361.46/136.57 MAYBE 361.46/136.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 361.46/136.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 361.46/136.58 361.46/136.58 361.46/136.58 Outermost Termination of the given OTRS could not be shown: 361.46/136.58 361.46/136.58 (0) OTRS 361.46/136.58 (1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms] 361.46/136.58 (2) QTRS 361.46/136.58 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 361.46/136.58 (4) QDP 361.46/136.58 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (6) AND 361.46/136.58 (7) QDP 361.46/136.58 (8) UsableRulesProof [EQUIVALENT, 0 ms] 361.46/136.58 (9) QDP 361.46/136.58 (10) QReductionProof [EQUIVALENT, 0 ms] 361.46/136.58 (11) QDP 361.46/136.58 (12) MRRProof [EQUIVALENT, 15 ms] 361.46/136.58 (13) QDP 361.46/136.58 (14) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (15) AND 361.46/136.58 (16) QDP 361.46/136.58 (17) UsableRulesProof [EQUIVALENT, 0 ms] 361.46/136.58 (18) QDP 361.46/136.58 (19) QReductionProof [EQUIVALENT, 0 ms] 361.46/136.58 (20) QDP 361.46/136.58 (21) UsableRulesReductionPairsProof [EQUIVALENT, 5 ms] 361.46/136.58 (22) QDP 361.46/136.58 (23) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (24) TRUE 361.46/136.58 (25) QDP 361.46/136.58 (26) UsableRulesProof [EQUIVALENT, 0 ms] 361.46/136.58 (27) QDP 361.46/136.58 (28) QReductionProof [EQUIVALENT, 0 ms] 361.46/136.58 (29) QDP 361.46/136.58 (30) UsableRulesReductionPairsProof [EQUIVALENT, 8 ms] 361.46/136.58 (31) QDP 361.46/136.58 (32) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (33) QDP 361.46/136.58 (34) QReductionProof [EQUIVALENT, 0 ms] 361.46/136.58 (35) QDP 361.46/136.58 (36) MRRProof [EQUIVALENT, 7 ms] 361.46/136.58 (37) QDP 361.46/136.58 (38) PisEmptyProof [EQUIVALENT, 0 ms] 361.46/136.58 (39) YES 361.46/136.58 (40) QDP 361.46/136.58 (41) UsableRulesProof [EQUIVALENT, 0 ms] 361.46/136.58 (42) QDP 361.46/136.58 (43) QReductionProof [EQUIVALENT, 0 ms] 361.46/136.58 (44) QDP 361.46/136.58 (45) TransformationProof [EQUIVALENT, 0 ms] 361.46/136.58 (46) QDP 361.46/136.58 (47) UsableRulesProof [EQUIVALENT, 0 ms] 361.46/136.58 (48) QDP 361.46/136.58 (49) QReductionProof [EQUIVALENT, 0 ms] 361.46/136.58 (50) QDP 361.46/136.58 (51) Trivial-Transformation [SOUND, 0 ms] 361.46/136.58 (52) QTRS 361.46/136.58 (53) DependencyPairsProof [EQUIVALENT, 0 ms] 361.46/136.58 (54) QDP 361.46/136.58 (55) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (56) AND 361.46/136.58 (57) QDP 361.46/136.58 (58) UsableRulesProof [EQUIVALENT, 0 ms] 361.46/136.58 (59) QDP 361.46/136.58 (60) QDPSizeChangeProof [EQUIVALENT, 0 ms] 361.46/136.58 (61) YES 361.46/136.58 (62) QDP 361.46/136.58 (63) UsableRulesProof [EQUIVALENT, 0 ms] 361.46/136.58 (64) QDP 361.46/136.58 (65) QDPSizeChangeProof [EQUIVALENT, 0 ms] 361.46/136.58 (66) YES 361.46/136.58 (67) QDP 361.46/136.58 (68) TransformationProof [EQUIVALENT, 0 ms] 361.46/136.58 (69) QDP 361.46/136.58 (70) QDPOrderProof [EQUIVALENT, 28 ms] 361.46/136.58 (71) QDP 361.46/136.58 (72) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] 361.46/136.58 (73) QTRS 361.46/136.58 (74) AAECC Innermost [EQUIVALENT, 0 ms] 361.46/136.58 (75) QTRS 361.46/136.58 (76) DependencyPairsProof [EQUIVALENT, 25 ms] 361.46/136.58 (77) QDP 361.46/136.58 (78) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (79) AND 361.46/136.58 (80) QDP 361.46/136.58 (81) UsableRulesProof [EQUIVALENT, 0 ms] 361.46/136.58 (82) QDP 361.46/136.58 (83) QReductionProof [EQUIVALENT, 0 ms] 361.46/136.58 (84) QDP 361.46/136.58 (85) QDPSizeChangeProof [EQUIVALENT, 0 ms] 361.46/136.58 (86) YES 361.46/136.58 (87) QDP 361.46/136.58 (88) UsableRulesProof [EQUIVALENT, 0 ms] 361.46/136.58 (89) QDP 361.46/136.58 (90) QReductionProof [EQUIVALENT, 2 ms] 361.46/136.58 (91) QDP 361.46/136.58 (92) TransformationProof [EQUIVALENT, 0 ms] 361.46/136.58 (93) QDP 361.46/136.58 (94) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (95) QDP 361.46/136.58 (96) TransformationProof [EQUIVALENT, 0 ms] 361.46/136.58 (97) QDP 361.46/136.58 (98) TransformationProof [EQUIVALENT, 0 ms] 361.46/136.58 (99) QDP 361.46/136.58 (100) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (101) QDP 361.46/136.58 (102) TransformationProof [EQUIVALENT, 0 ms] 361.46/136.58 (103) QDP 361.46/136.58 (104) TransformationProof [EQUIVALENT, 0 ms] 361.46/136.58 (105) QDP 361.46/136.58 (106) TransformationProof [EQUIVALENT, 0 ms] 361.46/136.58 (107) QDP 361.46/136.58 (108) TransformationProof [EQUIVALENT, 0 ms] 361.46/136.58 (109) QDP 361.46/136.58 (110) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (111) QDP 361.46/136.58 (112) TransformationProof [EQUIVALENT, 0 ms] 361.46/136.58 (113) QDP 361.46/136.58 (114) QDPOrderProof [EQUIVALENT, 15 ms] 361.46/136.58 (115) QDP 361.46/136.58 (116) QDPOrderProof [EQUIVALENT, 17 ms] 361.46/136.58 (117) QDP 361.46/136.58 (118) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (119) QDP 361.46/136.58 (120) MNOCProof [EQUIVALENT, 2 ms] 361.46/136.58 (121) QDP 361.46/136.58 (122) SplitQDPProof [EQUIVALENT, 0 ms] 361.46/136.58 (123) AND 361.46/136.58 (124) QDP 361.46/136.58 (125) SemLabProof [SOUND, 0 ms] 361.46/136.58 (126) QDP 361.46/136.58 (127) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (128) QDP 361.46/136.58 (129) UsableRulesReductionPairsProof [EQUIVALENT, 6 ms] 361.46/136.58 (130) QDP 361.46/136.58 (131) MRRProof [EQUIVALENT, 9 ms] 361.46/136.58 (132) QDP 361.46/136.58 (133) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (134) QDP 361.46/136.58 (135) MRRProof [EQUIVALENT, 4 ms] 361.46/136.58 (136) QDP 361.46/136.58 (137) MRRProof [EQUIVALENT, 0 ms] 361.46/136.58 (138) QDP 361.46/136.58 (139) MRRProof [EQUIVALENT, 17 ms] 361.46/136.58 (140) QDP 361.46/136.58 (141) PisEmptyProof [SOUND, 0 ms] 361.46/136.58 (142) TRUE 361.46/136.58 (143) QDP 361.46/136.58 (144) SplitQDPProof [EQUIVALENT, 0 ms] 361.46/136.58 (145) AND 361.46/136.58 (146) QDP 361.46/136.58 (147) SemLabProof [SOUND, 0 ms] 361.46/136.58 (148) QDP 361.46/136.58 (149) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (150) QDP 361.46/136.58 (151) MRRProof [EQUIVALENT, 7 ms] 361.46/136.58 (152) QDP 361.46/136.58 (153) DependencyGraphProof [EQUIVALENT, 0 ms] 361.46/136.58 (154) QDP 361.46/136.58 (155) MRRProof [EQUIVALENT, 0 ms] 361.46/136.58 (156) QDP 361.46/136.58 (157) MRRProof [EQUIVALENT, 8 ms] 361.46/136.58 (158) QDP 361.46/136.58 (159) PisEmptyProof [SOUND, 0 ms] 361.46/136.58 (160) TRUE 361.46/136.58 (161) QDP 361.46/136.58 361.46/136.58 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (0) 361.46/136.58 Obligation: 361.46/136.58 Term rewrite system R: 361.46/136.58 The TRS R consists of the following rules: 361.46/136.58 361.46/136.58 a(a(a(x))) -> c 361.46/136.58 b(u(x)) -> b(d(x)) 361.46/136.58 d(a(x)) -> a(d(x)) 361.46/136.58 d(b(x)) -> u(a(b(x))) 361.46/136.58 a(u(x)) -> u(a(x)) 361.46/136.58 361.46/136.58 361.46/136.58 361.46/136.58 Outermost Strategy. 361.46/136.58 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (1) Thiemann-SpecialC-Transformation (EQUIVALENT) 361.46/136.58 We applied the Thiemann-SpecialC transformation to transform the outermost TRS to an innermost TRS. 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (2) 361.46/136.58 Obligation: 361.46/136.58 Q restricted rewrite system: 361.46/136.58 The TRS R consists of the following rules: 361.46/136.58 361.46/136.58 top(go_up(x)) -> top(reduce(x)) 361.46/136.58 reduce(a(x_1)) -> check_a(redex_a(x_1)) 361.46/136.58 reduce(b(x_1)) -> check_b(redex_b(x_1)) 361.46/136.58 reduce(d(x_1)) -> check_d(redex_d(x_1)) 361.46/136.58 redex_a(a(a(x))) -> result_a(c) 361.46/136.58 redex_b(u(x)) -> result_b(b(d(x))) 361.46/136.58 redex_d(a(x)) -> result_d(a(d(x))) 361.46/136.58 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.58 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.58 check_a(result_a(x)) -> go_up(x) 361.46/136.58 check_b(result_b(x)) -> go_up(x) 361.46/136.58 check_d(result_d(x)) -> go_up(x) 361.46/136.58 check_a(redex_a(x_1)) -> in_a_1(reduce(x_1)) 361.46/136.58 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 361.46/136.58 check_d(redex_d(x_1)) -> in_d_1(reduce(x_1)) 361.46/136.58 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 361.46/136.58 in_a_1(go_up(x_1)) -> go_up(a(x_1)) 361.46/136.58 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 361.46/136.58 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 361.46/136.58 in_d_1(go_up(x_1)) -> go_up(d(x_1)) 361.46/136.58 361.46/136.58 The set Q consists of the following terms: 361.46/136.58 361.46/136.58 top(go_up(x0)) 361.46/136.58 reduce(a(x0)) 361.46/136.58 reduce(b(x0)) 361.46/136.58 reduce(d(x0)) 361.46/136.58 redex_a(a(a(x0))) 361.46/136.58 redex_b(u(x0)) 361.46/136.58 redex_d(a(x0)) 361.46/136.58 redex_d(b(x0)) 361.46/136.58 redex_a(u(x0)) 361.46/136.58 check_a(result_a(x0)) 361.46/136.58 check_b(result_b(x0)) 361.46/136.58 check_d(result_d(x0)) 361.46/136.58 check_a(redex_a(x0)) 361.46/136.58 check_b(redex_b(x0)) 361.46/136.58 check_d(redex_d(x0)) 361.46/136.58 reduce(u(x0)) 361.46/136.58 in_a_1(go_up(x0)) 361.46/136.58 in_b_1(go_up(x0)) 361.46/136.58 in_u_1(go_up(x0)) 361.46/136.58 in_d_1(go_up(x0)) 361.46/136.58 361.46/136.58 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (3) DependencyPairsProof (EQUIVALENT) 361.46/136.58 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (4) 361.46/136.58 Obligation: 361.46/136.58 Q DP problem: 361.46/136.58 The TRS P consists of the following rules: 361.46/136.58 361.46/136.58 TOP(go_up(x)) -> TOP(reduce(x)) 361.46/136.58 TOP(go_up(x)) -> REDUCE(x) 361.46/136.58 REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1)) 361.46/136.58 REDUCE(a(x_1)) -> REDEX_A(x_1) 361.46/136.58 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 361.46/136.58 REDUCE(b(x_1)) -> REDEX_B(x_1) 361.46/136.58 REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1)) 361.46/136.58 REDUCE(d(x_1)) -> REDEX_D(x_1) 361.46/136.58 CHECK_A(redex_a(x_1)) -> IN_A_1(reduce(x_1)) 361.46/136.58 CHECK_A(redex_a(x_1)) -> REDUCE(x_1) 361.46/136.58 CHECK_B(redex_b(x_1)) -> IN_B_1(reduce(x_1)) 361.46/136.58 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 361.46/136.58 CHECK_D(redex_d(x_1)) -> IN_D_1(reduce(x_1)) 361.46/136.58 CHECK_D(redex_d(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(u(x_1)) -> IN_U_1(reduce(x_1)) 361.46/136.58 REDUCE(u(x_1)) -> REDUCE(x_1) 361.46/136.58 361.46/136.58 The TRS R consists of the following rules: 361.46/136.58 361.46/136.58 top(go_up(x)) -> top(reduce(x)) 361.46/136.58 reduce(a(x_1)) -> check_a(redex_a(x_1)) 361.46/136.58 reduce(b(x_1)) -> check_b(redex_b(x_1)) 361.46/136.58 reduce(d(x_1)) -> check_d(redex_d(x_1)) 361.46/136.58 redex_a(a(a(x))) -> result_a(c) 361.46/136.58 redex_b(u(x)) -> result_b(b(d(x))) 361.46/136.58 redex_d(a(x)) -> result_d(a(d(x))) 361.46/136.58 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.58 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.58 check_a(result_a(x)) -> go_up(x) 361.46/136.58 check_b(result_b(x)) -> go_up(x) 361.46/136.58 check_d(result_d(x)) -> go_up(x) 361.46/136.58 check_a(redex_a(x_1)) -> in_a_1(reduce(x_1)) 361.46/136.58 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 361.46/136.58 check_d(redex_d(x_1)) -> in_d_1(reduce(x_1)) 361.46/136.58 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 361.46/136.58 in_a_1(go_up(x_1)) -> go_up(a(x_1)) 361.46/136.58 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 361.46/136.58 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 361.46/136.58 in_d_1(go_up(x_1)) -> go_up(d(x_1)) 361.46/136.58 361.46/136.58 The set Q consists of the following terms: 361.46/136.58 361.46/136.58 top(go_up(x0)) 361.46/136.58 reduce(a(x0)) 361.46/136.58 reduce(b(x0)) 361.46/136.58 reduce(d(x0)) 361.46/136.58 redex_a(a(a(x0))) 361.46/136.58 redex_b(u(x0)) 361.46/136.58 redex_d(a(x0)) 361.46/136.58 redex_d(b(x0)) 361.46/136.58 redex_a(u(x0)) 361.46/136.58 check_a(result_a(x0)) 361.46/136.58 check_b(result_b(x0)) 361.46/136.58 check_d(result_d(x0)) 361.46/136.58 check_a(redex_a(x0)) 361.46/136.58 check_b(redex_b(x0)) 361.46/136.58 check_d(redex_d(x0)) 361.46/136.58 reduce(u(x0)) 361.46/136.58 in_a_1(go_up(x0)) 361.46/136.58 in_b_1(go_up(x0)) 361.46/136.58 in_u_1(go_up(x0)) 361.46/136.58 in_d_1(go_up(x0)) 361.46/136.58 361.46/136.58 We have to consider all minimal (P,Q,R)-chains. 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (5) DependencyGraphProof (EQUIVALENT) 361.46/136.58 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes. 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (6) 361.46/136.58 Complex Obligation (AND) 361.46/136.58 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (7) 361.46/136.58 Obligation: 361.46/136.58 Q DP problem: 361.46/136.58 The TRS P consists of the following rules: 361.46/136.58 361.46/136.58 CHECK_A(redex_a(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1)) 361.46/136.58 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 361.46/136.58 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1)) 361.46/136.58 CHECK_D(redex_d(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(u(x_1)) -> REDUCE(x_1) 361.46/136.58 361.46/136.58 The TRS R consists of the following rules: 361.46/136.58 361.46/136.58 top(go_up(x)) -> top(reduce(x)) 361.46/136.58 reduce(a(x_1)) -> check_a(redex_a(x_1)) 361.46/136.58 reduce(b(x_1)) -> check_b(redex_b(x_1)) 361.46/136.58 reduce(d(x_1)) -> check_d(redex_d(x_1)) 361.46/136.58 redex_a(a(a(x))) -> result_a(c) 361.46/136.58 redex_b(u(x)) -> result_b(b(d(x))) 361.46/136.58 redex_d(a(x)) -> result_d(a(d(x))) 361.46/136.58 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.58 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.58 check_a(result_a(x)) -> go_up(x) 361.46/136.58 check_b(result_b(x)) -> go_up(x) 361.46/136.58 check_d(result_d(x)) -> go_up(x) 361.46/136.58 check_a(redex_a(x_1)) -> in_a_1(reduce(x_1)) 361.46/136.58 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 361.46/136.58 check_d(redex_d(x_1)) -> in_d_1(reduce(x_1)) 361.46/136.58 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 361.46/136.58 in_a_1(go_up(x_1)) -> go_up(a(x_1)) 361.46/136.58 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 361.46/136.58 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 361.46/136.58 in_d_1(go_up(x_1)) -> go_up(d(x_1)) 361.46/136.58 361.46/136.58 The set Q consists of the following terms: 361.46/136.58 361.46/136.58 top(go_up(x0)) 361.46/136.58 reduce(a(x0)) 361.46/136.58 reduce(b(x0)) 361.46/136.58 reduce(d(x0)) 361.46/136.58 redex_a(a(a(x0))) 361.46/136.58 redex_b(u(x0)) 361.46/136.58 redex_d(a(x0)) 361.46/136.58 redex_d(b(x0)) 361.46/136.58 redex_a(u(x0)) 361.46/136.58 check_a(result_a(x0)) 361.46/136.58 check_b(result_b(x0)) 361.46/136.58 check_d(result_d(x0)) 361.46/136.58 check_a(redex_a(x0)) 361.46/136.58 check_b(redex_b(x0)) 361.46/136.58 check_d(redex_d(x0)) 361.46/136.58 reduce(u(x0)) 361.46/136.58 in_a_1(go_up(x0)) 361.46/136.58 in_b_1(go_up(x0)) 361.46/136.58 in_u_1(go_up(x0)) 361.46/136.58 in_d_1(go_up(x0)) 361.46/136.58 361.46/136.58 We have to consider all minimal (P,Q,R)-chains. 