x)) -> go_up.0(x) 275.82/137.32 check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) 275.82/137.32 in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) 275.82/137.32 in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) 275.82/137.32 redex_0.0(f.0(x)) -> result_0.1(1.0(x)) 275.82/137.32 redex_0.0(f.1(x)) -> result_0.1(1.1(x)) 275.82/137.32 check_0.0(result_0.1(x)) -> go_up.1(x) 275.82/137.32 check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) 275.82/137.32 in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) 275.82/137.32 in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce.0(f.0(x0)) 275.82/137.32 reduce.0(f.1(x0)) 275.82/137.32 reduce.1(0.0(x0)) 275.82/137.32 reduce.1(0.1(x0)) 275.82/137.32 reduce.1(1.0(x0)) 275.82/137.32 reduce.1(1.1(x0)) 275.82/137.32 reduce.1(a.) 275.82/137.32 redex_f.0(f.0(x0)) 275.82/137.32 redex_f.0(f.1(x0)) 275.82/137.32 redex_0.0(f.0(x0)) 275.82/137.32 redex_0.0(f.1(x0)) 275.82/137.32 redex_1.0(f.0(x0)) 275.82/137.32 redex_1.0(f.1(x0)) 275.82/137.32 check_f.0(result_f.0(x0)) 275.82/137.32 check_f.0(result_f.1(x0)) 275.82/137.32 check_0.0(result_0.0(x0)) 275.82/137.32 check_0.0(result_0.1(x0)) 275.82/137.32 check_1.0(result_1.0(x0)) 275.82/137.32 check_1.0(result_1.1(x0)) 275.82/137.32 check_f.0(redex_f.0(x0)) 275.82/137.32 check_f.0(redex_f.1(x0)) 275.82/137.32 check_0.0(redex_0.0(x0)) 275.82/137.32 check_0.0(redex_0.1(x0)) 275.82/137.32 check_1.0(redex_1.0(x0)) 275.82/137.32 check_1.0(redex_1.1(x0)) 275.82/137.32 in_f_1.0(go_up.0(x0)) 275.82/137.32 in_f_1.0(go_up.1(x0)) 275.82/137.32 in_0_1.0(go_up.0(x0)) 275.82/137.32 in_0_1.0(go_up.1(x0)) 275.82/137.32 in_1_1.0(go_up.0(x0)) 275.82/137.32 in_1_1.0(go_up.1(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (46) QDPOrderProof (EQUIVALENT) 275.82/137.32 We use the reduction pair processor [LPAR04,JAR06]. 275.82/137.32 275.82/137.32 275.82/137.32 The following pairs can be oriented strictly and are deleted. 275.82/137.32 275.82/137.32 TOP.0(go_up.1(0.0(x0))) -> TOP.0(check_0.0(redex_0.0(x0))) 275.82/137.32 The remaining pairs can at least be oriented weakly. 275.82/137.32 Used ordering: Polynomial interpretation [POLO]: 275.82/137.32 275.82/137.32 POL(0.0(x_1)) = 1 275.82/137.32 POL(0.1(x_1)) = 1 275.82/137.32 POL(1.0(x_1)) = 0 275.82/137.32 POL(1.1(x_1)) = 0 275.82/137.32 POL(TOP.0(x_1)) = x_1 275.82/137.32 POL(a.) = 0 275.82/137.32 POL(check_0.0(x_1)) = x_1 275.82/137.32 POL(check_1.0(x_1)) = 0 275.82/137.32 POL(check_f.0(x_1)) = x_1 275.82/137.32 POL(f.0(x_1)) = x_1 275.82/137.32 POL(f.1(x_1)) = 0 275.82/137.32 POL(go_up.0(x_1)) = 0 275.82/137.32 POL(go_up.1(x_1)) = x_1 275.82/137.32 POL(in_0_1.0(x_1)) = 1 275.82/137.32 POL(in_1_1.0(x_1)) = 0 275.82/137.32 POL(in_f_1.0(x_1)) = 0 275.82/137.32 POL(redex_0.0(x_1)) = 0 275.82/137.32 POL(redex_0.1(x_1)) = 1 275.82/137.32 POL(redex_1.0(x_1)) = 0 275.82/137.32 POL(redex_1.1(x_1)) = x_1 275.82/137.32 POL(redex_f.1(x_1)) = 0 275.82/137.32 POL(reduce.1(x_1)) = x_1 275.82/137.32 POL(result_0.1(x_1)) = x_1 275.82/137.32 POL(result_1.0(x_1)) = 0 275.82/137.32 275.82/137.32 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 275.82/137.32 275.82/137.32 check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) 275.82/137.32 redex_0.0(f.0(x)) -> result_0.1(1.0(x)) 275.82/137.32 redex_0.0(f.1(x)) -> result_0.1(1.1(x)) 275.82/137.32 check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) 275.82/137.32 check_0.0(result_0.1(x)) -> go_up.1(x) 275.82/137.32 check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) 275.82/137.32 check_1.0(result_1.0(x)) -> go_up.0(x) 275.82/137.32 reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) 275.82/137.32 reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) 275.82/137.32 reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) 275.82/137.32 reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) 275.82/137.32 in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) 275.82/137.32 in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) 275.82/137.32 in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) 275.82/137.32 in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) 275.82/137.32 in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) 275.82/137.32 in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (47) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) 275.82/137.32 TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) 275.82/137.32 TOP.0(go_up.1(1.0(x0))) -> TOP.0(check_1.0(redex_1.0(x0))) 275.82/137.32 TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) 275.82/137.32 reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) 275.82/137.32 reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) 275.82/137.32 reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) 275.82/137.32 reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) 275.82/137.32 reduce.1(a.) -> go_up.0(f.1(a.)) 275.82/137.32 in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) 275.82/137.32 in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) 275.82/137.32 redex_1.0(f.0(x)) -> result_1.0(f.1(0.0(x))) 275.82/137.32 redex_1.0(f.1(x)) -> result_1.0(f.1(0.1(x))) 275.82/137.32 check_1.0(result_1.0(x)) -> go_up.0(x) 275.82/137.32 check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) 275.82/137.32 in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) 275.82/137.32 in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) 275.82/137.32 redex_0.0(f.0(x)) -> result_0.1(1.0(x)) 275.82/137.32 redex_0.0(f.1(x)) -> result_0.1(1.1(x)) 275.82/137.32 check_0.0(result_0.1(x)) -> go_up.1(x) 275.82/137.32 check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) 275.82/137.32 in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) 275.82/137.32 in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce.0(f.0(x0)) 275.82/137.32 reduce.0(f.1(x0)) 275.82/137.32 reduce.1(0.0(x0)) 275.82/137.32 reduce.1(0.1(x0)) 275.82/137.32 reduce.1(1.0(x0)) 275.82/137.32 reduce.1(1.1(x0)) 275.82/137.32 reduce.1(a.) 275.82/137.32 redex_f.0(f.0(x0)) 275.82/137.32 redex_f.0(f.1(x0)) 275.82/137.32 redex_0.0(f.0(x0)) 275.82/137.32 redex_0.0(f.1(x0)) 275.82/137.32 redex_1.0(f.0(x0)) 275.82/137.32 redex_1.0(f.1(x0)) 275.82/137.32 check_f.0(result_f.0(x0)) 275.82/137.32 check_f.0(result_f.1(x0)) 275.82/137.32 check_0.0(result_0.0(x0)) 275.82/137.32 check_0.0(result_0.1(x0)) 275.82/137.32 check_1.0(result_1.0(x0)) 275.82/137.32 check_1.0(result_1.1(x0)) 275.82/137.32 check_f.0(redex_f.0(x0)) 275.82/137.32 check_f.0(redex_f.1(x0)) 275.82/137.32 check_0.0(redex_0.0(x0)) 275.82/137.32 check_0.0(redex_0.1(x0)) 275.82/137.32 check_1.0(redex_1.0(x0)) 275.82/137.32 check_1.0(redex_1.1(x0)) 275.82/137.32 in_f_1.0(go_up.0(x0)) 275.82/137.32 in_f_1.0(go_up.1(x0)) 275.82/137.32 in_0_1.0(go_up.0(x0)) 275.82/137.32 in_0_1.0(go_up.1(x0)) 275.82/137.32 in_1_1.0(go_up.0(x0)) 275.82/137.32 in_1_1.0(go_up.1(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (48) QDPOrderProof (EQUIVALENT) 275.82/137.32 We use the reduction pair processor [LPAR04,JAR06]. 275.82/137.32 275.82/137.32 275.82/137.32 The following pairs can be oriented strictly and are deleted. 275.82/137.32 275.82/137.32 TOP.0(go_up.1(1.0(x0))) -> TOP.0(check_1.0(redex_1.0(x0))) 275.82/137.32 The remaining pairs can at least be oriented weakly. 275.82/137.32 Used ordering: Polynomial interpretation [POLO]: 275.82/137.32 275.82/137.32 POL(0.0(x_1)) = 0 275.82/137.32 POL(0.1(x_1)) = 0 275.82/137.32 POL(1.0(x_1)) = 1 275.82/137.32 POL(1.1(x_1)) = 1 275.82/137.32 POL(TOP.0(x_1)) = x_1 275.82/137.32 POL(a.) = 0 275.82/137.32 POL(check_0.0(x_1)) = x_1 275.82/137.32 POL(check_1.0(x_1)) = x_1 275.82/137.32 POL(check_f.0(x_1)) = x_1 275.82/137.32 POL(f.0(x_1)) = x_1 275.82/137.32 POL(f.1(x_1)) = 0 275.82/137.32 POL(go_up.0(x_1)) = 0 275.82/137.32 POL(go_up.1(x_1)) = x_1 275.82/137.32 POL(in_0_1.0(x_1)) = 0 275.82/137.32 POL(in_1_1.0(x_1)) = 1 275.82/137.32 POL(in_f_1.0(x_1)) = 0 275.82/137.32 POL(redex_0.0(x_1)) = 1 275.82/137.32 POL(redex_0.1(x_1)) = 0 275.82/137.32 POL(redex_1.0(x_1)) = 0 275.82/137.32 POL(redex_1.1(x_1)) = 1 275.82/137.32 POL(redex_f.1(x_1)) = 0 275.82/137.32 POL(reduce.1(x_1)) = 1 275.82/137.32 POL(result_0.1(x_1)) = x_1 275.82/137.32 POL(result_1.0(x_1)) = 0 275.82/137.32 275.82/137.32 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 275.82/137.32 275.82/137.32 check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) 275.82/137.32 check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) 275.82/137.32 redex_1.0(f.0(x)) -> result_1.0(f.1(0.0(x))) 275.82/137.32 redex_1.0(f.1(x)) -> result_1.0(f.1(0.1(x))) 275.82/137.32 check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) 275.82/137.32 check_1.0(result_1.0(x)) -> go_up.0(x) 275.82/137.32 reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) 275.82/137.32 reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) 275.82/137.32 reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) 275.82/137.32 reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) 275.82/137.32 in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) 275.82/137.32 in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) 275.82/137.32 redex_0.0(f.0(x)) -> result_0.1(1.0(x)) 275.82/137.32 redex_0.0(f.1(x)) -> result_0.1(1.1(x)) 275.82/137.32 check_0.0(result_0.1(x)) -> go_up.1(x) 275.82/137.32 in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) 275.82/137.32 in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) 275.82/137.32 in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) 275.82/137.32 in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (49) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) 275.82/137.32 TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) 275.82/137.32 TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) 275.82/137.32 reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) 275.82/137.32 reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) 275.82/137.32 reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) 275.82/137.32 reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) 275.82/137.32 reduce.1(a.) -> go_up.0(f.1(a.)) 275.82/137.32 in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) 275.82/137.32 in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) 275.82/137.32 redex_1.0(f.0(x)) -> result_1.0(f.1(0.0(x))) 275.82/137.32 redex_1.0(f.1(x)) -> result_1.0(f.1(0.1(x))) 275.82/137.32 check_1.0(result_1.0(x)) -> go_up.0(x) 275.82/137.32 check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) 275.82/137.32 in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) 275.82/137.32 in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) 275.82/137.32 redex_0.0(f.0(x)) -> result_0.1(1.0(x)) 275.82/137.32 redex_0.0(f.1(x)) -> result_0.1(1.1(x)) 275.82/137.32 check_0.0(result_0.1(x)) -> go_up.1(x) 275.82/137.32 check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) 275.82/137.32 in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) 275.82/137.32 in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce.0(f.0(x0)) 275.82/137.32 reduce.0(f.1(x0)) 275.82/137.32 reduce.1(0.0(x0)) 275.82/137.32 reduce.1(0.1(x0)) 275.82/137.32 reduce.1(1.0(x0)) 275.82/137.32 reduce.1(1.1(x0)) 275.82/137.32 reduce.1(a.) 275.82/137.32 redex_f.0(f.0(x0)) 275.82/137.32 redex_f.0(f.1(x0)) 275.82/137.32 redex_0.0(f.0(x0)) 275.82/137.32 redex_0.0(f.1(x0)) 275.82/137.32 redex_1.0(f.0(x0)) 275.82/137.32 redex_1.0(f.1(x0)) 275.82/137.32 check_f.0(result_f.0(x0)) 275.82/137.32 check_f.0(result_f.1(x0)) 275.82/137.32 check_0.0(result_0.0(x0)) 275.82/137.32 check_0.0(result_0.1(x0)) 275.82/137.32 check_1.0(result_1.0(x0)) 275.82/137.32 check_1.0(result_1.1(x0)) 275.82/137.32 check_f.0(redex_f.0(x0)) 275.82/137.32 check_f.0(redex_f.1(x0)) 275.82/137.32 check_0.0(redex_0.0(x0)) 275.82/137.32 check_0.0(redex_0.1(x0)) 275.82/137.32 check_1.0(redex_1.0(x0)) 275.82/137.32 check_1.0(redex_1.1(x0)) 275.82/137.32 in_f_1.0(go_up.0(x0)) 275.82/137.32 in_f_1.0(go_up.1(x0)) 275.82/137.32 in_0_1.0(go_up.0(x0)) 275.82/137.32 in_0_1.0(go_up.1(x0)) 275.82/137.32 in_1_1.0(go_up.0(x0)) 275.82/137.32 in_1_1.0(go_up.1(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (50) PisEmptyProof (SOUND) 275.82/137.32 The TRS P is empty. Hence, there is no (P,Q,R) chain. