/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Outermost Termination of the given OTRS could not be shown: (0) OTRS (1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) QDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) QDP (10) QReductionProof [EQUIVALENT, 0 ms] (11) QDP (12) MRRProof [EQUIVALENT, 15 ms] (13) QDP (14) DependencyGraphProof [EQUIVALENT, 0 ms] (15) QDP (16) UsableRulesProof [EQUIVALENT, 0 ms] (17) QDP (18) QReductionProof [EQUIVALENT, 0 ms] (19) QDP (20) UsableRulesReductionPairsProof [EQUIVALENT, 7 ms] (21) QDP (22) DependencyGraphProof [EQUIVALENT, 0 ms] (23) TRUE (24) QDP (25) UsableRulesProof [EQUIVALENT, 0 ms] (26) QDP (27) QReductionProof [EQUIVALENT, 0 ms] (28) QDP (29) TransformationProof [EQUIVALENT, 0 ms] (30) QDP (31) UsableRulesProof [EQUIVALENT, 0 ms] (32) QDP (33) QDPOrderProof [EQUIVALENT, 13 ms] (34) QDP (35) SplitQDPProof [EQUIVALENT, 0 ms] (36) AND (37) QDP (38) SemLabProof [SOUND, 0 ms] (39) QDP (40) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (41) QDP (42) MRRProof [EQUIVALENT, 6 ms] (43) QDP (44) QDPOrderProof [EQUIVALENT, 0 ms] (45) QDP (46) QDPOrderProof [EQUIVALENT, 0 ms] (47) QDP (48) QDPOrderProof [EQUIVALENT, 0 ms] (49) QDP (50) QDPOrderProof [EQUIVALENT, 0 ms] (51) QDP (52) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (53) QDP (54) PisEmptyProof [SOUND, 0 ms] (55) TRUE (56) QDP (57) TransformationProof [SOUND, 0 ms] (58) QDP (59) UsableRulesProof [EQUIVALENT, 0 ms] (60) QDP (61) QReductionProof [EQUIVALENT, 0 ms] (62) QDP (63) UsableRulesProof [EQUIVALENT, 0 ms] (64) QDP (65) QReductionProof [EQUIVALENT, 0 ms] (66) QDP (67) Trivial-Transformation [SOUND, 0 ms] (68) QTRS (69) DependencyPairsProof [EQUIVALENT, 0 ms] (70) QDP (71) DependencyGraphProof [EQUIVALENT, 2 ms] (72) AND (73) QDP (74) UsableRulesProof [EQUIVALENT, 0 ms] (75) QDP (76) QDPSizeChangeProof [EQUIVALENT, 0 ms] (77) YES (78) QDP (79) UsableRulesProof [EQUIVALENT, 0 ms] (80) QDP (81) NonTerminationLoopProof [COMPLETE, 6 ms] (82) NO (83) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] (84) QTRS (85) AAECC Innermost [EQUIVALENT, 0 ms] (86) QTRS (87) DependencyPairsProof [EQUIVALENT, 0 ms] (88) QDP (89) DependencyGraphProof [EQUIVALENT, 0 ms] (90) AND (91) QDP (92) UsableRulesProof [EQUIVALENT, 0 ms] (93) QDP (94) QReductionProof [EQUIVALENT, 0 ms] (95) QDP (96) QDPSizeChangeProof [EQUIVALENT, 0 ms] (97) YES (98) QDP (99) UsableRulesProof [EQUIVALENT, 0 ms] (100) QDP (101) QReductionProof [EQUIVALENT, 0 ms] (102) QDP (103) TransformationProof [EQUIVALENT, 32 ms] (104) QDP (105) DependencyGraphProof [EQUIVALENT, 0 ms] (106) QDP (107) UsableRulesProof [EQUIVALENT, 0 ms] (108) QDP (109) TransformationProof [EQUIVALENT, 0 ms] (110) QDP (111) TransformationProof [EQUIVALENT, 0 ms] (112) QDP (113) TransformationProof [EQUIVALENT, 0 ms] (114) QDP (115) TransformationProof [EQUIVALENT, 0 ms] (116) QDP (117) TransformationProof [EQUIVALENT, 0 ms] (118) QDP (119) DependencyGraphProof [EQUIVALENT, 0 ms] (120) QDP (121) TransformationProof [EQUIVALENT, 0 ms] (122) QDP (123) TransformationProof [EQUIVALENT, 0 ms] (124) QDP (125) TransformationProof [EQUIVALENT, 0 ms] (126) QDP (127) TransformationProof [EQUIVALENT, 0 ms] (128) QDP (129) TransformationProof [EQUIVALENT, 0 ms] (130) QDP (131) QDPOrderProof [EQUIVALENT, 23 ms] (132) QDP (133) DependencyGraphProof [EQUIVALENT, 0 ms] (134) QDP (135) QDPOrderProof [EQUIVALENT, 25 ms] (136) QDP (137) DependencyGraphProof [EQUIVALENT, 0 ms] (138) QDP (139) QDPOrderProof [EQUIVALENT, 0 ms] (140) QDP (141) SplitQDPProof [EQUIVALENT, 0 ms] (142) AND (143) QDP (144) SemLabProof [SOUND, 0 ms] (145) QDP (146) UsableRulesReductionPairsProof [EQUIVALENT, 12 ms] (147) QDP (148) MRRProof [EQUIVALENT, 0 ms] (149) QDP (150) DependencyGraphProof [EQUIVALENT, 0 ms] (151) QDP (152) PisEmptyProof [SOUND, 0 ms] (153) TRUE (154) QDP (155) QReductionProof [EQUIVALENT, 0 ms] (156) QDP (157) SplitQDPProof [EQUIVALENT, 0 ms] (158) AND (159) QDP (160) SemLabProof [SOUND, 0 ms] (161) QDP (162) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (163) QDP (164) MRRProof [EQUIVALENT, 3 ms] (165) QDP (166) DependencyGraphProof [EQUIVALENT, 0 ms] (167) QDP (168) PisEmptyProof [SOUND, 0 ms] (169) TRUE (170) QDP (171) QReductionProof [EQUIVALENT, 0 ms] (172) QDP ---------------------------------------- (0) Obligation: Term rewrite system R: The TRS R consists of the following rules: a -> f(a) f(f(x)) -> b 0(f(x)) -> 1(x) 1(f(x)) -> 2(x) 2(f(x)) -> f(0(x)) Outermost Strategy. ---------------------------------------- (1) Thiemann-SpecialC-Transformation (EQUIVALENT) We applied the Thiemann-SpecialC transformation to transform the outermost TRS to an innermost TRS. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) redex_f(f(x)) -> result_f(b) redex_0(f(x)) -> result_0(1(x)) redex_1(f(x)) -> result_1(2(x)) redex_2(f(x)) -> result_2(f(0(x))) check_f(result_f(x)) -> go_up(x) check_0(result_0(x)) -> go_up(x) check_1(result_1(x)) -> go_up(x) check_2(result_2(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) TOP(go_up(x)) -> REDUCE(x) REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) REDUCE(f(x_1)) -> REDEX_F(x_1) REDUCE(0(x_1)) -> CHECK_0(redex_0(x_1)) REDUCE(0(x_1)) -> REDEX_0(x_1) REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) REDUCE(1(x_1)) -> REDEX_1(x_1) REDUCE(2(x_1)) -> CHECK_2(redex_2(x_1)) REDUCE(2(x_1)) -> REDEX_2(x_1) CHECK_F(redex_f(x_1)) -> IN_F_1(reduce(x_1)) CHECK_F(redex_f(x_1)) -> REDUCE(x_1) CHECK_0(redex_0(x_1)) -> IN_0_1(reduce(x_1)) CHECK_0(redex_0(x_1)) -> REDUCE(x_1) CHECK_1(redex_1(x_1)) -> IN_1_1(reduce(x_1)) CHECK_1(redex_1(x_1)) -> REDUCE(x_1) CHECK_2(redex_2(x_1)) -> IN_2_1(reduce(x_1)) CHECK_2(redex_2(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) redex_f(f(x)) -> result_f(b) redex_0(f(x)) -> result_0(1(x)) redex_1(f(x)) -> result_1(2(x)) redex_2(f(x)) -> result_2(f(0(x))) check_f(result_f(x)) -> go_up(x) check_0(result_0(x)) -> go_up(x) check_1(result_1(x)) -> go_up(x) check_2(result_2(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 11 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: REDUCE(0(x_1)) -> CHECK_0(redex_0(x_1)) CHECK_0(redex_0(x_1)) -> REDUCE(x_1) REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) CHECK_1(redex_1(x_1)) -> REDUCE(x_1) REDUCE(2(x_1)) -> CHECK_2(redex_2(x_1)) CHECK_2(redex_2(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) redex_f(f(x)) -> result_f(b) redex_0(f(x)) -> result_0(1(x)) redex_1(f(x)) -> result_1(2(x)) redex_2(f(x)) -> result_2(f(0(x))) check_f(result_f(x)) -> go_up(x) check_0(result_0(x)) -> go_up(x) check_1(result_1(x)) -> go_up(x) check_2(result_2(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: REDUCE(0(x_1)) -> CHECK_0(redex_0(x_1)) CHECK_0(redex_0(x_1)) -> REDUCE(x_1) REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) CHECK_1(redex_1(x_1)) -> REDUCE(x_1) REDUCE(2(x_1)) -> CHECK_2(redex_2(x_1)) CHECK_2(redex_2(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: redex_2(f(x)) -> result_2(f(0(x))) redex_1(f(x)) -> result_1(2(x)) redex_0(f(x)) -> result_0(1(x)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(go_up(x0)) reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: REDUCE(0(x_1)) -> CHECK_0(redex_0(x_1)) CHECK_0(redex_0(x_1)) -> REDUCE(x_1) REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) CHECK_1(redex_1(x_1)) -> REDUCE(x_1) REDUCE(2(x_1)) -> CHECK_2(redex_2(x_1)) CHECK_2(redex_2(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: redex_2(f(x)) -> result_2(f(0(x))) redex_1(f(x)) -> result_1(2(x)) redex_0(f(x)) -> result_0(1(x)) The set Q consists of the following terms: redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: REDUCE(0(x_1)) -> CHECK_0(redex_0(x_1)) REDUCE(2(x_1)) -> CHECK_2(redex_2(x_1)) CHECK_2(redex_2(x_1)) -> REDUCE(x_1) Strictly oriented rules of the TRS R: redex_0(f(x)) -> result_0(1(x)) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + 2*x_1 POL(1(x_1)) = 2*x_1 POL(2(x_1)) = 2 + 2*x_1 POL(CHECK_0(x_1)) = 1 + 2*x_1 POL(CHECK_1(x_1)) = 1 + 2*x_1 POL(CHECK_2(x_1)) = 2 + x_1 POL(REDUCE(x_1)) = 1 + 2*x_1 POL(f(x_1)) = 2 + 2*x_1 POL(redex_0(x_1)) = 2*x_1 POL(redex_1(x_1)) = 2*x_1 POL(redex_2(x_1)) = 2*x_1 POL(result_0(x_1)) = 2*x_1 POL(result_1(x_1)) = 2*x_1 POL(result_2(x_1)) = x_1 ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_0(redex_0(x_1)) -> REDUCE(x_1) REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) CHECK_1(redex_1(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: redex_2(f(x)) -> result_2(f(0(x))) redex_1(f(x)) -> result_1(2(x)) The set Q consists of the following terms: redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_1(redex_1(x_1)) -> REDUCE(x_1) REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) The TRS R consists of the following rules: redex_2(f(x)) -> result_2(f(0(x))) redex_1(f(x)) -> result_1(2(x)) The set Q consists of the following terms: redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_1(redex_1(x_1)) -> REDUCE(x_1) REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) The TRS R consists of the following rules: redex_1(f(x)) -> result_1(2(x)) The set Q consists of the following terms: redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. redex_0(f(x0)) redex_2(f(x0)) ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_1(redex_1(x_1)) -> REDUCE(x_1) REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) The TRS R consists of the following rules: redex_1(f(x)) -> result_1(2(x)) The set Q consists of the following terms: redex_1(f(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) UsableRulesReductionPairsProof (EQUIVALENT) 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. The following dependency pairs can be deleted: REDUCE(1(x_1)) -> CHECK_1(redex_1(x_1)) The following rules are removed from R: redex_1(f(x)) -> result_1(2(x)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(1(x_1)) = 2*x_1 POL(2(x_1)) = x_1 POL(CHECK_1(x_1)) = x_1 POL(REDUCE(x_1)) = 2*x_1 POL(f(x_1)) = 2*x_1 POL(redex_1(x_1)) = 2*x_1 POL(result_1(x_1)) = 2*x_1 ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_1(redex_1(x_1)) -> REDUCE(x_1) R is empty. The set Q consists of the following terms: redex_1(f(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (23) TRUE ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) redex_f(f(x)) -> result_f(b) redex_0(f(x)) -> result_0(1(x)) redex_1(f(x)) -> result_1(2(x)) redex_2(f(x)) -> result_2(f(0(x))) check_f(result_f(x)) -> go_up(x) check_0(result_0(x)) -> go_up(x) check_1(result_1(x)) -> go_up(x) check_2(result_2(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) redex_2(f(x)) -> result_2(f(0(x))) check_2(result_2(x)) -> go_up(x) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) redex_1(f(x)) -> result_1(2(x)) check_1(result_1(x)) -> go_up(x) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) redex_0(f(x)) -> result_0(1(x)) check_0(result_0(x)) -> go_up(x) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) redex_f(f(x)) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(go_up(x0)) ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) redex_2(f(x)) -> result_2(f(0(x))) check_2(result_2(x)) -> go_up(x) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) redex_1(f(x)) -> result_1(2(x)) check_1(result_1(x)) -> go_up(x) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) redex_0(f(x)) -> result_0(1(x)) check_0(result_0(x)) -> go_up(x) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) redex_f(f(x)) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]: (TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))),TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))) (TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))),TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0)))) (TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))),TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0)))) (TOP(go_up(2(x0))) -> TOP(check_2(redex_2(x0))),TOP(go_up(2(x0))) -> TOP(check_2(redex_2(x0)))) (TOP(go_up(a)) -> TOP(go_up(f(a))),TOP(go_up(a)) -> TOP(go_up(f(a)))) ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) TOP(go_up(2(x0))) -> TOP(check_2(redex_2(x0))) TOP(go_up(a)) -> TOP(go_up(f(a))) The TRS R consists of the following rules: reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) redex_2(f(x)) -> result_2(f(0(x))) check_2(result_2(x)) -> go_up(x) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) redex_1(f(x)) -> result_1(2(x)) check_1(result_1(x)) -> go_up(x) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) redex_0(f(x)) -> result_0(1(x)) check_0(result_0(x)) -> go_up(x) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) redex_f(f(x)) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) TOP(go_up(2(x0))) -> TOP(check_2(redex_2(x0))) TOP(go_up(a)) -> TOP(go_up(f(a))) The TRS R consists of the following rules: redex_2(f(x)) -> result_2(f(0(x))) check_2(result_2(x)) -> go_up(x) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) redex_1(f(x)) -> result_1(2(x)) check_1(result_1(x)) -> go_up(x) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) redex_0(f(x)) -> result_0(1(x)) check_0(result_0(x)) -> go_up(x) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) redex_f(f(x)) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP(go_up(a)) -> TOP(go_up(f(a))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(TOP(x_1)) = x_1 POL(a) = 1 POL(b) = 0 POL(check_0(x_1)) = x_1 POL(check_1(x_1)) = x_1 POL(check_2(x_1)) = x_1 POL(check_f(x_1)) = x_1 POL(f(x_1)) = 0 POL(go_up(x_1)) = x_1 POL(in_0_1(x_1)) = 0 POL(in_1_1(x_1)) = 0 POL(in_2_1(x_1)) = 0 POL(in_f_1(x_1)) = 0 POL(redex_0(x_1)) = 0 POL(redex_1(x_1)) = 0 POL(redex_2(x_1)) = 0 POL(redex_f(x_1)) = 0 POL(reduce(x_1)) = 0 POL(result_0(x_1)) = x_1 POL(result_1(x_1)) = x_1 POL(result_2(x_1)) = x_1 POL(result_f(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: redex_f(f(x)) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) redex_0(f(x)) -> result_0(1(x)) check_0(result_0(x)) -> go_up(x) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) redex_1(f(x)) -> result_1(2(x)) check_1(result_1(x)) -> go_up(x) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) redex_2(f(x)) -> result_2(f(0(x))) check_2(result_2(x)) -> go_up(x) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) TOP(go_up(2(x0))) -> TOP(check_2(redex_2(x0))) The TRS R consists of the following rules: redex_2(f(x)) -> result_2(f(0(x))) check_2(result_2(x)) -> go_up(x) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) redex_1(f(x)) -> result_1(2(x)) check_1(result_1(x)) -> go_up(x) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) redex_0(f(x)) -> result_0(1(x)) check_0(result_0(x)) -> go_up(x) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) redex_f(f(x)) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) SplitQDPProof (EQUIVALENT) We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem ---------------------------------------- (36) Complex Obligation (AND) ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) TOP(go_up(2(x0))) -> TOP(check_2(redex_2(x0))) The