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (8) UsableRulesProof (EQUIVALENT) 361.46/136.58 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. 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (9) 361.46/136.58 Obligation: 361.46/136.58 Q DP problem: 361.46/136.58 The TRS P consists of the following rules: 361.46/136.58 361.46/136.58 CHECK_A(redex_a(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1)) 361.46/136.58 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 361.46/136.58 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1)) 361.46/136.58 CHECK_D(redex_d(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(u(x_1)) -> REDUCE(x_1) 361.46/136.58 361.46/136.58 The TRS R consists of the following rules: 361.46/136.58 361.46/136.58 redex_d(a(x)) -> result_d(a(d(x))) 361.46/136.58 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.58 redex_b(u(x)) -> result_b(b(d(x))) 361.46/136.58 redex_a(a(a(x))) -> result_a(c) 361.46/136.58 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.58 361.46/136.58 The set Q consists of the following terms: 361.46/136.58 361.46/136.58 top(go_up(x0)) 361.46/136.58 reduce(a(x0)) 361.46/136.58 reduce(b(x0)) 361.46/136.58 reduce(d(x0)) 361.46/136.58 redex_a(a(a(x0))) 361.46/136.58 redex_b(u(x0)) 361.46/136.58 redex_d(a(x0)) 361.46/136.58 redex_d(b(x0)) 361.46/136.58 redex_a(u(x0)) 361.46/136.58 check_a(result_a(x0)) 361.46/136.58 check_b(result_b(x0)) 361.46/136.58 check_d(result_d(x0)) 361.46/136.58 check_a(redex_a(x0)) 361.46/136.58 check_b(redex_b(x0)) 361.46/136.58 check_d(redex_d(x0)) 361.46/136.58 reduce(u(x0)) 361.46/136.58 in_a_1(go_up(x0)) 361.46/136.58 in_b_1(go_up(x0)) 361.46/136.58 in_u_1(go_up(x0)) 361.46/136.58 in_d_1(go_up(x0)) 361.46/136.58 361.46/136.58 We have to consider all minimal (P,Q,R)-chains. 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (10) QReductionProof (EQUIVALENT) 361.46/136.58 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 361.46/136.58 361.46/136.58 top(go_up(x0)) 361.46/136.58 reduce(a(x0)) 361.46/136.58 reduce(b(x0)) 361.46/136.58 reduce(d(x0)) 361.46/136.58 check_a(result_a(x0)) 361.46/136.58 check_b(result_b(x0)) 361.46/136.58 check_d(result_d(x0)) 361.46/136.58 check_a(redex_a(x0)) 361.46/136.58 check_b(redex_b(x0)) 361.46/136.58 check_d(redex_d(x0)) 361.46/136.58 reduce(u(x0)) 361.46/136.58 in_a_1(go_up(x0)) 361.46/136.58 in_b_1(go_up(x0)) 361.46/136.58 in_u_1(go_up(x0)) 361.46/136.58 in_d_1(go_up(x0)) 361.46/136.58 361.46/136.58 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (11) 361.46/136.58 Obligation: 361.46/136.58 Q DP problem: 361.46/136.58 The TRS P consists of the following rules: 361.46/136.58 361.46/136.58 CHECK_A(redex_a(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1)) 361.46/136.58 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 361.46/136.58 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1)) 361.46/136.58 CHECK_D(redex_d(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(u(x_1)) -> REDUCE(x_1) 361.46/136.58 361.46/136.58 The TRS R consists of the following rules: 361.46/136.58 361.46/136.58 redex_d(a(x)) -> result_d(a(d(x))) 361.46/136.58 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.58 redex_b(u(x)) -> result_b(b(d(x))) 361.46/136.58 redex_a(a(a(x))) -> result_a(c) 361.46/136.58 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.58 361.46/136.58 The set Q consists of the following terms: 361.46/136.58 361.46/136.58 redex_a(a(a(x0))) 361.46/136.58 redex_b(u(x0)) 361.46/136.58 redex_d(a(x0)) 361.46/136.58 redex_d(b(x0)) 361.46/136.58 redex_a(u(x0)) 361.46/136.58 361.46/136.58 We have to consider all minimal (P,Q,R)-chains. 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (12) MRRProof (EQUIVALENT) 361.46/136.58 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. 361.46/136.58 361.46/136.58 Strictly oriented dependency pairs: 361.46/136.58 361.46/136.58 REDUCE(u(x_1)) -> REDUCE(x_1) 361.46/136.58 361.46/136.58 Strictly oriented rules of the TRS R: 361.46/136.58 361.46/136.58 redex_d(a(x)) -> result_d(a(d(x))) 361.46/136.58 redex_b(u(x)) -> result_b(b(d(x))) 361.46/136.58 361.46/136.58 Used ordering: Polynomial interpretation [POLO]: 361.46/136.58 361.46/136.58 POL(CHECK_A(x_1)) = 1 + x_1 361.46/136.58 POL(CHECK_B(x_1)) = 1 + x_1 361.46/136.58 POL(CHECK_D(x_1)) = x_1 361.46/136.58 POL(REDUCE(x_1)) = 1 + x_1 361.46/136.58 POL(a(x_1)) = x_1 361.46/136.58 POL(b(x_1)) = x_1 361.46/136.58 POL(c) = 0 361.46/136.58 POL(d(x_1)) = 2*x_1 361.46/136.58 POL(redex_a(x_1)) = x_1 361.46/136.58 POL(redex_b(x_1)) = x_1 361.46/136.58 POL(redex_d(x_1)) = 1 + 2*x_1 361.46/136.58 POL(result_a(x_1)) = x_1 361.46/136.58 POL(result_b(x_1)) = x_1 361.46/136.58 POL(result_d(x_1)) = x_1 361.46/136.58 POL(u(x_1)) = 1 + 2*x_1 361.46/136.58 361.46/136.58 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (13) 361.46/136.58 Obligation: 361.46/136.58 Q DP problem: 361.46/136.58 The TRS P consists of the following rules: 361.46/136.58 361.46/136.58 CHECK_A(redex_a(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1)) 361.46/136.58 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 361.46/136.58 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1)) 361.46/136.58 CHECK_D(redex_d(x_1)) -> REDUCE(x_1) 361.46/136.58 361.46/136.58 The TRS R consists of the following rules: 361.46/136.58 361.46/136.58 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.58 redex_a(a(a(x))) -> result_a(c) 361.46/136.58 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.58 361.46/136.58 The set Q consists of the following terms: 361.46/136.58 361.46/136.58 redex_a(a(a(x0))) 361.46/136.58 redex_b(u(x0)) 361.46/136.58 redex_d(a(x0)) 361.46/136.58 redex_d(b(x0)) 361.46/136.58 redex_a(u(x0)) 361.46/136.58 361.46/136.58 We have to consider all minimal (P,Q,R)-chains. 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (14) DependencyGraphProof (EQUIVALENT) 361.46/136.58 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (15) 361.46/136.58 Complex Obligation (AND) 361.46/136.58 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (16) 361.46/136.58 Obligation: 361.46/136.58 Q DP problem: 361.46/136.58 The TRS P consists of the following rules: 361.46/136.58 361.46/136.58 CHECK_D(redex_d(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1)) 361.46/136.58 361.46/136.58 The TRS R consists of the following rules: 361.46/136.58 361.46/136.58 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.58 redex_a(a(a(x))) -> result_a(c) 361.46/136.58 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.58 361.46/136.58 The set Q consists of the following terms: 361.46/136.58 361.46/136.58 redex_a(a(a(x0))) 361.46/136.58 redex_b(u(x0)) 361.46/136.58 redex_d(a(x0)) 361.46/136.58 redex_d(b(x0)) 361.46/136.58 redex_a(u(x0)) 361.46/136.58 361.46/136.58 We have to consider all minimal (P,Q,R)-chains. 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (17) UsableRulesProof (EQUIVALENT) 361.46/136.58 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. 361.46/136.58 ---------------------------------------- 361.46/136.58 361.46/136.58 (18) 361.46/136.58 Obligation: 361.46/136.58 Q DP problem: 361.46/136.58 The TRS P consists of the following rules: 361.46/136.58 361.46/136.58 CHECK_D(redex_d(x_1)) -> REDUCE(x_1) 361.46/136.58 REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1)) 361.46/136.58 361.46/136.58 The TRS R consists of the following rules: 361.46/136.58 361.46/136.58 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.58 361.46/136.58 The set Q consists of the following terms: 361.46/136.58 361.46/136.58 redex_a(a(a(x0))) 361.46/136.58 redex_b(u(x0)) 361.46/136.58 redex_d(a(x0)) 361.46/136.58 redex_d(b(x0)) 361.46/136.58 redex_a(u(x0)) 361.46/136.58 361.46/136.58 We have to consider all minimal (P,Q,R)-chains. 361.46/136.58 ---------------------------------------- 361.46/136.59 361.46/136.59 (19) QReductionProof (EQUIVALENT) 361.46/136.59 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 361.46/136.59 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_b(u(x0)) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (20) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 CHECK_D(redex_d(x_1)) -> REDUCE(x_1) 361.46/136.59 REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1)) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 redex_d(a(x0)) 361.46/136.59 redex_d(b(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (21) UsableRulesReductionPairsProof (EQUIVALENT) 361.46/136.59 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. 361.46/136.59 361.46/136.59 The following dependency pairs can be deleted: 361.46/136.59 361.46/136.59 REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1)) 361.46/136.59 No rules are removed from R. 361.46/136.59 361.46/136.59 Used ordering: POLO with Polynomial interpretation [POLO]: 361.46/136.59 361.46/136.59 POL(CHECK_D(x_1)) = x_1 361.46/136.59 POL(REDUCE(x_1)) = 2*x_1 361.46/136.59 POL(a(x_1)) = 2*x_1 361.46/136.59 POL(b(x_1)) = 2*x_1 361.46/136.59 POL(d(x_1)) = 2*x_1 361.46/136.59 POL(redex_d(x_1)) = 2*x_1 361.46/136.59 POL(result_d(x_1)) = x_1 361.46/136.59 POL(u(x_1)) = x_1 361.46/136.59 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (22) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 CHECK_D(redex_d(x_1)) -> REDUCE(x_1) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 redex_d(a(x0)) 361.46/136.59 redex_d(b(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (23) DependencyGraphProof (EQUIVALENT) 361.46/136.59 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (24) 361.46/136.59 TRUE 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (25) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1)) 361.46/136.59 CHECK_A(redex_a(x_1)) -> REDUCE(x_1) 361.46/136.59 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 361.46/136.59 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.59 redex_a(a(a(x))) -> result_a(c) 361.46/136.59 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_b(u(x0)) 361.46/136.59 redex_d(a(x0)) 361.46/136.59 redex_d(b(x0)) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (26) UsableRulesProof (EQUIVALENT) 361.46/136.59 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. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (27) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1)) 361.46/136.59 CHECK_A(redex_a(x_1)) -> REDUCE(x_1) 361.46/136.59 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 361.46/136.59 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 redex_a(a(a(x))) -> result_a(c) 361.46/136.59 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_b(u(x0)) 361.46/136.59 redex_d(a(x0)) 361.46/136.59 redex_d(b(x0)) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (28) QReductionProof (EQUIVALENT) 361.46/136.59 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 361.46/136.59 361.46/136.59 redex_d(a(x0)) 361.46/136.59 redex_d(b(x0)) 361.46/136.59 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (29) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1)) 361.46/136.59 CHECK_A(redex_a(x_1)) -> REDUCE(x_1) 361.46/136.59 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 361.46/136.59 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 redex_a(a(a(x))) -> result_a(c) 361.46/136.59 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_b(u(x0)) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (30) UsableRulesReductionPairsProof (EQUIVALENT) 361.46/136.59 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. 361.46/136.59 361.46/136.59 The following dependency pairs can be deleted: 361.46/136.59 361.46/136.59 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 361.46/136.59 No rules are removed from R. 361.46/136.59 361.46/136.59 Used ordering: POLO with Polynomial interpretation [POLO]: 361.46/136.59 361.46/136.59 POL(CHECK_A(x_1)) = x_1 361.46/136.59 POL(CHECK_B(x_1)) = 2*x_1 361.46/136.59 POL(REDUCE(x_1)) = 2*x_1 361.46/136.59 POL(a(x_1)) = 2*x_1 361.46/136.59 POL(b(x_1)) = 2*x_1 361.46/136.59 POL(c) = 0 361.46/136.59 POL(redex_a(x_1)) = 2*x_1 361.46/136.59 POL(redex_b(x_1)) = x_1 361.46/136.59 POL(result_a(x_1)) = x_1 361.46/136.59 POL(u(x_1)) = 2*x_1 361.46/136.59 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (31) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1)) 361.46/136.59 CHECK_A(redex_a(x_1)) -> REDUCE(x_1) 361.46/136.59 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 redex_a(a(a(x))) -> result_a(c) 361.46/136.59 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_b(u(x0)) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (32) DependencyGraphProof (EQUIVALENT) 361.46/136.59 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (33) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 CHECK_A(redex_a(x_1)) -> REDUCE(x_1) 361.46/136.59 REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1)) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 redex_a(a(a(x))) -> result_a(c) 361.46/136.59 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_b(u(x0)) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (34) QReductionProof (EQUIVALENT) 361.46/136.59 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 361.46/136.59 361.46/136.59 redex_b(u(x0)) 361.46/136.59 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (35) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 CHECK_A(redex_a(x_1)) -> REDUCE(x_1) 361.46/136.59 REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1)) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 redex_a(a(a(x))) -> result_a(c) 361.46/136.59 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (36) MRRProof (EQUIVALENT) 361.46/136.59 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. 