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (51) 275.82/137.32 TRUE 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (52) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 275.82/137.32 TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) 275.82/137.32 TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 reduce(a) -> go_up(f(a)) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (53) TransformationProof (SOUND) 275.82/137.32 By rewriting [LPAR04] the rule TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) at position [0] we obtained the following new rules [LPAR04]: 275.82/137.32 275.82/137.32 (TOP(go_up(f(x0))) -> TOP(in_f_1(reduce(x0))),TOP(go_up(f(x0))) -> TOP(in_f_1(reduce(x0)))) 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (54) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) 275.82/137.32 TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) 275.82/137.32 TOP(go_up(f(x0))) -> TOP(in_f_1(reduce(x0))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 reduce(a) -> go_up(f(a)) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (55) UsableRulesProof (EQUIVALENT) 275.82/137.32 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. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (56) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.32 The TRS P consists of the following rules: 275.82/137.32 275.82/137.32 TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) 275.82/137.32 TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) 275.82/137.32 TOP(go_up(f(x0))) -> TOP(in_f_1(reduce(x0))) 275.82/137.32 275.82/137.32 The TRS R consists of the following rules: 275.82/137.32 275.82/137.32 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.32 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.32 reduce(a) -> go_up(f(a)) 275.82/137.32 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.32 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.32 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.32 check_1(result_1(x)) -> go_up(x) 275.82/137.32 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.32 redex_0(f(x)) -> result_0(1(x)) 275.82/137.32 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.32 check_0(result_0(x)) -> go_up(x) 275.82/137.32 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.32 275.82/137.32 The set Q consists of the following terms: 275.82/137.32 275.82/137.32 reduce(f(x0)) 275.82/137.32 reduce(0(x0)) 275.82/137.32 reduce(1(x0)) 275.82/137.32 reduce(a) 275.82/137.32 redex_f(f(x0)) 275.82/137.32 redex_0(f(x0)) 275.82/137.32 redex_1(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_0(result_0(x0)) 275.82/137.32 check_1(result_1(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 check_0(redex_0(x0)) 275.82/137.32 check_1(redex_1(x0)) 275.82/137.32 in_f_1(go_up(x0)) 275.82/137.32 in_0_1(go_up(x0)) 275.82/137.32 in_1_1(go_up(x0)) 275.82/137.32 275.82/137.32 We have to consider all minimal (P,Q,R)-chains. 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (57) QReductionProof (EQUIVALENT) 275.82/137.32 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 275.82/137.32 275.82/137.32 redex_f(f(x0)) 275.82/137.32 check_f(result_f(x0)) 275.82/137.32 check_f(redex_f(x0)) 275.82/137.32 275.82/137.32 275.82/137.32 ---------------------------------------- 275.82/137.32 275.82/137.32 (58) 275.82/137.32 Obligation: 275.82/137.32 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) 275.82/137.33 TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) 275.82/137.33 TOP(go_up(f(x0))) -> TOP(in_f_1(reduce(x0))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.33 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.33 reduce(a) -> go_up(f(a)) 275.82/137.33 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.33 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.33 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.33 check_1(result_1(x)) -> go_up(x) 275.82/137.33 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.33 redex_0(f(x)) -> result_0(1(x)) 275.82/137.33 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.33 check_0(result_0(x)) -> go_up(x) 275.82/137.33 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 reduce(f(x0)) 275.82/137.33 reduce(0(x0)) 275.82/137.33 reduce(1(x0)) 275.82/137.33 reduce(a) 275.82/137.33 redex_0(f(x0)) 275.82/137.33 redex_1(f(x0)) 275.82/137.33 check_0(result_0(x0)) 275.82/137.33 check_1(result_1(x0)) 275.82/137.33 check_0(redex_0(x0)) 275.82/137.33 check_1(redex_1(x0)) 275.82/137.33 in_f_1(go_up(x0)) 275.82/137.33 in_0_1(go_up(x0)) 275.82/137.33 in_1_1(go_up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (59) UsableRulesProof (EQUIVALENT) 275.82/137.33 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. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (60) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(go_up(x)) -> TOP(reduce(x)) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 reduce(f(x_1)) -> check_f(redex_f(x_1)) 275.82/137.33 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.33 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.33 reduce(a) -> go_up(f(a)) 275.82/137.33 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.33 check_1(result_1(x)) -> go_up(x) 275.82/137.33 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.33 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.33 redex_0(f(x)) -> result_0(1(x)) 275.82/137.33 check_0(result_0(x)) -> go_up(x) 275.82/137.33 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.33 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.33 redex_f(f(x)) -> result_f(b) 275.82/137.33 check_f(result_f(x)) -> go_up(x) 275.82/137.33 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.33 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 top(go_up(x0)) 275.82/137.33 reduce(f(x0)) 275.82/137.33 reduce(0(x0)) 275.82/137.33 reduce(1(x0)) 275.82/137.33 reduce(a) 275.82/137.33 redex_f(f(x0)) 275.82/137.33 redex_0(f(x0)) 275.82/137.33 redex_1(f(x0)) 275.82/137.33 check_f(result_f(x0)) 275.82/137.33 check_0(result_0(x0)) 275.82/137.33 check_1(result_1(x0)) 275.82/137.33 check_f(redex_f(x0)) 275.82/137.33 check_0(redex_0(x0)) 275.82/137.33 check_1(redex_1(x0)) 275.82/137.33 in_f_1(go_up(x0)) 275.82/137.33 in_0_1(go_up(x0)) 275.82/137.33 in_1_1(go_up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (61) QReductionProof (EQUIVALENT) 275.82/137.33 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 275.82/137.33 275.82/137.33 top(go_up(x0)) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (62) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(go_up(x)) -> TOP(reduce(x)) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 reduce(f(x_1)) -> check_f(redex_f(x_1)) 275.82/137.33 reduce(0(x_1)) -> check_0(redex_0(x_1)) 275.82/137.33 reduce(1(x_1)) -> check_1(redex_1(x_1)) 275.82/137.33 reduce(a) -> go_up(f(a)) 275.82/137.33 redex_1(f(x)) -> result_1(f(0(x))) 275.82/137.33 check_1(result_1(x)) -> go_up(x) 275.82/137.33 check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) 275.82/137.33 in_1_1(go_up(x_1)) -> go_up(1(x_1)) 275.82/137.33 redex_0(f(x)) -> result_0(1(x)) 275.82/137.33 check_0(result_0(x)) -> go_up(x) 275.82/137.33 check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) 275.82/137.33 in_0_1(go_up(x_1)) -> go_up(0(x_1)) 275.82/137.33 redex_f(f(x)) -> result_f(b) 275.82/137.33 check_f(result_f(x)) -> go_up(x) 275.82/137.33 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 275.82/137.33 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 reduce(f(x0)) 275.82/137.33 reduce(0(x0)) 275.82/137.33 reduce(1(x0)) 275.82/137.33 reduce(a) 275.82/137.33 redex_f(f(x0)) 275.82/137.33 redex_0(f(x0)) 275.82/137.33 redex_1(f(x0)) 275.82/137.33 check_f(result_f(x0)) 275.82/137.33 check_0(result_0(x0)) 275.82/137.33 check_1(result_1(x0)) 275.82/137.33 check_f(redex_f(x0)) 275.82/137.33 check_0(redex_0(x0)) 275.82/137.33 check_1(redex_1(x0)) 275.82/137.33 in_f_1(go_up(x0)) 275.82/137.33 in_0_1(go_up(x0)) 275.82/137.33 in_1_1(go_up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (63) Trivial-Transformation (SOUND) 275.82/137.33 We applied the Trivial transformation to transform the outermost TRS to a standard TRS. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (64) 275.82/137.33 Obligation: 275.82/137.33 Q restricted rewrite system: 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 a -> f(a) 275.82/137.33 f(f(x)) -> b 275.82/137.33 0(f(x)) -> 1(x) 275.82/137.33 1(f(x)) -> f(0(x)) 275.82/137.33 275.82/137.33 Q is empty. 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (65) DependencyPairsProof (EQUIVALENT) 275.82/137.33 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (66) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 A -> F(a) 275.82/137.33 A -> A 275.82/137.33 0^1(f(x)) -> 1^1(x) 275.82/137.33 1^1(f(x)) -> F(0(x)) 275.82/137.33 1^1(f(x)) -> 0^1(x) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 a -> f(a) 275.82/137.33 f(f(x)) -> b 275.82/137.33 0(f(x)) -> 1(x) 275.82/137.33 1(f(x)) -> f(0(x)) 275.82/137.33 275.82/137.33 Q is empty. 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (67) DependencyGraphProof (EQUIVALENT) 275.82/137.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (68) 275.82/137.33 Complex Obligation (AND) 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (69) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 1^1(f(x)) -> 0^1(x) 275.82/137.33 0^1(f(x)) -> 1^1(x) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 a -> f(a) 275.82/137.33 f(f(x)) -> b 275.82/137.33 0(f(x)) -> 1(x) 275.82/137.33 1(f(x)) -> f(0(x)) 275.82/137.33 275.82/137.33 Q is empty. 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (70) UsableRulesProof (EQUIVALENT) 275.82/137.33 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. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (71) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 1^1(f(x)) -> 0^1(x) 275.82/137.33 0^1(f(x)) -> 1^1(x) 275.82/137.33 275.82/137.33 R is empty. 275.82/137.33 Q is empty. 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (72) QDPSizeChangeProof (EQUIVALENT) 275.82/137.33 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. 275.82/137.33 275.82/137.33 From the DPs we obtained the following set of size-change graphs: 275.82/137.33 *0^1(f(x)) -> 1^1(x) 275.82/137.33 The graph contains the following edges 1 > 1 275.82/137.33 275.82/137.33 275.82/137.33 *1^1(f(x)) -> 0^1(x) 275.82/137.33 The graph contains the following edges 1 > 1 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (73) 275.82/137.33 YES 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (74) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 A -> A 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 a -> f(a) 275.82/137.33 f(f(x)) -> b 275.82/137.33 0(f(x)) -> 1(x) 275.82/137.33 1(f(x)) -> f(0(x)) 275.82/137.33 275.82/137.33 Q is empty. 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (75) UsableRulesProof (EQUIVALENT) 275.82/137.33 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. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (76) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 A -> A 275.82/137.33 275.82/137.33 R is empty. 275.82/137.33 Q is empty. 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (77) NonTerminationLoopProof (COMPLETE) 275.82/137.33 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 275.82/137.33 Found a loop by semiunifying a rule from P directly. 