TRS R consists of the following rules: redex_2(f(x)) -> result_2(f(0(x))) check_2(result_2(x)) -> go_up(x) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) redex_1(f(x)) -> result_1(2(x)) check_1(result_1(x)) -> go_up(x) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) redex_0(f(x)) -> result_0(1(x)) check_0(result_0(x)) -> go_up(x) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) redex_f(f(x)) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (38) SemLabProof (SOUND) We found the following model for the rules of the TRSs R and P. Interpretation over the domain with elements from 0 to 1. a: 1 reduce: 0 redex_0: 0 check_1: 0 redex_f: 0 2: 1 result_2: 0 in_1_1: 0 TOP: 0 in_0_1: 0 result_0: 0 f: 0 check_0: 0 result_1: 0 result_f: 0 redex_2: 0 check_2: 0 check_f: 0 redex_1: 0 go_up: 0 in_f_1: 0 b: 0 in_2_1: 0 0: 1 1: 1 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(go_up.0(f.0(x0))) -> TOP.0(check_f.0(redex_f.0(x0))) TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) TOP.0(go_up.1(0.0(x0))) -> TOP.0(check_0.0(redex_0.0(x0))) TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) TOP.0(go_up.1(1.0(x0))) -> TOP.0(check_1.0(redex_1.0(x0))) TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) TOP.0(go_up.1(2.0(x0))) -> TOP.0(check_2.0(redex_2.0(x0))) TOP.0(go_up.1(2.1(x0))) -> TOP.0(check_2.0(redex_2.1(x0))) The TRS R consists of the following rules: redex_2.0(f.0(x)) -> result_2.0(f.1(0.0(x))) redex_2.0(f.1(x)) -> result_2.0(f.1(0.1(x))) check_2.0(result_2.0(x)) -> go_up.0(x) check_2.0(result_2.1(x)) -> go_up.1(x) check_2.0(redex_2.0(x_1)) -> in_2_1.0(reduce.0(x_1)) check_2.0(redex_2.1(x_1)) -> in_2_1.0(reduce.1(x_1)) reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) reduce.1(2.0(x_1)) -> check_2.0(redex_2.0(x_1)) reduce.1(2.1(x_1)) -> check_2.0(redex_2.1(x_1)) reduce.1(a.) -> go_up.0(f.1(a.)) in_2_1.0(go_up.0(x_1)) -> go_up.1(2.0(x_1)) in_2_1.0(go_up.1(x_1)) -> go_up.1(2.1(x_1)) redex_1.0(f.0(x)) -> result_1.1(2.0(x)) redex_1.0(f.1(x)) -> result_1.1(2.1(x)) check_1.0(result_1.0(x)) -> go_up.0(x) check_1.0(result_1.1(x)) -> go_up.1(x) check_1.0(redex_1.0(x_1)) -> in_1_1.0(reduce.0(x_1)) check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) redex_0.0(f.0(x)) -> result_0.1(1.0(x)) redex_0.0(f.1(x)) -> result_0.1(1.1(x)) check_0.0(result_0.0(x)) -> go_up.0(x) check_0.0(result_0.1(x)) -> go_up.1(x) check_0.0(redex_0.0(x_1)) -> in_0_1.0(reduce.0(x_1)) check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) redex_f.0(f.0(x)) -> result_f.0(b.) redex_f.0(f.1(x)) -> result_f.0(b.) check_f.0(result_f.0(x)) -> go_up.0(x) check_f.0(result_f.1(x)) -> go_up.1(x) check_f.0(redex_f.0(x_1)) -> in_f_1.0(reduce.0(x_1)) check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) The set Q consists of the following terms: reduce.0(f.0(x0)) reduce.0(f.1(x0)) reduce.1(0.0(x0)) reduce.1(0.1(x0)) reduce.1(1.0(x0)) reduce.1(1.1(x0)) reduce.1(2.0(x0)) reduce.1(2.1(x0)) reduce.1(a.) redex_f.0(f.0(x0)) redex_f.0(f.1(x0)) redex_0.0(f.0(x0)) redex_0.0(f.1(x0)) redex_1.0(f.0(x0)) redex_1.0(f.1(x0)) redex_2.0(f.0(x0)) redex_2.0(f.1(x0)) check_f.0(result_f.0(x0)) check_f.0(result_f.1(x0)) check_0.0(result_0.0(x0)) check_0.0(result_0.1(x0)) check_1.0(result_1.0(x0)) check_1.0(result_1.1(x0)) check_2.0(result_2.0(x0)) check_2.0(result_2.1(x0)) check_f.0(redex_f.0(x0)) check_f.0(redex_f.1(x0)) check_0.0(redex_0.0(x0)) check_0.0(redex_0.1(x0)) check_1.0(redex_1.0(x0)) check_1.0(redex_1.1(x0)) check_2.0(redex_2.0(x0)) check_2.0(redex_2.1(x0)) in_f_1.0(go_up.0(x0)) in_f_1.0(go_up.1(x0)) in_0_1.0(go_up.0(x0)) in_0_1.0(go_up.1(x0)) in_1_1.0(go_up.0(x0)) in_1_1.0(go_up.1(x0)) in_2_1.0(go_up.0(x0)) in_2_1.0(go_up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (40) UsableRulesReductionPairsProof (EQUIVALENT) 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. No dependency pairs are removed. The following rules are removed from R: check_2.0(result_2.1(x)) -> go_up.1(x) check_1.0(result_1.0(x)) -> go_up.0(x) check_0.0(result_0.0(x)) -> go_up.0(x) check_f.0(result_f.1(x)) -> go_up.1(x) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0.0(x_1)) = x_1 POL(0.1(x_1)) = x_1 POL(1.0(x_1)) = x_1 POL(1.1(x_1)) = x_1 POL(2.0(x_1)) = x_1 POL(2.1(x_1)) = x_1 POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(check_0.0(x_1)) = x_1 POL(check_1.0(x_1)) = x_1 POL(check_2.0(x_1)) = x_1 POL(check_f.0(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = x_1 POL(go_up.0(x_1)) = x_1 POL(go_up.1(x_1)) = x_1 POL(in_0_1.0(x_1)) = x_1 POL(in_1_1.0(x_1)) = x_1 POL(in_2_1.0(x_1)) = x_1 POL(in_f_1.0(x_1)) = x_1 POL(redex_0.0(x_1)) = x_1 POL(redex_0.1(x_1)) = x_1 POL(redex_1.0(x_1)) = x_1 POL(redex_1.1(x_1)) = x_1 POL(redex_2.0(x_1)) = x_1 POL(redex_2.1(x_1)) = x_1 POL(redex_f.0(x_1)) = x_1 POL(redex_f.1(x_1)) = x_1 POL(reduce.0(x_1)) = x_1 POL(reduce.1(x_1)) = x_1 POL(result_0.0(x_1)) = x_1 POL(result_0.1(x_1)) = x_1 POL(result_1.0(x_1)) = x_1 POL(result_1.1(x_1)) = x_1 POL(result_2.0(x_1)) = x_1 POL(result_2.1(x_1)) = x_1 POL(result_f.0(x_1)) = x_1 POL(result_f.1(x_1)) = x_1 ---------------------------------------- (41) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(go_up.0(f.0(x0))) -> TOP.0(check_f.0(redex_f.0(x0))) TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) TOP.0(go_up.1(0.0(x0))) -> TOP.0(check_0.0(redex_0.0(x0))) TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) TOP.0(go_up.1(1.0(x0))) -> TOP.0(check_1.0(redex_1.0(x0))) TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) TOP.0(go_up.1(2.0(x0))) -> TOP.0(check_2.0(redex_2.0(x0))) TOP.0(go_up.1(2.1(x0))) -> TOP.0(check_2.0(redex_2.1(x0))) The TRS R consists of the following rules: check_2.0(redex_2.1(x_1)) -> in_2_1.0(reduce.1(x_1)) reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) reduce.1(2.0(x_1)) -> check_2.0(redex_2.0(x_1)) reduce.1(2.1(x_1)) -> check_2.0(redex_2.1(x_1)) reduce.1(a.) -> go_up.0(f.1(a.)) in_2_1.0(go_up.0(x_1)) -> go_up.1(2.0(x_1)) in_2_1.0(go_up.1(x_1)) -> go_up.1(2.1(x_1)) redex_2.0(f.0(x)) -> result_2.0(f.1(0.0(x))) redex_2.0(f.1(x)) -> result_2.0(f.1(0.1(x))) check_2.0(result_2.0(x)) -> go_up.0(x) check_2.0(redex_2.0(x_1)) -> in_2_1.0(reduce.0(x_1)) check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) redex_1.0(f.0(x)) -> result_1.1(2.0(x)) redex_1.0(f.1(x)) -> result_1.1(2.1(x)) check_1.0(result_1.1(x)) -> go_up.1(x) check_1.0(redex_1.0(x_1)) -> in_1_1.0(reduce.0(x_1)) check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) redex_0.0(f.0(x)) -> result_0.1(1.0(x)) redex_0.0(f.1(x)) -> result_0.1(1.1(x)) check_0.0(result_0.1(x)) -> go_up.1(x) check_0.0(redex_0.0(x_1)) -> in_0_1.0(reduce.0(x_1)) check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) redex_f.0(f.0(x)) -> result_f.0(b.) redex_f.0(f.1(x)) -> result_f.0(b.) check_f.0(result_f.0(x)) -> go_up.0(x) check_f.0(redex_f.0(x_1)) -> in_f_1.0(reduce.0(x_1)) The set Q consists of the following terms: reduce.0(f.0(x0)) reduce.0(f.1(x0)) reduce.1(0.0(x0)) reduce.1(0.1(x0)) reduce.1(1.0(x0)) reduce.1(1.1(x0)) reduce.1(2.0(x0)) reduce.1(2.1(x0)) reduce.1(a.) redex_f.0(f.0(x0)) redex_f.0(f.1(x0)) redex_0.0(f.0(x0)) redex_0.0(f.1(x0)) redex_1.0(f.0(x0)) redex_1.0(f.1(x0)) redex_2.0(f.0(x0)) redex_2.0(f.1(x0)) check_f.0(result_f.0(x0)) check_f.0(result_f.1(x0)) check_0.0(result_0.0(x0)) check_0.0(result_0.1(x0)) check_1.0(result_1.0(x0)) check_1.0(result_1.1(x0)) check_2.0(result_2.0(x0)) check_2.0(result_2.1(x0)) check_f.0(redex_f.0(x0)) check_f.0(redex_f.1(x0)) check_0.0(redex_0.0(x0)) check_0.0(redex_0.1(x0)) check_1.0(redex_1.0(x0)) check_1.0(redex_1.1(x0)) check_2.0(redex_2.0(x0)) check_2.0(redex_2.1(x0)) in_f_1.0(go_up.0(x0)) in_f_1.0(go_up.1(x0)) in_0_1.0(go_up.0(x0)) in_0_1.0(go_up.1(x0)) in_1_1.0(go_up.0(x0)) in_1_1.0(go_up.1(x0)) in_2_1.0(go_up.0(x0)) in_2_1.0(go_up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (42) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: check_2.0(redex_2.0(x_1)) -> in_2_1.0(reduce.0(x_1)) check_1.0(redex_1.0(x_1)) -> in_1_1.0(reduce.0(x_1)) check_0.0(redex_0.0(x_1)) -> in_0_1.0(reduce.0(x_1)) check_f.0(redex_f.0(x_1)) -> in_f_1.0(reduce.0(x_1)) Used ordering: Polynomial interpretation [POLO]: POL(0.0(x_1)) = 1 + x_1 POL(0.1(x_1)) = 1 + x_1 POL(1.0(x_1)) = 1 + x_1 POL(1.1(x_1)) = 1 + x_1 POL(2.0(x_1)) = 1 + x_1 POL(2.1(x_1)) = 1 + x_1 POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(check_0.0(x_1)) = 1 + x_1 POL(check_1.0(x_1)) = 1 + x_1 POL(check_2.0(x_1)) = 1 + x_1 POL(check_f.0(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = x_1 POL(go_up.0(x_1)) = 1 + x_1 POL(go_up.1(x_1)) = 1 + x_1 POL(in_0_1.0(x_1)) = 1 + x_1 POL(in_1_1.0(x_1)) = 1 + x_1 POL(in_2_1.0(x_1)) = 1 + x_1 POL(in_f_1.0(x_1)) = x_1 POL(redex_0.0(x_1)) = 1 + x_1 POL(redex_0.1(x_1)) = 1 + x_1 POL(redex_1.0(x_1)) = 1 + x_1 POL(redex_1.1(x_1)) = 1 + x_1 POL(redex_2.0(x_1)) = 1 + x_1 POL(redex_2.1(x_1)) = 1 + x_1 POL(redex_f.0(x_1)) = 1 + x_1 POL(redex_f.1(x_1)) = 1 + x_1 POL(reduce.0(x_1)) = x_1 POL(reduce.1(x_1)) = 1 + x_1 POL(result_0.1(x_1)) = x_1 POL(result_1.1(x_1)) = x_1 POL(result_2.0(x_1)) = x_1 POL(result_f.0(x_1)) = 1 + x_1 ---------------------------------------- (43) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(go_up.0(f.0(x0))) -> TOP.0(check_f.0(redex_f.0(x0))) TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) TOP.0(go_up.1(0.0(x0))) -> TOP.0(check_0.0(redex_0.0(x0))) TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) TOP.0(go_up.1(1.0(x0))) -> TOP.0(check_1.0(redex_1.0(x0))) TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) TOP.0(go_up.1(2.0(x0))) -> TOP.0(check_2.0(redex_2.0(x0))) TOP.0(go_up.1(2.1(x0))) -> TOP.0(check_2.0(redex_2.1(x0))) The TRS R consists of the following rules: check_2.0(redex_2.1(x_1)) -> in_2_1.0(reduce.1(x_1)) reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) reduce.1(2.0(x_1)) -> check_2.0(redex_2.0(x_1)) reduce.1(2.1(x_1)) -> check_2.0(redex_2.1(x_1)) reduce.1(a.) -> go_up.0(f.1(a.)) in_2_1.0(go_up.0(x_1)) -> go_up.1(2.0(x_1)) in_2_1.0(go_up.1(x_1)) -> go_up.1(2.1(x_1)) redex_2.0(f.0(x)) -> result_2.0(f.1(0.0(x))) redex_2.0(f.1(x)) -> result_2.0(f.1(0.1(x))) check_2.0(result_2.0(x)) -> go_up.0(x) check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) redex_1.0(f.0(x)) -> result_1.1(2.0(x)) redex_1.0(f.1(x)) -> result_1.1(2.1(x)) check_1.0(result_1.1(x)) -> go_up.1(x) check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) redex_0.0(f.0(x)) -> result_0.1(1.0(x)) redex_0.0(f.1(x)) -> result_0.1(1.1(x)) check_0.0(result_0.1(x)) -> go_up.1(x) check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) redex_f.0(f.0(x)) -> result_f.0(b.) redex_f.0(f.1(x)) -> result_f.0(b.) check_f.0(result_f.0(x)) -> go_up.0(x) The set Q consists of the following terms: reduce.0(f.0(x0)) reduce.0(f.1(x0)) reduce.1(0.0(x0)) reduce.1(0.1(x0)) reduce.1(1.0(x0)) reduce.1(1.1(x0)) reduce.1(2.0(x0)) reduce.1(2.1(x0)) reduce.1(a.) redex_f.0(f.0(x0)) redex_f.0(f.1(x0)) redex_0.0(f.0(x0)) redex_0.0(f.1(x0)) redex_1.0(f.0(x0)) redex_1.0(f.1(x0)) redex_2.0(f.0(x0)) redex_2.0(f.1(x0)) check_f.0(result_f.0(x0)) check_f.0(result_f.1(x0)) check_0.0(result_0.0(x0)) check_0.0(result_0.1(x0)) check_1.0(result_1.0(x0)) check_1.0(result_1.1(x0)) check_2.0(result_2.0(x0)) check_2.0(result_2.1(x0)) check_f.0(redex_f.0(x0)) check_f.0(redex_f.1(x0)) check_0.0(redex_0.0(x0)) check_0.0(redex_0.1(x0)) check_1.0(redex_1.0(x0)) check_1.0(redex_1.1(x0)) check_2.0(redex_2.0(x0)) check_2.0(redex_2.1(x0)) in_f_1.0(go_up.0(x0)) in_f_1.0(go_up.1(x0)) in_0_1.0(go_up.0(x0)) in_0_1.0(go_up.1(x0)) in_1_1.0(go_up.0(x0)) in_1_1.0(go_up.1(x0)) in_2_1.0(go_up.0(x0)) in_2_1.0(go_up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (44) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP.0(go_up.1(2.0(x0))) -> TOP.0(check_2.0(redex_2.0(x0))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0.0(x_1)) = 1 POL(0.1(x_1)) = 1 + x_1 POL(1.0(x_1)) = 1 POL(1.1(x_1)) = 1 + x_1 POL(2.0(x_1)) = 0 POL(2.1(x_1)) = 1 + x_1 POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 1 POL(check_0.0(x_1)) = 1 POL(check_1.0(x_1)) = 1 POL(check_2.0(x_1)) = x_1 POL(check_f.0(x_1)) = 0 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 0 POL(go_up.0(x_1)) = 0 POL(go_up.1(x_1)) = 1 POL(in_0_1.0(x_1)) = 1 POL(in_1_1.0(x_1)) = 1 POL(in_2_1.0(x_1)) = 1 POL(in_f_1.0(x_1)) = 0 POL(redex_0.0(x_1)) = 0 POL(redex_0.1(x_1)) = x_1 POL(redex_1.0(x_1)) = 0 POL(redex_1.1(x_1)) = x_1 POL(redex_2.0(x_1)) = 0 POL(redex_2.1(x_1)) = 1 POL(redex_f.0(x_1)) = x_1 POL(redex_f.1(x_1)) = x_1 POL(reduce.1(x_1)) = x_1 POL(result_0.1(x_1)) = 0 POL(result_1.1(x_1)) = 0 POL(result_2.0(x_1)) = 0 POL(result_f.0(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) check_f.0(result_f.0(x)) -> go_up.0(x) check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) check_0.0(result_0.1(x)) -> go_up.1(x) check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) check_1.0(result_1.1(x)) -> go_up.1(x) redex_2.0(f.0(x)) -> result_2.0(f.1(0.0(x))) redex_2.0(f.1(x)) -> result_2.0(f.1(0.1(x))) check_2.0(redex_2.1(x_1)) -> in_2_1.0(reduce.1(x_1)) check_2.0(result_2.0(x)) -> go_up.0(x) reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) reduce.1(2.0(x_1)) -> check_2.0(redex_2.0(x_1)) reduce.1(2.1(x_1)) -> check_2.0(redex_2.1(x_1)) in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) in_2_1.0(go_up.0(x_1)) -> go_up.1(2.0(x_1)) in_2_1.0(go_up.1(x_1)) -> go_up.1(2.1(x_1)) ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(go_up.0(f.0(x0))) -> TOP.0(check_f.0(redex_f.0(x0))) TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) TOP.0(go_up.1(0.0(x0))) -> TOP.0(check_0.0(redex_0.0(x0))) TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) TOP.0(go_up.1(1.0(x0))) -> TOP.0(check_1.0(redex_1.0(x0))) TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) TOP.0(go_up.1(2.1(x0))) -> TOP.0(check_2.0(redex_2.1(x0))) The TRS R consists of the following rules: check_2.0(redex_2.1(x_1)) -> in_2_1.0(reduce.1(x_1)) reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) reduce.1(2.0(x_1)) -> check_2.0(redex_2.0(x_1)) reduce.1(2.1(x_1)) -> check_2.0(redex_2.1(x_1)) reduce.1(a.) -> go_up.0(f.1(a.)) in_2_1.0(go_up.0(x_1)) -> go_up.1(2.0(x_1)) in_2_1.0(go_up.1(x_1)) -> go_up.1(2.1(x_1)) redex_2.0(f.0(x)) -> result_2.0(f.1(0.0(x))) redex_2.0(f.1(x)) -> result_2.0(f.1(0.1(x))) check_2.0(result_2.0(x)) -> go_up.0(x) check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) redex_1.0(f.0(x)) -> result_1.1(2.0(x)) redex_1.0(f.1(x)) -> result_1.1(2.1(x)) check_1.0(result_1.1(x)) -> go_up.1(x) check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) redex_0.0(f.0(x)) -> result_0.1(1.0(x)) redex_0.0(f.1(x)) -> result_0.1(1.1(x)) check_0.0(result_0.1(x)) -> go_up.1(x) check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) redex_f.0(f.0(x)) -> result_f.0(b.) redex_f.0(f.1(x)) -> result_f.0(b.) check_f.0(result_f.0(x)) -> go_up.0(x) The set Q consists of the following terms: reduce.0(f.0(x0)) reduce.0(f.1(x0)) reduce.1(0.0(x0)) reduce.1(0.1(x0)) reduce.1(1.0(x0)) reduce.1(1.1(x0)) reduce.1(2.0(x0)) reduce.1(2.1(x0)) reduce.1(a.) redex_f.0(f.0(x0)) redex_f.0(f.1(x0)) redex_0.0(f.0(x0)) redex_0.0(f.1(x0)) redex_1.0(f.0(x0)) redex_1.0(f.1(x0)) redex_2.0(f.0(x0)) redex_2.0(f.1(x0)) check_f.0(result_f.0(x0)) check_f.0(result_f.1(x0)) check_0.0(result_0.0(x0)) check_0.0(result_0.1(x0)) check_1.0(result_1.0(x0)) check_1.0(result_1.1(x0)) check_2.0(result_2.0(x0)) check_2.0(result_2.1(x0)) check_f.0(redex_f.0(x0)) check_f.0(redex_f.1(x0)) check_0.0(redex_0.0(x0)) check_0.0(redex_0.1(x0)) check_1.0(redex_1.0(x0)) check_1.0(redex_1.1(x0)) check_2.0(redex_2.0(x0)) check_2.0(redex_2.1(x0)) in_f_1.0(go_up.0(x0)) in_f_1.0(go_up.1(x0)) in_0_1.0(go_up.0(x0)) in_0_1.0(go_up.1(x0)) in_1_1.0(go_up.0(x0)) in_1_1.0(go_up.1(x0)) in_2_1.0(go_up.0(x0)) in_2_1.0(go_up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (46) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP.0(go_up.1(0.0(x0))) -> TOP.0(check_0.0(redex_0.0(x0))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0.0(x_1)) = 1 POL(0.1(x_1)) = 1 POL(1.0(x_1)) = 0 POL(1.1(x_1)) = 0 POL(2.0(x_1)) = 0 POL(2.1(x_1)) = 0 POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 1 POL(check_0.0(x_1)) = x_1 POL(check_1.0(x_1)) = x_1 POL(check_2.0(x_1)) = 0 POL(check_f.0(x_1)) = 0 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 0 POL(go_up.0(x_1)) = 0 POL(go_up.1(x_1)) = x_1 POL(in_0_1.0(x_1)) = 1 POL(in_1_1.0(x_1)) = 0 POL(in_2_1.0(x_1)) = 0 POL(in_f_1.0(x_1)) = 0 POL(redex_0.0(x_1)) = 0 POL(redex_0.1(x_1)) = 1 POL(redex_1.0(x_1)) = 0 POL(redex_1.1(x_1)) = 0 POL(redex_2.0(x_1)) = 0 POL(redex_2.1(x_1)) = x_1 POL(redex_f.0(x_1)) = x_1 POL(redex_f.1(x_1)) = x_1 POL(reduce.1(x_1)) = x_1 POL(result_0.1(x_1)) = x_1 POL(result_1.1(x_1)) = x_1 POL(result_2.0(x_1)) = 0 POL(result_f.0(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) check_f.0(result_f.0(x)) -> go_up.0(x) redex_0.0(f.0(x)) -> result_0.1(1.0(x)) redex_0.0(f.1(x)) -> result_0.1(1.1(x)) check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) check_0.0(result_0.1(x)) -> go_up.1(x) redex_1.0(f.0(x)) -> result_1.1(2.0(x)) redex_1.0(f.1(x)) -> result_1.1(2.1(x)) check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) check_1.0(result_1.1(x)) -> go_up.1(x) check_2.0(redex_2.1(x_1)) -> in_2_1.0(reduce.1(x_1)) reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) reduce.1(2.0(x_1)) -> check_2.0(redex_2.0(x_1)) reduce.1(2.1(x_1)) -> check_2.0(redex_2.1(x_1)) in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) check_2.0(result_2.0(x)) -> go_up.0(x) in_2_1.0(go_up.0(x_1)) -> go_up.1(2.0(x_1)) in_2_1.0(go_up.1(x_1)) -> go_up.1(2.1(x_1)) ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(go_up.0(f.0(x0))) -> TOP.0(check_f.0(redex_f.0(x0))) TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) TOP.0(go_up.1(1.0(x0))) -> TOP.0(check_1.0(redex_1.0(x0))) TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) TOP.0(go_up.1(2.1(x0))) -> TOP.0(check_2.0(redex_2.1(x0))) The TRS R consists of the following rules: check_2.0(redex_2.1(x_1)) -> in_2_1.0(reduce.1(x_1)) reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) reduce.1(2.0(x_1)) -> check_2.0(redex_2.0(x_1)) reduce.1(2.1(x_1)) -> check_2.0(redex_2.1(x_1)) reduce.1(a.) -> go_up.0(f.1(a.)) in_2_1.0(go_up.0(x_1)) -> go_up.1(2.0(x_1)) in_2_1.0(go_up.1(x_1)) -> go_up.1(2.1(x_1)) redex_2.0(f.0(x)) -> result_2.0(f.1(0.0(x))) redex_2.0(f.1(x)) -> result_2.0(f.1(0.1(x))) check_2.0(result_2.0(x)) -> go_up.0(x) check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) redex_1.0(f.0(x)) -> result_1.1(2.0(x)) redex_1.0(f.1(x)) -> result_1.1(2.1(x)) check_1.0(result_1.1(x)) -> go_up.1(x) check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) redex_0.0(f.0(x)) -> result_0.1(1.0(x)) redex_0.0(f.1(x)) -> result_0.1(1.1(x)) check_0.0(result_0.1(x)) -> go_up.1(x) check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) redex_f.0(f.0(x)) -> result_f.0(b.) redex_f.0(f.1(x)) -> result_f.0(b.) check_f.0(result_f.0(x)) -> go_up.0(x) The set Q consists of the following terms: reduce.0(f.0(x0)) reduce.0(f.1(x0)) reduce.1(0.0(x0)) reduce.1(0.1(x0)) reduce.1(1.0(x0)) reduce.1(1.1(x0)) reduce.1(2.0(x0)) reduce.1(2.1(x0)) reduce.1(a.) redex_f.0(f.0(x0)) redex_f.0(f.1(x0)) redex_0.0(f.0(x0)) redex_0.0(f.1(x0)) redex_1.0(f.0(x0)) redex_1.0(f.1(x0)) redex_2.0(f.0(x0)) redex_2.0(f.1(x0)) check_f.0(result_f.0(x0)) check_f.0(result_f.1(x0)) check_0.0(result_0.0(x0)) check_0.0(result_0.1(x0)) check_1.0(result_1.0(x0)) check_1.0(result_1.1(x0)) check_2.0(result_2.0(x0)) check_2.0(result_2.1(x0)) check_f.0(redex_f.0(x0)) check_f.0(redex_f.1(x0)) check_0.0(redex_0.0(x0)) check_0.0(redex_0.1(x0)) check_1.0(redex_1.0(x0)) check_1.0(redex_1.1(x0)) check_2.0(redex_2.0(x0)) check_2.0(redex_2.1(x0)) in_f_1.0(go_up.0(x0)) in_f_1.0(go_up.1(x0)) in_0_1.0(go_up.0(x0)) in_0_1.0(go_up.1(x0)) in_1_1.0(go_up.0(x0)) in_1_1.0(go_up.1(x0)) in_2_1.0(go_up.0(x0)) in_2_1.0(go_up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP.0(go_up.1(1.0(x0))) -> TOP.0(check_1.0(redex_1.0(x0))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0.0(x_1)) = 1 POL(0.1(x_1)) = 1 POL(1.0(x_1)) = 1 POL(1.1(x_1)) = 1 POL(2.0(x_1)) = 0 POL(2.1(x_1)) = 0 POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 1 POL(check_0.0(x_1)) = x_1 POL(check_1.0(x_1)) = x_1 POL(check_2.0(x_1)) = 0 POL(check_f.0(x_1)) = 0 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 0 POL(go_up.0(x_1)) = 0 POL(go_up.1(x_1)) = x_1 POL(in_0_1.0(x_1)) = 1 POL(in_1_1.0(x_1)) = 1 POL(in_2_1.0(x_1)) = 0 POL(in_f_1.0(x_1)) = 0 POL(redex_0.0(x_1)) = 1 POL(redex_0.1(x_1)) = 1 POL(redex_1.0(x_1)) = 0 POL(redex_1.1(x_1)) = 1 POL(redex_2.0(x_1)) = 0 POL(redex_2.1(x_1)) = x_1 POL(redex_f.0(x_1)) = x_1 POL(redex_f.1(x_1)) = x_1 POL(reduce.1(x_1)) = x_1 POL(result_0.1(x_1)) = x_1 POL(result_1.1(x_1)) = x_1 POL(result_2.0(x_1)) = 0 POL(result_f.0(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) check_f.0(result_f.0(x)) -> go_up.0(x) check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) redex_1.0(f.0(x)) -> result_1.1(2.0(x)) redex_1.0(f.1(x)) -> result_1.1(2.1(x)) check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) check_1.0(result_1.1(x)) -> go_up.1(x) check_2.0(redex_2.1(x_1)) -> in_2_1.0(reduce.1(x_1)) reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) reduce.1(2.0(x_1)) -> check_2.0(redex_2.0(x_1)) reduce.1(2.1(x_1)) -> check_2.0(redex_2.1(x_1)) in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) redex_0.0(f.0(x)) -> result_0.1(1.0(x)) redex_0.0(f.1(x)) -> result_0.1(1.1(x)) check_0.0(result_0.1(x)) -> go_up.1(x) in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) check_2.0(result_2.0(x)) -> go_up.0(x) in_2_1.0(go_up.0(x_1)) -> go_up.1(2.0(x_1)) in_2_1.0(go_up.1(x_1)) -> go_up.1(2.1(x_1)) ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(go_up.0(f.0(x0))) -> TOP.0(check_f.0(redex_f.0(x0))) TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) TOP.0(go_up.1(2.1(x0))) -> TOP.0(check_2.0(redex_2.1(x0))) The TRS R consists of the following rules: check_2.0(redex_2.1(x_1)) -> in_2_1.0(reduce.1(x_1)) reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) reduce.1(2.0(x_1)) -> check_2.0(redex_2.0(x_1)) reduce.1(2.1(x_1)) -> check_2.0(redex_2.1(x_1)) reduce.1(a.) -> go_up.0(f.1(a.)) in_2_1.0(go_up.0(x_1)) -> go_up.1(2.0(x_1)) in_2_1.0(go_up.1(x_1)) -> go_up.1(2.1(x_1)) redex_2.0(f.0(x)) -> result_2.0(f.1(0.0(x))) redex_2.0(f.1(x)) -> result_2.0(f.1(0.1(x))) check_2.0(result_2.0(x)) -> go_up.0(x) check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) redex_1.0(f.0(x)) -> result_1.1(2.0(x)) redex_1.0(f.1(x)) -> result_1.1(2.1(x)) check_1.0(result_1.1(x)) -> go_up.1(x) check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) redex_0.0(f.0(x)) -> result_0.1(1.0(x)) redex_0.0(f.1(x)) -> result_0.1(1.1(x)) check_0.0(result_0.1(x)) -> go_up.1(x) check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) redex_f.0(f.0(x)) -> result_f.0(b.) redex_f.0(f.1(x)) -> result_f.0(b.) check_f.0(result_f.0(x)) -> go_up.0(x) The set Q consists of the following terms: reduce.0(f.0(x0)) reduce.0(f.1(x0)) reduce.1(0.0(x0)) reduce.1(0.1(x0)) reduce.1(1.0(x0)) reduce.1(1.1(x0)) reduce.1(2.0(x0)) reduce.1(2.1(x0)) reduce.1(a.) redex_f.0(f.0(x0)) redex_f.0(f.1(x0)) redex_0.0(f.0(x0)) redex_0.0(f.1(x0)) redex_1.0(f.0(x0)) redex_1.0(f.1(x0)) redex_2.0(f.0(x0)) redex_2.0(f.1(x0)) check_f.0(result_f.0(x0)) check_f.0(result_f.1(x0)) check_0.0(result_0.0(x0)) check_0.0(result_0.1(x0)) check_1.0(result_1.0(x0)) check_1.0(result_1.1(x0)) check_2.0(result_2.0(x0)) check_2.0(result_2.1(x0)) check_f.0(redex_f.0(x0)) check_f.0(redex_f.1(x0)) check_0.0(redex_0.0(x0)) check_0.0(redex_0.1(x0)) check_1.0(redex_1.0(x0)) check_1.0(redex_1.1(x0)) check_2.0(redex_2.0(x0)) check_2.0(redex_2.1(x0)) in_f_1.0(go_up.0(x0)) in_f_1.0(go_up.1(x0)) in_0_1.0(go_up.0(x0)) in_0_1.0(go_up.1(x0)) in_1_1.0(go_up.0(x0)) in_1_1.0(go_up.1(x0)) in_2_1.0(go_up.0(x0)) in_2_1.0(go_up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (50) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP.0(go_up.0(f.0(x0))) -> TOP.0(check_f.0(redex_f.0(x0))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0.0(x_1)) = x_1 POL(0.1(x_1)) = x_1 POL(1.0(x_1)) = x_1 POL(1.1(x_1)) = x_1 POL(2.0(x_1)) = x_1 POL(2.1(x_1)) = x_1 POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(check_0.0(x_1)) = 0 POL(check_1.0(x_1)) = 0 POL(check_2.0(x_1)) = x_1 POL(check_f.0(x_1)) = x_1 POL(f.0(x_1)) = 1 POL(f.1(x_1)) = 1 POL(go_up.0(x_1)) = x_1 POL(go_up.1(x_1)) = 0 POL(in_0_1.0(x_1)) = 0 POL(in_1_1.0(x_1)) = 0 POL(in_2_1.0(x_1)) = 0 POL(in_f_1.0(x_1)) = 1 POL(redex_0.0(x_1)) = 0 POL(redex_0.1(x_1)) = x_1 POL(redex_1.0(x_1)) = 0 POL(redex_1.1(x_1)) = x_1 POL(redex_2.0(x_1)) = x_1 POL(redex_2.1(x_1)) = 0 POL(redex_f.0(x_1)) = 0 POL(redex_f.1(x_1)) = 1 POL(reduce.1(x_1)) = x_1 POL(result_0.1(x_1)) = 0 POL(result_1.1(x_1)) = 0 POL(result_2.0(x_1)) = x_1 POL(result_f.0(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: redex_f.0(f.0(x)) -> result_f.0(b.) redex_f.0(f.1(x)) -> result_f.0(b.) check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) check_f.0(result_f.0(x)) -> go_up.0(x) check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) check_2.0(redex_2.1(x_1)) -> in_2_1.0(reduce.1(x_1)) reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) reduce.1(2.0(x_1)) -> check_2.0(redex_2.0(x_1)) reduce.1(2.1(x_1)) -> check_2.0(redex_2.1(x_1)) in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) check_0.0(result_0.1(x)) -> go_up.1(x) in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) check_1.0(result_1.1(x)) -> go_up.1(x) in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) redex_2.0(f.0(x)) -> result_2.0(f.1(0.0(x))) redex_2.0(f.1(x)) -> result_2.0(f.1(0.1(x))) check_2.0(result_2.0(x)) -> go_up.0(x) in_2_1.0(go_up.0(x_1)) -> go_up.1(2.0(x_1)) in_2_1.0(go_up.1(x_1)) -> go_up.1(2.1(x_1)) ---------------------------------------- (51) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) TOP.0(go_up.1(2.1(x0))) -> TOP.0(check_2.0(redex_2.1(x0))) The TRS R consists of the following rules: check_2.0(redex_2.1(x_1)) -> in_2_1.0(reduce.1(x_1)) reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) reduce.1(2.0(x_1)) -> check_2.0(redex_2.0(x_1)) reduce.1(2.1(x_1)) -> check_2.0(redex_2.1(x_1)) reduce.1(a.) -> go_up.0(f.1(a.)) in_2_1.0(go_up.0(x_1)) -> go_up.1(2.0(x_1)) in_2_1.0(go_up.1(x_1)) -> go_up.1(2.1(x_1)) redex_2.0(f.0(x)) -> result_2.0(f.1(0.0(x))) redex_2.0(f.1(x)) -> result_2.0(f.1(0.1(x))) check_2.0(result_2.0(x)) -> go_up.0(x) check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) redex_1.0(f.0(x)) -> result_1.1(2.0(x)) redex_1.0(f.1(x)) -> result_1.1(2.1(x)) check_1.0(result_1.1(x)) -> go_up.1(x) check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) redex_0.0(f.0(x)) -> result_0.1(1.0(x)) redex_0.0(f.1(x)) -> result_0.1(1.1(x)) check_0.0(result_0.1(x)) -> go_up.1(x) check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) redex_f.0(f.0(x)) -> result_f.0(b.) redex_f.0(f.1(x)) -> result_f.0(b.) check_f.0(result_f.0(x)) -> go_up.0(x) The set Q consists of the following terms: reduce.0(f.0(x0)) reduce.0(f.1(x0)) reduce.1(0.0(x0)) reduce.1(0.1(x0)) reduce.1(1.0(x0)) reduce.1(1.1(x0)) reduce.1(2.0(x0)) reduce.1(2.1(x0)) reduce.1(a.) redex_f.0(f.0(x0)) redex_f.0(f.1(x0)) redex_0.0(f.0(x0)) redex_0.0(f.1(x0)) redex_1.0(f.0(x0)) redex_1.0(f.1(x0)) redex_2.0(f.0(x0)) redex_2.0(f.1(x0)) check_f.0(result_f.0(x0)) check_f.0(result_f.1(x0)) check_0.0(result_0.0(x0)) check_0.0(result_0.1(x0)) check_1.0(result_1.0(x0)) check_1.0(result_1.1(x0)) check_2.0(result_2.0(x0)) check_2.0(result_2.1(x0)) check_f.0(redex_f.0(x0)) check_f.0(redex_f.1(x0)) check_0.0(redex_0.0(x0)) check_0.0(redex_0.1(x0)) check_1.0(redex_1.0(x0)) check_1.0(redex_1.1(x0)) check_2.0(redex_2.0(x0)) check_2.0(redex_2.1(x0)) in_f_1.0(go_up.0(x0)) in_f_1.0(go_up.1(x0)) in_0_1.0(go_up.0(x0)) in_0_1.0(go_up.1(x0)) in_1_1.0(go_up.0(x0)) in_1_1.0(go_up.1(x0)) in_2_1.0(go_up.0(x0)) in_2_1.0(go_up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (52) UsableRulesReductionPairsProof (EQUIVALENT) 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. No dependency pairs are removed. The following rules are removed from R: redex_f.0(f.0(x)) -> result_f.0(b.) redex_f.0(f.1(x)) -> result_f.0(b.) check_f.0(result_f.0(x)) -> go_up.0(x) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0.0(x_1)) = x_1 POL(0.1(x_1)) = x_1 POL(1.0(x_1)) = x_1 POL(1.1(x_1)) = x_1 POL(2.0(x_1)) = x_1 POL(2.1(x_1)) = x_1 POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(check_0.0(x_1)) = x_1 POL(check_1.0(x_1)) = x_1 POL(check_2.0(x_1)) = x_1 POL(check_f.0(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = x_1 POL(go_up.0(x_1)) = x_1 POL(go_up.1(x_1)) = x_1 POL(in_0_1.0(x_1)) = x_1 POL(in_1_1.0(x_1)) = x_1 POL(in_2_1.0(x_1)) = x_1 POL(in_f_1.0(x_1)) = x_1 POL(redex_0.0(x_1)) = x_1 POL(redex_0.1(x_1)) = x_1 POL(redex_1.0(x_1)) = x_1 POL(redex_1.1(x_1)) = x_1 POL(redex_2.0(x_1)) = x_1 POL(redex_2.1(x_1)) = x_1 POL(redex_f.1(x_1)) = x_1 POL(reduce.1(x_1)) = x_1 POL(result_0.1(x_1)) = x_1 POL(result_1.1(x_1)) = x_1 POL(result_2.0(x_1)) = x_1 ---------------------------------------- (53) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(go_up.0(f.1(x0))) -> TOP.0(check_f.0(redex_f.1(x0))) TOP.0(go_up.1(0.1(x0))) -> TOP.0(check_0.0(redex_0.1(x0))) TOP.0(go_up.1(1.1(x0))) -> TOP.0(check_1.0(redex_1.1(x0))) TOP.0(go_up.1(2.1(x0))) -> TOP.0(check_2.0(redex_2.1(x0))) The TRS R consists of the following rules: check_2.0(redex_2.1(x_1)) -> in_2_1.0(reduce.1(x_1)) reduce.1(0.0(x_1)) -> check_0.0(redex_0.0(x_1)) reduce.1(0.1(x_1)) -> check_0.0(redex_0.1(x_1)) reduce.1(1.0(x_1)) -> check_1.0(redex_1.0(x_1)) reduce.1(1.1(x_1)) -> check_1.0(redex_1.1(x_1)) reduce.1(2.0(x_1)) -> check_2.0(redex_2.0(x_1)) reduce.1(2.1(x_1)) -> check_2.0(redex_2.1(x_1)) reduce.1(a.) -> go_up.0(f.1(a.)) in_2_1.0(go_up.0(x_1)) -> go_up.1(2.0(x_1)) in_2_1.0(go_up.1(x_1)) -> go_up.1(2.1(x_1)) redex_2.0(f.0(x)) -> result_2.0(f.1(0.0(x))) redex_2.0(f.1(x)) -> result_2.0(f.1(0.1(x))) check_2.0(result_2.0(x)) -> go_up.0(x) check_1.0(redex_1.1(x_1)) -> in_1_1.0(reduce.1(x_1)) in_1_1.0(go_up.0(x_1)) -> go_up.1(1.0(x_1)) in_1_1.0(go_up.1(x_1)) -> go_up.1(1.1(x_1)) redex_1.0(f.0(x)) -> result_1.1(2.0(x)) redex_1.0(f.1(x)) -> result_1.1(2.1(x)) check_1.0(result_1.1(x)) -> go_up.1(x) check_0.0(redex_0.1(x_1)) -> in_0_1.0(reduce.1(x_1)) in_0_1.0(go_up.0(x_1)) -> go_up.1(0.0(x_1)) in_0_1.0(go_up.1(x_1)) -> go_up.1(0.1(x_1)) redex_0.0(f.0(x)) -> result_0.1(1.0(x)) redex_0.0(f.1(x)) -> result_0.1(1.1(x)) check_0.0(result_0.1(x)) -> go_up.1(x) check_f.0(redex_f.1(x_1)) -> in_f_1.0(reduce.1(x_1)) in_f_1.0(go_up.0(x_1)) -> go_up.0(f.0(x_1)) in_f_1.0(go_up.1(x_1)) -> go_up.0(f.1(x_1)) The set Q consists of the following terms: reduce.0(f.0(x0)) reduce.0(f.1(x0)) reduce.1(0.0(x0)) reduce.1(0.1(x0)) reduce.1(1.0(x0)) reduce.1(1.1(x0)) reduce.1(2.0(x0)) reduce.1(2.1(x0)) reduce.1(a.) redex_f.0(f.0(x0)) redex_f.0(f.1(x0)) redex_0.0(f.0(x0)) redex_0.0(f.1(x0)) redex_1.0(f.0(x0)) redex_1.0(f.1(x0)) redex_2.0(f.0(x0)) redex_2.0(f.1(x0)) check_f.0(result_f.0(x0)) check_f.0(result_f.1(x0)) check_0.0(result_0.0(x0)) check_0.0(result_0.1(x0)) check_1.0(result_1.0(x0)) check_1.0(result_1.1(x0)) check_2.0(result_2.0(x0)) check_2.0(result_2.1(x0)) check_f.0(redex_f.0(x0)) check_f.0(redex_f.1(x0)) check_0.0(redex_0.0(x0)) check_0.0(redex_0.1(x0)) check_1.0(redex_1.0(x0)) check_1.0(redex_1.1(x0)) check_2.0(redex_2.0(x0)) check_2.0(redex_2.1(x0)) in_f_1.0(go_up.0(x0)) in_f_1.0(go_up.1(x0)) in_0_1.0(go_up.0(x0)) in_0_1.0(go_up.1(x0)) in_1_1.0(go_up.0(x0)) in_1_1.0(go_up.1(x0)) in_2_1.0(go_up.0(x0)) in_2_1.0(go_up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (54) PisEmptyProof (SOUND) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (55) TRUE ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) TOP(go_up(2(x0))) -> TOP(check_2(redex_2(x0))) The TRS R consists of the following rules: check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) redex_2(f(x)) -> result_2(f(0(x))) check_2(result_2(x)) -> go_up(x) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) redex_1(f(x)) -> result_1(2(x)) check_1(result_1(x)) -> go_up(x) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) redex_0(f(x)) -> result_0(1(x)) check_0(result_0(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) TransformationProof (SOUND) 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]: (TOP(go_up(f(x0))) -> TOP(in_f_1(reduce(x0))),TOP(go_up(f(x0))) -> TOP(in_f_1(reduce(x0)))) ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) TOP(go_up(2(x0))) -> TOP(check_2(redex_2(x0))) TOP(go_up(f(x0))) -> TOP(in_f_1(reduce(x0))) The TRS R consists of the following rules: check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) redex_2(f(x)) -> result_2(f(0(x))) check_2(result_2(x)) -> go_up(x) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) redex_1(f(x)) -> result_1(2(x)) check_1(result_1(x)) -> go_up(x) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) redex_0(f(x)) -> result_0(1(x)) check_0(result_0(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) TOP(go_up(2(x0))) -> TOP(check_2(redex_2(x0))) TOP(go_up(f(x0))) -> TOP(in_f_1(reduce(x0))) The TRS R consists of the following rules: reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) redex_2(f(x)) -> result_2(f(0(x))) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) check_2(result_2(x)) -> go_up(x) in_2_1(go_up(x_1)) -> go_up(2(x_1)) redex_1(f(x)) -> result_1(2(x)) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) check_1(result_1(x)) -> go_up(x) in_1_1(go_up(x_1)) -> go_up(1(x_1)) redex_0(f(x)) -> result_0(1(x)) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) check_0(result_0(x)) -> go_up(x) in_0_1(go_up(x_1)) -> go_up(0(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. redex_f(f(x0)) check_f(result_f(x0)) check_f(redex_f(x0)) ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(0(x0))) -> TOP(check_0(redex_0(x0))) TOP(go_up(1(x0))) -> TOP(check_1(redex_1(x0))) TOP(go_up(2(x0))) -> TOP(check_2(redex_2(x0))) TOP(go_up(f(x0))) -> TOP(in_f_1(reduce(x0))) The TRS R consists of the following rules: reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) redex_2(f(x)) -> result_2(f(0(x))) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) check_2(result_2(x)) -> go_up(x) in_2_1(go_up(x_1)) -> go_up(2(x_1)) redex_1(f(x)) -> result_1(2(x)) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) check_1(result_1(x)) -> go_up(x) in_1_1(go_up(x_1)) -> go_up(1(x_1)) redex_0(f(x)) -> result_0(1(x)) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) check_0(result_0(x)) -> go_up(x) in_0_1(go_up(x_1)) -> go_up(0(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) redex_2(f(x)) -> result_2(f(0(x))) check_2(result_2(x)) -> go_up(x) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) redex_1(f(x)) -> result_1(2(x)) check_1(result_1(x)) -> go_up(x) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) redex_0(f(x)) -> result_0(1(x)) check_0(result_0(x)) -> go_up(x) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) redex_f(f(x)) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(go_up(x0)) ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(0(x_1)) -> check_0(redex_0(x_1)) reduce(1(x_1)) -> check_1(redex_1(x_1)) reduce(2(x_1)) -> check_2(redex_2(x_1)) reduce(a) -> go_up(f(a)) redex_2(f(x)) -> result_2(f(0(x))) check_2(result_2(x)) -> go_up(x) check_2(redex_2(x_1)) -> in_2_1(reduce(x_1)) in_2_1(go_up(x_1)) -> go_up(2(x_1)) redex_1(f(x)) -> result_1(2(x)) check_1(result_1(x)) -> go_up(x) check_1(redex_1(x_1)) -> in_1_1(reduce(x_1)) in_1_1(go_up(x_1)) -> go_up(1(x_1)) redex_0(f(x)) -> result_0(1(x)) check_0(result_0(x)) -> go_up(x) check_0(redex_0(x_1)) -> in_0_1(reduce(x_1)) in_0_1(go_up(x_1)) -> go_up(0(x_1)) redex_f(f(x)) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(0(x0)) reduce(1(x0)) reduce(2(x0)) reduce(a) redex_f(f(x0)) redex_0(f(x0)) redex_1(f(x0)) redex_2(f(x0)) check_f(result_f(x0)) check_0(result_0(x0)) check_1(result_1(x0)) check_2(result_2(x0)) check_f(redex_f(x0)) check_0(redex_0(x0)) check_1(redex_1(x0)) check_2(redex_2(x0)) in_f_1(go_up(x0)) in_0_1(go_up(x0)) in_1_1(go_up(x0)) in_2_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) Trivial-Transformation (SOUND) We applied the Trivial transformation to transform the outermost TRS to a standard TRS. ---------------------------------------- (68) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a -> f(a) f(f(x)) -> b 0(f(x)) -> 1(x) 1(f(x)) -> 2(x) 2(f(x)) -> f(0(x)) Q is empty. ---------------------------------------- (69) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: A -> F(a) A -> A 0^1(f(x)) -> 1^1(x) 1^1(f(x)) -> 2^1(x) 2^1(f(x)) -> F(0(x)) 2^1(f(x)) -> 0^1(x) The TRS R consists of the following rules: a -> f(a) f(f(x)) -> b 0(f(x)) -> 1(x) 1(f(x)) -> 2(x) 2(f(x)) -> f(0(x)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. ---------------------------------------- (72) Complex Obligation (AND) ---------------------------------------- (73) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(f(x)) -> 2^1(x) 2^1(f(x)) -> 0^1(x) 0^1(f(x)) -> 1^1(x) The TRS R consists of the following rules: a -> f(a) f(f(x)) -> b 0(f(x)) -> 1(x) 1(f(x)) -> 2(x) 2(f(x)) -> f(0(x)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (74) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (75) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(f(x)) -> 2^1(x) 2^1(f(x)) -> 0^1(x) 0^1(f(x)) -> 1^1(x) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (76) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *2^1(f(x)) -> 0^1(x) The graph contains the following edges 1 > 1 *0^1(f(x)) -> 1^1(x) The graph contains the following edges 1 > 1 *1^1(f(x)) -> 2^1(x) The graph contains the following edges 1 > 1 ---------------------------------------- (77) YES ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: A -> A The TRS R consists of the following rules: a -> f(a) f(f(x)) -> b 0(f(x)) -> 1(x) 1(f(x)) -> 2(x) 2(f(x)) -> f(0(x)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: A -> A R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = A evaluates to t =A Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from A to A. ---------------------------------------- (82) NO ---------------------------------------- (83) Raffelsieper-Zantema-Transformation (SOUND) We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. ---------------------------------------- (84) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(x))) -> up(b) down(0(f(x))) -> up(1(x)) down(1(f(x))) -> up(2(x)) down(2(f(x))) -> up(f(0(x))) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(0(y6))) -> f_flat(down(0(y6))) down(f(1(y7))) -> f_flat(down(1(y7))) down(f(2(y8))) -> f_flat(down(2(y8))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 0_flat(up(x_1)) -> up(0(x_1)) 1_flat(up(x_1)) -> up(1(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) Q is empty. ---------------------------------------- (85) AAECC Innermost (EQUIVALENT) We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(0(y6))) -> f_flat(down(0(y6))) down(f(1(y7))) -> f_flat(down(1(y7))) down(f(2(y8))) -> f_flat(down(2(y8))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 0_flat(up(x_1)) -> up(0(x_1)) 1_flat(up(x_1)) -> up(1(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(a) -> up(f(a)) down(f(f(x))) -> up(b) down(0(f(x))) -> up(1(x)) down(1(f(x))) -> up(2(x)) down(2(f(x))) -> up(f(0(x))) The TRS R 2 is top(up(x)) -> top(down(x)) The signature Sigma is {top_1} ---------------------------------------- (86) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(x))) -> up(b) down(0(f(x))) -> up(1(x)) down(1(f(x))) -> up(2(x)) down(2(f(x))) -> up(f(0(x))) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(0(y6))) -> f_flat(down(0(y6))) down(f(1(y7))) -> f_flat(down(1(y7))) down(f(2(y8))) -> f_flat(down(2(y8))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 0_flat(up(x_1)) -> up(0(x_1)) 1_flat(up(x_1)) -> up(1(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) top(up(x0)) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) ---------------------------------------- (87) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) TOP(up(x)) -> DOWN(x) DOWN(f(a)) -> F_FLAT(down(a)) DOWN(f(a)) -> DOWN(a) DOWN(f(b)) -> F_FLAT(down(b)) DOWN(f(b)) -> DOWN(b) DOWN(f(0(y6))) -> F_FLAT(down(0(y6))) DOWN(f(0(y6))) -> DOWN(0(y6)) DOWN(f(1(y7))) -> F_FLAT(down(1(y7))) DOWN(f(1(y7))) -> DOWN(1(y7)) DOWN(f(2(y8))) -> F_FLAT(down(2(y8))) DOWN(f(2(y8))) -> DOWN(2(y8)) DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant)) DOWN(f(fresh_constant)) -> DOWN(fresh_constant) DOWN(0(a)) -> 0_FLAT(down(a)) DOWN(0(a)) -> DOWN(a) DOWN(0(b)) -> 0_FLAT(down(b)) DOWN(0(b)) -> DOWN(b) DOWN(0(0(y11))) -> 0_FLAT(down(0(y11))) DOWN(0(0(y11))) -> DOWN(0(y11)) DOWN(0(1(y12))) -> 0_FLAT(down(1(y12))) DOWN(0(1(y12))) -> DOWN(1(y12)) DOWN(0(2(y13))) -> 0_FLAT(down(2(y13))) DOWN(0(2(y13))) -> DOWN(2(y13)) DOWN(0(fresh_constant)) -> 0_FLAT(down(fresh_constant)) DOWN(0(fresh_constant)) -> DOWN(fresh_constant) DOWN(1(a)) -> 1_FLAT(down(a)) DOWN(1(a)) -> DOWN(a) DOWN(1(b)) -> 1_FLAT(down(b)) DOWN(1(b)) -> DOWN(b) DOWN(1(0(y16))) -> 1_FLAT(down(0(y16))) DOWN(1(0(y16))) -> DOWN(0(y16)) DOWN(1(1(y17))) -> 1_FLAT(down(1(y17))) DOWN(1(1(y17))) -> DOWN(1(y17)) DOWN(1(2(y18))) -> 1_FLAT(down(2(y18))) DOWN(1(2(y18))) -> DOWN(2(y18)) DOWN(1(fresh_constant)) -> 1_FLAT(down(fresh_constant)) DOWN(1(fresh_constant)) -> DOWN(fresh_constant) DOWN(2(a)) -> 2_FLAT(down(a)) DOWN(2(a)) -> DOWN(a) DOWN(2(b)) -> 2_FLAT(down(b)) DOWN(2(b)) -> DOWN(b) DOWN(2(0(y21))) -> 2_FLAT(down(0(y21))) DOWN(2(0(y21))) -> DOWN(0(y21)) DOWN(2(1(y22))) -> 2_FLAT(down(1(y22))) DOWN(2(1(y22))) -> DOWN(1(y22)) DOWN(2(2(y23))) -> 2_FLAT(down(2(y23))) DOWN(2(2(y23))) -> DOWN(2(y23)) DOWN(2(fresh_constant)) -> 2_FLAT(down(fresh_constant)) DOWN(2(fresh_constant)) -> DOWN(fresh_constant) The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(x))) -> up(b) down(0(f(x))) -> up(1(x)) down(1(f(x))) -> up(2(x)) down(2(f(x))) -> up(f(0(x))) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(0(y6))) -> f_flat(down(0(y6))) down(f(1(y7))) -> f_flat(down(1(y7))) down(f(2(y8))) -> f_flat(down(2(y8))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 0_flat(up(x_1)) -> up(0(x_1)) 1_flat(up(x_1)) -> up(1(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) top(up(x0)) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 40 less nodes. ---------------------------------------- (90) Complex Obligation (AND) ---------------------------------------- (91) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(0(1(y12))) -> DOWN(1(y12)) DOWN(1(0(y16))) -> DOWN(0(y16)) DOWN(0(0(y11))) -> DOWN(0(y11)) DOWN(0(2(y13))) -> DOWN(2(y13)) DOWN(2(0(y21))) -> DOWN(0(y21)) DOWN(2(1(y22))) -> DOWN(1(y22)) DOWN(1(1(y17))) -> DOWN(1(y17)) DOWN(1(2(y18))) -> DOWN(2(y18)) DOWN(2(2(y23))) -> DOWN(2(y23)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(x))) -> up(b) down(0(f(x))) -> up(1(x)) down(1(f(x))) -> up(2(x)) down(2(f(x))) -> up(f(0(x))) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(0(y6))) -> f_flat(down(0(y6))) down(f(1(y7))) -> f_flat(down(1(y7))) down(f(2(y8))) -> f_flat(down(2(y8))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 0_flat(up(x_1)) -> up(0(x_1)) 1_flat(up(x_1)) -> up(1(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) top(up(x0)) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (92) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (93) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(0(1(y12))) -> DOWN(1(y12)) DOWN(1(0(y16))) -> DOWN(0(y16)) DOWN(0(0(y11))) -> DOWN(0(y11)) DOWN(0(2(y13))) -> DOWN(2(y13)) DOWN(2(0(y21))) -> DOWN(0(y21)) DOWN(2(1(y22))) -> DOWN(1(y22)) DOWN(1(1(y17))) -> DOWN(1(y17)) DOWN(1(2(y18))) -> DOWN(2(y18)) DOWN(2(2(y23))) -> DOWN(2(y23)) R is empty. The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) top(up(x0)) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (94) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) top(up(x0)) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) ---------------------------------------- (95) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(0(1(y12))) -> DOWN(1(y12)) DOWN(1(0(y16))) -> DOWN(0(y16)) DOWN(0(0(y11))) -> DOWN(0(y11)) DOWN(0(2(y13))) -> DOWN(2(y13)) DOWN(2(0(y21))) -> DOWN(0(y21)) DOWN(2(1(y22))) -> DOWN(1(y22)) DOWN(1(1(y17))) -> DOWN(1(y17)) DOWN(1(2(y18))) -> DOWN(2(y18)) DOWN(2(2(y23))) -> DOWN(2(y23)) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (96) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *DOWN(1(0(y16))) -> DOWN(0(y16)) The graph contains the following edges 1 > 1 *DOWN(1(1(y17))) -> DOWN(1(y17)) The graph contains the following edges 1 > 1 *DOWN(1(2(y18))) -> DOWN(2(y18)) The graph contains the following edges 1 > 1 *DOWN(0(0(y11))) -> DOWN(0(y11)) The graph contains the following edges 1 > 1 *DOWN(2(0(y21))) -> DOWN(0(y21)) The graph contains the following edges 1 > 1 *DOWN(0(1(y12))) -> DOWN(1(y12)) The graph contains the