361.46/136.59 361.46/136.59 Strictly oriented dependency pairs: 361.46/136.59 361.46/136.59 CHECK_A(redex_a(x_1)) -> REDUCE(x_1) 361.46/136.59 REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1)) 361.46/136.59 361.46/136.59 Strictly oriented rules of the TRS R: 361.46/136.59 361.46/136.59 redex_a(a(a(x))) -> result_a(c) 361.46/136.59 361.46/136.59 Used ordering: Polynomial interpretation [POLO]: 361.46/136.59 361.46/136.59 POL(CHECK_A(x_1)) = 1 + x_1 361.46/136.59 POL(REDUCE(x_1)) = 2*x_1 361.46/136.59 POL(a(x_1)) = 2 + 2*x_1 361.46/136.59 POL(c) = 0 361.46/136.59 POL(redex_a(x_1)) = 2 + 2*x_1 361.46/136.59 POL(result_a(x_1)) = x_1 361.46/136.59 POL(u(x_1)) = x_1 361.46/136.59 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (37) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 P is empty. 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (38) PisEmptyProof (EQUIVALENT) 361.46/136.59 The TRS P is empty. Hence, there is no (P,Q,R) chain. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (39) 361.46/136.59 YES 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (40) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 TOP(go_up(x)) -> TOP(reduce(x)) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 top(go_up(x)) -> top(reduce(x)) 361.46/136.59 reduce(a(x_1)) -> check_a(redex_a(x_1)) 361.46/136.59 reduce(b(x_1)) -> check_b(redex_b(x_1)) 361.46/136.59 reduce(d(x_1)) -> check_d(redex_d(x_1)) 361.46/136.59 redex_a(a(a(x))) -> result_a(c) 361.46/136.59 redex_b(u(x)) -> result_b(b(d(x))) 361.46/136.59 redex_d(a(x)) -> result_d(a(d(x))) 361.46/136.59 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.59 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.59 check_a(result_a(x)) -> go_up(x) 361.46/136.59 check_b(result_b(x)) -> go_up(x) 361.46/136.59 check_d(result_d(x)) -> go_up(x) 361.46/136.59 check_a(redex_a(x_1)) -> in_a_1(reduce(x_1)) 361.46/136.59 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 361.46/136.59 check_d(redex_d(x_1)) -> in_d_1(reduce(x_1)) 361.46/136.59 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 361.46/136.59 in_a_1(go_up(x_1)) -> go_up(a(x_1)) 361.46/136.59 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 361.46/136.59 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 361.46/136.59 in_d_1(go_up(x_1)) -> go_up(d(x_1)) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 top(go_up(x0)) 361.46/136.59 reduce(a(x0)) 361.46/136.59 reduce(b(x0)) 361.46/136.59 reduce(d(x0)) 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_b(u(x0)) 361.46/136.59 redex_d(a(x0)) 361.46/136.59 redex_d(b(x0)) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 check_a(result_a(x0)) 361.46/136.59 check_b(result_b(x0)) 361.46/136.59 check_d(result_d(x0)) 361.46/136.59 check_a(redex_a(x0)) 361.46/136.59 check_b(redex_b(x0)) 361.46/136.59 check_d(redex_d(x0)) 361.46/136.59 reduce(u(x0)) 361.46/136.59 in_a_1(go_up(x0)) 361.46/136.59 in_b_1(go_up(x0)) 361.46/136.59 in_u_1(go_up(x0)) 361.46/136.59 in_d_1(go_up(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (41) UsableRulesProof (EQUIVALENT) 361.46/136.59 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. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (42) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 TOP(go_up(x)) -> TOP(reduce(x)) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 reduce(a(x_1)) -> check_a(redex_a(x_1)) 361.46/136.59 reduce(b(x_1)) -> check_b(redex_b(x_1)) 361.46/136.59 reduce(d(x_1)) -> check_d(redex_d(x_1)) 361.46/136.59 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 361.46/136.59 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 361.46/136.59 redex_d(a(x)) -> result_d(a(d(x))) 361.46/136.59 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.59 check_d(result_d(x)) -> go_up(x) 361.46/136.59 check_d(redex_d(x_1)) -> in_d_1(reduce(x_1)) 361.46/136.59 in_d_1(go_up(x_1)) -> go_up(d(x_1)) 361.46/136.59 redex_b(u(x)) -> result_b(b(d(x))) 361.46/136.59 check_b(result_b(x)) -> go_up(x) 361.46/136.59 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 361.46/136.59 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 361.46/136.59 redex_a(a(a(x))) -> result_a(c) 361.46/136.59 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.59 check_a(result_a(x)) -> go_up(x) 361.46/136.59 check_a(redex_a(x_1)) -> in_a_1(reduce(x_1)) 361.46/136.59 in_a_1(go_up(x_1)) -> go_up(a(x_1)) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 top(go_up(x0)) 361.46/136.59 reduce(a(x0)) 361.46/136.59 reduce(b(x0)) 361.46/136.59 reduce(d(x0)) 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_b(u(x0)) 361.46/136.59 redex_d(a(x0)) 361.46/136.59 redex_d(b(x0)) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 check_a(result_a(x0)) 361.46/136.59 check_b(result_b(x0)) 361.46/136.59 check_d(result_d(x0)) 361.46/136.59 check_a(redex_a(x0)) 361.46/136.59 check_b(redex_b(x0)) 361.46/136.59 check_d(redex_d(x0)) 361.46/136.59 reduce(u(x0)) 361.46/136.59 in_a_1(go_up(x0)) 361.46/136.59 in_b_1(go_up(x0)) 361.46/136.59 in_u_1(go_up(x0)) 361.46/136.59 in_d_1(go_up(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (43) QReductionProof (EQUIVALENT) 361.46/136.59 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 361.46/136.59 361.46/136.59 top(go_up(x0)) 361.46/136.59 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (44) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 TOP(go_up(x)) -> TOP(reduce(x)) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 reduce(a(x_1)) -> check_a(redex_a(x_1)) 361.46/136.59 reduce(b(x_1)) -> check_b(redex_b(x_1)) 361.46/136.59 reduce(d(x_1)) -> check_d(redex_d(x_1)) 361.46/136.59 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 361.46/136.59 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 361.46/136.59 redex_d(a(x)) -> result_d(a(d(x))) 361.46/136.59 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.59 check_d(result_d(x)) -> go_up(x) 361.46/136.59 check_d(redex_d(x_1)) -> in_d_1(reduce(x_1)) 361.46/136.59 in_d_1(go_up(x_1)) -> go_up(d(x_1)) 361.46/136.59 redex_b(u(x)) -> result_b(b(d(x))) 361.46/136.59 check_b(result_b(x)) -> go_up(x) 361.46/136.59 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 361.46/136.59 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 361.46/136.59 redex_a(a(a(x))) -> result_a(c) 361.46/136.59 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.59 check_a(result_a(x)) -> go_up(x) 361.46/136.59 check_a(redex_a(x_1)) -> in_a_1(reduce(x_1)) 361.46/136.59 in_a_1(go_up(x_1)) -> go_up(a(x_1)) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 reduce(a(x0)) 361.46/136.59 reduce(b(x0)) 361.46/136.59 reduce(d(x0)) 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_b(u(x0)) 361.46/136.59 redex_d(a(x0)) 361.46/136.59 redex_d(b(x0)) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 check_a(result_a(x0)) 361.46/136.59 check_b(result_b(x0)) 361.46/136.59 check_d(result_d(x0)) 361.46/136.59 check_a(redex_a(x0)) 361.46/136.59 check_b(redex_b(x0)) 361.46/136.59 check_d(redex_d(x0)) 361.46/136.59 reduce(u(x0)) 361.46/136.59 in_a_1(go_up(x0)) 361.46/136.59 in_b_1(go_up(x0)) 361.46/136.59 in_u_1(go_up(x0)) 361.46/136.59 in_d_1(go_up(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (45) TransformationProof (EQUIVALENT) 361.46/136.59 By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]: 361.46/136.59 361.46/136.59 (TOP(go_up(a(x0))) -> TOP(check_a(redex_a(x0))),TOP(go_up(a(x0))) -> TOP(check_a(redex_a(x0)))) 361.46/136.59 (TOP(go_up(b(x0))) -> TOP(check_b(redex_b(x0))),TOP(go_up(b(x0))) -> TOP(check_b(redex_b(x0)))) 361.46/136.59 (TOP(go_up(d(x0))) -> TOP(check_d(redex_d(x0))),TOP(go_up(d(x0))) -> TOP(check_d(redex_d(x0)))) 361.46/136.59 (TOP(go_up(u(x0))) -> TOP(in_u_1(reduce(x0))),TOP(go_up(u(x0))) -> TOP(in_u_1(reduce(x0)))) 361.46/136.59 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (46) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 TOP(go_up(a(x0))) -> TOP(check_a(redex_a(x0))) 361.46/136.59 TOP(go_up(b(x0))) -> TOP(check_b(redex_b(x0))) 361.46/136.59 TOP(go_up(d(x0))) -> TOP(check_d(redex_d(x0))) 361.46/136.59 TOP(go_up(u(x0))) -> TOP(in_u_1(reduce(x0))) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 reduce(a(x_1)) -> check_a(redex_a(x_1)) 361.46/136.59 reduce(b(x_1)) -> check_b(redex_b(x_1)) 361.46/136.59 reduce(d(x_1)) -> check_d(redex_d(x_1)) 361.46/136.59 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 361.46/136.59 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 361.46/136.59 redex_d(a(x)) -> result_d(a(d(x))) 361.46/136.59 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.59 check_d(result_d(x)) -> go_up(x) 361.46/136.59 check_d(redex_d(x_1)) -> in_d_1(reduce(x_1)) 361.46/136.59 in_d_1(go_up(x_1)) -> go_up(d(x_1)) 361.46/136.59 redex_b(u(x)) -> result_b(b(d(x))) 361.46/136.59 check_b(result_b(x)) -> go_up(x) 361.46/136.59 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 361.46/136.59 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 361.46/136.59 redex_a(a(a(x))) -> result_a(c) 361.46/136.59 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.59 check_a(result_a(x)) -> go_up(x) 361.46/136.59 check_a(redex_a(x_1)) -> in_a_1(reduce(x_1)) 361.46/136.59 in_a_1(go_up(x_1)) -> go_up(a(x_1)) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 reduce(a(x0)) 361.46/136.59 reduce(b(x0)) 361.46/136.59 reduce(d(x0)) 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_b(u(x0)) 361.46/136.59 redex_d(a(x0)) 361.46/136.59 redex_d(b(x0)) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 check_a(result_a(x0)) 361.46/136.59 check_b(result_b(x0)) 361.46/136.59 check_d(result_d(x0)) 361.46/136.59 check_a(redex_a(x0)) 361.46/136.59 check_b(redex_b(x0)) 361.46/136.59 check_d(redex_d(x0)) 361.46/136.59 reduce(u(x0)) 361.46/136.59 in_a_1(go_up(x0)) 361.46/136.59 in_b_1(go_up(x0)) 361.46/136.59 in_u_1(go_up(x0)) 361.46/136.59 in_d_1(go_up(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (47) UsableRulesProof (EQUIVALENT) 361.46/136.59 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. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (48) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 TOP(go_up(x)) -> TOP(reduce(x)) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 reduce(a(x_1)) -> check_a(redex_a(x_1)) 361.46/136.59 reduce(b(x_1)) -> check_b(redex_b(x_1)) 361.46/136.59 reduce(d(x_1)) -> check_d(redex_d(x_1)) 361.46/136.59 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 361.46/136.59 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 361.46/136.59 redex_d(a(x)) -> result_d(a(d(x))) 361.46/136.59 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.59 check_d(result_d(x)) -> go_up(x) 361.46/136.59 check_d(redex_d(x_1)) -> in_d_1(reduce(x_1)) 361.46/136.59 in_d_1(go_up(x_1)) -> go_up(d(x_1)) 361.46/136.59 redex_b(u(x)) -> result_b(b(d(x))) 361.46/136.59 check_b(result_b(x)) -> go_up(x) 361.46/136.59 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 361.46/136.59 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 361.46/136.59 redex_a(a(a(x))) -> result_a(c) 361.46/136.59 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.59 check_a(result_a(x)) -> go_up(x) 361.46/136.59 check_a(redex_a(x_1)) -> in_a_1(reduce(x_1)) 361.46/136.59 in_a_1(go_up(x_1)) -> go_up(a(x_1)) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 top(go_up(x0)) 361.46/136.59 reduce(a(x0)) 361.46/136.59 reduce(b(x0)) 361.46/136.59 reduce(d(x0)) 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_b(u(x0)) 361.46/136.59 redex_d(a(x0)) 361.46/136.59 redex_d(b(x0)) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 check_a(result_a(x0)) 361.46/136.59 check_b(result_b(x0)) 361.46/136.59 check_d(result_d(x0)) 361.46/136.59 check_a(redex_a(x0)) 361.46/136.59 check_b(redex_b(x0)) 361.46/136.59 check_d(redex_d(x0)) 361.46/136.59 reduce(u(x0)) 361.46/136.59 in_a_1(go_up(x0)) 361.46/136.59 in_b_1(go_up(x0)) 361.46/136.59 in_u_1(go_up(x0)) 361.46/136.59 in_d_1(go_up(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (49) QReductionProof (EQUIVALENT) 361.46/136.59 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 361.46/136.59 361.46/136.59 top(go_up(x0)) 361.46/136.59 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (50) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 TOP(go_up(x)) -> TOP(reduce(x)) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 reduce(a(x_1)) -> check_a(redex_a(x_1)) 361.46/136.59 reduce(b(x_1)) -> check_b(redex_b(x_1)) 361.46/136.59 reduce(d(x_1)) -> check_d(redex_d(x_1)) 361.46/136.59 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 361.46/136.59 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 361.46/136.59 redex_d(a(x)) -> result_d(a(d(x))) 361.46/136.59 redex_d(b(x)) -> result_d(u(a(b(x)))) 361.46/136.59 check_d(result_d(x)) -> go_up(x) 361.46/136.59 check_d(redex_d(x_1)) -> in_d_1(reduce(x_1)) 361.46/136.59 in_d_1(go_up(x_1)) -> go_up(d(x_1)) 361.46/136.59 redex_b(u(x)) -> result_b(b(d(x))) 361.46/136.59 check_b(result_b(x)) -> go_up(x) 361.46/136.59 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 361.46/136.59 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 361.46/136.59 redex_a(a(a(x))) -> result_a(c) 361.46/136.59 redex_a(u(x)) -> result_a(u(a(x))) 361.46/136.59 check_a(result_a(x)) -> go_up(x) 361.46/136.59 check_a(redex_a(x_1)) -> in_a_1(reduce(x_1)) 361.46/136.59 in_a_1(go_up(x_1)) -> go_up(a(x_1)) 361.46/136.59 361.46/136.59 The set Q consists of the following terms: 361.46/136.59 361.46/136.59 reduce(a(x0)) 361.46/136.59 reduce(b(x0)) 361.46/136.59 reduce(d(x0)) 361.46/136.59 redex_a(a(a(x0))) 361.46/136.59 redex_b(u(x0)) 361.46/136.59 redex_d(a(x0)) 361.46/136.59 redex_d(b(x0)) 361.46/136.59 redex_a(u(x0)) 361.46/136.59 check_a(result_a(x0)) 361.46/136.59 check_b(result_b(x0)) 361.46/136.59 check_d(result_d(x0)) 361.46/136.59 check_a(redex_a(x0)) 361.46/136.59 check_b(redex_b(x0)) 361.46/136.59 check_d(redex_d(x0)) 361.46/136.59 reduce(u(x0)) 361.46/136.59 in_a_1(go_up(x0)) 361.46/136.59 in_b_1(go_up(x0)) 361.46/136.59 in_u_1(go_up(x0)) 361.46/136.59 in_d_1(go_up(x0)) 361.46/136.59 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (51) Trivial-Transformation (SOUND) 361.46/136.59 We applied the Trivial transformation to transform the outermost TRS to a standard TRS. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (52) 361.46/136.59 Obligation: 361.46/136.59 Q restricted rewrite system: 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 a(a(a(x))) -> c 361.46/136.59 b(u(x)) -> b(d(x)) 361.46/136.59 d(a(x)) -> a(d(x)) 361.46/136.59 d(b(x)) -> u(a(b(x))) 361.46/136.59 a(u(x)) -> u(a(x)) 361.46/136.59 361.46/136.59 Q is empty. 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (53) DependencyPairsProof (EQUIVALENT) 361.46/136.59 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (54) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 B(u(x)) -> B(d(x)) 361.46/136.59 B(u(x)) -> D(x) 361.46/136.59 D(a(x)) -> A(d(x)) 361.46/136.59 D(a(x)) -> D(x) 361.46/136.59 D(b(x)) -> A(b(x)) 361.46/136.59 A(u(x)) -> A(x) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 a(a(a(x))) -> c 361.46/136.59 b(u(x)) -> b(d(x)) 361.46/136.59 d(a(x)) -> a(d(x)) 361.46/136.59 d(b(x)) -> u(a(b(x))) 361.46/136.59 a(u(x)) -> u(a(x)) 361.46/136.59 361.46/136.59 Q is empty. 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (55) DependencyGraphProof (EQUIVALENT) 361.46/136.59 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 3 less nodes. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (56) 361.46/136.59 Complex Obligation (AND) 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (57) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 A(u(x)) -> A(x) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 a(a(a(x))) -> c 361.46/136.59 b(u(x)) -> b(d(x)) 361.46/136.59 d(a(x)) -> a(d(x)) 361.46/136.59 d(b(x)) -> u(a(b(x))) 361.46/136.59 a(u(x)) -> u(a(x)) 361.46/136.59 361.46/136.59 Q is empty. 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (58) UsableRulesProof (EQUIVALENT) 361.46/136.59 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. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (59) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 A(u(x)) -> A(x) 361.46/136.59 361.46/136.59 R is empty. 361.46/136.59 Q is empty. 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (60) QDPSizeChangeProof (EQUIVALENT) 361.46/136.59 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. 361.46/136.59 361.46/136.59 From the DPs we obtained the following set of size-change graphs: 361.46/136.59 *A(u(x)) -> A(x) 361.46/136.59 The graph contains the following edges 1 > 1 361.46/136.59 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (61) 361.46/136.59 YES 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (62) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 D(a(x)) -> D(x) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 a(a(a(x))) -> c 361.46/136.59 b(u(x)) -> b(d(x)) 361.46/136.59 d(a(x)) -> a(d(x)) 361.46/136.59 d(b(x)) -> u(a(b(x))) 361.46/136.59 a(u(x)) -> u(a(x)) 361.46/136.59 361.46/136.59 Q is empty. 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (63) UsableRulesProof (EQUIVALENT) 361.46/136.59 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. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (64) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 D(a(x)) -> D(x) 361.46/136.59 361.46/136.59 R is empty. 361.46/136.59 Q is empty. 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (65) QDPSizeChangeProof (EQUIVALENT) 361.46/136.59 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. 361.46/136.59 361.46/136.59 From the DPs we obtained the following set of size-change graphs: 361.46/136.59 *D(a(x)) -> D(x) 361.46/136.59 The graph contains the following edges 1 > 1 361.46/136.59 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (66) 361.46/136.59 YES 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (67) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 B(u(x)) -> B(d(x)) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 a(a(a(x))) -> c 361.46/136.59 b(u(x)) -> b(d(x)) 361.46/136.59 d(a(x)) -> a(d(x)) 361.46/136.59 d(b(x)) -> u(a(b(x))) 361.46/136.59 a(u(x)) -> u(a(x)) 361.46/136.59 361.46/136.59 Q is empty. 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (68) TransformationProof (EQUIVALENT) 361.46/136.59 By narrowing [LPAR04] the rule B(u(x)) -> B(d(x)) at position [0] we obtained the following new rules [LPAR04]: 361.46/136.59 361.46/136.59 (B(u(a(x0))) -> B(a(d(x0))),B(u(a(x0))) -> B(a(d(x0)))) 361.46/136.59 (B(u(b(x0))) -> B(u(a(b(x0)))),B(u(b(x0))) -> B(u(a(b(x0))))) 361.46/136.59 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (69) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 B(u(a(x0))) -> B(a(d(x0))) 361.46/136.59 B(u(b(x0))) -> B(u(a(b(x0)))) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 a(a(a(x))) -> c 361.46/136.59 b(u(x)) -> b(d(x)) 361.46/136.59 d(a(x)) -> a(d(x)) 361.46/136.59 d(b(x)) -> u(a(b(x))) 361.46/136.59 a(u(x)) -> u(a(x)) 361.46/136.59 361.46/136.59 Q is empty. 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (70) QDPOrderProof (EQUIVALENT) 361.46/136.59 We use the reduction pair processor [LPAR04,JAR06]. 361.46/136.59 361.46/136.59 361.46/136.59 The following pairs can be oriented strictly and are deleted. 361.46/136.59 361.46/136.59 B(u(b(x0))) -> B(u(a(b(x0)))) 361.46/136.59 The remaining pairs can at least be oriented weakly. 361.46/136.59 Used ordering: Matrix interpretation [MATRO]: 361.46/136.59 361.46/136.59 Non-tuple symbols: 361.46/136.59 <<< 361.46/136.59 M( a_1(x_1) ) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 361.46/136.59 >>> 361.46/136.59 361.46/136.59 <<< 361.46/136.59 M( b_1(x_1) ) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 361.46/136.59 >>> 361.46/136.59 361.46/136.59 <<< 361.46/136.59 M( c ) = [[0], [0]] 361.46/136.59 >>> 361.46/136.59 361.46/136.59 <<< 361.46/136.59 M( d_1(x_1) ) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 361.46/136.59 >>> 361.46/136.59 361.46/136.59 <<< 361.46/136.59 M( u_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 361.46/136.59 >>> 361.46/136.59 361.46/136.59 Tuple symbols: 361.46/136.59 <<< 361.46/136.59 M( B_1(x_1) ) = [[0]] + [[1, 0]] * x_1 361.46/136.59 >>> 361.46/136.59 361.46/136.59 361.46/136.59 361.46/136.59 Matrix type: 361.46/136.59 361.46/136.59 We used a basic matrix type which is not further parametrizeable. 361.46/136.59 361.46/136.59 361.46/136.59 361.46/136.59 361.46/136.59 361.46/136.59 As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. 361.46/136.59 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 361.46/136.59 361.46/136.59 a(a(a(x))) -> c 361.46/136.59 a(u(x)) -> u(a(x)) 361.46/136.59 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (71) 361.46/136.59 Obligation: 361.46/136.59 Q DP problem: 361.46/136.59 The TRS P consists of the following rules: 361.46/136.59 361.46/136.59 B(u(a(x0))) -> B(a(d(x0))) 361.46/136.59 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 a(a(a(x))) -> c 361.46/136.59 b(u(x)) -> b(d(x)) 361.46/136.59 d(a(x)) -> a(d(x)) 361.46/136.59 d(b(x)) -> u(a(b(x))) 361.46/136.59 a(u(x)) -> u(a(x)) 361.46/136.59 361.46/136.59 Q is empty. 361.46/136.59 We have to consider all minimal (P,Q,R)-chains. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (72) Raffelsieper-Zantema-Transformation (SOUND) 361.46/136.59 We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (73) 361.46/136.59 Obligation: 361.46/136.59 Q restricted rewrite system: 361.46/136.59 The TRS R consists of the following rules: 361.46/136.59 361.46/136.59 down(a(a(a(x)))) -> up(c) 361.46/136.59 down(b(u(x))) -> up(b(d(x))) 361.46/136.59 down(d(a(x))) -> up(a(d(x))) 361.46/136.59 down(d(b(x))) -> up(u(a(b(x)))) 361.46/136.59 down(a(u(x))) -> up(u(a(x))) 361.46/136.59 top(up(x)) -> top(down(x)) 361.46/136.59 down(u(y2)) -> u_flat(down(y2)) 361.46/136.59 down(a(c)) -> a_flat(down(c)) 361.46/136.59 down(a(b(y6))) -> a_flat(down(b(y6))) 361.46/136.59 down(a(d(y8))) -> a_flat(down(d(y8))) 361.46/136.59 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.46/136.59 down(b(a(y10))) -> b_flat(down(a(y10))) 361.46/136.59 down(b(c)) -> b_flat(down(c)) 361.46/136.59 down(b(b(y11))) -> b_flat(down(b(y11))) 361.46/136.59 down(b(d(y13))) -> b_flat(down(d(y13))) 361.46/136.59 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.46/136.59 down(d(c)) -> d_flat(down(c)) 361.46/136.59 down(d(u(y17))) -> d_flat(down(u(y17))) 361.46/136.59 down(d(d(y18))) -> d_flat(down(d(y18))) 361.46/136.59 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.46/136.59 down(a(a(c))) -> a_flat(down(a(c))) 361.46/136.59 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.46/136.59 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.46/136.59 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.46/136.59 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.46/136.59 a_flat(up(x_1)) -> up(a(x_1)) 361.46/136.59 b_flat(up(x_1)) -> up(b(x_1)) 361.46/136.59 u_flat(up(x_1)) -> up(u(x_1)) 361.46/136.59 d_flat(up(x_1)) -> up(d(x_1)) 361.46/136.59 361.46/136.59 Q is empty. 361.46/136.59 361.46/136.59 ---------------------------------------- 361.46/136.59 361.46/136.59 (74) AAECC Innermost (EQUIVALENT) 361.46/136.59 We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 361.46/136.59 down(u(y2)) -> u_flat(down(y2)) 361.46/136.59 down(a(c)) -> a_flat(down(c)) 361.46/136.59 down(a(b(y6))) -> a_flat(down(b(y6))) 361.46/136.59 down(a(d(y8))) -> a_flat(down(d(y8))) 361.46/136.59 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.46/136.59 down(b(a(y10))) -> b_flat(down(a(y10))) 361.46/136.59 down(b(c)) -> b_flat(down(c)) 361.46/136.59 down(b(b(y11))) -> b_flat(down(b(y11))) 361.46/136.59 down(b(d(y13))) -> b_flat(down(d(y13))) 361.46/136.59 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.46/136.59 down(d(c)) -> d_flat(down(c)) 361.46/136.59 down(d(u(y17))) -> d_flat(down(u(y17))) 361.46/136.59 down(d(d(y18))) -> d_flat(down(d(y18))) 361.46/136.59 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.46/136.59 down(a(a(c))) -> a_flat(down(a(c))) 361.46/136.59 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.46/136.59 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.46/136.59 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.46/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.46/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.46/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.46/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.46/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.46/136.60 down(a(a(a(x)))) -> up(c) 361.46/136.60 down(b(u(x))) -> up(b(d(x))) 361.46/136.60 down(d(a(x))) -> up(a(d(x))) 361.46/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.46/136.60 down(a(u(x))) -> up(u(a(x))) 361.46/136.60 361.46/136.60 The TRS R 2 is 361.46/136.60 top(up(x)) -> top(down(x)) 361.46/136.60 361.46/136.60 The signature Sigma is {top_1} 361.46/136.60 ---------------------------------------- 361.46/136.60 361.46/136.60 (75) 361.46/136.60 Obligation: 361.46/136.60 Q restricted rewrite system: 361.46/136.60 The TRS R consists of the following rules: 361.46/136.60 361.46/136.60 down(a(a(a(x)))) -> up(c) 361.46/136.60 down(b(u(x))) -> up(b(d(x))) 361.46/136.60 down(d(a(x))) -> up(a(d(x))) 361.46/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.46/136.60 down(a(u(x))) -> up(u(a(x))) 361.46/136.60 top(up(x)) -> top(down(x)) 361.46/136.60 down(u(y2)) -> u_flat(down(y2)) 361.46/136.60 down(a(c)) -> a_flat(down(c)) 361.46/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.46/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.46/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.46/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.46/136.60 down(b(c)) -> b_flat(down(c)) 361.46/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.46/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.46/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.46/136.60 down(d(c)) -> d_flat(down(c)) 361.46/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.46/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.46/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.46/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.46/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.46/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.46/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.46/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.46/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.46/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.46/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.46/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.46/136.60 361.46/136.60 The set Q consists of the following terms: 361.46/136.60 361.46/136.60 down(a(a(a(x0)))) 361.46/136.60 down(b(u(x0))) 361.46/136.60 down(d(a(x0))) 