275.82/137.33 275.82/137.33 s = A evaluates to t =A 275.82/137.33 275.82/137.33 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 275.82/137.33 * Matcher: [ ] 275.82/137.33 * Semiunifier: [ ] 275.82/137.33 275.82/137.33 -------------------------------------------------------------------------------- 275.82/137.33 Rewriting sequence 275.82/137.33 275.82/137.33 The DP semiunifies directly so there is only one rewrite step from A to A. 275.82/137.33 275.82/137.33 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (78) 275.82/137.33 NO 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (79) Raffelsieper-Zantema-Transformation (SOUND) 275.82/137.33 We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (80) 275.82/137.33 Obligation: 275.82/137.33 Q restricted rewrite system: 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 down(f(f(x))) -> up(b) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 top(up(x)) -> top(down(x)) 275.82/137.33 down(f(a)) -> f_flat(down(a)) 275.82/137.33 down(f(b)) -> f_flat(down(b)) 275.82/137.33 down(f(0(y5))) -> f_flat(down(0(y5))) 275.82/137.33 down(f(1(y6))) -> f_flat(down(1(y6))) 275.82/137.33 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 275.82/137.33 Q is empty. 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (81) AAECC Innermost (EQUIVALENT) 275.82/137.33 We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 275.82/137.33 down(f(a)) -> f_flat(down(a)) 275.82/137.33 down(f(b)) -> f_flat(down(b)) 275.82/137.33 down(f(0(y5))) -> f_flat(down(0(y5))) 275.82/137.33 down(f(1(y6))) -> f_flat(down(1(y6))) 275.82/137.33 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 down(f(f(x))) -> up(b) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 275.82/137.33 The TRS R 2 is 275.82/137.33 top(up(x)) -> top(down(x)) 275.82/137.33 275.82/137.33 The signature Sigma is {top_1} 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (82) 275.82/137.33 Obligation: 275.82/137.33 Q restricted rewrite system: 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 down(f(f(x))) -> up(b) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 top(up(x)) -> top(down(x)) 275.82/137.33 down(f(a)) -> f_flat(down(a)) 275.82/137.33 down(f(b)) -> f_flat(down(b)) 275.82/137.33 down(f(0(y5))) -> f_flat(down(0(y5))) 275.82/137.33 down(f(1(y6))) -> f_flat(down(1(y6))) 275.82/137.33 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 top(up(x0)) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (83) DependencyPairsProof (EQUIVALENT) 275.82/137.33 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (84) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(x)) -> TOP(down(x)) 275.82/137.33 TOP(up(x)) -> DOWN(x) 275.82/137.33 DOWN(f(a)) -> F_FLAT(down(a)) 275.82/137.33 DOWN(f(a)) -> DOWN(a) 275.82/137.33 DOWN(f(b)) -> F_FLAT(down(b)) 275.82/137.33 DOWN(f(b)) -> DOWN(b) 275.82/137.33 DOWN(f(0(y5))) -> F_FLAT(down(0(y5))) 275.82/137.33 DOWN(f(0(y5))) -> DOWN(0(y5)) 275.82/137.33 DOWN(f(1(y6))) -> F_FLAT(down(1(y6))) 275.82/137.33 DOWN(f(1(y6))) -> DOWN(1(y6)) 275.82/137.33 DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant)) 275.82/137.33 DOWN(f(fresh_constant)) -> DOWN(fresh_constant) 275.82/137.33 DOWN(0(a)) -> 0_FLAT(down(a)) 275.82/137.33 DOWN(0(a)) -> DOWN(a) 275.82/137.33 DOWN(0(b)) -> 0_FLAT(down(b)) 275.82/137.33 DOWN(0(b)) -> DOWN(b) 275.82/137.33 DOWN(0(0(y9))) -> 0_FLAT(down(0(y9))) 275.82/137.33 DOWN(0(0(y9))) -> DOWN(0(y9)) 275.82/137.33 DOWN(0(1(y10))) -> 0_FLAT(down(1(y10))) 275.82/137.33 DOWN(0(1(y10))) -> DOWN(1(y10)) 275.82/137.33 DOWN(0(fresh_constant)) -> 0_FLAT(down(fresh_constant)) 275.82/137.33 DOWN(0(fresh_constant)) -> DOWN(fresh_constant) 275.82/137.33 DOWN(1(a)) -> 1_FLAT(down(a)) 275.82/137.33 DOWN(1(a)) -> DOWN(a) 275.82/137.33 DOWN(1(b)) -> 1_FLAT(down(b)) 275.82/137.33 DOWN(1(b)) -> DOWN(b) 275.82/137.33 DOWN(1(0(y13))) -> 1_FLAT(down(0(y13))) 275.82/137.33 DOWN(1(0(y13))) -> DOWN(0(y13)) 275.82/137.33 DOWN(1(1(y14))) -> 1_FLAT(down(1(y14))) 275.82/137.33 DOWN(1(1(y14))) -> DOWN(1(y14)) 275.82/137.33 DOWN(1(fresh_constant)) -> 1_FLAT(down(fresh_constant)) 275.82/137.33 DOWN(1(fresh_constant)) -> DOWN(fresh_constant) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 down(f(f(x))) -> up(b) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 top(up(x)) -> top(down(x)) 275.82/137.33 down(f(a)) -> f_flat(down(a)) 275.82/137.33 down(f(b)) -> f_flat(down(b)) 275.82/137.33 down(f(0(y5))) -> f_flat(down(0(y5))) 275.82/137.33 down(f(1(y6))) -> f_flat(down(1(y6))) 275.82/137.33 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 top(up(x0)) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (85) DependencyGraphProof (EQUIVALENT) 275.82/137.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 27 less nodes. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (86) 275.82/137.33 Complex Obligation (AND) 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (87) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 DOWN(0(1(y10))) -> DOWN(1(y10)) 275.82/137.33 DOWN(1(0(y13))) -> DOWN(0(y13)) 275.82/137.33 DOWN(0(0(y9))) -> DOWN(0(y9)) 275.82/137.33 DOWN(1(1(y14))) -> DOWN(1(y14)) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 down(f(f(x))) -> up(b) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 top(up(x)) -> top(down(x)) 275.82/137.33 down(f(a)) -> f_flat(down(a)) 275.82/137.33 down(f(b)) -> f_flat(down(b)) 275.82/137.33 down(f(0(y5))) -> f_flat(down(0(y5))) 275.82/137.33 down(f(1(y6))) -> f_flat(down(1(y6))) 275.82/137.33 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 top(up(x0)) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (88) UsableRulesProof (EQUIVALENT) 275.82/137.33 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. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (89) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 DOWN(0(1(y10))) -> DOWN(1(y10)) 275.82/137.33 DOWN(1(0(y13))) -> DOWN(0(y13)) 275.82/137.33 DOWN(0(0(y9))) -> DOWN(0(y9)) 275.82/137.33 DOWN(1(1(y14))) -> DOWN(1(y14)) 275.82/137.33 275.82/137.33 R is empty. 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 top(up(x0)) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (90) QReductionProof (EQUIVALENT) 275.82/137.33 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 top(up(x0)) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (91) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 DOWN(0(1(y10))) -> DOWN(1(y10)) 275.82/137.33 DOWN(1(0(y13))) -> DOWN(0(y13)) 275.82/137.33 DOWN(0(0(y9))) -> DOWN(0(y9)) 275.82/137.33 DOWN(1(1(y14))) -> DOWN(1(y14)) 275.82/137.33 275.82/137.33 R is empty. 275.82/137.33 Q is empty. 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (92) QDPSizeChangeProof (EQUIVALENT) 275.82/137.33 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. 275.82/137.33 275.82/137.33 From the DPs we obtained the following set of size-change graphs: 275.82/137.33 *DOWN(1(0(y13))) -> DOWN(0(y13)) 275.82/137.33 The graph contains the following edges 1 > 1 275.82/137.33 275.82/137.33 275.82/137.33 *DOWN(1(1(y14))) -> DOWN(1(y14)) 275.82/137.33 The graph contains the following edges 1 > 1 275.82/137.33 275.82/137.33 275.82/137.33 *DOWN(0(0(y9))) -> DOWN(0(y9)) 275.82/137.33 The graph contains the following edges 1 > 1 275.82/137.33 275.82/137.33 275.82/137.33 *DOWN(0(1(y10))) -> DOWN(1(y10)) 275.82/137.33 The graph contains the following edges 1 > 1 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (93) 275.82/137.33 YES 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (94) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(x)) -> TOP(down(x)) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 down(f(f(x))) -> up(b) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 top(up(x)) -> top(down(x)) 275.82/137.33 down(f(a)) -> f_flat(down(a)) 275.82/137.33 down(f(b)) -> f_flat(down(b)) 275.82/137.33 down(f(0(y5))) -> f_flat(down(0(y5))) 275.82/137.33 down(f(1(y6))) -> f_flat(down(1(y6))) 275.82/137.33 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 top(up(x0)) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (95) UsableRulesProof (EQUIVALENT) 275.82/137.33 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. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (96) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(x)) -> TOP(down(x)) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 down(f(f(x))) -> up(b) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(f(a)) -> f_flat(down(a)) 275.82/137.33 down(f(b)) -> f_flat(down(b)) 275.82/137.33 down(f(0(y5))) -> f_flat(down(0(y5))) 275.82/137.33 down(f(1(y6))) -> f_flat(down(1(y6))) 275.82/137.33 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 top(up(x0)) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (97) QReductionProof (EQUIVALENT) 275.82/137.33 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 275.82/137.33 275.82/137.33 top(up(x0)) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (98) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(x)) -> TOP(down(x)) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 down(f(f(x))) -> up(b) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(f(a)) -> f_flat(down(a)) 275.82/137.33 down(f(b)) -> f_flat(down(b)) 275.82/137.33 down(f(0(y5))) -> f_flat(down(0(y5))) 275.82/137.33 down(f(1(y6))) -> f_flat(down(1(y6))) 275.82/137.33 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (99) TransformationProof (EQUIVALENT) 275.82/137.33 By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]: 275.82/137.33 275.82/137.33 (TOP(up(a)) -> TOP(up(f(a))),TOP(up(a)) -> TOP(up(f(a)))) 275.82/137.33 (TOP(up(f(f(x0)))) -> TOP(up(b)),TOP(up(f(f(x0)))) -> TOP(up(b))) 275.82/137.33 (TOP(up(0(f(x0)))) -> TOP(up(1(x0))),TOP(up(0(f(x0)))) -> TOP(up(1(x0)))) 275.82/137.33 (TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))),TOP(up(1(f(x0)))) -> TOP(up(f(0(x0))))) 275.82/137.33 (TOP(up(f(a))) -> TOP(f_flat(down(a))),TOP(up(f(a))) -> TOP(f_flat(down(a)))) 275.82/137.33 (TOP(up(f(b))) -> TOP(f_flat(down(b))),TOP(up(f(b))) -> TOP(f_flat(down(b)))) 275.82/137.33 (TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))),TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0))))) 275.82/137.33 (TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))),TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0))))) 275.82/137.33 (TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))),TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant)))) 275.82/137.33 (TOP(up(0(a))) -> TOP(0_flat(down(a))),TOP(up(0(a))) -> TOP(0_flat(down(a)))) 275.82/137.33 (TOP(up(0(b))) -> TOP(0_flat(down(b))),TOP(up(0(b))) -> TOP(0_flat(down(b)))) 275.82/137.33 (TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))),TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0))))) 275.82/137.33 (TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))),TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0))))) 275.82/137.33 (TOP(up(0(fresh_constant))) -> TOP(0_flat(down(fresh_constant))),TOP(up(0(fresh_constant))) -> TOP(0_flat(down(fresh_constant)))) 275.82/137.33 (TOP(up(1(a))) -> TOP(1_flat(down(a))),TOP(up(1(a))) -> TOP(1_flat(down(a)))) 275.82/137.33 (TOP(up(1(b))) -> TOP(1_flat(down(b))),TOP(up(1(b))) -> TOP(1_flat(down(b)))) 275.82/137.33 (TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))),TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0))))) 275.82/137.33 (TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))),TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0))))) 275.82/137.33 (TOP(up(1(fresh_constant))) -> TOP(1_flat(down(fresh_constant))),TOP(up(1(fresh_constant))) -> TOP(1_flat(down(fresh_constant)))) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (100) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(a)) -> TOP(up(f(a))) 275.82/137.33 TOP(up(f(f(x0)))) -> TOP(up(b)) 275.82/137.33 TOP(up(0(f(x0)))) -> TOP(up(1(x0))) 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 TOP(up(f(a))) -> TOP(f_flat(down(a))) 275.82/137.33 TOP(up(f(b))) -> TOP(f_flat(down(b))) 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))) 275.82/137.33 TOP(up(0(a))) -> TOP(0_flat(down(a))) 275.82/137.33 TOP(up(0(b))) -> TOP(0_flat(down(b))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(0(fresh_constant))) -> TOP(0_flat(down(fresh_constant))) 275.82/137.33 TOP(up(1(a))) -> TOP(1_flat(down(a))) 275.82/137.33 TOP(up(1(b))) -> TOP(1_flat(down(b))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(fresh_constant))) -> TOP(1_flat(down(fresh_constant))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 down(f(f(x))) -> up(b) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(f(a)) -> f_flat(down(a)) 275.82/137.33 down(f(b)) -> f_flat(down(b)) 275.82/137.33 down(f(0(y5))) -> f_flat(down(0(y5))) 275.82/137.33 down(f(1(y6))) -> f_flat(down(1(y6))) 275.82/137.33 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (101) DependencyGraphProof (EQUIVALENT) 275.82/137.