following edges 1 > 1 *DOWN(0(2(y13))) -> DOWN(2(y13)) The graph contains the following edges 1 > 1 *DOWN(2(1(y22))) -> DOWN(1(y22)) The graph contains the following edges 1 > 1 *DOWN(2(2(y23))) -> DOWN(2(y23)) The graph contains the following edges 1 > 1 ---------------------------------------- (97) YES ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(x))) -> up(b) down(0(f(x))) -> up(1(x)) down(1(f(x))) -> up(2(x)) down(2(f(x))) -> up(f(0(x))) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(0(y6))) -> f_flat(down(0(y6))) down(f(1(y7))) -> f_flat(down(1(y7))) down(f(2(y8))) -> f_flat(down(2(y8))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 0_flat(up(x_1)) -> up(0(x_1)) 1_flat(up(x_1)) -> up(1(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) top(up(x0)) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (99) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(x))) -> up(b) down(0(f(x))) -> up(1(x)) down(1(f(x))) -> up(2(x)) down(2(f(x))) -> up(f(0(x))) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(0(y6))) -> f_flat(down(0(y6))) down(f(1(y7))) -> f_flat(down(1(y7))) down(f(2(y8))) -> f_flat(down(2(y8))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) 2_flat(up(x_1)) -> up(2(x_1)) 1_flat(up(x_1)) -> up(1(x_1)) 0_flat(up(x_1)) -> up(0(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) top(up(x0)) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(up(x0)) ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(x))) -> up(b) down(0(f(x))) -> up(1(x)) down(1(f(x))) -> up(2(x)) down(2(f(x))) -> up(f(0(x))) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(0(y6))) -> f_flat(down(0(y6))) down(f(1(y7))) -> f_flat(down(1(y7))) down(f(2(y8))) -> f_flat(down(2(y8))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) 2_flat(up(x_1)) -> up(2(x_1)) 1_flat(up(x_1)) -> up(1(x_1)) 0_flat(up(x_1)) -> up(0(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(a)) -> TOP(up(f(a))),TOP(up(a)) -> TOP(up(f(a)))) (TOP(up(f(f(x0)))) -> TOP(up(b)),TOP(up(f(f(x0)))) -> TOP(up(b))) (TOP(up(0(f(x0)))) -> TOP(up(1(x0))),TOP(up(0(f(x0)))) -> TOP(up(1(x0)))) (TOP(up(1(f(x0)))) -> TOP(up(2(x0))),TOP(up(1(f(x0)))) -> TOP(up(2(x0)))) (TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))),TOP(up(2(f(x0)))) -> TOP(up(f(0(x0))))) (TOP(up(f(a))) -> TOP(f_flat(down(a))),TOP(up(f(a))) -> TOP(f_flat(down(a)))) (TOP(up(f(b))) -> TOP(f_flat(down(b))),TOP(up(f(b))) -> TOP(f_flat(down(b)))) (TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))),TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0))))) (TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))),TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0))))) (TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))),TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0))))) (TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))),TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant)))) (TOP(up(0(a))) -> TOP(0_flat(down(a))),TOP(up(0(a))) -> TOP(0_flat(down(a)))) (TOP(up(0(b))) -> TOP(0_flat(down(b))),TOP(up(0(b))) -> TOP(0_flat(down(b)))) (TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))),TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0))))) (TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))),TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0))))) (TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))),TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0))))) (TOP(up(0(fresh_constant))) -> TOP(0_flat(down(fresh_constant))),TOP(up(0(fresh_constant))) -> TOP(0_flat(down(fresh_constant)))) (TOP(up(1(a))) -> TOP(1_flat(down(a))),TOP(up(1(a))) -> TOP(1_flat(down(a)))) (TOP(up(1(b))) -> TOP(1_flat(down(b))),TOP(up(1(b))) -> TOP(1_flat(down(b)))) (TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))),TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0))))) (TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))),TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0))))) (TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))),TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0))))) (TOP(up(1(fresh_constant))) -> TOP(1_flat(down(fresh_constant))),TOP(up(1(fresh_constant))) -> TOP(1_flat(down(fresh_constant)))) (TOP(up(2(a))) -> TOP(2_flat(down(a))),TOP(up(2(a))) -> TOP(2_flat(down(a)))) (TOP(up(2(b))) -> TOP(2_flat(down(b))),TOP(up(2(b))) -> TOP(2_flat(down(b)))) (TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))),TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0))))) (TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))),TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0))))) (TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))),TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0))))) (TOP(up(2(fresh_constant))) -> TOP(2_flat(down(fresh_constant))),TOP(up(2(fresh_constant))) -> TOP(2_flat(down(fresh_constant)))) ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(a)) -> TOP(up(f(a))) TOP(up(f(f(x0)))) -> TOP(up(b)) TOP(up(0(f(x0)))) -> TOP(up(1(x0))) TOP(up(1(f(x0)))) -> TOP(up(2(x0))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(f(a))) -> TOP(f_flat(down(a))) TOP(up(f(b))) -> TOP(f_flat(down(b))) TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))) TOP(up(0(a))) -> TOP(0_flat(down(a))) TOP(up(0(b))) -> TOP(0_flat(down(b))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(0(fresh_constant))) -> TOP(0_flat(down(fresh_constant))) TOP(up(1(a))) -> TOP(1_flat(down(a))) TOP(up(1(b))) -> TOP(1_flat(down(b))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(1(fresh_constant))) -> TOP(1_flat(down(fresh_constant))) TOP(up(2(a))) -> TOP(2_flat(down(a))) TOP(up(2(b))) -> TOP(2_flat(down(b))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(2(fresh_constant))) -> TOP(2_flat(down(fresh_constant))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(x))) -> up(b) down(0(f(x))) -> up(1(x)) down(1(f(x))) -> up(2(x)) down(2(f(x))) -> up(f(0(x))) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(0(y6))) -> f_flat(down(0(y6))) down(f(1(y7))) -> f_flat(down(1(y7))) down(f(2(y8))) -> f_flat(down(2(y8))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) 2_flat(up(x_1)) -> up(2(x_1)) 1_flat(up(x_1)) -> up(1(x_1)) 0_flat(up(x_1)) -> up(0(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes. ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(a))) -> TOP(f_flat(down(a))) TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(0(f(x0)))) -> TOP(up(1(x0))) TOP(up(1(f(x0)))) -> TOP(up(2(x0))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(2(a))) -> TOP(2_flat(down(a))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(0(a))) -> TOP(0_flat(down(a))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(a))) -> TOP(1_flat(down(a))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(f(f(x))) -> up(b) down(0(f(x))) -> up(1(x)) down(1(f(x))) -> up(2(x)) down(2(f(x))) -> up(f(0(x))) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(0(y6))) -> f_flat(down(0(y6))) down(f(1(y7))) -> f_flat(down(1(y7))) down(f(2(y8))) -> f_flat(down(2(y8))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) 2_flat(up(x_1)) -> up(2(x_1)) 1_flat(up(x_1)) -> up(1(x_1)) 0_flat(up(x_1)) -> up(0(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(a))) -> TOP(f_flat(down(a))) TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(0(f(x0)))) -> TOP(up(1(x0))) TOP(up(1(f(x0)))) -> TOP(up(2(x0))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(2(a))) -> TOP(2_flat(down(a))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(0(a))) -> TOP(0_flat(down(a))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(a))) -> TOP(1_flat(down(a))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) TransformationProof (EQUIVALENT) 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]: (TOP(up(f(a))) -> TOP(f_flat(up(f(a)))),TOP(up(f(a))) -> TOP(f_flat(up(f(a))))) ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(0(f(x0)))) -> TOP(up(1(x0))) TOP(up(1(f(x0)))) -> TOP(up(2(x0))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(2(a))) -> TOP(2_flat(down(a))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(0(a))) -> TOP(0_flat(down(a))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(a))) -> TOP(1_flat(down(a))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(2(a))) -> TOP(2_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(2(a))) -> TOP(2_flat(up(f(a)))),TOP(up(2(a))) -> TOP(2_flat(up(f(a))))) ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(0(f(x0)))) -> TOP(up(1(x0))) TOP(up(1(f(x0)))) -> TOP(up(2(x0))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(0(a))) -> TOP(0_flat(down(a))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(a))) -> TOP(1_flat(down(a))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(2(a))) -> TOP(2_flat(up(f(a)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (113) TransformationProof (EQUIVALENT) 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]: (TOP(up(0(a))) -> TOP(0_flat(up(f(a)))),TOP(up(0(a))) -> TOP(0_flat(up(f(a))))) ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(0(f(x0)))) -> TOP(up(1(x0))) TOP(up(1(f(x0)))) -> TOP(up(2(x0))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(a))) -> TOP(1_flat(down(a))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(2(a))) -> TOP(2_flat(up(f(a)))) TOP(up(0(a))) -> TOP(0_flat(up(f(a)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (115) TransformationProof (EQUIVALENT) 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]: (TOP(up(1(a))) -> TOP(1_flat(up(f(a)))),TOP(up(1(a))) -> TOP(1_flat(up(f(a))))) ---------------------------------------- (116) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(0(f(x0)))) -> TOP(up(1(x0))) TOP(up(1(f(x0)))) -> TOP(up(2(x0))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(2(a))) -> TOP(2_flat(up(f(a)))) TOP(up(0(a))) -> TOP(0_flat(up(f(a)))) TOP(up(1(a))) -> TOP(1_flat(up(f(a)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (117) TransformationProof (EQUIVALENT) 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]: (TOP(up(f(a))) -> TOP(up(f(f(a)))),TOP(up(f(a))) -> TOP(up(f(f(a))))) ---------------------------------------- (118) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(0(f(x0)))) -> TOP(up(1(x0))) TOP(up(1(f(x0)))) -> TOP(up(2(x0))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(2(a))) -> TOP(2_flat(up(f(a)))) TOP(up(0(a))) -> TOP(0_flat(up(f(a)))) TOP(up(1(a))) -> TOP(1_flat(up(f(a)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (119) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (120) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(0(f(x0)))) -> TOP(up(1(x0))) TOP(up(1(f(x0)))) -> TOP(up(2(x0))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(2(a))) -> TOP(2_flat(up(f(a)))) TOP(up(0(a))) -> TOP(0_flat(up(f(a)))) TOP(up(1(a))) -> TOP(1_flat(up(f(a)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (121) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(2(a))) -> TOP(2_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(2(a))) -> TOP(up(2(f(a)))),TOP(up(2(a))) -> TOP(up(2(f(a))))) ---------------------------------------- (122) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(0(f(x0)))) -> TOP(up(1(x0))) TOP(up(1(f(x0)))) -> TOP(up(2(x0))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(0(a))) -> TOP(0_flat(up(f(a)))) TOP(up(1(a))) -> TOP(1_flat(up(f(a)))) TOP(up(2(a))) -> TOP(up(2(f(a)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (123) TransformationProof (EQUIVALENT) 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]: (TOP(up(0(a))) -> TOP(up(0(f(a)))),TOP(up(0(a))) -> TOP(up(0(f(a))))) ---------------------------------------- (124) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(0(f(x0)))) -> TOP(up(1(x0))) TOP(up(1(f(x0)))) -> TOP(up(2(x0))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(1(a))) -> TOP(1_flat(up(f(a)))) TOP(up(2(a))) -> TOP(up(2(f(a)))) TOP(up(0(a))) -> TOP(up(0(f(a)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (125) TransformationProof (EQUIVALENT) 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]: (TOP(up(1(a))) -> TOP(up(1(f(a)))),TOP(up(1(a))) -> TOP(up(1(f(a))))) ---------------------------------------- (126) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(0(f(x0)))) -> TOP(up(1(x0))) TOP(up(1(f(x0)))) -> TOP(up(2(x0))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(2(a))) -> TOP(up(2(f(a)))) TOP(up(0(a))) -> TOP(up(0(f(a)))) TOP(up(1(a))) -> TOP(up(1(f(a)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (127) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule TOP(up(0(f(x0)))) -> TOP(up(1(x0))) we obtained the following new rules [LPAR04]: (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))))) (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))))) (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))))) (TOP(up(0(f(2(y_0))))) -> TOP(up(1(2(y_0)))),TOP(up(0(f(2(y_0))))) -> TOP(up(1(2(y_0))))) (TOP(up(0(f(a)))) -> TOP(up(1(a))),TOP(up(0(f(a)))) -> TOP(up(1(a)))) ---------------------------------------- (128) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(1(f(x0)))) -> TOP(up(2(x0))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(2(a))) -> TOP(up(2(f(a)))) TOP(up(0(a))) -> TOP(up(0(f(a)))) TOP(up(1(a))) -> TOP(up(1(f(a)))) TOP(up(0(f(f(y_0))))) -> TOP(up(1(f(y_0)))) TOP(up(0(f(0(y_0))))) -> TOP(up(1(0(y_0)))) TOP(up(0(f(1(y_0))))) -> TOP(up(1(1(y_0)))) TOP(up(0(f(2(y_0))))) -> TOP(up(1(2(y_0)))) TOP(up(0(f(a)))) -> TOP(up(1(a))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (129) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule TOP(up(1(f(x0)))) -> TOP(up(2(x0))) we obtained the following new rules [LPAR04]: (TOP(up(1(f(f(y_0))))) -> TOP(up(2(f(y_0)))),TOP(up(1(f(f(y_0))))) -> TOP(up(2(f(y_0))))) (TOP(up(1(f(0(y_0))))) -> TOP(up(2(0(y_0)))),TOP(up(1(f(0(y_0))))) -> TOP(up(2(0(y_0))))) (TOP(up(1(f(1(y_0))))) -> TOP(up(2(1(y_0)))),TOP(up(1(f(1(y_0))))) -> TOP(up(2(1(y_0))))) (TOP(up(1(f(2(y_0))))) -> TOP(up(2(2(y_0)))),TOP(up(1(f(2(y_0))))) -> TOP(up(2(2(y_0))))) (TOP(up(1(f(a)))) -> TOP(up(2(a))),TOP(up(1(f(a)))) -> TOP(up(2(a)))) ---------------------------------------- (130) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(2(a))) -> TOP(up(2(f(a)))) TOP(up(0(a))) -> TOP(up(0(f(a)))) TOP(up(1(a))) -> TOP(up(1(f(a)))) TOP(up(0(f(f(y_0))))) -> TOP(up(1(f(y_0)))) TOP(up(0(f(0(y_0))))) -> TOP(up(1(0(y_0)))) TOP(up(0(f(1(y_0))))) -> TOP(up(1(1(y_0)))) TOP(up(0(f(2(y_0))))) -> TOP(up(1(2(y_0)))) TOP(up(0(f(a)))) -> TOP(up(1(a))) TOP(up(1(f(f(y_0))))) -> TOP(up(2(f(y_0)))) TOP(up(1(f(0(y_0))))) -> TOP(up(2(0(y_0)))) TOP(up(1(f(1(y_0))))) -> TOP(up(2(1(y_0)))) TOP(up(1(f(2(y_0))))) -> TOP(up(2(2(y_0)))) TOP(up(1(f(a)))) -> TOP(up(2(a))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (131) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP(up(1(f(f(y_0))))) -> TOP(up(2(f(y_0)))) TOP(up(1(f(0(y_0))))) -> TOP(up(2(0(y_0)))) TOP(up(1(f(1(y_0))))) -> TOP(up(2(1(y_0)))) TOP(up(1(f(2(y_0))))) -> TOP(up(2(2(y_0)))) TOP(up(1(f(a)))) -> TOP(up(2(a))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(0_flat(x_1)) = 