361.46/136.60 down(d(b(x0))) 361.46/136.60 down(a(u(x0))) 361.46/136.60 top(up(x0)) 361.46/136.60 down(u(x0)) 361.46/136.60 down(a(c)) 361.46/136.60 down(a(b(x0))) 361.46/136.60 down(a(d(x0))) 361.46/136.60 down(a(fresh_constant)) 361.46/136.60 down(b(a(x0))) 361.46/136.60 down(b(c)) 361.46/136.60 down(b(b(x0))) 361.46/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (76) DependencyPairsProof (EQUIVALENT) 361.56/136.60 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (77) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(x)) -> TOP(down(x)) 361.56/136.60 TOP(up(x)) -> DOWN(x) 361.56/136.60 DOWN(u(y2)) -> U_FLAT(down(y2)) 361.56/136.60 DOWN(u(y2)) -> DOWN(y2) 361.56/136.60 DOWN(a(c)) -> A_FLAT(down(c)) 361.56/136.60 DOWN(a(c)) -> DOWN(c) 361.56/136.60 DOWN(a(b(y6))) -> A_FLAT(down(b(y6))) 361.56/136.60 DOWN(a(b(y6))) -> DOWN(b(y6)) 361.56/136.60 DOWN(a(d(y8))) -> A_FLAT(down(d(y8))) 361.56/136.60 DOWN(a(d(y8))) -> DOWN(d(y8)) 361.56/136.60 DOWN(a(fresh_constant)) -> A_FLAT(down(fresh_constant)) 361.56/136.60 DOWN(a(fresh_constant)) -> DOWN(fresh_constant) 361.56/136.60 DOWN(b(a(y10))) -> B_FLAT(down(a(y10))) 361.56/136.60 DOWN(b(a(y10))) -> DOWN(a(y10)) 361.56/136.60 DOWN(b(c)) -> B_FLAT(down(c)) 361.56/136.60 DOWN(b(c)) -> DOWN(c) 361.56/136.60 DOWN(b(b(y11))) -> B_FLAT(down(b(y11))) 361.56/136.60 DOWN(b(b(y11))) -> DOWN(b(y11)) 361.56/136.60 DOWN(b(d(y13))) -> B_FLAT(down(d(y13))) 361.56/136.60 DOWN(b(d(y13))) -> DOWN(d(y13)) 361.56/136.60 DOWN(b(fresh_constant)) -> B_FLAT(down(fresh_constant)) 361.56/136.60 DOWN(b(fresh_constant)) -> DOWN(fresh_constant) 361.56/136.60 DOWN(d(c)) -> D_FLAT(down(c)) 361.56/136.60 DOWN(d(c)) -> DOWN(c) 361.56/136.60 DOWN(d(u(y17))) -> D_FLAT(down(u(y17))) 361.56/136.60 DOWN(d(u(y17))) -> DOWN(u(y17)) 361.56/136.60 DOWN(d(d(y18))) -> D_FLAT(down(d(y18))) 361.56/136.60 DOWN(d(d(y18))) -> DOWN(d(y18)) 361.56/136.60 DOWN(d(fresh_constant)) -> D_FLAT(down(fresh_constant)) 361.56/136.60 DOWN(d(fresh_constant)) -> DOWN(fresh_constant) 361.56/136.60 DOWN(a(a(c))) -> A_FLAT(down(a(c))) 361.56/136.60 DOWN(a(a(c))) -> DOWN(a(c)) 361.56/136.60 DOWN(a(a(b(y21)))) -> A_FLAT(down(a(b(y21)))) 361.56/136.60 DOWN(a(a(b(y21)))) -> DOWN(a(b(y21))) 361.56/136.60 DOWN(a(a(u(y22)))) -> A_FLAT(down(a(u(y22)))) 361.56/136.60 DOWN(a(a(u(y22)))) -> DOWN(a(u(y22))) 361.56/136.60 DOWN(a(a(d(y23)))) -> A_FLAT(down(a(d(y23)))) 361.56/136.60 DOWN(a(a(d(y23)))) -> DOWN(a(d(y23))) 361.56/136.60 DOWN(a(a(fresh_constant))) -> A_FLAT(down(a(fresh_constant))) 361.56/136.60 DOWN(a(a(fresh_constant))) -> DOWN(a(fresh_constant)) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 top(up(x)) -> top(down(x)) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 top(up(x0)) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (78) DependencyGraphProof (EQUIVALENT) 361.56/136.60 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 29 less nodes. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (79) 361.56/136.60 Complex Obligation (AND) 361.56/136.60 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (80) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 DOWN(a(b(y6))) -> DOWN(b(y6)) 361.56/136.60 DOWN(b(a(y10))) -> DOWN(a(y10)) 361.56/136.60 DOWN(a(d(y8))) -> DOWN(d(y8)) 361.56/136.60 DOWN(d(u(y17))) -> DOWN(u(y17)) 361.56/136.60 DOWN(u(y2)) -> DOWN(y2) 361.56/136.60 DOWN(b(b(y11))) -> DOWN(b(y11)) 361.56/136.60 DOWN(b(d(y13))) -> DOWN(d(y13)) 361.56/136.60 DOWN(d(d(y18))) -> DOWN(d(y18)) 361.56/136.60 DOWN(a(a(b(y21)))) -> DOWN(a(b(y21))) 361.56/136.60 DOWN(a(a(d(y23)))) -> DOWN(a(d(y23))) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 top(up(x)) -> top(down(x)) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 top(up(x0)) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (81) UsableRulesProof (EQUIVALENT) 361.56/136.60 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. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (82) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 DOWN(a(b(y6))) -> DOWN(b(y6)) 361.56/136.60 DOWN(b(a(y10))) -> DOWN(a(y10)) 361.56/136.60 DOWN(a(d(y8))) -> DOWN(d(y8)) 361.56/136.60 DOWN(d(u(y17))) -> DOWN(u(y17)) 361.56/136.60 DOWN(u(y2)) -> DOWN(y2) 361.56/136.60 DOWN(b(b(y11))) -> DOWN(b(y11)) 361.56/136.60 DOWN(b(d(y13))) -> DOWN(d(y13)) 361.56/136.60 DOWN(d(d(y18))) -> DOWN(d(y18)) 361.56/136.60 DOWN(a(a(b(y21)))) -> DOWN(a(b(y21))) 361.56/136.60 DOWN(a(a(d(y23)))) -> DOWN(a(d(y23))) 361.56/136.60 361.56/136.60 R is empty. 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 top(up(x0)) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (83) QReductionProof (EQUIVALENT) 361.56/136.60 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 top(up(x0)) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (84) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 DOWN(a(b(y6))) -> DOWN(b(y6)) 361.56/136.60 DOWN(b(a(y10))) -> DOWN(a(y10)) 361.56/136.60 DOWN(a(d(y8))) -> DOWN(d(y8)) 361.56/136.60 DOWN(d(u(y17))) -> DOWN(u(y17)) 361.56/136.60 DOWN(u(y2)) -> DOWN(y2) 361.56/136.60 DOWN(b(b(y11))) -> DOWN(b(y11)) 361.56/136.60 DOWN(b(d(y13))) -> DOWN(d(y13)) 361.56/136.60 DOWN(d(d(y18))) -> DOWN(d(y18)) 361.56/136.60 DOWN(a(a(b(y21)))) -> DOWN(a(b(y21))) 361.56/136.60 DOWN(a(a(d(y23)))) -> DOWN(a(d(y23))) 361.56/136.60 361.56/136.60 R is empty. 361.56/136.60 Q is empty. 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (85) QDPSizeChangeProof (EQUIVALENT) 361.56/136.60 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. 361.56/136.60 361.56/136.60 From the DPs we obtained the following set of size-change graphs: 361.56/136.60 *DOWN(b(a(y10))) -> DOWN(a(y10)) 361.56/136.60 The graph contains the following edges 1 > 1 361.56/136.60 361.56/136.60 361.56/136.60 *DOWN(b(b(y11))) -> DOWN(b(y11)) 361.56/136.60 The graph contains the following edges 1 > 1 361.56/136.60 361.56/136.60 361.56/136.60 *DOWN(b(d(y13))) -> DOWN(d(y13)) 361.56/136.60 The graph contains the following edges 1 > 1 361.56/136.60 361.56/136.60 361.56/136.60 *DOWN(a(a(b(y21)))) -> DOWN(a(b(y21))) 361.56/136.60 The graph contains the following edges 1 > 1 361.56/136.60 361.56/136.60 361.56/136.60 *DOWN(u(y2)) -> DOWN(y2) 361.56/136.60 The graph contains the following edges 1 > 1 361.56/136.60 361.56/136.60 361.56/136.60 *DOWN(a(b(y6))) -> DOWN(b(y6)) 361.56/136.60 The graph contains the following edges 1 > 1 361.56/136.60 361.56/136.60 361.56/136.60 *DOWN(a(a(d(y23)))) -> DOWN(a(d(y23))) 361.56/136.60 The graph contains the following edges 1 > 1 361.56/136.60 361.56/136.60 361.56/136.60 *DOWN(a(d(y8))) -> DOWN(d(y8)) 361.56/136.60 The graph contains the following edges 1 > 1 361.56/136.60 361.56/136.60 361.56/136.60 *DOWN(d(d(y18))) -> DOWN(d(y18)) 361.56/136.60 The graph contains the following edges 1 > 1 361.56/136.60 361.56/136.60 361.56/136.60 *DOWN(d(u(y17))) -> DOWN(u(y17)) 361.56/136.60 The graph contains the following edges 1 > 1 361.56/136.60 361.56/136.60 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (86) 361.56/136.60 YES 361.56/136.60 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (87) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(x)) -> TOP(down(x)) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 top(up(x)) -> top(down(x)) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 top(up(x0)) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (88) UsableRulesProof (EQUIVALENT) 361.56/136.60 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. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (89) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(x)) -> TOP(down(x)) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 top(up(x0)) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (90) QReductionProof (EQUIVALENT) 361.56/136.60 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 361.56/136.60 361.56/136.60 top(up(x0)) 361.56/136.60 361.56/136.60 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (91) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(x)) -> TOP(down(x)) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (92) TransformationProof (EQUIVALENT) 361.56/136.60 By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]: 361.56/136.60 361.56/136.60 (TOP(up(a(a(a(x0))))) -> TOP(up(c)),TOP(up(a(a(a(x0))))) -> TOP(up(c))) 361.56/136.60 (TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))),TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))) 361.56/136.60 (TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))),TOP(up(d(a(x0)))) -> TOP(up(a(d(x0))))) 361.56/136.60 (TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))),TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0)))))) 361.56/136.60 (TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))),TOP(up(a(u(x0)))) -> TOP(up(u(a(x0))))) 361.56/136.60 (TOP(up(u(x0))) -> TOP(u_flat(down(x0))),TOP(up(u(x0))) -> TOP(u_flat(down(x0)))) 361.56/136.60 (TOP(up(a(c))) -> TOP(a_flat(down(c))),TOP(up(a(c))) -> TOP(a_flat(down(c)))) 361.56/136.60 (TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))),TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))) 361.56/136.60 (TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))),TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))) 361.56/136.60 (TOP(up(a(fresh_constant))) -> TOP(a_flat(down(fresh_constant))),TOP(up(a(fresh_constant))) -> TOP(a_flat(down(fresh_constant)))) 361.56/136.60 (TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))),TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))) 361.56/136.60 (TOP(up(b(c))) -> TOP(b_flat(down(c))),TOP(up(b(c))) -> TOP(b_flat(down(c)))) 361.56/136.60 (TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))),TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))) 361.56/136.60 (TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))),TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))) 361.56/136.60 (TOP(up(b(fresh_constant))) -> TOP(b_flat(down(fresh_constant))),TOP(up(b(fresh_constant))) -> TOP(b_flat(down(fresh_constant)))) 361.56/136.60 (TOP(up(d(c))) -> TOP(d_flat(down(c))),TOP(up(d(c))) -> TOP(d_flat(down(c)))) 361.56/136.60 (TOP(up(d(u(x0)))) -> TOP(d_flat(down(u(x0)))),TOP(up(d(u(x0)))) -> TOP(d_flat(down(u(x0))))) 361.56/136.60 (TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))),TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))) 361.56/136.60 (TOP(up(d(fresh_constant))) -> TOP(d_flat(down(fresh_constant))),TOP(up(d(fresh_constant))) -> TOP(d_flat(down(fresh_constant)))) 361.56/136.60 (TOP(up(a(a(c)))) -> TOP(a_flat(down(a(c)))),TOP(up(a(a(c)))) -> TOP(a_flat(down(a(c))))) 361.56/136.60 (TOP(up(a(a(b(x0))))) -> TOP(a_flat(down(a(b(x0))))),TOP(up(a(a(b(x0))))) -> TOP(a_flat(down(a(b(x0)))))) 361.56/136.60 (TOP(up(a(a(u(x0))))) -> TOP(a_flat(down(a(u(x0))))),TOP(up(a(a(u(x0))))) -> TOP(a_flat(down(a(u(x0)))))) 361.56/136.60 (TOP(up(a(a(d(x0))))) -> TOP(a_flat(down(a(d(x0))))),TOP(up(a(a(d(x0))))) -> TOP(a_flat(down(a(d(x0)))))) 361.56/136.60 (TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(down(a(fresh_constant)))),TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(down(a(fresh_constant))))) 361.56/136.60 361.56/136.60 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (93) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(a(a(a(x0))))) -> TOP(up(c)) 361.56/136.60 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.60 TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))) 361.56/136.60 TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))) 361.56/136.60 TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))) 361.56/136.60 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.60 TOP(up(a(c))) -> TOP(a_flat(down(c))) 361.56/136.60 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.60 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.60 TOP(up(a(fresh_constant))) -> TOP(a_flat(down(fresh_constant))) 361.56/136.60 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.60 TOP(up(b(c))) -> TOP(b_flat(down(c))) 361.56/136.60 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.60 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.60 TOP(up(b(fresh_constant))) -> TOP(b_flat(down(fresh_constant))) 361.56/136.60 TOP(up(d(c))) -> TOP(d_flat(down(c))) 361.56/136.60 TOP(up(d(u(x0)))) -> TOP(d_flat(down(u(x0)))) 361.56/136.60 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(fresh_constant))) -> TOP(d_flat(down(fresh_constant))) 361.56/136.60 TOP(up(a(a(c)))) -> TOP(a_flat(down(a(c)))) 361.56/136.60 TOP(up(a(a(b(x0))))) -> TOP(a_flat(down(a(b(x0))))) 361.56/136.60 TOP(up(a(a(u(x0))))) -> TOP(a_flat(down(a(u(x0))))) 361.56/136.60 TOP(up(a(a(d(x0))))) -> TOP(a_flat(down(a(d(x0))))) 361.56/136.60 TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(down(a(fresh_constant)))) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (94) DependencyGraphProof (EQUIVALENT) 361.56/136.60 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (95) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.60 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))) 361.56/136.60 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))) 361.56/136.60 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.60 TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))) 361.56/136.60 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.60 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.60 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.60 TOP(up(d(u(x0)))) -> TOP(d_flat(down(u(x0)))) 361.56/136.60 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.60 TOP(up(a(a(c)))) -> TOP(a_flat(down(a(c)))) 361.56/136.60 TOP(up(a(a(b(x0))))) -> TOP(a_flat(down(a(b(x0))))) 361.56/136.60 TOP(up(a(a(u(x0))))) -> TOP(a_flat(down(a(u(x0))))) 361.56/136.60 TOP(up(a(a(d(x0))))) -> TOP(a_flat(down(a(d(x0))))) 361.56/136.60 TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(down(a(fresh_constant)))) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (96) TransformationProof (EQUIVALENT) 361.56/136.60 By rewriting [LPAR04] the rule TOP(up(d(u(x0)))) -> TOP(d_flat(down(u(x0)))) at position [0,0] we obtained the following new rules [LPAR04]: 361.56/136.60 361.56/136.60 (TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))),TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0))))) 361.56/136.60 361.56/136.60 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (97) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.60 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))) 361.56/136.60 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))) 361.56/136.60 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.60 TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))) 361.56/136.60 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.60 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.60 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.60 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.60 TOP(up(a(a(c)))) -> TOP(a_flat(down(a(c)))) 361.56/136.60 TOP(up(a(a(b(x0))))) -> TOP(a_flat(down(a(b(x0))))) 361.56/136.60 TOP(up(a(a(u(x0))))) -> TOP(a_flat(down(a(u(x0))))) 361.56/136.60 TOP(up(a(a(d(x0))))) -> TOP(a_flat(down(a(d(x0))))) 361.56/136.60 TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(down(a(fresh_constant)))) 361.56/136.60 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (98) TransformationProof (EQUIVALENT) 361.56/136.60 By rewriting [LPAR04] the rule TOP(up(a(a(c)))) -> TOP(a_flat(down(a(c)))) at position [0,0] we obtained the following new rules [LPAR04]: 361.56/136.60 361.56/136.60 (TOP(up(a(a(c)))) -> TOP(a_flat(a_flat(down(c)))),TOP(up(a(a(c)))) -> TOP(a_flat(a_flat(down(c))))) 361.56/136.60 361.56/136.60 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (99) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.60 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))) 361.56/136.60 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))) 361.56/136.60 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.60 TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))) 361.56/136.60 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.60 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.60 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.60 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.60 TOP(up(a(a(b(x0))))) -> TOP(a_flat(down(a(b(x0))))) 361.56/136.60 TOP(up(a(a(u(x0))))) -> TOP(a_flat(down(a(u(x0))))) 361.56/136.60 TOP(up(a(a(d(x0))))) -> TOP(a_flat(down(a(d(x0))))) 361.56/136.60 TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(down(a(fresh_constant)))) 361.56/136.60 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.60 TOP(up(a(a(c)))) -> TOP(a_flat(a_flat(down(c)))) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (100) DependencyGraphProof (EQUIVALENT) 361.56/136.60 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (101) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.60 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.60 TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))) 361.56/136.60 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))) 361.56/136.60 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.60 TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))) 361.56/136.60 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.60 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.60 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.60 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.60 TOP(up(a(a(b(x0))))) -> TOP(a_flat(down(a(b(x0))))) 361.56/136.60 TOP(up(a(a(u(x0))))) -> TOP(a_flat(down(a(u(x0))))) 361.56/136.60 TOP(up(a(a(d(x0))))) -> TOP(a_flat(down(a(d(x0))))) 361.56/136.60 TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(down(a(fresh_constant)))) 361.56/136.60 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (102) TransformationProof (EQUIVALENT) 361.56/136.60 By rewriting [LPAR04] the rule TOP(up(a(a(b(x0))))) -> TOP(a_flat(down(a(b(x0))))) at position [0,0] we obtained the following new rules [LPAR04]: 361.56/136.60 361.56/136.60 (TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))),TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0)))))) 361.56/136.60 361.56/136.60 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (103) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.60 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.60 TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))) 361.56/136.60 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))) 361.56/136.60 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.60 TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))) 361.56/136.60 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.60 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.60 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.60 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.60 TOP(up(a(a(u(x0))))) -> TOP(a_flat(down(a(u(x0))))) 361.56/136.60 TOP(up(a(a(d(x0))))) -> TOP(a_flat(down(a(d(x0))))) 361.56/136.60 TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(down(a(fresh_constant)))) 361.56/136.60 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.60 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (104) TransformationProof (EQUIVALENT) 361.56/136.60 By rewriting [LPAR04] the rule TOP(up(a(a(u(x0))))) -> TOP(a_flat(down(a(u(x0))))) at position [0,0] we obtained the following new rules [LPAR04]: 361.56/136.60 361.56/136.60 (TOP(up(a(a(u(x0))))) -> TOP(a_flat(up(u(a(x0))))),TOP(up(a(a(u(x0))))) -> TOP(a_flat(up(u(a(x0)))))) 361.56/136.60 361.56/136.60 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (105) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.60 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.60 TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))) 361.56/136.60 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))) 361.56/136.60 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.60 TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))) 361.56/136.60 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.60 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.60 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.60 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.60 TOP(up(a(a(d(x0))))) -> TOP(a_flat(down(a(d(x0))))) 361.56/136.60 TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(down(a(fresh_constant)))) 361.56/136.60 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.60 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.60 TOP(up(a(a(u(x0))))) -> TOP(a_flat(up(u(a(x0))))) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (106) TransformationProof (EQUIVALENT) 361.56/136.60 By rewriting [LPAR04] the rule TOP(up(a(a(d(x0))))) -> TOP(a_flat(down(a(d(x0))))) at position [0,0] we obtained the following new rules [LPAR04]: 361.56/136.60 361.56/136.60 (TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0))))),TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0)))))) 361.56/136.60 361.56/136.60 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (107) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.60 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.60 TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))) 361.56/136.60 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))) 361.56/136.60 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.60 TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))) 361.56/136.60 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.60 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.60 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.60 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.60 TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(down(a(fresh_constant)))) 361.56/136.60 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.60 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.60 TOP(up(a(a(u(x0))))) -> TOP(a_flat(up(u(a(x0))))) 361.56/136.60 TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0))))) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (108) TransformationProof (EQUIVALENT) 361.56/136.60 By rewriting [LPAR04] the rule TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(down(a(fresh_constant)))) at position [0,0] we obtained the following new rules [LPAR04]: 361.56/136.60 361.56/136.60 (TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(a_flat(down(fresh_constant)))),TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(a_flat(down(fresh_constant))))) 361.56/136.60 361.56/136.60 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (109) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.60 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.60 TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))) 361.56/136.60 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))) 361.56/136.60 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.60 TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))) 361.56/136.60 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.60 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.60 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.60 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.60 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.60 TOP(up(a(a(u(x0))))) -> TOP(a_flat(up(u(a(x0))))) 361.56/136.60 TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0))))) 361.56/136.60 TOP(up(a(a(fresh_constant)))) -> TOP(a_flat(a_flat(down(fresh_constant)))) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (110) DependencyGraphProof (EQUIVALENT) 361.56/136.60 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 361.56/136.60 ---------------------------------------- 361.56/136.60 361.56/136.60 (111) 361.56/136.60 Obligation: 361.56/136.60 Q DP problem: 361.56/136.60 The TRS P consists of the following rules: 361.56/136.60 361.56/136.60 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.60 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))) 361.56/136.60 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))) 361.56/136.60 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.60 TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))) 361.56/136.60 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.60 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.60 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.60 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.60 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.60 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.60 TOP(up(a(a(u(x0))))) -> TOP(a_flat(up(u(a(x0))))) 361.56/136.60 TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0))))) 361.56/136.60 361.56/136.60 The TRS R consists of the following rules: 361.56/136.60 361.56/136.60 down(a(a(a(x)))) -> up(c) 361.56/136.60 down(b(u(x))) -> up(b(d(x))) 361.56/136.60 down(d(a(x))) -> up(a(d(x))) 361.56/136.60 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.60 down(a(u(x))) -> up(u(a(x))) 361.56/136.60 down(u(y2)) -> u_flat(down(y2)) 361.56/136.60 down(a(c)) -> a_flat(down(c)) 361.56/136.60 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.60 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.60 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.60 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.60 down(b(c)) -> b_flat(down(c)) 361.56/136.60 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.60 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.60 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.60 down(d(c)) -> d_flat(down(c)) 361.56/136.60 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.60 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.60 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.60 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.60 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.60 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.60 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.60 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.60 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.60 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.60 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.60 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.60 361.56/136.60 The set Q consists of the following terms: 361.56/136.60 361.56/136.60 down(a(a(a(x0)))) 361.56/136.60 down(b(u(x0))) 361.56/136.60 down(d(a(x0))) 361.56/136.60 down(d(b(x0))) 361.56/136.60 down(a(u(x0))) 361.56/136.60 down(u(x0)) 361.56/136.60 down(a(c)) 361.56/136.60 down(a(b(x0))) 361.56/136.60 down(a(d(x0))) 361.56/136.60 down(a(fresh_constant)) 361.56/136.60 down(b(a(x0))) 361.56/136.60 down(b(c)) 361.56/136.60 down(b(b(x0))) 361.56/136.60 down(b(d(x0))) 361.56/136.60 down(b(fresh_constant)) 361.56/136.60 down(d(c)) 361.56/136.60 down(d(u(x0))) 361.56/136.60 down(d(d(x0))) 361.56/136.60 down(d(fresh_constant)) 361.56/136.60 down(a(a(c))) 361.56/136.60 down(a(a(b(x0)))) 361.56/136.60 down(a(a(u(x0)))) 361.56/136.60 down(a(a(d(x0)))) 361.56/136.60 down(a(a(fresh_constant))) 361.56/136.60 a_flat(up(x0)) 361.56/136.60 b_flat(up(x0)) 361.56/136.60 u_flat(up(x0)) 361.56/136.60 d_flat(up(x0)) 361.56/136.60 361.56/136.60 We have to consider all minimal (P,Q,R)-chains. 