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (102) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(f(a))) -> TOP(f_flat(down(a))) 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(f(x0)))) -> TOP(up(1(x0))) 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 TOP(up(1(a))) -> TOP(1_flat(down(a))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(a))) -> TOP(0_flat(down(a))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 down(f(f(x))) -> up(b) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(f(a)) -> f_flat(down(a)) 275.82/137.33 down(f(b)) -> f_flat(down(b)) 275.82/137.33 down(f(0(y5))) -> f_flat(down(0(y5))) 275.82/137.33 down(f(1(y6))) -> f_flat(down(1(y6))) 275.82/137.33 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (103) UsableRulesProof (EQUIVALENT) 275.82/137.33 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. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (104) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(f(a))) -> TOP(f_flat(down(a))) 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(f(x0)))) -> TOP(up(1(x0))) 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 TOP(up(1(a))) -> TOP(1_flat(down(a))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(a))) -> TOP(0_flat(down(a))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (105) TransformationProof (EQUIVALENT) 275.82/137.33 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]: 275.82/137.33 275.82/137.33 (TOP(up(f(a))) -> TOP(f_flat(up(f(a)))),TOP(up(f(a))) -> TOP(f_flat(up(f(a))))) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (106) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(f(x0)))) -> TOP(up(1(x0))) 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 TOP(up(1(a))) -> TOP(1_flat(down(a))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(a))) -> TOP(0_flat(down(a))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (107) TransformationProof (EQUIVALENT) 275.82/137.33 By rewriting [LPAR04] the rule TOP(up(1(a))) -> TOP(1_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]: 275.82/137.33 275.82/137.33 (TOP(up(1(a))) -> TOP(1_flat(up(f(a)))),TOP(up(1(a))) -> TOP(1_flat(up(f(a))))) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (108) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(f(x0)))) -> TOP(up(1(x0))) 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(a))) -> TOP(0_flat(down(a))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) 275.82/137.33 TOP(up(1(a))) -> TOP(1_flat(up(f(a)))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (109) TransformationProof (EQUIVALENT) 275.82/137.33 By rewriting [LPAR04] the rule TOP(up(0(a))) -> TOP(0_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]: 275.82/137.33 275.82/137.33 (TOP(up(0(a))) -> TOP(0_flat(up(f(a)))),TOP(up(0(a))) -> TOP(0_flat(up(f(a))))) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (110) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(f(x0)))) -> TOP(up(1(x0))) 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) 275.82/137.33 TOP(up(1(a))) -> TOP(1_flat(up(f(a)))) 275.82/137.33 TOP(up(0(a))) -> TOP(0_flat(up(f(a)))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (111) TransformationProof (EQUIVALENT) 275.82/137.33 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]: 275.82/137.33 275.82/137.33 (TOP(up(f(a))) -> TOP(up(f(f(a)))),TOP(up(f(a))) -> TOP(up(f(f(a))))) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (112) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(f(x0)))) -> TOP(up(1(x0))) 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(a))) -> TOP(1_flat(up(f(a)))) 275.82/137.33 TOP(up(0(a))) -> TOP(0_flat(up(f(a)))) 275.82/137.33 TOP(up(f(a))) -> TOP(up(f(f(a)))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (113) DependencyGraphProof (EQUIVALENT) 275.82/137.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (114) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(0(f(x0)))) -> TOP(up(1(x0))) 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(a))) -> TOP(1_flat(up(f(a)))) 275.82/137.33 TOP(up(0(a))) -> TOP(0_flat(up(f(a)))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (115) TransformationProof (EQUIVALENT) 275.82/137.33 By rewriting [LPAR04] the rule TOP(up(1(a))) -> TOP(1_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]: 275.82/137.33 275.82/137.33 (TOP(up(1(a))) -> TOP(up(1(f(a)))),TOP(up(1(a))) -> TOP(up(1(f(a))))) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (116) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(0(f(x0)))) -> TOP(up(1(x0))) 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(0(a))) -> TOP(0_flat(up(f(a)))) 275.82/137.33 TOP(up(1(a))) -> TOP(up(1(f(a)))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (117) TransformationProof (EQUIVALENT) 275.82/137.33 By rewriting [LPAR04] the rule TOP(up(0(a))) -> TOP(0_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]: 275.82/137.33 275.82/137.33 (TOP(up(0(a))) -> TOP(up(0(f(a)))),TOP(up(0(a))) -> TOP(up(0(f(a))))) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (118) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(0(f(x0)))) -> TOP(up(1(x0))) 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(a))) -> TOP(up(1(f(a)))) 275.82/137.33 TOP(up(0(a))) -> TOP(up(0(f(a)))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (119) TransformationProof (EQUIVALENT) 275.82/137.33 By forward instantiating [JAR06] the rule TOP(up(0(f(x0)))) -> TOP(up(1(x0))) we obtained the following new rules [LPAR04]: 275.82/137.33 275.82/137.33 (TOP(up(0(f(f(y_0))))) -> TOP(up(1(f(y_0)))),TOP(up(0(f(f(y_0))))) -> TOP(up(1(f(y_0))))) 275.82/137.33 (TOP(up(0(f(0(y_0))))) -> TOP(up(1(0(y_0)))),TOP(up(0(f(0(y_0))))) -> TOP(up(1(0(y_0))))) 275.82/137.33 (TOP(up(0(f(1(y_0))))) -> TOP(up(1(1(y_0)))),TOP(up(0(f(1(y_0))))) -> TOP(up(1(1(y_0))))) 275.82/137.33 (TOP(up(0(f(a)))) -> TOP(up(1(a))),TOP(up(0(f(a)))) -> TOP(up(1(a)))) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (120) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(a))) -> TOP(up(1(f(a)))) 275.82/137.33 TOP(up(0(a))) -> TOP(up(0(f(a)))) 275.82/137.33 TOP(up(0(f(f(y_0))))) -> TOP(up(1(f(y_0)))) 275.82/137.33 TOP(up(0(f(0(y_0))))) -> TOP(up(1(0(y_0)))) 275.82/137.33 TOP(up(0(f(1(y_0))))) -> TOP(up(1(1(y_0)))) 275.82/137.33 TOP(up(0(f(a)))) -> TOP(up(1(a))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (121) QDPOrderProof (EQUIVALENT) 275.82/137.33 We use the reduction pair processor [LPAR04,JAR06]. 275.82/137.33 275.82/137.33 275.82/137.33 The following pairs can be oriented strictly and are deleted. 275.82/137.33 275.82/137.33 TOP(up(0(f(f(y_0))))) -> TOP(up(1(f(y_0)))) 275.82/137.33 TOP(up(0(f(0(y_0))))) -> TOP(up(1(0(y_0)))) 275.82/137.33 TOP(up(0(f(1(y_0))))) -> TOP(up(1(1(y_0)))) 275.82/137.33 TOP(up(0(f(a)))) -> TOP(up(1(a))) 275.82/137.33 The remaining pairs can at least be oriented weakly. 275.82/137.33 Used ordering: Polynomial interpretation [POLO]: 275.82/137.33 275.82/137.33 POL(0(x_1)) = 1 275.82/137.33 POL(0_flat(x_1)) = 1 275.82/137.33 POL(1(x_1)) = 0 275.82/137.33 POL(1_flat(x_1)) = 0 275.82/137.33 POL(TOP(x_1)) = x_1 275.82/137.33 POL(a) = 0 275.82/137.33 POL(b) = 0 275.82/137.33 POL(down(x_1)) = 0 275.82/137.33 POL(f(x_1)) = 0 275.82/137.33 POL(f_flat(x_1)) = 0 275.82/137.33 POL(fresh_constant) = 0 275.82/137.33 POL(up(x_1)) = x_1 275.82/137.33 275.82/137.33 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 275.82/137.33 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (122) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(a))) -> TOP(up(1(f(a)))) 275.82/137.33 TOP(up(0(a))) -> TOP(up(0(f(a)))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (123) DependencyGraphProof (EQUIVALENT) 275.82/137.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (124) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(a))) -> TOP(up(1(f(a)))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (125) QDPOrderProof (EQUIVALENT) 275.82/137.33 We use the reduction pair processor [LPAR04,JAR06]. 275.82/137.33 275.82/137.33 275.82/137.33 The following pairs can be oriented strictly and are deleted. 275.82/137.33 275.82/137.33 TOP(up(1(f(x0)))) -> TOP(up(f(0(x0)))) 275.82/137.33 The remaining pairs can at least be oriented weakly. 275.82/137.33 Used ordering: Polynomial interpretation [POLO]: 275.82/137.33 275.82/137.33 POL(0(x_1)) = 0 275.82/137.33 POL(0_flat(x_1)) = 0 275.82/137.33 POL(1(x_1)) = 1 275.82/137.33 POL(1_flat(x_1)) = 1 275.82/137.33 POL(TOP(x_1)) = x_1 275.82/137.33 POL(a) = 0 275.82/137.33 POL(b) = 0 275.82/137.33 POL(down(x_1)) = 0 275.82/137.33 POL(f(x_1)) = 0 275.82/137.33 POL(f_flat(x_1)) = 0 275.82/137.33 POL(fresh_constant) = 0 275.82/137.33 POL(up(x_1)) = x_1 275.82/137.33 275.82/137.33 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 275.82/137.33 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (126) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(a))) -> TOP(up(1(f(a)))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (127) DependencyGraphProof (EQUIVALENT) 275.82/137.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (128) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.33 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.33 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.33 275.82/137.33 The TRS R consists of the following rules: 275.82/137.33 275.82/137.33 down(1(f(x))) -> up(f(0(x))) 275.82/137.33 down(1(a)) -> 1_flat(down(a)) 275.82/137.33 down(1(b)) -> 1_flat(down(b)) 275.82/137.33 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.33 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.33 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.33 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.33 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.33 down(0(f(x))) -> up(1(x)) 275.82/137.33 down(0(a)) -> 0_flat(down(a)) 275.82/137.33 down(0(b)) -> 0_flat(down(b)) 275.82/137.33 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.33 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.33 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.33 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.33 down(a) -> up(f(a)) 275.82/137.33 275.82/137.33 The set Q consists of the following terms: 275.82/137.33 275.82/137.33 down(a) 275.82/137.33 down(f(f(x0))) 275.82/137.33 down(0(f(x0))) 275.82/137.33 down(1(f(x0))) 275.82/137.33 down(f(a)) 275.82/137.33 down(f(b)) 275.82/137.33 down(f(0(x0))) 275.82/137.33 down(f(1(x0))) 275.82/137.33 down(f(fresh_constant)) 275.82/137.33 down(0(a)) 275.82/137.33 down(0(b)) 275.82/137.33 down(0(0(x0))) 275.82/137.33 down(0(1(x0))) 275.82/137.33 down(0(fresh_constant)) 275.82/137.33 down(1(a)) 275.82/137.33 down(1(b)) 275.82/137.33 down(1(0(x0))) 275.82/137.33 down(1(1(x0))) 275.82/137.33 down(1(fresh_constant)) 275.82/137.33 f_flat(up(x0)) 275.82/137.33 0_flat(up(x0)) 275.82/137.33 1_flat(up(x0)) 275.82/137.33 275.82/137.33 We have to consider all minimal (P,Q,R)-chains. 