1 POL(1(x_1)) = 1 POL(1_flat(x_1)) = 1 POL(2(x_1)) = 0 POL(2_flat(x_1)) = 0 POL(TOP(x_1)) = x_1 POL(a) = 0 POL(b) = 0 POL(down(x_1)) = 0 POL(f(x_1)) = 0 POL(f_flat(x_1)) = 0 POL(fresh_constant) = 0 POL(up(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: f_flat(up(x_1)) -> up(f(x_1)) 0_flat(up(x_1)) -> up(0(x_1)) 1_flat(up(x_1)) -> up(1(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) ---------------------------------------- (132) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(2(a))) -> TOP(up(2(f(a)))) TOP(up(0(a))) -> TOP(up(0(f(a)))) TOP(up(1(a))) -> TOP(up(1(f(a)))) TOP(up(0(f(f(y_0))))) -> TOP(up(1(f(y_0)))) TOP(up(0(f(0(y_0))))) -> TOP(up(1(0(y_0)))) TOP(up(0(f(1(y_0))))) -> TOP(up(1(1(y_0)))) TOP(up(0(f(2(y_0))))) -> TOP(up(1(2(y_0)))) TOP(up(0(f(a)))) -> TOP(up(1(a))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (133) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. ---------------------------------------- (134) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(2(a))) -> TOP(up(2(f(a)))) TOP(up(0(f(0(y_0))))) -> TOP(up(1(0(y_0)))) TOP(up(0(f(1(y_0))))) -> TOP(up(1(1(y_0)))) TOP(up(0(f(2(y_0))))) -> TOP(up(1(2(y_0)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (135) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP(up(2(f(x0)))) -> TOP(up(f(0(x0)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(0_flat(x_1)) = 0 POL(1(x_1)) = 0 POL(1_flat(x_1)) = 0 POL(2(x_1)) = 1 POL(2_flat(x_1)) = 1 POL(TOP(x_1)) = x_1 POL(a) = 0 POL(b) = 0 POL(down(x_1)) = 0 POL(f(x_1)) = 0 POL(f_flat(x_1)) = 0 POL(fresh_constant) = 0 POL(up(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: f_flat(up(x_1)) -> up(f(x_1)) 0_flat(up(x_1)) -> up(0(x_1)) 1_flat(up(x_1)) -> up(1(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) ---------------------------------------- (136) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(2(a))) -> TOP(up(2(f(a)))) TOP(up(0(f(0(y_0))))) -> TOP(up(1(0(y_0)))) TOP(up(0(f(1(y_0))))) -> TOP(up(1(1(y_0)))) TOP(up(0(f(2(y_0))))) -> TOP(up(1(2(y_0)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (137) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (138) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) TOP(up(0(f(0(y_0))))) -> TOP(up(1(0(y_0)))) TOP(up(0(f(1(y_0))))) -> TOP(up(1(1(y_0)))) TOP(up(0(f(2(y_0))))) -> TOP(up(1(2(y_0)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (139) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP(up(0(f(0(y_0))))) -> TOP(up(1(0(y_0)))) TOP(up(0(f(1(y_0))))) -> TOP(up(1(1(y_0)))) TOP(up(0(f(2(y_0))))) -> TOP(up(1(2(y_0)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(0_flat(x_1)) = 1 POL(1(x_1)) = 0 POL(1_flat(x_1)) = 0 POL(2(x_1)) = 0 POL(2_flat(x_1)) = 0 POL(TOP(x_1)) = x_1 POL(a) = 0 POL(b) = 0 POL(down(x_1)) = 0 POL(f(x_1)) = 0 POL(f_flat(x_1)) = 0 POL(fresh_constant) = 0 POL(up(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: f_flat(up(x_1)) -> up(f(x_1)) 0_flat(up(x_1)) -> up(0(x_1)) 1_flat(up(x_1)) -> up(1(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) ---------------------------------------- (140) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (141) SplitQDPProof (EQUIVALENT) We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem ---------------------------------------- (142) Complex Obligation (AND) ---------------------------------------- (143) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) The TRS R consists of the following rules: down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(2(fresh_constant)) -> 2_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) 2_flat(up(x_1)) -> up(2(x_1)) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) down(1(fresh_constant)) -> 1_flat(down(fresh_constant)) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) down(0(fresh_constant)) -> 0_flat(down(fresh_constant)) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (144) SemLabProof (SOUND) We found the following model for the rules of the TRSs R and P. Interpretation over the domain with elements from 0 to 1. 2_flat: 0 a: 0 2: 0 TOP: 0 b: 0 down: 0 f: 0 0: 0 fresh_constant: 1 up: 0 1: 0 1_flat: 0 0_flat: 0 f_flat: 0 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. ---------------------------------------- (145) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) TOP.0(up.0(f.0(0.1(x0)))) -> TOP.0(f_flat.0(down.0(0.1(x0)))) TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) TOP.0(up.0(f.0(1.1(x0)))) -> TOP.0(f_flat.0(down.0(1.1(x0)))) TOP.0(up.0(f.0(2.0(x0)))) -> TOP.0(f_flat.0(down.0(2.0(x0)))) TOP.0(up.0(f.0(2.1(x0)))) -> TOP.0(f_flat.0(down.0(2.1(x0)))) TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) TOP.0(up.0(0.0(0.1(x0)))) -> TOP.0(0_flat.0(down.0(0.1(x0)))) TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) TOP.0(up.0(0.0(1.1(x0)))) -> TOP.0(0_flat.0(down.0(1.1(x0)))) TOP.0(up.0(0.0(2.0(x0)))) -> TOP.0(0_flat.0(down.0(2.0(x0)))) TOP.0(up.0(0.0(2.1(x0)))) -> TOP.0(0_flat.0(down.0(2.1(x0)))) TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) TOP.0(up.0(1.0(0.1(x0)))) -> TOP.0(1_flat.0(down.0(0.1(x0)))) TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) TOP.0(up.0(1.0(1.1(x0)))) -> TOP.0(1_flat.0(down.0(1.1(x0)))) TOP.0(up.0(1.0(2.0(x0)))) -> TOP.0(1_flat.0(down.0(2.0(x0)))) TOP.0(up.0(1.0(2.1(x0)))) -> TOP.0(1_flat.0(down.0(2.1(x0)))) TOP.0(up.0(2.0(0.0(x0)))) -> TOP.0(2_flat.0(down.0(0.0(x0)))) TOP.0(up.0(2.0(0.1(x0)))) -> TOP.0(2_flat.0(down.0(0.1(x0)))) TOP.0(up.0(2.0(1.0(x0)))) -> TOP.0(2_flat.0(down.0(1.0(x0)))) TOP.0(up.0(2.0(1.1(x0)))) -> TOP.0(2_flat.0(down.0(1.1(x0)))) TOP.0(up.0(2.0(2.0(x0)))) -> TOP.0(2_flat.0(down.0(2.0(x0)))) TOP.0(up.0(2.0(2.1(x0)))) -> TOP.0(2_flat.0(down.0(2.1(x0)))) The TRS R consists of the following rules: down.0(2.0(f.0(x))) -> up.0(f.0(0.0(x))) down.0(2.0(f.1(x))) -> up.0(f.0(0.1(x))) down.0(2.0(a.)) -> 2_flat.0(down.0(a.)) down.0(2.0(b.)) -> 2_flat.0(down.0(b.)) down.0(2.0(0.0(y21))) -> 2_flat.0(down.0(0.0(y21))) down.0(2.0(0.1(y21))) -> 2_flat.0(down.0(0.1(y21))) down.0(2.0(1.0(y22))) -> 2_flat.0(down.0(1.0(y22))) down.0(2.0(1.1(y22))) -> 2_flat.0(down.0(1.1(y22))) down.0(2.0(2.0(y23))) -> 2_flat.0(down.0(2.0(y23))) down.0(2.0(2.1(y23))) -> 2_flat.0(down.0(2.1(y23))) down.0(2.1(fresh_constant.)) -> 2_flat.0(down.1(fresh_constant.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 2_flat.0(up.0(x_1)) -> up.0(2.0(x_1)) 2_flat.0(up.1(x_1)) -> up.0(2.1(x_1)) down.0(1.0(f.0(x))) -> up.0(2.0(x)) down.0(1.0(f.1(x))) -> up.0(2.1(x)) down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) down.0(1.0(b.)) -> 1_flat.0(down.0(b.)) down.0(1.0(0.0(y16))) -> 1_flat.0(down.0(0.0(y16))) down.0(1.0(0.1(y16))) -> 1_flat.0(down.0(0.1(y16))) down.0(1.0(1.0(y17))) -> 1_flat.0(down.0(1.0(y17))) down.0(1.0(1.1(y17))) -> 1_flat.0(down.0(1.1(y17))) down.0(1.0(2.0(y18))) -> 1_flat.0(down.0(2.0(y18))) down.0(1.0(2.1(y18))) -> 1_flat.0(down.0(2.1(y18))) down.0(1.1(fresh_constant.)) -> 1_flat.0(down.1(fresh_constant.)) 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) 1_flat.0(up.1(x_1)) -> up.0(1.1(x_1)) down.0(0.0(f.0(x))) -> up.0(1.0(x)) down.0(0.0(f.1(x))) -> up.0(1.1(x)) down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) down.0(0.0(b.)) -> 0_flat.0(down.0(b.)) down.0(0.0(0.0(y11))) -> 0_flat.0(down.0(0.0(y11))) down.0(0.0(0.1(y11))) -> 0_flat.0(down.0(0.1(y11))) down.0(0.0(1.0(y12))) -> 0_flat.0(down.0(1.0(y12))) down.0(0.0(1.1(y12))) -> 0_flat.0(down.0(1.1(y12))) down.0(0.0(2.0(y13))) -> 0_flat.0(down.0(2.0(y13))) down.0(0.0(2.1(y13))) -> 0_flat.0(down.0(2.1(y13))) down.0(0.1(fresh_constant.)) -> 0_flat.0(down.1(fresh_constant.)) 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) 0_flat.0(up.1(x_1)) -> up.0(0.1(x_1)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(f.0(f.0(x0))) down.0(f.0(f.1(x0))) down.0(0.0(f.0(x0))) down.0(0.0(f.1(x0))) down.0(1.0(f.0(x0))) down.0(1.0(f.1(x0))) down.0(2.0(f.0(x0))) down.0(2.0(f.1(x0))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(0.0(x0))) down.0(f.0(0.1(x0))) down.0(f.0(1.0(x0))) down.0(f.0(1.1(x0))) down.0(f.0(2.0(x0))) down.0(f.0(2.1(x0))) down.0(f.1(fresh_constant.)) down.0(0.0(a.)) down.0(0.0(b.)) down.0(0.0(0.0(x0))) down.0(0.0(0.1(x0))) down.0(0.0(1.0(x0))) down.0(0.0(1.1(x0))) down.0(0.0(2.0(x0))) down.0(0.0(2.1(x0))) down.0(0.1(fresh_constant.)) down.0(1.0(a.)) down.0(1.0(b.)) down.0(1.0(0.0(x0))) down.0(1.0(0.1(x0))) down.0(1.0(1.0(x0))) down.0(1.0(1.1(x0))) down.0(1.0(2.0(x0))) down.0(1.0(2.1(x0))) down.0(1.1(fresh_constant.)) down.0(2.0(a.)) down.0(2.0(b.)) down.0(2.0(0.0(x0))) down.0(2.0(0.1(x0))) down.0(2.0(1.0(x0))) down.0(2.0(1.1(x0))) down.0(2.0(2.0(x0))) down.0(2.0(2.1(x0))) down.0(2.1(fresh_constant.)) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) 0_flat.0(up.0(x0)) 0_flat.0(up.1(x0)) 1_flat.0(up.0(x0)) 1_flat.0(up.1(x0)) 2_flat.0(up.0(x0)) 2_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (146) UsableRulesReductionPairsProof (EQUIVALENT) 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. No dependency pairs are removed. The following rules are removed from R: down.0(2.0(f.1(x))) -> up.0(f.0(0.1(x))) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 2_flat.0(up.1(x_1)) -> up.0(2.1(x_1)) down.0(1.0(f.1(x))) -> up.0(2.1(x)) 1_flat.0(up.1(x_1)) -> up.0(1.1(x_1)) down.0(0.0(f.1(x))) -> up.0(1.1(x)) 0_flat.0(up.1(x_1)) -> up.0(0.1(x_1)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0.0(x_1)) = 1 + x_1 POL(0.1(x_1)) = 1 + x_1 POL(0_flat.0(x_1)) = 1 + x_1 POL(1.0(x_1)) = 1 + x_1 POL(1.1(x_1)) = 1 + x_1 POL(1_flat.0(x_1)) = 1 + x_1 POL(2.0(x_1)) = 1 + x_1 POL(2.1(x_1)) = 1 + x_1 POL(2_flat.0(x_1)) = 1 + x_1 POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(down.0(x_1)) = x_1 POL(down.1(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 1 + x_1 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(up.0(x_1)) = x_1 POL(up.1(x_1)) = 1 + x_1 ---------------------------------------- (147) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) TOP.0(up.0(f.0(0.1(x0)))) -> TOP.0(f_flat.0(down.0(0.1(x0)))) TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) TOP.0(up.0(f.0(1.1(x0)))) -> TOP.0(f_flat.0(down.0(1.1(x0)))) TOP.0(up.0(f.0(2.0(x0)))) -> TOP.0(f_flat.0(down.0(2.0(x0)))) TOP.0(up.0(f.0(2.1(x0)))) -> TOP.0(f_flat.0(down.0(2.1(x0)))) TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) TOP.0(up.0(0.0(0.1(x0)))) -> TOP.0(0_flat.0(down.0(0.1(x0)))) TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) TOP.0(up.0(0.0(1.1(x0)))) -> TOP.0(0_flat.0(down.0(1.1(x0)))) TOP.0(up.0(0.0(2.0(x0)))) -> TOP.0(0_flat.0(down.0(2.0(x0)))) TOP.0(up.0(0.0(2.1(x0)))) -> TOP.0(0_flat.0(down.0(2.1(x0)))) TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) TOP.0(up.0(1.0(0.1(x0)))) -> TOP.0(1_flat.0(down.0(0.1(x0)))) TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) TOP.0(up.0(1.0(1.1(x0)))) -> TOP.0(1_flat.0(down.0(1.1(x0)))) TOP.0(up.0(1.0(2.0(x0)))) -> TOP.0(1_flat.0(down.0(2.0(x0)))) TOP.0(up.0(1.0(2.1(x0)))) -> TOP.0(1_flat.0(down.0(2.1(x0)))) TOP.0(up.0(2.0(0.0(x0)))) -> TOP.0(2_flat.0(down.0(0.0(x0)))) TOP.0(up.0(2.0(0.1(x0)))) -> TOP.0(2_flat.0(down.0(0.1(x0)))) TOP.0(up.0(2.0(1.0(x0)))) -> TOP.0(2_flat.0(down.0(1.0(x0)))) TOP.0(up.0(2.0(1.1(x0)))) -> TOP.0(2_flat.0(down.0(1.1(x0)))) TOP.0(up.0(2.0(2.0(x0)))) -> TOP.0(2_flat.0(down.0(2.0(x0)))) TOP.0(up.0(2.0(2.1(x0)))) -> TOP.0(2_flat.0(down.0(2.1(x0)))) The TRS R consists of the following rules: down.0(2.1(fresh_constant.)) -> 2_flat.0(down.1(fresh_constant.)) 2_flat.0(up.0(x_1)) -> up.0(2.0(x_1)) down.0(2.0(f.0(x))) -> up.0(f.0(0.0(x))) down.0(2.0(a.)) -> 2_flat.0(down.0(a.)) down.0(2.0(b.)) -> 2_flat.0(down.0(b.)) down.0(2.0(0.0(y21))) -> 2_flat.0(down.0(0.0(y21))) down.0(2.0(0.1(y21))) -> 2_flat.0(down.0(0.1(y21))) down.0(2.0(1.0(y22))) -> 2_flat.0(down.0(1.0(y22))) down.0(2.0(1.1(y22))) -> 2_flat.0(down.0(1.1(y22))) down.0(2.0(2.0(y23))) -> 2_flat.0(down.0(2.0(y23))) down.0(2.0(2.1(y23))) -> 2_flat.0(down.0(2.1(y23))) down.0(1.1(fresh_constant.)) -> 1_flat.0(down.1(fresh_constant.)) down.0(1.0(f.0(x))) -> up.0(2.0(x)) down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) down.0(1.0(b.)) -> 1_flat.0(down.0(b.)) down.0(1.0(0.0(y16))) -> 1_flat.0(down.0(0.0(y16))) down.0(1.0(0.1(y16))) -> 1_flat.0(down.0(0.1(y16))) down.0(1.0(1.0(y17))) -> 1_flat.0(down.0(1.0(y17))) down.0(1.0(1.1(y17))) -> 1_flat.0(down.0(1.1(y17))) down.0(1.0(2.0(y18))) -> 1_flat.0(down.0(2.0(y18))) down.0(1.0(2.1(y18))) -> 1_flat.0(down.0(2.1(y18))) 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) down.0(0.1(fresh_constant.)) -> 0_flat.0(down.1(fresh_constant.)) down.0(0.0(f.0(x))) -> up.0(1.0(x)) down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) down.0(0.0(b.)) -> 0_flat.0(down.0(b.)) down.0(0.0(0.0(y11))) -> 0_flat.0(down.0(0.0(y11))) down.0(0.0(0.1(y11))) -> 0_flat.0(down.0(0.1(y11))) down.0(0.0(1.0(y12))) -> 0_flat.0(down.0(1.0(y12))) down.0(0.0(1.1(y12))) -> 0_flat.0(down.0(1.1(y12))) down.0(0.0(2.0(y13))) -> 0_flat.0(down.0(2.0(y13))) down.0(0.0(2.1(y13))) -> 0_flat.0(down.0(2.1(y13))) 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) down.0(a.) -> up.0(f.0(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) The set Q consists of the following terms: down.0(a.) down.0(f.0(f.0(x0))) down.0(f.0(f.1(x0))) down.0(0.0(f.0(x0))) down.0(0.0(f.1(x0))) down.0(1.0(f.0(x0))) down.0(1.0(f.1(x0))) down.0(2.0(f.0(x0))) down.0(2.0(f.1(x0))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(0.0(x0))) down.0(f.0(0.1(x0))) down.0(f.0(1.0(x0))) down.0(f.0(1.1(x0))) down.0(f.0(2.0(x0))) down.0(f.0(2.1(x0))) down.0(f.1(fresh_constant.)) down.0(0.0(a.)) down.0(0.0(b.)) down.0(0.0(0.0(x0))) down.0(0.0(0.1(x0))) down.0(0.0(1.0(x0))) down.0(0.0(1.1(x0))) down.0(0.0(2.0(x0))) down.0(0.0(2.1(x0))) down.0(0.1(fresh_constant.)) down.0(1.0(a.)) down.0(1.0(b.)) down.0(1.0(0.0(x0))) down.0(1.0(0.1(x0))) down.0(1.0(1.0(x0))) down.0(1.0(1.1(x0))) down.0(1.0(2.0(x0))) down.0(1.0(2.1(x0))) down.0(1.1(fresh_constant.)) down.0(2.0(a.)) down.0(2.0(b.)) down.0(2.0(0.0(x0))) down.0(2.0(0.1(x0))) down.0(2.0(1.0(x0))) down.0(2.0(1.1(x0))) down.0(2.0(2.0(x0))) down.0(2.0(2.1(x0))) down.0(2.1(fresh_constant.)) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) 0_flat.0(up.0(x0)) 0_flat.0(up.1(x0)) 1_flat.0(up.0(x0)) 1_flat.0(up.1(x0)) 2_flat.0(up.0(x0)) 2_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (148) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: down.0(2.1(fresh_constant.)) -> 2_flat.0(down.1(fresh_constant.)) down.0(1.1(fresh_constant.)) -> 1_flat.0(down.1(fresh_constant.)) down.0(0.1(fresh_constant.)) -> 0_flat.0(down.1(fresh_constant.)) Used ordering: Polynomial interpretation [POLO]: POL(0.0(x_1)) = x_1 POL(0.1(x_1)) = 1 + x_1 POL(0_flat.0(x_1)) = x_1 POL(1.0(x_1)) = x_1 POL(1.1(x_1)) = 1 + x_1 POL(1_flat.0(x_1)) = x_1 POL(2.0(x_1)) = x_1 POL(2.1(x_1)) = 1 + x_1 POL(2_flat.0(x_1)) = x_1 POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(down.0(x_1)) = x_1 POL(down.1(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(up.0(x_1)) = x_1 ---------------------------------------- (149) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) TOP.0(up.0(f.0(0.1(x0)))) -> TOP.0(f_flat.0(down.0(0.1(x0)))) TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) TOP.0(up.0(f.0(1.1(x0)))) -> TOP.0(f_flat.0(down.0(1.1(x0)))) TOP.0(up.0(f.0(2.0(x0)))) -> TOP.0(f_flat.0(down.0(2.0(x0)))) TOP.0(up.0(f.0(2.1(x0)))) -> TOP.0(f_flat.0(down.0(2.1(x0)))) TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) TOP.0(up.0(0.0(0.1(x0)))) -> TOP.0(0_flat.0(down.0(0.1(x0)))) TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) TOP.0(up.0(0.0(1.1(x0)))) -> TOP.0(0_flat.0(down.0(1.1(x0)))) TOP.0(up.0(0.0(2.0(x0)))) -> TOP.0(0_flat.0(down.0(2.0(x0)))) TOP.0(up.0(0.0(2.1(x0)))) -> TOP.0(0_flat.0(down.0(2.1(x0)))) TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) TOP.0(up.0(1.0(0.1(x0)))) -> TOP.0(1_flat.0(down.0(0.1(x0)))) TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) TOP.0(up.0(1.0(1.1(x0)))) -> TOP.0(1_flat.0(down.0(1.1(x0)))) TOP.0(up.0(1.0(2.0(x0)))) -> TOP.0(1_flat.0(down.0(2.0(x0)))) TOP.0(up.0(1.0(2.1(x0)))) -> TOP.0(1_flat.0(down.0(2.1(x0)))) TOP.0(up.0(2.0(0.0(x0)))) -> TOP.0(2_flat.0(down.0(0.0(x0)))) TOP.0(up.0(2.0(0.1(x0)))) -> TOP.0(2_flat.0(down.0(0.1(x0)))) TOP.0(up.0(2.0(1.0(x0)))) -> TOP.0(2_flat.0(down.0(1.0(x0)))) TOP.0(up.0(2.0(1.1(x0)))) -> TOP.0(2_flat.0(down.0(1.1(x0)))) TOP.0(up.0(2.0(2.0(x0)))) -> TOP.0(2_flat.0(down.0(2.0(x0)))) TOP.0(up.0(2.0(2.1(x0)))) -> TOP.0(2_flat.0(down.0(2.1(x0)))) The TRS R consists of the following rules: 2_flat.0(up.0(x_1)) -> up.0(2.0(x_1)) down.0(2.0(f.0(x))) -> up.0(f.0(0.0(x))) down.0(2.0(a.)) -> 2_flat.0(down.0(a.)) down.0(2.0(b.)) -> 2_flat.0(down.0(b.)) down.0(2.0(0.0(y21))) -> 2_flat.0(down.0(0.0(y21))) down.0(2.0(0.1(y21))) -> 2_flat.0(down.0(0.1(y21))) down.0(2.0(1.0(y22))) -> 2_flat.0(down.0(1.0(y22))) down.0(2.0(1.1(y22))) -> 2_flat.0(down.0(1.1(y22))) down.0(2.0(2.0(y23))) -> 2_flat.0(down.0(2.0(y23))) down.0(2.0(2.1(y23))) -> 2_flat.0(down.0(2.1(y23))) down.0(1.0(f.0(x))) -> up.0(2.0(x)) down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) down.0(1.0(b.)) -> 1_flat.0(down.0(b.)) down.0(1.0(0.0(y16))) -> 1_flat.0(down.0(0.0(y16))) down.0(1.0(0.1(y16))) -> 1_flat.0(down.0(0.1(y16))) down.0(1.0(1.0(y17))) -> 1_flat.0(down.0(1.0(y17))) down.0(1.0(1.1(y17))) -> 1_flat.0(down.0(1.1(y17))) down.0(1.0(2.0(y18))) -> 1_flat.0(down.0(2.0(y18))) down.0(1.0(2.1(y18))) -> 1_flat.0(down.0(2.1(y18))) 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) down.0(0.0(f.0(x))) -> up.0(1.0(x)) down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) down.0(0.0(b.)) -> 0_flat.0(down.0(b.)) down.0(0.0(0.0(y11))) -> 0_flat.0(down.0(0.0(y11))) down.0(0.0(0.1(y11))) -> 0_flat.0(down.0(0.1(y11))) down.0(0.0(1.0(y12))) -> 0_flat.0(down.0(1.0(y12))) down.0(0.0(1.1(y12))) -> 0_flat.0(down.0(1.1(y12))) down.0(0.0(2.0(y13))) -> 0_flat.0(down.0(2.0(y13))) down.0(0.0(2.1(y13))) -> 0_flat.0(down.0(2.1(y13))) 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) down.0(a.) -> up.0(f.0(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) The set Q consists of the following terms: down.0(a.) down.0(f.0(f.0(x0))) down.0(f.0(f.1(x0))) down.0(0.0(f.0(x0))) down.0(0.0(f.1(x0))) down.0(1.0(f.0(x0))) down.0(1.0(f.1(x0))) down.0(2.0(f.0(x0))) down.0(2.0(f.1(x0))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(0.0(x0))) down.0(f.0(0.1(x0))) down.0(f.0(1.0(x0))) down.0(f.0(1.1(x0))) down.0(f.0(2.0(x0))) down.0(f.0(2.1(x0))) down.0(f.1(fresh_constant.)) down.0(0.0(a.)) down.0(0.0(b.)) down.0(0.0(0.0(x0))) down.0(0.0(0.1(x0))) down.0(0.0(1.0(x0))) down.0(0.0(1.1(x0))) down.0(0.0(2.0(x0))) down.0(0.0(2.1(x0))) down.0(0.1(fresh_constant.)) down.0(1.0(a.)) down.0(1.0(b.)) down.0(1.0(0.0(x0))) down.0(1.0(0.1(x0))) down.0(1.0(1.0(x0))) down.0(1.0(1.1(x0))) down.0(1.0(2.0(x0))) down.0(1.0(2.1(x0))) down.0(1.1(fresh_constant.)) down.0(2.0(a.)) down.0(2.0(b.)) down.0(2.0(0.0(x0))) down.0(2.0(0.1(x0))) down.0(2.0(1.0(x0))) down.0(2.0(1.1(x0))) down.0(2.0(2.0(x0))) down.0(2.0(2.1(x0))) down.0(2.1(fresh_constant.)) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) 0_flat.0(up.0(x0)) 0_flat.0(up.1(x0)) 1_flat.0(up.0(x0)) 1_flat.0(up.1(x0)) 2_flat.0(up.0(x0)) 2_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (150) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 12 less nodes. ---------------------------------------- (151) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) TOP.0(up.0(f.0(2.0(x0)))) -> TOP.0(f_flat.0(down.0(2.0(x0)))) TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) TOP.0(up.0(0.0(2.0(x0)))) -> TOP.0(0_flat.0(down.0(2.0(x0)))) TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) TOP.0(up.0(1.0(2.0(x0)))) -> TOP.0(1_flat.0(down.0(2.0(x0)))) TOP.0(up.0(2.0(0.0(x0)))) -> TOP.0(2_flat.0(down.0(0.0(x0)))) TOP.0(up.0(2.0(1.0(x0)))) -> TOP.0(2_flat.0(down.0(1.0(x0)))) TOP.0(up.0(2.0(2.0(x0)))) -> TOP.0(2_flat.0(down.0(2.0(x0)))) The TRS R consists of the following rules: 2_flat.0(up.0(x_1)) -> up.0(2.0(x_1)) down.0(2.0(f.0(x))) -> up.0(f.0(0.0(x))) down.0(2.0(a.)) -> 2_flat.0(down.0(a.)) down.0(2.0(b.)) -> 2_flat.0(down.0(b.)) down.0(2.0(0.0(y21))) -> 2_flat.0(down.0(0.0(y21))) down.0(2.0(0.1(y21))) -> 2_flat.0(down.0(0.1(y21))) down.0(2.0(1.0(y22))) -> 2_flat.0(down.0(1.0(y22))) down.0(2.0(1.1(y22))) -> 2_flat.0(down.0(1.1(y22))) down.0(2.0(2.0(y23))) -> 2_flat.0(down.0(2.0(y23))) down.0(2.0(2.1(y23))) -> 2_flat.0(down.0(2.1(y23))) down.0(1.0(f.0(x))) -> up.0(2.0(x)) down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) down.0(1.0(b.)) -> 1_flat.0(down.0(b.)) down.0(1.0(0.0(y16))) -> 1_flat.0(down.0(0.0(y16))) down.0(1.0(0.1(y16))) -> 1_flat.0(down.0(0.1(y16))) down.0(1.0(1.0(y17))) -> 1_flat.0(down.0(1.0(y17))) down.0(1.0(1.1(y17))) -> 1_flat.0(down.0(1.1(y17))) down.0(1.0(2.0(y18))) -> 1_flat.0(down.0(2.0(y18))) down.0(1.0(2.1(y18))) -> 1_flat.0(down.0(2.1(y18))) 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) down.0(0.0(f.0(x))) -> up.0(1.0(x)) down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) down.0(0.0(b.)) -> 0_flat.0(down.0(b.)) down.0(0.0(0.0(y11))) -> 0_flat.0(down.0(0.0(y11))) down.0(0.0(0.1(y11))) -> 0_flat.0(down.0(0.1(y11))) down.0(0.0(1.0(y12))) -> 0_flat.0(down.0(1.0(y12))) down.0(0.0(1.1(y12))) -> 0_flat.0(down.0(1.1(y12))) down.0(0.0(2.0(y13))) -> 0_flat.0(down.0(2.0(y13))) down.0(0.0(2.1(y13))) -> 0_flat.0(down.0(2.1(y13))) 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) down.0(a.) -> up.0(f.0(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) The set Q consists of the following terms: down.0(a.) down.0(f.0(f.0(x0))) down.0(f.0(f.1(x0))) down.0(0.0(f.0(x0))) down.0(0.0(f.1(x0))) down.0(1.0(f.0(x0))) down.0(1.0(f.1(x0))) down.0(2.0(f.0(x0))) down.0(2.0(f.1(x0))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(0.0(x0))) down.0(f.0(0.1(x0))) down.0(f.0(1.0(x0))) down.0(f.0(1.1(x0))) down.0(f.0(2.0(x0))) down.0(f.0(2.1(x0))) down.0(f.1(fresh_constant.)) down.0(0.0(a.)) down.0(0.0(b.)) down.0(0.0(0.0(x0))) down.0(0.0(0.1(x0))) down.0(0.0(1.0(x0))) down.0(0.0(1.1(x0))) down.0(0.0(2.0(x0))) down.0(0.0(2.1(x0))) down.0(0.1(fresh_constant.)) down.0(1.0(a.)) down.0(1.0(b.)) down.0(1.0(0.0(x0))) down.0(1.0(0.1(x0))) down.0(1.0(1.0(x0))) down.0(1.0(1.1(x0))) down.0(1.0(2.0(x0))) down.0(1.0(2.1(x0))) down.0(1.1(fresh_constant.)) down.0(2.0(a.)) down.0(2.0(b.)) down.0(2.0(0.0(x0))) down.0(2.0(0.1(x0))) down.0(2.0(1.0(x0))) down.0(2.0(1.1(x0))) down.0(2.0(2.0(x0))) down.0(2.0(2.1(x0))) down.0(2.1(fresh_constant.)) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) 0_flat.0(up.0(x0)) 0_flat.0(up.1(x0)) 1_flat.0(up.0(x0)) 1_flat.0(up.1(x0)) 2_flat.0(up.0(x0)) 2_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (152) PisEmptyProof (SOUND) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (153) TRUE ---------------------------------------- (154) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) The TRS R consists of the following rules: 2_flat(up(x_1)) -> up(2(x_1)) down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (155) QReductionProof (EQUIVALENT) We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN]. down(f(fresh_constant)) down(0(fresh_constant)) down(1(fresh_constant)) down(2(fresh_constant)) ---------------------------------------- (156) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) The TRS R consists of the following rules: 2_flat(up(x_1)) -> up(2(x_1)) down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all (P,Q,R)-chains. ---------------------------------------- (157) SplitQDPProof (EQUIVALENT) We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem ---------------------------------------- (158) Complex Obligation (AND) ---------------------------------------- (159) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) The TRS R consists of the following rules: 2_flat(up(x_1)) -> up(2(x_1)) down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(b)) -> 2_flat(down(b)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(b)) -> 1_flat(down(b)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(b)) -> 0_flat(down(b)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (160) SemLabProof (SOUND) We found the following model for the rules of the TRSs R and P. Interpretation over the domain with elements from 0 to 1. 2_flat: 0 a: 0 2: 0 TOP: 0 b: 1 down: 0 f: 0 fresh_constant: 0 0: 0 up: 0 1: 0 1_flat: 0 0_flat: 0 f_flat: 0 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. ---------------------------------------- (161) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) TOP.0(up.0(f.0(0.1(x0)))) -> TOP.0(f_flat.0(down.0(0.1(x0)))) TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) TOP.0(up.0(f.0(1.1(x0)))) -> TOP.0(f_flat.0(down.0(1.1(x0)))) TOP.0(up.0(f.0(2.0(x0)))) -> TOP.0(f_flat.0(down.0(2.0(x0)))) TOP.0(up.0(f.0(2.1(x0)))) -> TOP.0(f_flat.0(down.0(2.1(x0)))) TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) TOP.0(up.0(0.0(0.1(x0)))) -> TOP.0(0_flat.0(down.0(0.1(x0)))) TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) TOP.0(up.0(0.0(1.1(x0)))) -> TOP.0(0_flat.0(down.0(1.1(x0)))) TOP.0(up.0(0.0(2.0(x0)))) -> TOP.0(0_flat.0(down.0(2.0(x0)))) TOP.0(up.0(0.0(2.1(x0)))) -> TOP.0(0_flat.0(down.0(2.1(x0)))) TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) TOP.0(up.0(1.0(0.1(x0)))) -> TOP.0(1_flat.0(down.0(0.1(x0)))) TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) TOP.0(up.0(1.0(1.1(x0)))) -> TOP.0(1_flat.0(down.0(1.1(x0)))) TOP.0(up.0(1.0(2.0(x0)))) -> TOP.0(1_flat.0(down.0(2.0(x0)))) TOP.0(up.0(1.0(2.1(x0)))) -> TOP.0(1_flat.0(down.0(2.1(x0)))) TOP.0(up.0(2.0(0.0(x0)))) -> TOP.0(2_flat.0(down.0(0.0(x0)))) TOP.0(up.0(2.0(0.1(x0)))) -> TOP.0(2_flat.0(down.0(0.1(x0)))) TOP.0(up.0(2.0(1.0(x0)))) -> TOP.0(2_flat.0(down.0(1.0(x0)))) TOP.0(up.0(2.0(1.1(x0)))) -> TOP.0(2_flat.0(down.0(1.1(x0)))) TOP.0(up.0(2.0(2.0(x0)))) -> TOP.0(2_flat.0(down.0(2.0(x0)))) TOP.0(up.0(2.0(2.1(x0)))) -> TOP.0(2_flat.0(down.0(2.1(x0)))) The TRS R consists of the following rules: 2_flat.0(up.0(x_1)) -> up.0(2.0(x_1)) 2_flat.0(up.1(x_1)) -> up.0(2.1(x_1)) down.0(2.0(f.0(x))) -> up.0(f.0(0.0(x))) down.0(2.0(f.1(x))) -> up.0(f.0(0.1(x))) down.0(2.0(a.)) -> 2_flat.0(down.0(a.)) down.0(2.1(b.)) -> 2_flat.0(down.1(b.)) down.0(2.0(0.0(y21))) -> 2_flat.0(down.0(0.0(y21))) down.0(2.0(0.1(y21))) -> 2_flat.0(down.0(0.1(y21))) down.0(2.0(1.0(y22))) -> 2_flat.0(down.0(1.0(y22))) down.0(2.0(1.1(y22))) -> 2_flat.0(down.0(1.1(y22))) down.0(2.0(2.0(y23))) -> 2_flat.0(down.0(2.0(y23))) down.0(2.0(2.1(y23))) -> 2_flat.0(down.0(2.1(y23))) down.0(1.0(f.0(x))) -> up.0(2.0(x)) down.0(1.0(f.1(x))) -> up.0(2.1(x)) down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) down.0(1.1(b.)) -> 1_flat.0(down.1(b.)) down.0(1.0(0.0(y16))) -> 1_flat.0(down.0(0.0(y16))) down.0(1.0(0.1(y16))) -> 1_flat.0(down.0(0.1(y16))) down.0(1.0(1.0(y17))) -> 1_flat.0(down.0(1.0(y17))) down.0(1.0(1.1(y17))) -> 1_flat.0(down.0(1.1(y17))) down.0(1.0(2.0(y18))) -> 1_flat.0(down.0(2.0(y18))) down.0(1.0(2.1(y18))) -> 1_flat.0(down.0(2.1(y18))) 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) 1_flat.0(up.1(x_1)) -> up.0(1.1(x_1)) down.0(0.0(f.0(x))) -> up.0(1.0(x)) down.0(0.0(f.1(x))) -> up.0(1.1(x)) down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) down.0(0.1(b.)) -> 0_flat.0(down.1(b.)) down.0(0.0(0.0(y11))) -> 0_flat.0(down.0(0.0(y11))) down.0(0.0(0.1(y11))) -> 0_flat.0(down.0(0.1(y11))) down.0(0.0(1.0(y12))) -> 0_flat.0(down.0(1.0(y12))) down.0(0.0(1.1(y12))) -> 0_flat.0(down.0(1.1(y12))) down.0(0.0(2.0(y13))) -> 0_flat.0(down.0(2.0(y13))) down.0(0.0(2.1(y13))) -> 0_flat.0(down.0(2.1(y13))) 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) 0_flat.0(up.1(x_1)) -> up.0(0.1(x_1)) down.0(a.) -> up.0(f.0(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) The set Q consists of the following terms: down.0(a.) down.0(f.0(f.0(x0))) down.0(f.0(f.1(x0))) down.0(0.0(f.0(x0))) down.0(0.0(f.1(x0))) down.0(1.0(f.0(x0))) down.0(1.0(f.1(x0))) down.0(2.0(f.0(x0))) down.0(2.0(f.1(x0))) down.0(f.0(a.)) down.0(f.1(b.)) down.0(f.0(0.0(x0))) down.0(f.0(0.1(x0))) down.0(f.0(1.0(x0))) down.0(f.0(1.1(x0))) down.0(f.0(2.0(x0))) down.0(f.0(2.1(x0))) down.0(f.0(fresh_constant.)) down.0(0.0(a.)) down.0(0.1(b.)) down.0(0.0(0.0(x0))) down.0(0.0(0.1(x0))) down.0(0.0(1.0(x0))) down.0(0.0(1.1(x0))) down.0(0.0(2.0(x0))) down.0(0.0(2.1(x0))) down.0(0.0(fresh_constant.)) down.0(1.0(a.)) down.0(1.1(b.)) down.0(1.0(0.0(x0))) down.0(1.0(0.1(x0))) down.0(1.0(1.0(x0))) down.0(1.0(1.1(x0))) down.0(1.0(2.0(x0))) down.0(1.0(2.1(x0))) down.0(1.0(fresh_constant.)) down.0(2.0(a.)) down.0(2.1(b.)) down.0(2.0(0.0(x0))) down.0(2.0(0.1(x0))) down.0(2.0(1.0(x0))) down.0(2.0(1.1(x0))) down.0(2.0(2.0(x0))) down.0(2.0(2.1(x0))) down.0(2.0(fresh_constant.)) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) 0_flat.0(up.0(x0)) 0_flat.0(up.1(x0)) 1_flat.0(up.0(x0)) 1_flat.0(up.1(x0)) 2_flat.0(up.0(x0)) 2_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (162) UsableRulesReductionPairsProof (EQUIVALENT) 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. No dependency pairs are removed. The following rules are removed from R: 2_flat.0(up.1(x_1)) -> up.0(2.1(x_1)) down.0(2.0(f.1(x))) -> up.0(f.0(0.1(x))) down.0(1.0(f.1(x))) -> up.0(2.1(x)) 1_flat.0(up.1(x_1)) -> up.0(1.1(x_1)) down.0(0.0(f.1(x))) -> up.0(1.1(x)) 0_flat.0(up.1(x_1)) -> up.0(0.1(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0.0(x_1)) = 1 + x_1 POL(0.1(x_1)) = 1 + x_1 POL(0_flat.0(x_1)) = 1 + x_1 POL(1.0(x_1)) = 1 + x_1 POL(1.1(x_1)) = 1 + x_1 POL(1_flat.0(x_1)) = 1 + x_1 POL(2.0(x_1)) = 1 + x_1 POL(2.1(x_1)) = 1 + x_1 POL(2_flat.0(x_1)) = 1 + x_1 POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(down.0(x_1)) = x_1 POL(down.1(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 1 + x_1 POL(f_flat.0(x_1)) = x_1 POL(up.0(x_1)) = x_1 POL(up.1(x_1)) = 1 + x_1 ---------------------------------------- (163) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) TOP.0(up.0(f.0(0.1(x0)))) -> TOP.0(f_flat.0(down.0(0.1(x0)))) TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) TOP.0(up.0(f.0(1.1(x0)))) -> TOP.0(f_flat.0(down.0(1.1(x0)))) TOP.0(up.0(f.0(2.0(x0)))) -> TOP.0(f_flat.0(down.0(2.0(x0)))) TOP.0(up.0(f.0(2.1(x0)))) -> TOP.0(f_flat.0(down.0(2.1(x0)))) TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) TOP.0(up.0(0.0(0.1(x0)))) -> TOP.0(0_flat.0(down.0(0.1(x0)))) TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) TOP.0(up.0(0.0(1.1(x0)))) -> TOP.0(0_flat.0(down.0(1.1(x0)))) TOP.0(up.0(0.0(2.0(x0)))) -> TOP.0(0_flat.0(down.0(2.0(x0)))) TOP.0(up.0(0.0(2.1(x0)))) -> TOP.0(0_flat.0(down.0(2.1(x0)))) TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) TOP.0(up.0(1.0(0.1(x0)))) -> TOP.0(1_flat.0(down.0(0.1(x0)))) TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) TOP.0(up.0(1.0(1.1(x0)))) -> TOP.0(1_flat.0(down.0(1.1(x0)))) TOP.0(up.0(1.0(2.0(x0)))) -> TOP.0(1_flat.0(down.0(2.0(x0)))) TOP.0(up.0(1.0(2.1(x0)))) -> TOP.0(1_flat.0(down.0(2.1(x0)))) TOP.0(up.0(2.0(0.0(x0)))) -> TOP.0(2_flat.0(down.0(0.0(x0)))) TOP.0(up.0(2.0(0.1(x0)))) -> TOP.0(2_flat.0(down.0(0.1(x0)))) TOP.0(up.0(2.0(1.0(x0)))) -> TOP.0(2_flat.0(down.0(1.0(x0)))) TOP.0(up.0(2.0(1.1(x0)))) -> TOP.0(2_flat.0(down.0(1.1(x0)))) TOP.0(up.0(2.0(2.0(x0)))) -> TOP.0(2_flat.0(down.0(2.0(x0)))) TOP.0(up.0(2.0(2.1(x0)))) -> TOP.0(2_flat.0(down.0(2.1(x0)))) The TRS R consists of the following rules: down.0(2.1(b.)) -> 2_flat.0(down.1(b.)) 2_flat.0(up.0(x_1)) -> up.0(2.0(x_1)) down.0(2.0(f.0(x))) -> up.0(f.0(0.0(x))) down.0(2.0(a.)) -> 2_flat.0(down.0(a.)) down.0(2.0(0.0(y21))) -> 2_flat.0(down.0(0.0(y21))) down.0(2.0(0.1(y21))) -> 2_flat.0(down.0(0.1(y21))) down.0(2.0(1.0(y22))) -> 2_flat.0(down.0(1.0(y22))) down.0(2.0(1.1(y22))) -> 2_flat.0(down.0(1.1(y22))) down.0(2.0(2.0(y23))) -> 2_flat.0(down.0(2.0(y23))) down.0(2.0(2.1(y23))) -> 2_flat.0(down.0(2.1(y23))) down.0(1.1(b.)) -> 1_flat.0(down.1(b.)) down.0(1.0(f.0(x))) -> up.0(2.0(x)) down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) down.0(1.0(0.0(y16))) -> 1_flat.0(down.0(0.0(y16))) down.0(1.0(0.1(y16))) -> 1_flat.0(down.0(0.1(y16))) down.0(1.0(1.0(y17))) -> 1_flat.0(down.0(1.0(y17))) down.0(1.0(1.1(y17))) -> 1_flat.0(down.0(1.1(y17))) down.0(1.0(2.0(y18))) -> 1_flat.0(down.0(2.0(y18))) down.0(1.0(2.1(y18))) -> 1_flat.0(down.0(2.1(y18))) 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) down.0(0.1(b.)) -> 0_flat.0(down.1(b.)) down.0(0.0(f.0(x))) -> up.0(1.0(x)) down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) down.0(0.0(0.0(y11))) -> 0_flat.0(down.0(0.0(y11))) down.0(0.0(0.1(y11))) -> 0_flat.0(down.0(0.1(y11))) down.0(0.0(1.0(y12))) -> 0_flat.0(down.0(1.0(y12))) down.0(0.0(1.1(y12))) -> 0_flat.0(down.0(1.1(y12))) down.0(0.0(2.0(y13))) -> 0_flat.0(down.0(2.0(y13))) down.0(0.0(2.1(y13))) -> 0_flat.0(down.0(2.1(y13))) 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) down.0(a.) -> up.0(f.0(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) The set Q consists of the following terms: down.0(a.) down.0(f.0(f.0(x0))) down.0(f.0(f.1(x0))) down.0(0.0(f.0(x0))) down.0(0.0(f.1(x0))) down.0(1.0(f.0(x0))) down.0(1.0(f.1(x0))) down.0(2.0(f.0(x0))) down.0(2.0(f.1(x0))) down.0(f.0(a.)) down.0(f.1(b.)) down.0(f.0(0.0(x0))) down.0(f.0(0.1(x0))) down.0(f.0(1.0(x0))) down.0(f.0(1.1(x0))) down.0(f.0(2.0(x0))) down.0(f.0(2.1(x0))) down.0(f.0(fresh_constant.)) down.0(0.0(a.)) down.0(0.1(b.)) down.0(0.0(0.0(x0))) down.0(0.0(0.1(x0))) down.0(0.0(1.0(x0))) down.0(0.0(1.1(x0))) down.0(0.0(2.0(x0))) down.0(0.0(2.1(x0))) down.0(0.0(fresh_constant.)) down.0(1.0(a.)) down.0(1.1(b.)) down.0(1.0(0.0(x0))) down.0(1.0(0.1(x0))) down.0(1.0(1.0(x0))) down.0(1.0(1.1(x0))) down.0(1.0(2.0(x0))) down.0(1.0(2.1(x0))) down.0(1.0(fresh_constant.)) down.0(2.0(a.)) down.0(2.1(b.)) down.0(2.0(0.0(x0))) down.0(2.0(0.1(x0))) down.0(2.0(1.0(x0))) down.0(2.0(1.1(x0))) down.0(2.0(2.0(x0))) down.0(2.0(2.1(x0))) down.0(2.0(fresh_constant.)) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) 0_flat.0(up.0(x0)) 0_flat.0(up.1(x0)) 1_flat.0(up.0(x0)) 1_flat.0(up.1(x0)) 2_flat.0(up.0(x0)) 2_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (164) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: down.0(2.1(b.)) -> 2_flat.0(down.1(b.)) down.0(1.1(b.)) -> 1_flat.0(down.1(b.)) down.0(0.1(b.)) -> 0_flat.0(down.1(b.)) Used ordering: Polynomial interpretation [POLO]: POL(0.0(x_1)) = x_1 POL(0.1(x_1)) = 1 + x_1 POL(0_flat.0(x_1)) = x_1 POL(1.0(x_1)) = x_1 POL(1.1(x_1)) = 1 + x_1 POL(1_flat.0(x_1)) = x_1 POL(2.0(x_1)) = x_1 POL(2.1(x_1)) = 1 + x_1 POL(2_flat.0(x_1)) = x_1 POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(down.0(x_1)) = x_1 POL(down.1(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f_flat.0(x_1)) = x_1 POL(up.0(x_1)) = x_1 ---------------------------------------- (165) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) TOP.0(up.0(f.0(0.1(x0)))) -> TOP.0(f_flat.0(down.0(0.1(x0)))) TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) TOP.0(up.0(f.0(1.1(x0)))) -> TOP.0(f_flat.0(down.0(1.1(x0)))) TOP.0(up.0(f.0(2.0(x0)))) -> TOP.0(f_flat.0(down.0(2.0(x0)))) TOP.0(up.0(f.0(2.1(x0)))) -> TOP.0(f_flat.0(down.0(2.1(x0)))) TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) TOP.0(up.0(0.0(0.1(x0)))) -> TOP.0(0_flat.0(down.0(0.1(x0)))) TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) TOP.0(up.0(0.0(1.1(x0)))) -> TOP.0(0_flat.0(down.0(1.1(x0)))) TOP.0(up.0(0.0(2.0(x0)))) -> TOP.0(0_flat.0(down.0(2.0(x0)))) TOP.0(up.0(0.0(2.1(x0)))) -> TOP.0(0_flat.0(down.0(2.1(x0)))) TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) TOP.0(up.0(1.0(0.1(x0)))) -> TOP.0(1_flat.0(down.0(0.1(x0)))) TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) TOP.0(up.0(1.0(1.1(x0)))) -> TOP.0(1_flat.0(down.0(1.1(x0)))) TOP.0(up.0(1.0(2.0(x0)))) -> TOP.0(1_flat.0(down.0(2.0(x0)))) TOP.0(up.0(1.0(2.1(x0)))) -> TOP.0(1_flat.0(down.0(2.1(x0)))) TOP.0(up.0(2.0(0.0(x0)))) -> TOP.0(2_flat.0(down.0(0.0(x0)))) TOP.0(up.0(2.0(0.1(x0)))) -> TOP.0(2_flat.0(down.0(0.1(x0)))) TOP.0(up.0(2.0(1.0(x0)))) -> TOP.0(2_flat.0(down.0(1.0(x0)))) TOP.0(up.0(2.0(1.1(x0)))) -> TOP.0(2_flat.0(down.0(1.1(x0)))) TOP.0(up.0(2.0(2.0(x0)))) -> TOP.0(2_flat.0(down.0(2.0(x0)))) TOP.0(up.0(2.0(2.1(x0)))) -> TOP.0(2_flat.0(down.0(2.1(x0)))) The TRS R consists of the following rules: 2_flat.0(up.0(x_1)) -> up.0(2.0(x_1)) down.0(2.0(f.0(x))) -> up.0(f.0(0.0(x))) down.0(2.0(a.)) -> 2_flat.0(down.0(a.)) down.0(2.0(0.0(y21))) -> 2_flat.0(down.0(0.0(y21))) down.0(2.0(0.1(y21))) -> 2_flat.0(down.0(0.1(y21))) down.0(2.0(1.0(y22))) -> 2_flat.0(down.0(1.0(y22))) down.0(2.0(1.1(y22))) -> 2_flat.0(down.0(1.1(y22))) down.0(2.0(2.0(y23))) -> 2_flat.0(down.0(2.0(y23))) down.0(2.0(2.1(y23))) -> 2_flat.0(down.0(2.1(y23))) down.0(1.0(f.0(x))) -> up.0(2.0(x)) down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) down.0(1.0(0.0(y16))) -> 1_flat.0(down.0(0.0(y16))) down.0(1.0(0.1(y16))) -> 1_flat.0(down.0(0.1(y16))) down.0(1.0(1.0(y17))) -> 1_flat.0(down.0(1.0(y17))) down.0(1.0(1.1(y17))) -> 1_flat.0(down.0(1.1(y17))) down.0(1.0(2.0(y18))) -> 1_flat.0(down.0(2.0(y18))) down.0(1.0(2.1(y18))) -> 1_flat.0(down.0(2.1(y18))) 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) down.0(0.0(f.0(x))) -> up.0(1.0(x)) down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) down.0(0.0(0.0(y11))) -> 0_flat.0(down.0(0.0(y11))) down.0(0.0(0.1(y11))) -> 0_flat.0(down.0(0.1(y11))) down.0(0.0(1.0(y12))) -> 0_flat.0(down.0(1.0(y12))) down.0(0.0(1.1(y12))) -> 0_flat.0(down.0(1.1(y12))) down.0(0.0(2.0(y13))) -> 0_flat.0(down.0(2.0(y13))) down.0(0.0(2.1(y13))) -> 0_flat.0(down.0(2.1(y13))) 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) down.0(a.) -> up.0(f.0(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) The set Q consists of the following terms: down.0(a.) down.0(f.0(f.0(x0))) down.0(f.0(f.1(x0))) down.0(0.0(f.0(x0))) down.0(0.0(f.1(x0))) down.0(1.0(f.0(x0))) down.0(1.0(f.1(x0))) down.0(2.0(f.0(x0))) down.0(2.0(f.1(x0))) down.0(f.0(a.)) down.0(f.1(b.)) down.0(f.0(0.0(x0))) down.0(f.0(0.1(x0))) down.0(f.0(1.0(x0))) down.0(f.0(1.1(x0))) down.0(f.0(2.0(x0))) down.0(f.0(2.1(x0))) down.0(f.0(fresh_constant.)) down.0(0.0(a.)) down.0(0.1(b.)) down.0(0.0(0.0(x0))) down.0(0.0(0.1(x0))) down.0(0.0(1.0(x0))) down.0(0.0(1.1(x0))) down.0(0.0(2.0(x0))) down.0(0.0(2.1(x0))) down.0(0.0(fresh_constant.)) down.0(1.0(a.)) down.0(1.1(b.)) down.0(1.0(0.0(x0))) down.0(1.0(0.1(x0))) down.0(1.0(1.0(x0))) down.0(1.0(1.1(x0))) down.0(1.0(2.0(x0))) down.0(1.0(2.1(x0))) down.0(1.0(fresh_constant.)) down.0(2.0(a.)) down.0(2.1(b.)) down.0(2.0(0.0(x0))) down.0(2.0(0.1(x0))) down.0(2.0(1.0(x0))) down.0(2.0(1.1(x0))) down.0(2.0(2.0(x0))) down.0(2.0(2.1(x0))) down.0(2.0(fresh_constant.)) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) 0_flat.0(up.0(x0)) 0_flat.0(up.1(x0)) 1_flat.0(up.0(x0)) 1_flat.0(up.1(x0)) 2_flat.0(up.0(x0)) 2_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (166) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 12 less nodes. ---------------------------------------- (167) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(0.0(x0)))) -> TOP.0(f_flat.0(down.0(0.0(x0)))) TOP.0(up.0(f.0(1.0(x0)))) -> TOP.0(f_flat.0(down.0(1.0(x0)))) TOP.0(up.0(f.0(2.0(x0)))) -> TOP.0(f_flat.0(down.0(2.0(x0)))) TOP.0(up.0(0.0(0.0(x0)))) -> TOP.0(0_flat.0(down.0(0.0(x0)))) TOP.0(up.0(0.0(1.0(x0)))) -> TOP.0(0_flat.0(down.0(1.0(x0)))) TOP.0(up.0(0.0(2.0(x0)))) -> TOP.0(0_flat.0(down.0(2.0(x0)))) TOP.0(up.0(1.0(0.0(x0)))) -> TOP.0(1_flat.0(down.0(0.0(x0)))) TOP.0(up.0(1.0(1.0(x0)))) -> TOP.0(1_flat.0(down.0(1.0(x0)))) TOP.0(up.0(1.0(2.0(x0)))) -> TOP.0(1_flat.0(down.0(2.0(x0)))) TOP.0(up.0(2.0(0.0(x0)))) -> TOP.0(2_flat.0(down.0(0.0(x0)))) TOP.0(up.0(2.0(1.0(x0)))) -> TOP.0(2_flat.0(down.0(1.0(x0)))) TOP.0(up.0(2.0(2.0(x0)))) -> TOP.0(2_flat.0(down.0(2.0(x0)))) The TRS R consists of the following rules: 2_flat.0(up.0(x_1)) -> up.0(2.0(x_1)) down.0(2.0(f.0(x))) -> up.0(f.0(0.0(x))) down.0(2.0(a.)) -> 2_flat.0(down.0(a.)) down.0(2.0(0.0(y21))) -> 2_flat.0(down.0(0.0(y21))) down.0(2.0(0.1(y21))) -> 2_flat.0(down.0(0.1(y21))) down.0(2.0(1.0(y22))) -> 2_flat.0(down.0(1.0(y22))) down.0(2.0(1.1(y22))) -> 2_flat.0(down.0(1.1(y22))) down.0(2.0(2.0(y23))) -> 2_flat.0(down.0(2.0(y23))) down.0(2.0(2.1(y23))) -> 2_flat.0(down.0(2.1(y23))) down.0(1.0(f.0(x))) -> up.0(2.0(x)) down.0(1.0(a.)) -> 1_flat.0(down.0(a.)) down.0(1.0(0.0(y16))) -> 1_flat.0(down.0(0.0(y16))) down.0(1.0(0.1(y16))) -> 1_flat.0(down.0(0.1(y16))) down.0(1.0(1.0(y17))) -> 1_flat.0(down.0(1.0(y17))) down.0(1.0(1.1(y17))) -> 1_flat.0(down.0(1.1(y17))) down.0(1.0(2.0(y18))) -> 1_flat.0(down.0(2.0(y18))) down.0(1.0(2.1(y18))) -> 1_flat.0(down.0(2.1(y18))) 1_flat.0(up.0(x_1)) -> up.0(1.0(x_1)) down.0(0.0(f.0(x))) -> up.0(1.0(x)) down.0(0.0(a.)) -> 0_flat.0(down.0(a.)) down.0(0.0(0.0(y11))) -> 0_flat.0(down.0(0.0(y11))) down.0(0.0(0.1(y11))) -> 0_flat.0(down.0(0.1(y11))) down.0(0.0(1.0(y12))) -> 0_flat.0(down.0(1.0(y12))) down.0(0.0(1.1(y12))) -> 0_flat.0(down.0(1.1(y12))) down.0(0.0(2.0(y13))) -> 0_flat.0(down.0(2.0(y13))) down.0(0.0(2.1(y13))) -> 0_flat.0(down.0(2.1(y13))) 0_flat.0(up.0(x_1)) -> up.0(0.0(x_1)) down.0(a.) -> up.0(f.0(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) The set Q consists of the following terms: down.0(a.) down.0(f.0(f.0(x0))) down.0(f.0(f.1(x0))) down.0(0.0(f.0(x0))) down.0(0.0(f.1(x0))) down.0(1.0(f.0(x0))) down.0(1.0(f.1(x0))) down.0(2.0(f.0(x0))) down.0(2.0(f.1(x0))) down.0(f.0(a.)) down.0(f.1(b.)) down.0(f.0(0.0(x0))) down.0(f.0(0.1(x0))) down.0(f.0(1.0(x0))) down.0(f.0(1.1(x0))) down.0(f.0(2.0(x0))) down.0(f.0(2.1(x0))) down.0(f.0(fresh_constant.)) down.0(0.0(a.)) down.0(0.1(b.)) down.0(0.0(0.0(x0))) down.0(0.0(0.1(x0))) down.0(0.0(1.0(x0))) down.0(0.0(1.1(x0))) down.0(0.0(2.0(x0))) down.0(0.0(2.1(x0))) down.0(0.0(fresh_constant.)) down.0(1.0(a.)) down.0(1.1(b.)) down.0(1.0(0.0(x0))) down.0(1.0(0.1(x0))) down.0(1.0(1.0(x0))) down.0(1.0(1.1(x0))) down.0(1.0(2.0(x0))) down.0(1.0(2.1(x0))) down.0(1.0(fresh_constant.)) down.0(2.0(a.)) down.0(2.1(b.)) down.0(2.0(0.0(x0))) down.0(2.0(0.1(x0))) down.0(2.0(1.0(x0))) down.0(2.0(1.1(x0))) down.0(2.0(2.0(x0))) down.0(2.0(2.1(x0))) down.0(2.0(fresh_constant.)) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) 0_flat.0(up.0(x0)) 0_flat.0(up.1(x0)) 1_flat.0(up.0(x0)) 1_flat.0(up.1(x0)) 2_flat.0(up.0(x0)) 2_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (168) PisEmptyProof (SOUND) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (169) TRUE ---------------------------------------- (170) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) The TRS R consists of the following rules: 2_flat(up(x_1)) -> up(2(x_1)) down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(b)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(f(fresh_constant)) down(0(a)) down(0(b)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(0(fresh_constant)) down(1(a)) down(1(b)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(1(fresh_constant)) down(2(a)) down(2(b)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) down(2(fresh_constant)) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (171) QReductionProof (EQUIVALENT) We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN]. down(f(b)) down(f(fresh_constant)) down(0(b)) down(0(fresh_constant)) down(1(b)) down(1(fresh_constant)) down(2(b)) down(2(fresh_constant)) ---------------------------------------- (172) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(0(x0)))) -> TOP(f_flat(down(0(x0)))) TOP(up(f(1(x0)))) -> TOP(f_flat(down(1(x0)))) TOP(up(f(2(x0)))) -> TOP(f_flat(down(2(x0)))) TOP(up(0(0(x0)))) -> TOP(0_flat(down(0(x0)))) TOP(up(0(1(x0)))) -> TOP(0_flat(down(1(x0)))) TOP(up(0(2(x0)))) -> TOP(0_flat(down(2(x0)))) TOP(up(1(0(x0)))) -> TOP(1_flat(down(0(x0)))) TOP(up(1(1(x0)))) -> TOP(1_flat(down(1(x0)))) TOP(up(1(2(x0)))) -> TOP(1_flat(down(2(x0)))) TOP(up(2(0(x0)))) -> TOP(2_flat(down(0(x0)))) TOP(up(2(1(x0)))) -> TOP(2_flat(down(1(x0)))) TOP(up(2(2(x0)))) -> TOP(2_flat(down(2(x0)))) The TRS R consists of the following rules: 2_flat(up(x_1)) -> up(2(x_1)) down(2(f(x))) -> up(f(0(x))) down(2(a)) -> 2_flat(down(a)) down(2(0(y21))) -> 2_flat(down(0(y21))) down(2(1(y22))) -> 2_flat(down(1(y22))) down(2(2(y23))) -> 2_flat(down(2(y23))) down(1(f(x))) -> up(2(x)) down(1(a)) -> 1_flat(down(a)) down(1(0(y16))) -> 1_flat(down(0(y16))) down(1(1(y17))) -> 1_flat(down(1(y17))) down(1(2(y18))) -> 1_flat(down(2(y18))) 1_flat(up(x_1)) -> up(1(x_1)) down(0(f(x))) -> up(1(x)) down(0(a)) -> 0_flat(down(a)) down(0(0(y11))) -> 0_flat(down(0(y11))) down(0(1(y12))) -> 0_flat(down(1(y12))) down(0(2(y13))) -> 0_flat(down(2(y13))) 0_flat(up(x_1)) -> up(0(x_1)) down(a) -> up(f(a)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(f(f(x0))) down(0(f(x0))) down(1(f(x0))) down(2(f(x0))) down(f(a)) down(f(0(x0))) down(f(1(x0))) down(f(2(x0))) down(0(a)) down(0(0(x0))) down(0(1(x0))) down(0(2(x0))) down(1(a)) down(1(0(x0))) down(1(1(x0))) down(1(2(x0))) down(2(a)) down(2(0(x0))) down(2(1(x0))) down(2(2(x0))) f_flat(up(x0)) 0_flat(up(x0)) 1_flat(up(x0)) 2_flat(up(x0)) We have to consider all (P,Q,R)-chains.