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (129) SplitQDPProof (EQUIVALENT) 275.82/137.33 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (130) 275.82/137.33 Complex Obligation (AND) 275.82/137.33 275.82/137.33 ---------------------------------------- 275.82/137.33 275.82/137.33 (131) 275.82/137.33 Obligation: 275.82/137.33 Q DP problem: 275.82/137.33 The TRS P consists of the following rules: 275.82/137.33 275.82/137.33 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.82/137.33 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.82/137.33 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.82/137.33 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.82/137.34 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.82/137.34 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.82/137.34 275.82/137.34 The TRS R consists of the following rules: 275.82/137.34 275.82/137.34 down(1(f(x))) -> up(f(0(x))) 275.82/137.34 down(1(a)) -> 1_flat(down(a)) 275.82/137.34 down(1(b)) -> 1_flat(down(b)) 275.82/137.34 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.82/137.34 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.82/137.34 down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 275.82/137.34 f_flat(up(x_1)) -> up(f(x_1)) 275.82/137.34 1_flat(up(x_1)) -> up(1(x_1)) 275.82/137.34 down(0(f(x))) -> up(1(x)) 275.82/137.34 down(0(a)) -> 0_flat(down(a)) 275.82/137.34 down(0(b)) -> 0_flat(down(b)) 275.82/137.34 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.82/137.34 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.82/137.34 down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 275.82/137.34 0_flat(up(x_1)) -> up(0(x_1)) 275.82/137.34 down(a) -> up(f(a)) 275.82/137.34 275.82/137.34 The set Q consists of the following terms: 275.82/137.34 275.82/137.34 down(a) 275.82/137.34 down(f(f(x0))) 275.82/137.34 down(0(f(x0))) 275.82/137.34 down(1(f(x0))) 275.82/137.34 down(f(a)) 275.82/137.34 down(f(b)) 275.82/137.34 down(f(0(x0))) 275.82/137.34 down(f(1(x0))) 275.82/137.34 down(f(fresh_constant)) 275.82/137.34 down(0(a)) 275.82/137.34 down(0(b)) 275.82/137.34 down(0(0(x0))) 275.82/137.34 down(0(1(x0))) 275.82/137.34 down(0(fresh_constant)) 275.82/137.34 down(1(a)) 275.82/137.34 down(1(b)) 275.82/137.34 down(1(0(x0))) 275.82/137.34 down(1(1(x0))) 275.82/137.34 down(1(fresh_constant)) 275.82/137.34 f_flat(up(x0)) 275.82/137.34 0_flat(up(x0)) 275.82/137.34 1_flat(up(x0)) 275.82/137.34 275.82/137.34 We have to consider all minimal (P,Q,R)-chains. 275.82/137.34 ---------------------------------------- 275.82/137.34 275.82/137.34 (132) SemLabProof (SOUND) 275.82/137.34 We found the following model for the rules of the TRSs R and P. 275.82/137.34 Interpretation over the domain with elements from 0 to 1. 275.82/137.34 a: 0 275.82/137.34 b: 0 275.82/137.34 down: 0 275.82/137.34 f: 0 275.82/137.34 0: 0 275.82/137.34 fresh_constant: 1 275.82/137.34 up: 0 275.82/137.34 1: 0 275.82/137.34 1_flat: 0 275.82/137.34 0_flat: 0 275.82/137.34 f_flat: 0 275.82/137.34 TOP: 0 275.82/137.34 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 275.82/137.34 ---------------------------------------- 275.82/137.34 275.82/137.34 (133) 275.82/137.34 Obligation: 275.82/137.34 Q DP problem: 275.82/137.34 The TRS P consists of the following rules: 275.82/137.34 275.82/137.34 TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) 275.82/137.34 TOP.0(up.0(f.0(0.1(x0)))) -> TOP.0(f_flat.0(down.0(0.1(x0)))) 275.82/137.34 TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) 275.82/137.34 TOP.0(up.0(f.0(1.1(x0)))) -> TOP.0(f_flat.0(down.0(1.1(x0)))) 275.82/137.34 TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) 275.82/137.34 TOP.0(up.0(0.0(0.1(x0)))) -> TOP.0(0_flat.0(down.0(0.1(x0)))) 275.82/137.34 TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) 275.82/137.34 TOP.0(up.0(0.0(1.1(x0)))) -> TOP.0(0_flat.0(down.0(1.1(x0)))) 275.82/137.34 TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) 275.82/137.34 TOP.0(up.0(1.0(0.1(x0)))) -> TOP.0(1_flat.0(down.0(0.1(x0)))) 275.82/137.34 TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) 275.82/137.34 TOP.0(up.0(1.0(1.1(x0)))) -> TOP.0(1_flat.0(down.0(1.1(x0)))) 275.82/137.34 275.82/137.34 The TRS R consists of the following rules: 275.82/137.34 275.82/137.34 down.0(1.0(f.0(x))) -> up.0(f.0(0.0(x))) 275.82/137.34 down.0(1.0(f.1(x))) -> up.0(f.0(0.1(x))) 275.82/137.34 down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) 275.82/137.34 down.0(1.0(b.)) -> 1_flat.0(down.0(b.)) 275.82/137.34 down.0(1.0(0.0(y13))) -> 1_flat.0(down.0(0.0(y13))) 275.82/137.34 down.0(1.0(0.1(y13))) -> 1_flat.0(down.0(0.1(y13))) 275.82/137.34 down.0(1.0(1.0(y14))) -> 1_flat.0(down.0(1.0(y14))) 275.82/137.34 down.0(1.0(1.1(y14))) -> 1_flat.0(down.0(1.1(y14))) 275.82/137.34 down.0(1.1(fresh_constant.)) -> 1_flat.0(down.1(fresh_constant.)) 275.82/137.34 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 275.82/137.34 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 275.82/137.34 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) 275.82/137.34 1_flat.0(up.1(x_1)) -> up.0(1.1(x_1)) 275.82/137.34 down.0(0.0(f.0(x))) -> up.0(1.0(x)) 275.82/137.34 down.0(0.0(f.1(x))) -> up.0(1.1(x)) 275.82/137.34 down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) 275.82/137.34 down.0(0.0(b.)) -> 0_flat.0(down.0(b.)) 275.82/137.34 down.0(0.0(0.0(y9))) -> 0_flat.0(down.0(0.0(y9))) 275.82/137.34 down.0(0.0(0.1(y9))) -> 0_flat.0(down.0(0.1(y9))) 275.82/137.34 down.0(0.0(1.0(y10))) -> 0_flat.0(down.0(1.0(y10))) 275.82/137.34 down.0(0.0(1.1(y10))) -> 0_flat.0(down.0(1.1(y10))) 275.82/137.34 down.0(0.1(fresh_constant.)) -> 0_flat.0(down.1(fresh_constant.)) 275.82/137.34 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) 275.82/137.34 0_flat.0(up.1(x_1)) -> up.0(0.1(x_1)) 275.82/137.34 down.0(a.) -> up.0(f.0(a.)) 275.82/137.34 275.82/137.34 The set Q consists of the following terms: 275.82/137.34 275.82/137.34 down.0(a.) 275.82/137.34 down.0(f.0(f.0(x0))) 275.82/137.34 down.0(f.0(f.1(x0))) 275.82/137.34 down.0(0.0(f.0(x0))) 275.82/137.34 down.0(0.0(f.1(x0))) 275.82/137.34 down.0(1.0(f.0(x0))) 275.82/137.34 down.0(1.0(f.1(x0))) 275.82/137.34 down.0(f.0(a.)) 275.82/137.34 down.0(f.0(b.)) 275.82/137.34 down.0(f.0(0.0(x0))) 275.82/137.34 down.0(f.0(0.1(x0))) 275.82/137.34 down.0(f.0(1.0(x0))) 275.82/137.34 down.0(f.0(1.1(x0))) 275.82/137.34 down.0(f.1(fresh_constant.)) 275.82/137.34 down.0(0.0(a.)) 275.82/137.34 down.0(0.0(b.)) 275.82/137.34 down.0(0.0(0.0(x0))) 275.82/137.34 down.0(0.0(0.1(x0))) 275.82/137.34 down.0(0.0(1.0(x0))) 275.82/137.34 down.0(0.0(1.1(x0))) 275.82/137.34 down.0(0.1(fresh_constant.)) 275.82/137.34 down.0(1.0(a.)) 275.82/137.34 down.0(1.0(b.)) 275.82/137.34 down.0(1.0(0.0(x0))) 275.82/137.34 down.0(1.0(0.1(x0))) 275.82/137.34 down.0(1.0(1.0(x0))) 275.82/137.34 down.0(1.0(1.1(x0))) 275.82/137.34 down.0(1.1(fresh_constant.)) 275.82/137.34 f_flat.0(up.0(x0)) 275.82/137.34 f_flat.0(up.1(x0)) 275.82/137.34 0_flat.0(up.0(x0)) 275.82/137.34 0_flat.0(up.1(x0)) 275.82/137.34 1_flat.0(up.0(x0)) 275.82/137.34 1_flat.0(up.1(x0)) 275.82/137.34 275.82/137.34 We have to consider all minimal (P,Q,R)-chains. 275.82/137.34 ---------------------------------------- 275.82/137.34 275.82/137.34 (134) UsableRulesReductionPairsProof (EQUIVALENT) 275.82/137.34 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. 275.82/137.34 275.82/137.34 No dependency pairs are removed. 275.82/137.34 275.82/137.34 The following rules are removed from R: 275.82/137.34 275.82/137.34 down.0(1.0(f.1(x))) -> up.0(f.0(0.1(x))) 275.82/137.34 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 275.82/137.34 1_flat.0(up.1(x_1)) -> up.0(1.1(x_1)) 275.82/137.34 down.0(0.0(f.1(x))) -> up.0(1.1(x)) 275.82/137.34 0_flat.0(up.1(x_1)) -> up.0(0.1(x_1)) 275.82/137.34 Used ordering: POLO with Polynomial interpretation [POLO]: 275.82/137.34 275.82/137.34 POL(0.0(x_1)) = 1 + x_1 275.82/137.34 POL(0.1(x_1)) = 1 + x_1 275.82/137.34 POL(0_flat.0(x_1)) = 1 + x_1 275.82/137.34 POL(1.0(x_1)) = 1 + x_1 275.82/137.34 POL(1.1(x_1)) = 1 + x_1 275.82/137.34 POL(1_flat.0(x_1)) = 1 + x_1 275.82/137.34 POL(TOP.0(x_1)) = x_1 275.82/137.34 POL(a.) = 0 275.82/137.34 POL(b.) = 0 275.82/137.34 POL(down.0(x_1)) = x_1 275.82/137.34 POL(down.1(x_1)) = x_1 275.82/137.34 POL(f.0(x_1)) = x_1 275.82/137.34 POL(f.1(x_1)) = 1 + x_1 275.82/137.34 POL(f_flat.0(x_1)) = x_1 275.82/137.34 POL(fresh_constant.) = 0 275.82/137.34 POL(up.0(x_1)) = x_1 275.82/137.34 POL(up.1(x_1)) = 1 + x_1 275.82/137.34 275.82/137.34 275.82/137.34 ---------------------------------------- 275.82/137.34 275.82/137.34 (135) 275.82/137.34 Obligation: 275.82/137.34 Q DP problem: 275.82/137.34 The TRS P consists of the following rules: 275.82/137.34 275.82/137.34 TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) 275.82/137.34 TOP.0(up.0(f.0(0.1(x0)))) -> TOP.0(f_flat.0(down.0(0.1(x0)))) 275.82/137.34 TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(f.0(1.1(x0)))) -> TOP.0(f_flat.0(down.0(1.1(x0)))) 275.91/137.34 TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(0.1(x0)))) -> TOP.0(0_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(1.1(x0)))) -> TOP.0(0_flat.0(down.0(1.1(x0)))) 275.91/137.34 TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(0.1(x0)))) -> TOP.0(1_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(1.1(x0)))) -> TOP.0(1_flat.0(down.0(1.1(x0)))) 275.91/137.34 275.91/137.34 The TRS R consists of the following rules: 275.91/137.34 275.91/137.34 down.0(1.1(fresh_constant.)) -> 1_flat.0(down.1(fresh_constant.)) 275.91/137.34 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) 275.91/137.34 down.0(1.0(f.0(x))) -> up.0(f.0(0.0(x))) 275.91/137.34 down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) 275.91/137.34 down.0(1.0(b.)) -> 1_flat.0(down.0(b.)) 275.91/137.34 down.0(1.0(0.0(y13))) -> 1_flat.0(down.0(0.0(y13))) 275.91/137.34 down.0(1.0(0.1(y13))) -> 1_flat.0(down.0(0.1(y13))) 275.91/137.34 down.0(1.0(1.0(y14))) -> 1_flat.0(down.0(1.0(y14))) 275.91/137.34 down.0(1.0(1.1(y14))) -> 1_flat.0(down.0(1.1(y14))) 275.91/137.34 down.0(0.1(fresh_constant.)) -> 0_flat.0(down.1(fresh_constant.)) 275.91/137.34 down.0(0.0(f.0(x))) -> up.0(1.0(x)) 275.91/137.34 down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) 275.91/137.34 down.0(0.0(b.)) -> 0_flat.0(down.0(b.)) 275.91/137.34 down.0(0.0(0.0(y9))) -> 0_flat.0(down.0(0.0(y9))) 275.91/137.34 down.0(0.0(0.1(y9))) -> 0_flat.0(down.0(0.1(y9))) 275.91/137.34 down.0(0.0(1.0(y10))) -> 0_flat.0(down.0(1.0(y10))) 275.91/137.34 down.0(0.0(1.1(y10))) -> 0_flat.0(down.0(1.1(y10))) 275.91/137.34 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) 275.91/137.34 down.0(a.) -> up.0(f.0(a.)) 275.91/137.34 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 275.91/137.34 275.91/137.34 The set Q consists of the following terms: 275.91/137.34 275.91/137.34 down.0(a.) 275.91/137.34 down.0(f.0(f.0(x0))) 275.91/137.34 down.0(f.0(f.1(x0))) 275.91/137.34 down.0(0.0(f.0(x0))) 275.91/137.34 down.0(0.0(f.1(x0))) 275.91/137.34 down.0(1.0(f.0(x0))) 275.91/137.34 down.0(1.0(f.1(x0))) 275.91/137.34 down.0(f.0(a.)) 275.91/137.34 down.0(f.0(b.)) 275.91/137.34 down.0(f.0(0.0(x0))) 275.91/137.34 down.0(f.0(0.1(x0))) 275.91/137.34 down.0(f.0(1.0(x0))) 275.91/137.34 down.0(f.0(1.1(x0))) 275.91/137.34 down.0(f.1(fresh_constant.)) 275.91/137.34 down.0(0.0(a.)) 275.91/137.34 down.0(0.0(b.)) 275.91/137.34 down.0(0.0(0.0(x0))) 275.91/137.34 down.0(0.0(0.1(x0))) 275.91/137.34 down.0(0.0(1.0(x0))) 275.91/137.34 down.0(0.0(1.1(x0))) 275.91/137.34 down.0(0.1(fresh_constant.)) 275.91/137.34 down.0(1.0(a.)) 275.91/137.34 down.0(1.0(b.)) 275.91/137.34 down.0(1.0(0.0(x0))) 275.91/137.34 down.0(1.0(0.1(x0))) 275.91/137.34 down.0(1.0(1.0(x0))) 275.91/137.34 down.0(1.0(1.1(x0))) 275.91/137.34 down.0(1.1(fresh_constant.)) 275.91/137.34 f_flat.0(up.0(x0)) 275.91/137.34 f_flat.0(up.1(x0)) 275.91/137.34 0_flat.0(up.0(x0)) 275.91/137.34 0_flat.0(up.1(x0)) 275.91/137.34 1_flat.0(up.0(x0)) 275.91/137.34 1_flat.0(up.1(x0)) 275.91/137.34 275.91/137.34 We have to consider all minimal (P,Q,R)-chains. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (136) MRRProof (EQUIVALENT) 275.91/137.34 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. 275.91/137.34 275.91/137.34 275.91/137.34 Strictly oriented rules of the TRS R: 275.91/137.34 275.91/137.34 down.0(1.1(fresh_constant.)) -> 1_flat.0(down.1(fresh_constant.)) 275.91/137.34 down.0(0.1(fresh_constant.)) -> 0_flat.0(down.1(fresh_constant.)) 275.91/137.34 275.91/137.34 Used ordering: Polynomial interpretation [POLO]: 275.91/137.34 275.91/137.34 POL(0.0(x_1)) = x_1 275.91/137.34 POL(0.1(x_1)) = 1 + x_1 275.91/137.34 POL(0_flat.0(x_1)) = x_1 275.91/137.34 POL(1.0(x_1)) = x_1 275.91/137.34 POL(1.1(x_1)) = 1 + x_1 275.91/137.34 POL(1_flat.0(x_1)) = x_1 275.91/137.34 POL(TOP.0(x_1)) = x_1 275.91/137.34 POL(a.) = 0 275.91/137.34 POL(b.) = 0 275.91/137.34 POL(down.0(x_1)) = x_1 275.91/137.34 POL(down.1(x_1)) = x_1 275.91/137.34 POL(f.0(x_1)) = x_1 275.91/137.34 POL(f_flat.0(x_1)) = x_1 275.91/137.34 POL(fresh_constant.) = 0 275.91/137.34 POL(up.0(x_1)) = x_1 275.91/137.34 275.91/137.34 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (137) 275.91/137.34 Obligation: 275.91/137.34 Q DP problem: 275.91/137.34 The TRS P consists of the following rules: 275.91/137.34 275.91/137.34 TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(f.0(0.1(x0)))) -> TOP.0(f_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(f.0(1.1(x0)))) -> TOP.0(f_flat.0(down.0(1.1(x0)))) 275.91/137.34 TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(0.1(x0)))) -> TOP.0(0_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(1.1(x0)))) -> TOP.0(0_flat.0(down.0(1.1(x0)))) 275.91/137.34 TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(0.1(x0)))) -> TOP.0(1_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(1.1(x0)))) -> TOP.0(1_flat.0(down.0(1.1(x0)))) 275.91/137.34 275.91/137.34 The TRS R consists of the following rules: 275.91/137.34 275.91/137.34 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) 275.91/137.34 down.0(1.0(f.0(x))) -> up.0(f.0(0.0(x))) 275.91/137.34 down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) 275.91/137.34 down.0(1.0(b.)) -> 1_flat.0(down.0(b.)) 275.91/137.34 down.0(1.0(0.0(y13))) -> 1_flat.0(down.0(0.0(y13))) 275.91/137.34 down.0(1.0(0.1(y13))) -> 1_flat.0(down.0(0.1(y13))) 275.91/137.34 down.0(1.0(1.0(y14))) -> 1_flat.0(down.0(1.0(y14))) 275.91/137.34 down.0(1.0(1.1(y14))) -> 1_flat.0(down.0(1.1(y14))) 275.91/137.34 down.0(0.0(f.0(x))) -> up.0(1.0(x)) 275.91/137.34 down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) 275.91/137.34 down.0(0.0(b.)) -> 0_flat.0(down.0(b.)) 275.91/137.34 down.0(0.0(0.0(y9))) -> 0_flat.0(down.0(0.0(y9))) 275.91/137.34 down.0(0.0(0.1(y9))) -> 0_flat.0(down.0(0.1(y9))) 275.91/137.34 down.0(0.0(1.0(y10))) -> 0_flat.0(down.0(1.0(y10))) 275.91/137.34 down.0(0.0(1.1(y10))) -> 0_flat.0(down.0(1.1(y10))) 275.91/137.34 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) 275.91/137.34 down.0(a.) -> up.0(f.0(a.)) 275.91/137.34 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 275.91/137.34 275.91/137.34 The set Q consists of the following terms: 275.91/137.34 275.91/137.34 down.0(a.) 275.91/137.34 down.0(f.0(f.0(x0))) 275.91/137.34 down.0(f.0(f.1(x0))) 275.91/137.34 down.0(0.0(f.0(x0))) 275.91/137.34 down.0(0.0(f.1(x0))) 275.91/137.34 down.0(1.0(f.0(x0))) 275.91/137.34 down.0(1.0(f.1(x0))) 275.91/137.34 down.0(f.0(a.)) 275.91/137.34 down.0(f.0(b.)) 275.91/137.34 down.0(f.0(0.0(x0))) 275.91/137.34 down.0(f.0(0.1(x0))) 275.91/137.34 down.0(f.0(1.0(x0))) 275.91/137.34 down.0(f.0(1.1(x0))) 275.91/137.34 down.0(f.1(fresh_constant.)) 275.91/137.34 down.0(0.0(a.)) 275.91/137.34 down.0(0.0(b.)) 275.91/137.34 down.0(0.0(0.0(x0))) 275.91/137.34 down.0(0.0(0.1(x0))) 275.91/137.34 down.0(0.0(1.0(x0))) 275.91/137.34 down.0(0.0(1.1(x0))) 275.91/137.34 down.0(0.1(fresh_constant.)) 275.91/137.34 down.0(1.0(a.)) 275.91/137.34 down.0(1.0(b.)) 275.91/137.34 down.0(1.0(0.0(x0))) 275.91/137.34 down.0(1.0(0.1(x0))) 275.91/137.34 down.0(1.0(1.0(x0))) 275.91/137.34 down.0(1.0(1.1(x0))) 275.91/137.34 down.0(1.1(fresh_constant.)) 275.91/137.34 f_flat.0(up.0(x0)) 275.91/137.34 f_flat.0(up.1(x0)) 275.91/137.34 0_flat.0(up.0(x0)) 275.91/137.34 0_flat.0(up.1(x0)) 275.91/137.34 1_flat.0(up.0(x0)) 275.91/137.34 1_flat.0(up.1(x0)) 275.91/137.34 275.91/137.34 We have to consider all minimal (P,Q,R)-chains. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (138) DependencyGraphProof (EQUIVALENT) 275.91/137.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (139) 275.91/137.34 Obligation: 275.91/137.34 Q DP problem: 275.91/137.34 The TRS P consists of the following rules: 275.91/137.34 275.91/137.34 TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) 275.91/137.34 275.91/137.34 The TRS R consists of the following rules: 275.91/137.34 275.91/137.34 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) 275.91/137.34 down.0(1.0(f.0(x))) -> up.0(f.0(0.0(x))) 275.91/137.34 down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) 275.91/137.34 down.0(1.0(b.)) -> 1_flat.0(down.0(b.)) 275.91/137.34 down.0(1.0(0.0(y13))) -> 1_flat.0(down.0(0.0(y13))) 275.91/137.34 down.0(1.0(0.1(y13))) -> 1_flat.0(down.0(0.1(y13))) 275.91/137.34 down.0(1.0(1.0(y14))) -> 1_flat.0(down.0(1.0(y14))) 275.91/137.34 down.0(1.0(1.1(y14))) -> 1_flat.0(down.0(1.1(y14))) 275.91/137.34 down.0(0.0(f.0(x))) -> up.0(1.0(x)) 275.91/137.34 down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) 275.91/137.34 down.0(0.0(b.)) -> 0_flat.0(down.0(b.)) 275.91/137.34 down.0(0.0(0.0(y9))) -> 0_flat.0(down.0(0.0(y9))) 275.91/137.34 down.0(0.0(0.1(y9))) -> 0_flat.0(down.0(0.1(y9))) 275.91/137.34 down.0(0.0(1.0(y10))) -> 0_flat.0(down.0(1.0(y10))) 275.91/137.34 down.0(0.0(1.1(y10))) -> 0_flat.0(down.0(1.1(y10))) 275.91/137.34 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) 275.91/137.34 down.0(a.) -> up.0(f.0(a.)) 275.91/137.34 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 275.91/137.34 275.91/137.34 The set Q consists of the following terms: 275.91/137.34 275.91/137.34 down.0(a.) 275.91/137.34 down.0(f.0(f.0(x0))) 275.91/137.34 down.0(f.0(f.1(x0))) 275.91/137.34 down.0(0.0(f.0(x0))) 275.91/137.34 down.0(0.0(f.1(x0))) 275.91/137.34 down.0(1.0(f.0(x0))) 275.91/137.34 down.0(1.0(f.1(x0))) 275.91/137.34 down.0(f.0(a.)) 275.91/137.34 down.0(f.0(b.)) 275.91/137.34 down.0(f.0(0.0(x0))) 275.91/137.34 down.0(f.0(0.1(x0))) 275.91/137.34 down.0(f.0(1.0(x0))) 275.91/137.34 down.0(f.0(1.1(x0))) 275.91/137.34 down.0(f.1(fresh_constant.)) 275.91/137.34 down.0(0.0(a.)) 275.91/137.34 down.0(0.0(b.)) 275.91/137.34 down.0(0.0(0.0(x0))) 275.91/137.34 down.0(0.0(0.1(x0))) 275.91/137.34 down.0(0.0(1.0(x0))) 275.91/137.34 down.0(0.0(1.1(x0))) 275.91/137.34 down.0(0.1(fresh_constant.)) 275.91/137.34 down.0(1.0(a.)) 275.91/137.34 down.0(1.0(b.)) 275.91/137.34 down.0(1.0(0.0(x0))) 275.91/137.34 down.0(1.0(0.1(x0))) 275.91/137.34 down.0(1.0(1.0(x0))) 275.91/137.34 down.0(1.0(1.1(x0))) 275.91/137.34 down.0(1.1(fresh_constant.)) 275.91/137.34 f_flat.0(up.0(x0)) 275.91/137.34 f_flat.0(up.1(x0)) 275.91/137.34 0_flat.0(up.0(x0)) 275.91/137.34 0_flat.0(up.1(x0)) 275.91/137.34 1_flat.0(up.0(x0)) 275.91/137.34 1_flat.0(up.1(x0)) 275.91/137.34 275.91/137.34 We have to consider all minimal (P,Q,R)-chains. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (140) PisEmptyProof (SOUND) 275.91/137.34 The TRS P is empty. Hence, there is no (P,Q,R) chain. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (141) 275.91/137.34 TRUE 275.91/137.34 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (142) 275.91/137.34 Obligation: 275.91/137.34 Q DP problem: 275.91/137.34 The TRS P consists of the following rules: 275.91/137.34 275.91/137.34 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.91/137.34 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.91/137.34 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.91/137.34 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.91/137.34 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.91/137.34 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.91/137.34 275.91/137.34 The TRS R consists of the following rules: 275.91/137.34 275.91/137.34 1_flat(up(x_1)) -> up(1(x_1)) 275.91/137.34 down(1(f(x))) -> up(f(0(x))) 275.91/137.34 down(1(a)) -> 1_flat(down(a)) 275.91/137.34 down(1(b)) -> 1_flat(down(b)) 275.91/137.34 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.91/137.34 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.91/137.34 down(0(f(x))) -> up(1(x)) 275.91/137.34 down(0(a)) -> 0_flat(down(a)) 275.91/137.34 down(0(b)) -> 0_flat(down(b)) 275.91/137.34 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.91/137.34 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.91/137.34 0_flat(up(x_1)) -> up(0(x_1)) 275.91/137.34 down(a) -> up(f(a)) 275.91/137.34 f_flat(up(x_1)) -> up(f(x_1)) 275.91/137.34 275.91/137.34 The set Q consists of the following terms: 275.91/137.34 275.91/137.34 down(a) 275.91/137.34 down(f(f(x0))) 275.91/137.34 down(0(f(x0))) 275.91/137.34 down(1(f(x0))) 275.91/137.34 down(f(a)) 275.91/137.34 down(f(b)) 275.91/137.34 down(f(0(x0))) 275.91/137.34 down(f(1(x0))) 275.91/137.34 down(f(fresh_constant)) 275.91/137.34 down(0(a)) 275.91/137.34 down(0(b)) 275.91/137.34 down(0(0(x0))) 275.91/137.34 down(0(1(x0))) 275.91/137.34 down(0(fresh_constant)) 275.91/137.34 down(1(a)) 275.91/137.34 down(1(b)) 275.91/137.34 down(1(0(x0))) 275.91/137.34 down(1(1(x0))) 275.91/137.34 down(1(fresh_constant)) 275.91/137.34 f_flat(up(x0)) 275.91/137.34 0_flat(up(x0)) 275.91/137.34 1_flat(up(x0)) 275.91/137.34 275.91/137.34 We have to consider all minimal (P,Q,R)-chains. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (143) QReductionProof (EQUIVALENT) 275.91/137.34 We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN]. 275.91/137.34 275.91/137.34 down(f(fresh_constant)) 275.91/137.34 down(0(fresh_constant)) 275.91/137.34 down(1(fresh_constant)) 275.91/137.34 275.91/137.34 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (144) 275.91/137.34 Obligation: 275.91/137.34 Q DP problem: 275.91/137.34 The TRS P consists of the following rules: 275.91/137.34 275.91/137.34 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.91/137.34 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.91/137.34 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.91/137.34 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.91/137.34 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.91/137.34 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.91/137.34 275.91/137.34 The TRS R consists of the following rules: 275.91/137.34 275.91/137.34 1_flat(up(x_1)) -> up(1(x_1)) 275.91/137.34 down(1(f(x))) -> up(f(0(x))) 275.91/137.34 down(1(a)) -> 1_flat(down(a)) 275.91/137.34 down(1(b)) -> 1_flat(down(b)) 275.91/137.34 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.91/137.34 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.91/137.34 down(0(f(x))) -> up(1(x)) 275.91/137.34 down(0(a)) -> 0_flat(down(a)) 275.91/137.34 down(0(b)) -> 0_flat(down(b)) 275.91/137.34 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.91/137.34 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.91/137.34 0_flat(up(x_1)) -> up(0(x_1)) 275.91/137.34 down(a) -> up(f(a)) 275.91/137.34 f_flat(up(x_1)) -> up(f(x_1)) 275.91/137.34 275.91/137.34 The set Q consists of the following terms: 275.91/137.34 275.91/137.34 down(a) 275.91/137.34 down(f(f(x0))) 275.91/137.34 down(0(f(x0))) 275.91/137.34 down(1(f(x0))) 275.91/137.34 down(f(a)) 275.91/137.34 down(f(b)) 275.91/137.34 down(f(0(x0))) 275.91/137.34 down(f(1(x0))) 275.91/137.34 down(0(a)) 275.91/137.34 down(0(b)) 275.91/137.34 down(0(0(x0))) 275.91/137.34 down(0(1(x0))) 275.91/137.34 down(1(a)) 275.91/137.34 down(1(b)) 275.91/137.34 down(1(0(x0))) 275.91/137.34 down(1(1(x0))) 275.91/137.34 f_flat(up(x0)) 275.91/137.34 0_flat(up(x0)) 275.91/137.34 1_flat(up(x0)) 275.91/137.34 275.91/137.34 We have to consider all (P,Q,R)-chains. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (145) SplitQDPProof (EQUIVALENT) 275.91/137.34 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 275.91/137.34 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (146) 275.91/137.34 Complex Obligation (AND) 275.91/137.34 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (147) 275.91/137.34 Obligation: 275.91/137.34 Q DP problem: 275.91/137.34 The TRS P consists of the following rules: 275.91/137.34 275.91/137.34 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.91/137.34 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.91/137.34 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.91/137.34 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.91/137.34 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.91/137.34 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.91/137.34 275.91/137.34 The TRS R consists of the following rules: 275.91/137.34 275.91/137.34 1_flat(up(x_1)) -> up(1(x_1)) 275.91/137.34 down(1(f(x))) -> up(f(0(x))) 275.91/137.34 down(1(a)) -> 1_flat(down(a)) 275.91/137.34 down(1(b)) -> 1_flat(down(b)) 275.91/137.34 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.91/137.34 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.91/137.34 down(0(f(x))) -> up(1(x)) 275.91/137.34 down(0(a)) -> 0_flat(down(a)) 275.91/137.34 down(0(b)) -> 0_flat(down(b)) 275.91/137.34 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.91/137.34 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.91/137.34 0_flat(up(x_1)) -> up(0(x_1)) 275.91/137.34 down(a) -> up(f(a)) 275.91/137.34 f_flat(up(x_1)) -> up(f(x_1)) 275.91/137.34 275.91/137.34 The set Q consists of the following terms: 275.91/137.34 275.91/137.34 down(a) 275.91/137.34 down(f(f(x0))) 275.91/137.34 down(0(f(x0))) 275.91/137.34 down(1(f(x0))) 275.91/137.34 down(f(a)) 275.91/137.34 down(f(b)) 275.91/137.34 down(f(0(x0))) 275.91/137.34 down(f(1(x0))) 275.91/137.34 down(f(fresh_constant)) 275.91/137.34 down(0(a)) 275.91/137.34 down(0(b)) 275.91/137.34 down(0(0(x0))) 275.91/137.34 down(0(1(x0))) 275.91/137.34 down(0(fresh_constant)) 275.91/137.34 down(1(a)) 275.91/137.34 down(1(b)) 275.91/137.34 down(1(0(x0))) 275.91/137.34 down(1(1(x0))) 275.91/137.34 down(1(fresh_constant)) 275.91/137.34 f_flat(up(x0)) 275.91/137.34 0_flat(up(x0)) 275.91/137.34 1_flat(up(x0)) 275.91/137.34 275.91/137.34 We have to consider all minimal (P,Q,R)-chains. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (148) SemLabProof (SOUND) 275.91/137.34 We found the following model for the rules of the TRSs R and P. 275.91/137.34 Interpretation over the domain with elements from 0 to 1. 275.91/137.34 a: 0 275.91/137.34 b: 1 275.91/137.34 down: 0 275.91/137.34 f: 0 275.91/137.34 fresh_constant: 0 275.91/137.34 0: 0 275.91/137.34 up: 0 275.91/137.34 1: 0 275.91/137.34 1_flat: 0 275.91/137.34 0_flat: 0 275.91/137.34 f_flat: 0 275.91/137.34 TOP: 0 275.91/137.34 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (149) 275.91/137.34 Obligation: 275.91/137.34 Q DP problem: 275.91/137.34 The TRS P consists of the following rules: 275.91/137.34 275.91/137.34 TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(f.0(0.1(x0)))) -> TOP.0(f_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(f.0(1.1(x0)))) -> TOP.0(f_flat.0(down.0(1.1(x0)))) 275.91/137.34 TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(0.1(x0)))) -> TOP.0(0_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(1.1(x0)))) -> TOP.0(0_flat.0(down.0(1.1(x0)))) 275.91/137.34 TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(0.1(x0)))) -> TOP.0(1_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(1.1(x0)))) -> TOP.0(1_flat.0(down.0(1.1(x0)))) 275.91/137.34 275.91/137.34 The TRS R consists of the following rules: 275.91/137.34 275.91/137.34 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) 275.91/137.34 1_flat.0(up.1(x_1)) -> up.0(1.1(x_1)) 275.91/137.34 down.0(1.0(f.0(x))) -> up.0(f.0(0.0(x))) 275.91/137.34 down.0(1.0(f.1(x))) -> up.0(f.0(0.1(x))) 275.91/137.34 down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) 275.91/137.34 down.0(1.1(b.)) -> 1_flat.0(down.1(b.)) 275.91/137.34 down.0(1.0(0.0(y13))) -> 1_flat.0(down.0(0.0(y13))) 275.91/137.34 down.0(1.0(0.1(y13))) -> 1_flat.0(down.0(0.1(y13))) 275.91/137.34 down.0(1.0(1.0(y14))) -> 1_flat.0(down.0(1.0(y14))) 275.91/137.34 down.0(1.0(1.1(y14))) -> 1_flat.0(down.0(1.1(y14))) 275.91/137.34 down.0(0.0(f.0(x))) -> up.0(1.0(x)) 275.91/137.34 down.0(0.0(f.1(x))) -> up.0(1.1(x)) 275.91/137.34 down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) 275.91/137.34 down.0(0.1(b.)) -> 0_flat.0(down.1(b.)) 275.91/137.34 down.0(0.0(0.0(y9))) -> 0_flat.0(down.0(0.0(y9))) 275.91/137.34 down.0(0.0(0.1(y9))) -> 0_flat.0(down.0(0.1(y9))) 275.91/137.34 down.0(0.0(1.0(y10))) -> 0_flat.0(down.0(1.0(y10))) 275.91/137.34 down.0(0.0(1.1(y10))) -> 0_flat.0(down.0(1.1(y10))) 275.91/137.34 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) 275.91/137.34 0_flat.0(up.1(x_1)) -> up.0(0.1(x_1)) 275.91/137.34 down.0(a.) -> up.0(f.0(a.)) 275.91/137.34 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 275.91/137.34 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 275.91/137.34 275.91/137.34 The set Q consists of the following terms: 275.91/137.34 275.91/137.34 down.0(a.) 275.91/137.34 down.0(f.0(f.0(x0))) 275.91/137.34 down.0(f.0(f.1(x0))) 275.91/137.34 down.0(0.0(f.0(x0))) 275.91/137.34 down.0(0.0(f.1(x0))) 275.91/137.34 down.0(1.0(f.0(x0))) 275.91/137.34 down.0(1.0(f.1(x0))) 275.91/137.34 down.0(f.0(a.)) 275.91/137.34 down.0(f.1(b.)) 275.91/137.34 down.0(f.0(0.0(x0))) 275.91/137.34 down.0(f.0(0.1(x0))) 275.91/137.34 down.0(f.0(1.0(x0))) 275.91/137.34 down.0(f.0(1.1(x0))) 275.91/137.34 down.0(f.0(fresh_constant.)) 275.91/137.34 down.0(0.0(a.)) 275.91/137.34 down.0(0.1(b.)) 275.91/137.34 down.0(0.0(0.0(x0))) 275.91/137.34 down.0(0.0(0.1(x0))) 275.91/137.34 down.0(0.0(1.0(x0))) 275.91/137.34 down.0(0.0(1.1(x0))) 275.91/137.34 down.0(0.0(fresh_constant.)) 275.91/137.34 down.0(1.0(a.)) 275.91/137.34 down.0(1.1(b.)) 275.91/137.34 down.0(1.0(0.0(x0))) 275.91/137.34 down.0(1.0(0.1(x0))) 275.91/137.34 down.0(1.0(1.0(x0))) 275.91/137.34 down.0(1.0(1.1(x0))) 275.91/137.34 down.0(1.0(fresh_constant.)) 275.91/137.34 f_flat.0(up.0(x0)) 275.91/137.34 f_flat.0(up.1(x0)) 275.91/137.34 0_flat.0(up.0(x0)) 275.91/137.34 0_flat.0(up.1(x0)) 275.91/137.34 1_flat.0(up.0(x0)) 275.91/137.34 1_flat.0(up.1(x0)) 275.91/137.34 275.91/137.34 We have to consider all minimal (P,Q,R)-chains. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (150) UsableRulesReductionPairsProof (EQUIVALENT) 275.91/137.34 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. 275.91/137.34 275.91/137.34 No dependency pairs are removed. 275.91/137.34 275.91/137.34 The following rules are removed from R: 275.91/137.34 275.91/137.34 1_flat.0(up.1(x_1)) -> up.0(1.1(x_1)) 275.91/137.34 down.0(1.0(f.1(x))) -> up.0(f.0(0.1(x))) 275.91/137.34 down.0(0.0(f.1(x))) -> up.0(1.1(x)) 275.91/137.34 0_flat.0(up.1(x_1)) -> up.0(0.1(x_1)) 275.91/137.34 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 275.91/137.34 Used ordering: POLO with Polynomial interpretation [POLO]: 275.91/137.34 275.91/137.34 POL(0.0(x_1)) = 1 + x_1 275.91/137.34 POL(0.1(x_1)) = 1 + x_1 275.91/137.34 POL(0_flat.0(x_1)) = 1 + x_1 275.91/137.34 POL(1.0(x_1)) = 1 + x_1 275.91/137.34 POL(1.1(x_1)) = 1 + x_1 275.91/137.34 POL(1_flat.0(x_1)) = 1 + x_1 275.91/137.34 POL(TOP.0(x_1)) = x_1 275.91/137.34 POL(a.) = 0 275.91/137.34 POL(b.) = 0 275.91/137.34 POL(down.0(x_1)) = x_1 275.91/137.34 POL(down.1(x_1)) = x_1 275.91/137.34 POL(f.0(x_1)) = x_1 275.91/137.34 POL(f.1(x_1)) = 1 + x_1 275.91/137.34 POL(f_flat.0(x_1)) = x_1 275.91/137.34 POL(up.0(x_1)) = x_1 275.91/137.34 POL(up.1(x_1)) = 1 + x_1 275.91/137.34 275.91/137.34 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (151) 275.91/137.34 Obligation: 275.91/137.34 Q DP problem: 275.91/137.34 The TRS P consists of the following rules: 275.91/137.34 275.91/137.34 TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(f.0(0.1(x0)))) -> TOP.0(f_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(f.0(1.1(x0)))) -> TOP.0(f_flat.0(down.0(1.1(x0)))) 275.91/137.34 TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(0.1(x0)))) -> TOP.0(0_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(1.1(x0)))) -> TOP.0(0_flat.0(down.0(1.1(x0)))) 275.91/137.34 TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(0.1(x0)))) -> TOP.0(1_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(1.1(x0)))) -> TOP.0(1_flat.0(down.0(1.1(x0)))) 275.91/137.34 275.91/137.34 The TRS R consists of the following rules: 275.91/137.34 275.91/137.34 down.0(1.1(b.)) -> 1_flat.0(down.1(b.)) 275.91/137.34 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) 275.91/137.34 down.0(1.0(f.0(x))) -> up.0(f.0(0.0(x))) 275.91/137.34 down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) 275.91/137.34 down.0(1.0(0.0(y13))) -> 1_flat.0(down.0(0.0(y13))) 275.91/137.34 down.0(1.0(0.1(y13))) -> 1_flat.0(down.0(0.1(y13))) 275.91/137.34 down.0(1.0(1.0(y14))) -> 1_flat.0(down.0(1.0(y14))) 275.91/137.34 down.0(1.0(1.1(y14))) -> 1_flat.0(down.0(1.1(y14))) 275.91/137.34 down.0(0.1(b.)) -> 0_flat.0(down.1(b.)) 275.91/137.34 down.0(0.0(f.0(x))) -> up.0(1.0(x)) 275.91/137.34 down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) 275.91/137.34 down.0(0.0(0.0(y9))) -> 0_flat.0(down.0(0.0(y9))) 275.91/137.34 down.0(0.0(0.1(y9))) -> 0_flat.0(down.0(0.1(y9))) 275.91/137.34 down.0(0.0(1.0(y10))) -> 0_flat.0(down.0(1.0(y10))) 275.91/137.34 down.0(0.0(1.1(y10))) -> 0_flat.0(down.0(1.1(y10))) 275.91/137.34 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) 275.91/137.34 down.0(a.) -> up.0(f.0(a.)) 275.91/137.34 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 275.91/137.34 275.91/137.34 The set Q consists of the following terms: 275.91/137.34 275.91/137.34 down.0(a.) 275.91/137.34 down.0(f.0(f.0(x0))) 275.91/137.34 down.0(f.0(f.1(x0))) 275.91/137.34 down.0(0.0(f.0(x0))) 275.91/137.34 down.0(0.0(f.1(x0))) 275.91/137.34 down.0(1.0(f.0(x0))) 275.91/137.34 down.0(1.0(f.1(x0))) 275.91/137.34 down.0(f.0(a.)) 275.91/137.34 down.0(f.1(b.)) 275.91/137.34 down.0(f.0(0.0(x0))) 275.91/137.34 down.0(f.0(0.1(x0))) 275.91/137.34 down.0(f.0(1.0(x0))) 275.91/137.34 down.0(f.0(1.1(x0))) 275.91/137.34 down.0(f.0(fresh_constant.)) 275.91/137.34 down.0(0.0(a.)) 275.91/137.34 down.0(0.1(b.)) 275.91/137.34 down.0(0.0(0.0(x0))) 275.91/137.34 down.0(0.0(0.1(x0))) 275.91/137.34 down.0(0.0(1.0(x0))) 275.91/137.34 down.0(0.0(1.1(x0))) 275.91/137.34 down.0(0.0(fresh_constant.)) 275.91/137.34 down.0(1.0(a.)) 275.91/137.34 down.0(1.1(b.)) 275.91/137.34 down.0(1.0(0.0(x0))) 275.91/137.34 down.0(1.0(0.1(x0))) 275.91/137.34 down.0(1.0(1.0(x0))) 275.91/137.34 down.0(1.0(1.1(x0))) 275.91/137.34 down.0(1.0(fresh_constant.)) 275.91/137.34 f_flat.0(up.0(x0)) 275.91/137.34 f_flat.0(up.1(x0)) 275.91/137.34 0_flat.0(up.0(x0)) 275.91/137.34 0_flat.0(up.1(x0)) 275.91/137.34 1_flat.0(up.0(x0)) 275.91/137.34 1_flat.0(up.1(x0)) 275.91/137.34 275.91/137.34 We have to consider all minimal (P,Q,R)-chains. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (152) MRRProof (EQUIVALENT) 275.91/137.34 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. 275.91/137.34 275.91/137.34 275.91/137.34 Strictly oriented rules of the TRS R: 275.91/137.34 275.91/137.34 down.0(1.1(b.)) -> 1_flat.0(down.1(b.)) 275.91/137.34 down.0(0.1(b.)) -> 0_flat.0(down.1(b.)) 275.91/137.34 275.91/137.34 Used ordering: Polynomial interpretation [POLO]: 275.91/137.34 275.91/137.34 POL(0.0(x_1)) = x_1 275.91/137.34 POL(0.1(x_1)) = 1 + x_1 275.91/137.34 POL(0_flat.0(x_1)) = x_1 275.91/137.34 POL(1.0(x_1)) = x_1 275.91/137.34 POL(1.1(x_1)) = 1 + x_1 275.91/137.34 POL(1_flat.0(x_1)) = x_1 275.91/137.34 POL(TOP.0(x_1)) = x_1 275.91/137.34 POL(a.) = 0 275.91/137.34 POL(b.) = 0 275.91/137.34 POL(down.0(x_1)) = x_1 275.91/137.34 POL(down.1(x_1)) = x_1 275.91/137.34 POL(f.0(x_1)) = x_1 275.91/137.34 POL(f_flat.0(x_1)) = x_1 275.91/137.34 POL(up.0(x_1)) = x_1 275.91/137.34 275.91/137.34 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (153) 275.91/137.34 Obligation: 275.91/137.34 Q DP problem: 275.91/137.34 The TRS P consists of the following rules: 275.91/137.34 275.91/137.34 TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(f.0(0.1(x0)))) -> TOP.0(f_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(f.0(1.1(x0)))) -> TOP.0(f_flat.0(down.0(1.1(x0)))) 275.91/137.34 TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(0.1(x0)))) -> TOP.0(0_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(1.1(x0)))) -> TOP.0(0_flat.0(down.0(1.1(x0)))) 275.91/137.34 TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(0.1(x0)))) -> TOP.0(1_flat.0(down.0(0.1(x0)))) 275.91/137.34 TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(1.1(x0)))) -> TOP.0(1_flat.0(down.0(1.1(x0)))) 275.91/137.34 275.91/137.34 The TRS R consists of the following rules: 275.91/137.34 275.91/137.34 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) 275.91/137.34 down.0(1.0(f.0(x))) -> up.0(f.0(0.0(x))) 275.91/137.34 down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) 275.91/137.34 down.0(1.0(0.0(y13))) -> 1_flat.0(down.0(0.0(y13))) 275.91/137.34 down.0(1.0(0.1(y13))) -> 1_flat.0(down.0(0.1(y13))) 275.91/137.34 down.0(1.0(1.0(y14))) -> 1_flat.0(down.0(1.0(y14))) 275.91/137.34 down.0(1.0(1.1(y14))) -> 1_flat.0(down.0(1.1(y14))) 275.91/137.34 down.0(0.0(f.0(x))) -> up.0(1.0(x)) 275.91/137.34 down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) 275.91/137.34 down.0(0.0(0.0(y9))) -> 0_flat.0(down.0(0.0(y9))) 275.91/137.34 down.0(0.0(0.1(y9))) -> 0_flat.0(down.0(0.1(y9))) 275.91/137.34 down.0(0.0(1.0(y10))) -> 0_flat.0(down.0(1.0(y10))) 275.91/137.34 down.0(0.0(1.1(y10))) -> 0_flat.0(down.0(1.1(y10))) 275.91/137.34 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) 275.91/137.34 down.0(a.) -> up.0(f.0(a.)) 275.91/137.34 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 275.91/137.34 275.91/137.34 The set Q consists of the following terms: 275.91/137.34 275.91/137.34 down.0(a.) 275.91/137.34 down.0(f.0(f.0(x0))) 275.91/137.34 down.0(f.0(f.1(x0))) 275.91/137.34 down.0(0.0(f.0(x0))) 275.91/137.34 down.0(0.0(f.1(x0))) 275.91/137.34 down.0(1.0(f.0(x0))) 275.91/137.34 down.0(1.0(f.1(x0))) 275.91/137.34 down.0(f.0(a.)) 275.91/137.34 down.0(f.1(b.)) 275.91/137.34 down.0(f.0(0.0(x0))) 275.91/137.34 down.0(f.0(0.1(x0))) 275.91/137.34 down.0(f.0(1.0(x0))) 275.91/137.34 down.0(f.0(1.1(x0))) 275.91/137.34 down.0(f.0(fresh_constant.)) 275.91/137.34 down.0(0.0(a.)) 275.91/137.34 down.0(0.1(b.)) 275.91/137.34 down.0(0.0(0.0(x0))) 275.91/137.34 down.0(0.0(0.1(x0))) 275.91/137.34 down.0(0.0(1.0(x0))) 275.91/137.34 down.0(0.0(1.1(x0))) 275.91/137.34 down.0(0.0(fresh_constant.)) 275.91/137.34 down.0(1.0(a.)) 275.91/137.34 down.0(1.1(b.)) 275.91/137.34 down.0(1.0(0.0(x0))) 275.91/137.34 down.0(1.0(0.1(x0))) 275.91/137.34 down.0(1.0(1.0(x0))) 275.91/137.34 down.0(1.0(1.1(x0))) 275.91/137.34 down.0(1.0(fresh_constant.)) 275.91/137.34 f_flat.0(up.0(x0)) 275.91/137.34 f_flat.0(up.1(x0)) 275.91/137.34 0_flat.0(up.0(x0)) 275.91/137.34 0_flat.0(up.1(x0)) 275.91/137.34 1_flat.0(up.0(x0)) 275.91/137.34 1_flat.0(up.1(x0)) 275.91/137.34 275.91/137.34 We have to consider all minimal (P,Q,R)-chains. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (154) DependencyGraphProof (EQUIVALENT) 275.91/137.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (155) 275.91/137.34 Obligation: 275.91/137.34 Q DP problem: 275.91/137.34 The TRS P consists of the following rules: 275.91/137.34 275.91/137.34 TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) 275.91/137.34 TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) 275.91/137.34 275.91/137.34 The TRS R consists of the following rules: 275.91/137.34 275.91/137.34 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) 275.91/137.34 down.0(1.0(f.0(x))) -> up.0(f.0(0.0(x))) 275.91/137.34 down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) 275.91/137.34 down.0(1.0(0.0(y13))) -> 1_flat.0(down.0(0.0(y13))) 275.91/137.34 down.0(1.0(0.1(y13))) -> 1_flat.0(down.0(0.1(y13))) 275.91/137.34 down.0(1.0(1.0(y14))) -> 1_flat.0(down.0(1.0(y14))) 275.91/137.34 down.0(1.0(1.1(y14))) -> 1_flat.0(down.0(1.1(y14))) 275.91/137.34 down.0(0.0(f.0(x))) -> up.0(1.0(x)) 275.91/137.34 down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) 275.91/137.34 down.0(0.0(0.0(y9))) -> 0_flat.0(down.0(0.0(y9))) 275.91/137.34 down.0(0.0(0.1(y9))) -> 0_flat.0(down.0(0.1(y9))) 275.91/137.34 down.0(0.0(1.0(y10))) -> 0_flat.0(down.0(1.0(y10))) 275.91/137.34 down.0(0.0(1.1(y10))) -> 0_flat.0(down.0(1.1(y10))) 275.91/137.34 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) 275.91/137.34 down.0(a.) -> up.0(f.0(a.)) 275.91/137.34 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 275.91/137.34 275.91/137.34 The set Q consists of the following terms: 275.91/137.34 275.91/137.34 down.0(a.) 275.91/137.34 down.0(f.0(f.0(x0))) 275.91/137.34 down.0(f.0(f.1(x0))) 275.91/137.34 down.0(0.0(f.0(x0))) 275.91/137.34 down.0(0.0(f.1(x0))) 275.91/137.34 down.0(1.0(f.0(x0))) 275.91/137.34 down.0(1.0(f.1(x0))) 275.91/137.34 down.0(f.0(a.)) 275.91/137.34 down.0(f.1(b.)) 275.91/137.34 down.0(f.0(0.0(x0))) 275.91/137.34 down.0(f.0(0.1(x0))) 275.91/137.34 down.0(f.0(1.0(x0))) 275.91/137.34 down.0(f.0(1.1(x0))) 275.91/137.34 down.0(f.0(fresh_constant.)) 275.91/137.34 down.0(0.0(a.)) 275.91/137.34 down.0(0.1(b.)) 275.91/137.34 down.0(0.0(0.0(x0))) 275.91/137.34 down.0(0.0(0.1(x0))) 275.91/137.34 down.0(0.0(1.0(x0))) 275.91/137.34 down.0(0.0(1.1(x0))) 275.91/137.34 down.0(0.0(fresh_constant.)) 275.91/137.34 down.0(1.0(a.)) 275.91/137.34 down.0(1.1(b.)) 275.91/137.34 down.0(1.0(0.0(x0))) 275.91/137.34 down.0(1.0(0.1(x0))) 275.91/137.34 down.0(1.0(1.0(x0))) 275.91/137.34 down.0(1.0(1.1(x0))) 275.91/137.34 down.0(1.0(fresh_constant.)) 275.91/137.34 f_flat.0(up.0(x0)) 275.91/137.34 f_flat.0(up.1(x0)) 275.91/137.34 0_flat.0(up.0(x0)) 275.91/137.34 0_flat.0(up.1(x0)) 275.91/137.34 1_flat.0(up.0(x0)) 275.91/137.34 1_flat.0(up.1(x0)) 275.91/137.34 275.91/137.34 We have to consider all minimal (P,Q,R)-chains. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (156) PisEmptyProof (SOUND) 275.91/137.34 The TRS P is empty. Hence, there is no (P,Q,R) chain. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (157) 275.91/137.34 TRUE 275.91/137.34 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (158) 275.91/137.34 Obligation: 275.91/137.34 Q DP problem: 275.91/137.34 The TRS P consists of the following rules: 275.91/137.34 275.91/137.34 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.91/137.34 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.91/137.34 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.91/137.34 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.91/137.34 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.91/137.34 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.91/137.34 275.91/137.34 The TRS R consists of the following rules: 275.91/137.34 275.91/137.34 1_flat(up(x_1)) -> up(1(x_1)) 275.91/137.34 down(1(f(x))) -> up(f(0(x))) 275.91/137.34 down(1(a)) -> 1_flat(down(a)) 275.91/137.34 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.91/137.34 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.91/137.34 down(0(f(x))) -> up(1(x)) 275.91/137.34 down(0(a)) -> 0_flat(down(a)) 275.91/137.34 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.91/137.34 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.91/137.34 0_flat(up(x_1)) -> up(0(x_1)) 275.91/137.34 down(a) -> up(f(a)) 275.91/137.34 f_flat(up(x_1)) -> up(f(x_1)) 275.91/137.34 275.91/137.34 The set Q consists of the following terms: 275.91/137.34 275.91/137.34 down(a) 275.91/137.34 down(f(f(x0))) 275.91/137.34 down(0(f(x0))) 275.91/137.34 down(1(f(x0))) 275.91/137.34 down(f(a)) 275.91/137.34 down(f(b)) 275.91/137.34 down(f(0(x0))) 275.91/137.34 down(f(1(x0))) 275.91/137.34 down(f(fresh_constant)) 275.91/137.34 down(0(a)) 275.91/137.34 down(0(b)) 275.91/137.34 down(0(0(x0))) 275.91/137.34 down(0(1(x0))) 275.91/137.34 down(0(fresh_constant)) 275.91/137.34 down(1(a)) 275.91/137.34 down(1(b)) 275.91/137.34 down(1(0(x0))) 275.91/137.34 down(1(1(x0))) 275.91/137.34 down(1(fresh_constant)) 275.91/137.34 f_flat(up(x0)) 275.91/137.34 0_flat(up(x0)) 275.91/137.34 1_flat(up(x0)) 275.91/137.34 275.91/137.34 We have to consider all minimal (P,Q,R)-chains. 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (159) QReductionProof (EQUIVALENT) 275.91/137.34 We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN]. 275.91/137.34 275.91/137.34 down(f(b)) 275.91/137.34 down(f(fresh_constant)) 275.91/137.34 down(0(b)) 275.91/137.34 down(0(fresh_constant)) 275.91/137.34 down(1(b)) 275.91/137.34 down(1(fresh_constant)) 275.91/137.34 275.91/137.34 275.91/137.34 ---------------------------------------- 275.91/137.34 275.91/137.34 (160) 275.91/137.34 Obligation: 275.91/137.34 Q DP problem: 275.91/137.34 The TRS P consists of the following rules: 275.91/137.34 275.91/137.34 TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) 275.91/137.34 TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) 275.91/137.34 TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) 275.91/137.34 TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) 275.91/137.34 TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) 275.91/137.34 TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) 275.91/137.34 275.91/137.34 The TRS R consists of the following rules: 275.91/137.34 275.91/137.34 1_flat(up(x_1)) -> up(1(x_1)) 275.91/137.34 down(1(f(x))) -> up(f(0(x))) 275.91/137.34 down(1(a)) -> 1_flat(down(a)) 275.91/137.34 down(1(0(y13))) -> 1_flat(down(0(y13))) 275.91/137.34 down(1(1(y14))) -> 1_flat(down(1(y14))) 275.91/137.34 down(0(f(x))) -> up(1(x)) 275.91/137.34 down(0(a)) -> 0_flat(down(a)) 275.91/137.34 down(0(0(y9))) -> 0_flat(down(0(y9))) 275.91/137.34 down(0(1(y10))) -> 0_flat(down(1(y10))) 275.91/137.34 0_flat(up(x_1)) -> up(0(x_1)) 275.91/137.34 down(a) -> up(f(a)) 275.91/137.34 f_flat(up(x_1)) -> up(f(x_1)) 275.91/137.34 275.91/137.34 The set Q consists of the following terms: 275.91/137.34 275.91/137.34 down(a) 275.91/137.34 down(f(f(x0))) 275.91/137.34 down(0(f(x0))) 275.91/137.34 down(1(f(x0))) 275.91/137.34 down(f(a)) 275.91/137.34 down(f(0(x0))) 275.91/137.34 down(f(1(x0))) 275.91/137.34 down(0(a)) 275.91/137.34 down(0(0(x0))) 275.91/137.34 down(0(1(x0))) 275.91/137.34 down(1(a)) 275.91/137.34 down(1(0(x0))) 275.91/137.34 down(1(1(x0))) 275.91/137.34 f_flat(up(x0)) 275.91/137.34 0_flat(up(x0)) 275.91/137.34 1_flat(up(x0)) 275.91/137.34 275.91/137.34 We have to consider all (P,Q,R)-chains. 276.13/137.43 EOF