361.56/136.60 ---------------------------------------- 361.56/136.61 361.56/136.61 (112) TransformationProof (EQUIVALENT) 361.56/136.61 By rewriting [LPAR04] the rule TOP(up(a(a(u(x0))))) -> TOP(a_flat(up(u(a(x0))))) at position [0] we obtained the following new rules [LPAR04]: 361.56/136.61 361.56/136.61 (TOP(up(a(a(u(x0))))) -> TOP(up(a(u(a(x0))))),TOP(up(a(a(u(x0))))) -> TOP(up(a(u(a(x0)))))) 361.56/136.61 361.56/136.61 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (113) 361.56/136.61 Obligation: 361.56/136.61 Q DP problem: 361.56/136.61 The TRS P consists of the following rules: 361.56/136.61 361.56/136.61 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.61 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.61 TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))) 361.56/136.61 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.61 TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))) 361.56/136.61 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.61 TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))) 361.56/136.61 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.61 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.61 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.61 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.61 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.61 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.61 TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0))))) 361.56/136.61 TOP(up(a(a(u(x0))))) -> TOP(up(a(u(a(x0))))) 361.56/136.61 361.56/136.61 The TRS R consists of the following rules: 361.56/136.61 361.56/136.61 down(a(a(a(x)))) -> up(c) 361.56/136.61 down(b(u(x))) -> up(b(d(x))) 361.56/136.61 down(d(a(x))) -> up(a(d(x))) 361.56/136.61 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.61 down(a(u(x))) -> up(u(a(x))) 361.56/136.61 down(u(y2)) -> u_flat(down(y2)) 361.56/136.61 down(a(c)) -> a_flat(down(c)) 361.56/136.61 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.61 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.61 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.61 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.61 down(b(c)) -> b_flat(down(c)) 361.56/136.61 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.61 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.61 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.61 down(d(c)) -> d_flat(down(c)) 361.56/136.61 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.61 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.61 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.61 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.61 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.61 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.61 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.61 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.61 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.61 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.61 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.61 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.61 361.56/136.61 The set Q consists of the following terms: 361.56/136.61 361.56/136.61 down(a(a(a(x0)))) 361.56/136.61 down(b(u(x0))) 361.56/136.61 down(d(a(x0))) 361.56/136.61 down(d(b(x0))) 361.56/136.61 down(a(u(x0))) 361.56/136.61 down(u(x0)) 361.56/136.61 down(a(c)) 361.56/136.61 down(a(b(x0))) 361.56/136.61 down(a(d(x0))) 361.56/136.61 down(a(fresh_constant)) 361.56/136.61 down(b(a(x0))) 361.56/136.61 down(b(c)) 361.56/136.61 down(b(b(x0))) 361.56/136.61 down(b(d(x0))) 361.56/136.61 down(b(fresh_constant)) 361.56/136.61 down(d(c)) 361.56/136.61 down(d(u(x0))) 361.56/136.61 down(d(d(x0))) 361.56/136.61 down(d(fresh_constant)) 361.56/136.61 down(a(a(c))) 361.56/136.61 down(a(a(b(x0)))) 361.56/136.61 down(a(a(u(x0)))) 361.56/136.61 down(a(a(d(x0)))) 361.56/136.61 down(a(a(fresh_constant))) 361.56/136.61 a_flat(up(x0)) 361.56/136.61 b_flat(up(x0)) 361.56/136.61 u_flat(up(x0)) 361.56/136.61 d_flat(up(x0)) 361.56/136.61 361.56/136.61 We have to consider all minimal (P,Q,R)-chains. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (114) QDPOrderProof (EQUIVALENT) 361.56/136.61 We use the reduction pair processor [LPAR04,JAR06]. 361.56/136.61 361.56/136.61 361.56/136.61 The following pairs can be oriented strictly and are deleted. 361.56/136.61 361.56/136.61 TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))) 361.56/136.61 TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))) 361.56/136.61 The remaining pairs can at least be oriented weakly. 361.56/136.61 Used ordering: Polynomial interpretation [POLO]: 361.56/136.61 361.56/136.61 POL(TOP(x_1)) = x_1 361.56/136.61 POL(a(x_1)) = 0 361.56/136.61 POL(a_flat(x_1)) = 0 361.56/136.61 POL(b(x_1)) = 0 361.56/136.61 POL(b_flat(x_1)) = 0 361.56/136.61 POL(c) = 0 361.56/136.61 POL(d(x_1)) = 1 361.56/136.61 POL(d_flat(x_1)) = 1 361.56/136.61 POL(down(x_1)) = 1 361.56/136.61 POL(fresh_constant) = 0 361.56/136.61 POL(u(x_1)) = 0 361.56/136.61 POL(u_flat(x_1)) = 0 361.56/136.61 POL(up(x_1)) = x_1 361.56/136.61 361.56/136.61 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 361.56/136.61 361.56/136.61 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.61 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.61 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.61 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.61 361.56/136.61 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (115) 361.56/136.61 Obligation: 361.56/136.61 Q DP problem: 361.56/136.61 The TRS P consists of the following rules: 361.56/136.61 361.56/136.61 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.61 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.61 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.61 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.61 TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))) 361.56/136.61 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.61 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.61 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.61 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.61 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.61 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.61 TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0))))) 361.56/136.61 TOP(up(a(a(u(x0))))) -> TOP(up(a(u(a(x0))))) 361.56/136.61 361.56/136.61 The TRS R consists of the following rules: 361.56/136.61 361.56/136.61 down(a(a(a(x)))) -> up(c) 361.56/136.61 down(b(u(x))) -> up(b(d(x))) 361.56/136.61 down(d(a(x))) -> up(a(d(x))) 361.56/136.61 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.61 down(a(u(x))) -> up(u(a(x))) 361.56/136.61 down(u(y2)) -> u_flat(down(y2)) 361.56/136.61 down(a(c)) -> a_flat(down(c)) 361.56/136.61 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.61 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.61 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.61 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.61 down(b(c)) -> b_flat(down(c)) 361.56/136.61 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.61 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.61 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.61 down(d(c)) -> d_flat(down(c)) 361.56/136.61 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.61 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.61 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.61 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.61 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.61 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.61 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.61 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.61 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.61 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.61 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.61 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.61 361.56/136.61 The set Q consists of the following terms: 361.56/136.61 361.56/136.61 down(a(a(a(x0)))) 361.56/136.61 down(b(u(x0))) 361.56/136.61 down(d(a(x0))) 361.56/136.61 down(d(b(x0))) 361.56/136.61 down(a(u(x0))) 361.56/136.61 down(u(x0)) 361.56/136.61 down(a(c)) 361.56/136.61 down(a(b(x0))) 361.56/136.61 down(a(d(x0))) 361.56/136.61 down(a(fresh_constant)) 361.56/136.61 down(b(a(x0))) 361.56/136.61 down(b(c)) 361.56/136.61 down(b(b(x0))) 361.56/136.61 down(b(d(x0))) 361.56/136.61 down(b(fresh_constant)) 361.56/136.61 down(d(c)) 361.56/136.61 down(d(u(x0))) 361.56/136.61 down(d(d(x0))) 361.56/136.61 down(d(fresh_constant)) 361.56/136.61 down(a(a(c))) 361.56/136.61 down(a(a(b(x0)))) 361.56/136.61 down(a(a(u(x0)))) 361.56/136.61 down(a(a(d(x0)))) 361.56/136.61 down(a(a(fresh_constant))) 361.56/136.61 a_flat(up(x0)) 361.56/136.61 b_flat(up(x0)) 361.56/136.61 u_flat(up(x0)) 361.56/136.61 d_flat(up(x0)) 361.56/136.61 361.56/136.61 We have to consider all minimal (P,Q,R)-chains. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (116) QDPOrderProof (EQUIVALENT) 361.56/136.61 We use the reduction pair processor [LPAR04,JAR06]. 361.56/136.61 361.56/136.61 361.56/136.61 The following pairs can be oriented strictly and are deleted. 361.56/136.61 361.56/136.61 TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))) 361.56/136.61 The remaining pairs can at least be oriented weakly. 361.56/136.61 Used ordering: Polynomial interpretation [POLO]: 361.56/136.61 361.56/136.61 POL(TOP(x_1)) = x_1 361.56/136.61 POL(a(x_1)) = 1 361.56/136.61 POL(a_flat(x_1)) = 1 361.56/136.61 POL(b(x_1)) = 0 361.56/136.61 POL(b_flat(x_1)) = 0 361.56/136.61 POL(c) = 0 361.56/136.61 POL(d(x_1)) = 0 361.56/136.61 POL(d_flat(x_1)) = 0 361.56/136.61 POL(down(x_1)) = 1 + x_1 361.56/136.61 POL(fresh_constant) = 0 361.56/136.61 POL(u(x_1)) = 0 361.56/136.61 POL(u_flat(x_1)) = 0 361.56/136.61 POL(up(x_1)) = x_1 361.56/136.61 361.56/136.61 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 361.56/136.61 361.56/136.61 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.61 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.61 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.61 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.61 361.56/136.61 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (117) 361.56/136.61 Obligation: 361.56/136.61 Q DP problem: 361.56/136.61 The TRS P consists of the following rules: 361.56/136.61 361.56/136.61 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.61 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.61 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.61 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.61 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.61 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.61 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.61 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.61 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.61 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.61 TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0))))) 361.56/136.61 TOP(up(a(a(u(x0))))) -> TOP(up(a(u(a(x0))))) 361.56/136.61 361.56/136.61 The TRS R consists of the following rules: 361.56/136.61 361.56/136.61 down(a(a(a(x)))) -> up(c) 361.56/136.61 down(b(u(x))) -> up(b(d(x))) 361.56/136.61 down(d(a(x))) -> up(a(d(x))) 361.56/136.61 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.61 down(a(u(x))) -> up(u(a(x))) 361.56/136.61 down(u(y2)) -> u_flat(down(y2)) 361.56/136.61 down(a(c)) -> a_flat(down(c)) 361.56/136.61 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.61 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.61 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.61 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.61 down(b(c)) -> b_flat(down(c)) 361.56/136.61 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.61 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.61 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.61 down(d(c)) -> d_flat(down(c)) 361.56/136.61 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.61 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.61 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.61 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.61 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.61 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.61 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.61 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.61 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.61 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.61 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.61 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.61 361.56/136.61 The set Q consists of the following terms: 361.56/136.61 361.56/136.61 down(a(a(a(x0)))) 361.56/136.61 down(b(u(x0))) 361.56/136.61 down(d(a(x0))) 361.56/136.61 down(d(b(x0))) 361.56/136.61 down(a(u(x0))) 361.56/136.61 down(u(x0)) 361.56/136.61 down(a(c)) 361.56/136.61 down(a(b(x0))) 361.56/136.61 down(a(d(x0))) 361.56/136.61 down(a(fresh_constant)) 361.56/136.61 down(b(a(x0))) 361.56/136.61 down(b(c)) 361.56/136.61 down(b(b(x0))) 361.56/136.61 down(b(d(x0))) 361.56/136.61 down(b(fresh_constant)) 361.56/136.61 down(d(c)) 361.56/136.61 down(d(u(x0))) 361.56/136.61 down(d(d(x0))) 361.56/136.61 down(d(fresh_constant)) 361.56/136.61 down(a(a(c))) 361.56/136.61 down(a(a(b(x0)))) 361.56/136.61 down(a(a(u(x0)))) 361.56/136.61 down(a(a(d(x0)))) 361.56/136.61 down(a(a(fresh_constant))) 361.56/136.61 a_flat(up(x0)) 361.56/136.61 b_flat(up(x0)) 361.56/136.61 u_flat(up(x0)) 361.56/136.61 d_flat(up(x0)) 361.56/136.61 361.56/136.61 We have to consider all minimal (P,Q,R)-chains. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (118) DependencyGraphProof (EQUIVALENT) 361.56/136.61 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (119) 361.56/136.61 Obligation: 361.56/136.61 Q DP problem: 361.56/136.61 The TRS P consists of the following rules: 361.56/136.61 361.56/136.61 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.61 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.61 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.61 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.61 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.61 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.61 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.61 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.61 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.61 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.61 TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0))))) 361.56/136.61 361.56/136.61 The TRS R consists of the following rules: 361.56/136.61 361.56/136.61 down(a(a(a(x)))) -> up(c) 361.56/136.61 down(b(u(x))) -> up(b(d(x))) 361.56/136.61 down(d(a(x))) -> up(a(d(x))) 361.56/136.61 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.61 down(a(u(x))) -> up(u(a(x))) 361.56/136.61 down(u(y2)) -> u_flat(down(y2)) 361.56/136.61 down(a(c)) -> a_flat(down(c)) 361.56/136.61 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.61 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.61 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.61 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.61 down(b(c)) -> b_flat(down(c)) 361.56/136.61 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.61 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.61 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.61 down(d(c)) -> d_flat(down(c)) 361.56/136.61 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.61 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.61 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.61 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.61 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.61 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.61 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.61 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.61 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.61 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.61 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.61 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.61 361.56/136.61 The set Q consists of the following terms: 361.56/136.61 361.56/136.61 down(a(a(a(x0)))) 361.56/136.61 down(b(u(x0))) 361.56/136.61 down(d(a(x0))) 361.56/136.61 down(d(b(x0))) 361.56/136.61 down(a(u(x0))) 361.56/136.61 down(u(x0)) 361.56/136.61 down(a(c)) 361.56/136.61 down(a(b(x0))) 361.56/136.61 down(a(d(x0))) 361.56/136.61 down(a(fresh_constant)) 361.56/136.61 down(b(a(x0))) 361.56/136.61 down(b(c)) 361.56/136.61 down(b(b(x0))) 361.56/136.61 down(b(d(x0))) 361.56/136.61 down(b(fresh_constant)) 361.56/136.61 down(d(c)) 361.56/136.61 down(d(u(x0))) 361.56/136.61 down(d(d(x0))) 361.56/136.61 down(d(fresh_constant)) 361.56/136.61 down(a(a(c))) 361.56/136.61 down(a(a(b(x0)))) 361.56/136.61 down(a(a(u(x0)))) 361.56/136.61 down(a(a(d(x0)))) 361.56/136.61 down(a(a(fresh_constant))) 361.56/136.61 a_flat(up(x0)) 361.56/136.61 b_flat(up(x0)) 361.56/136.61 u_flat(up(x0)) 361.56/136.61 d_flat(up(x0)) 361.56/136.61 361.56/136.61 We have to consider all minimal (P,Q,R)-chains. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (120) MNOCProof (EQUIVALENT) 361.56/136.61 We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (121) 361.56/136.61 Obligation: 361.56/136.61 Q DP problem: 361.56/136.61 The TRS P consists of the following rules: 361.56/136.61 361.56/136.61 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.61 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.61 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.61 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.61 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.61 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.61 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.61 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.61 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.61 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.61 TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0))))) 361.56/136.61 361.56/136.61 The TRS R consists of the following rules: 361.56/136.61 361.56/136.61 down(a(a(a(x)))) -> up(c) 361.56/136.61 down(b(u(x))) -> up(b(d(x))) 361.56/136.61 down(d(a(x))) -> up(a(d(x))) 361.56/136.61 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.61 down(a(u(x))) -> up(u(a(x))) 361.56/136.61 down(u(y2)) -> u_flat(down(y2)) 361.56/136.61 down(a(c)) -> a_flat(down(c)) 361.56/136.61 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.61 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.61 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.61 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.61 down(b(c)) -> b_flat(down(c)) 361.56/136.61 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.61 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.61 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.61 down(d(c)) -> d_flat(down(c)) 361.56/136.61 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.61 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.61 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.61 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.61 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.61 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.61 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.61 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.61 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.61 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.61 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.61 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.61 361.56/136.61 Q is empty. 361.56/136.61 We have to consider all (P,Q,R)-chains. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (122) SplitQDPProof (EQUIVALENT) 361.56/136.61 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 361.56/136.61 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (123) 361.56/136.61 Complex Obligation (AND) 361.56/136.61 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (124) 361.56/136.61 Obligation: 361.56/136.61 Q DP problem: 361.56/136.61 The TRS P consists of the following rules: 361.56/136.61 361.56/136.61 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.61 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.61 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.61 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.61 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.61 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.61 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.61 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.61 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.61 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.61 TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0))))) 361.56/136.61 361.56/136.61 The TRS R consists of the following rules: 361.56/136.61 361.56/136.61 down(a(a(a(x)))) -> up(c) 361.56/136.61 down(b(u(x))) -> up(b(d(x))) 361.56/136.61 down(d(a(x))) -> up(a(d(x))) 361.56/136.61 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.61 down(a(u(x))) -> up(u(a(x))) 361.56/136.61 down(u(y2)) -> u_flat(down(y2)) 361.56/136.61 down(a(c)) -> a_flat(down(c)) 361.56/136.61 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.61 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.61 down(a(fresh_constant)) -> a_flat(down(fresh_constant)) 361.56/136.61 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.61 down(b(c)) -> b_flat(down(c)) 361.56/136.61 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.61 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.61 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 361.56/136.61 down(d(c)) -> d_flat(down(c)) 361.56/136.61 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.61 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.61 down(d(fresh_constant)) -> d_flat(down(fresh_constant)) 361.56/136.61 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.61 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.61 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.61 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.61 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.61 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.61 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.61 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.61 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.61 361.56/136.61 The set Q consists of the following terms: 361.56/136.61 361.56/136.61 down(a(a(a(x0)))) 361.56/136.61 down(b(u(x0))) 361.56/136.61 down(d(a(x0))) 361.56/136.61 down(d(b(x0))) 361.56/136.61 down(a(u(x0))) 361.56/136.61 down(u(x0)) 361.56/136.61 down(a(c)) 361.56/136.61 down(a(b(x0))) 361.56/136.61 down(a(d(x0))) 361.56/136.61 down(a(fresh_constant)) 361.56/136.61 down(b(a(x0))) 361.56/136.61 down(b(c)) 361.56/136.61 down(b(b(x0))) 361.56/136.61 down(b(d(x0))) 361.56/136.61 down(b(fresh_constant)) 361.56/136.61 down(d(c)) 361.56/136.61 down(d(u(x0))) 361.56/136.61 down(d(d(x0))) 361.56/136.61 down(d(fresh_constant)) 361.56/136.61 down(a(a(c))) 361.56/136.61 down(a(a(b(x0)))) 361.56/136.61 down(a(a(u(x0)))) 361.56/136.61 down(a(a(d(x0)))) 361.56/136.61 down(a(a(fresh_constant))) 361.56/136.61 a_flat(up(x0)) 361.56/136.61 b_flat(up(x0)) 361.56/136.61 u_flat(up(x0)) 361.56/136.61 d_flat(up(x0)) 361.56/136.61 361.56/136.61 We have to consider all minimal (P,Q,R)-chains. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (125) SemLabProof (SOUND) 361.56/136.61 We found the following model for the rules of the TRSs R and P. 361.56/136.61 Interpretation over the domain with elements from 0 to 1. 361.56/136.61 a: 0 361.56/136.61 c: 0 361.56/136.61 TOP: 0 361.56/136.61 u: 0 361.56/136.61 b: 0 361.56/136.61 d: 0 361.56/136.61 down: 0 361.56/136.61 d_flat: 0 361.56/136.61 fresh_constant: 1 361.56/136.61 up: 0 361.56/136.61 u_flat: 0 361.56/136.61 b_flat: 0 361.56/136.61 a_flat: 0 361.56/136.61 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (126) 361.56/136.61 Obligation: 361.56/136.61 Q DP problem: 361.56/136.61 The TRS P consists of the following rules: 361.56/136.61 361.56/136.61 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.61 TOP.0(up.0(u.1(x0))) -> TOP.0(u_flat.0(down.1(x0))) 361.56/136.61 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0)))) 361.56/136.61 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0)))) 361.56/136.61 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 361.56/136.61 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.61 TOP.0(up.0(d.0(u.1(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.1(x0)))) 361.56/136.61 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.61 TOP.0(up.0(a.0(a.0(b.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.1(x0))))) 361.56/136.61 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.61 TOP.0(up.0(a.0(a.0(d.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.1(x0))))) 361.56/136.61 361.56/136.61 The TRS R consists of the following rules: 361.56/136.61 361.56/136.61 down.0(a.0(a.0(a.0(x)))) -> up.0(c.) 361.56/136.61 down.0(a.0(a.0(a.1(x)))) -> up.0(c.) 361.56/136.61 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.61 down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x))) 361.56/136.61 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.61 down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x))) 361.56/136.61 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.61 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.61 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.61 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.61 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.61 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 361.56/136.61 down.0(a.0(c.)) -> a_flat.0(down.0(c.)) 361.56/136.61 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.61 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.61 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.61 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.61 down.0(a.1(fresh_constant.)) -> a_flat.0(down.1(fresh_constant.)) 361.56/136.61 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.61 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.61 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 361.56/136.61 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.61 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.61 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.61 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.61 down.0(b.1(fresh_constant.)) -> b_flat.0(down.1(fresh_constant.)) 361.56/136.61 down.0(d.0(c.)) -> d_flat.0(down.0(c.)) 361.56/136.61 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.61 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.61 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.61 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.61 down.0(d.1(fresh_constant.)) -> d_flat.0(down.1(fresh_constant.)) 361.56/136.61 down.0(a.0(a.0(c.))) -> a_flat.0(down.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.61 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.61 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.61 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.61 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.61 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) -> a_flat.0(down.0(a.1(fresh_constant.))) 361.56/136.61 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.61 a_flat.0(up.1(x_1)) -> up.0(a.1(x_1)) 361.56/136.61 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.61 d_flat.0(up.1(x_1)) -> up.0(d.1(x_1)) 361.56/136.61 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.61 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 361.56/136.61 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.61 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 361.56/136.61 361.56/136.61 The set Q consists of the following terms: 361.56/136.61 361.56/136.61 down.0(a.0(a.0(a.0(x0)))) 361.56/136.61 down.0(a.0(a.0(a.1(x0)))) 361.56/136.61 down.0(b.0(u.0(x0))) 361.56/136.61 down.0(b.0(u.1(x0))) 361.56/136.61 down.0(d.0(a.0(x0))) 361.56/136.61 down.0(d.0(a.1(x0))) 361.56/136.61 down.0(d.0(b.0(x0))) 361.56/136.61 down.0(d.0(b.1(x0))) 361.56/136.61 down.0(a.0(u.0(x0))) 361.56/136.61 down.0(a.0(u.1(x0))) 361.56/136.61 down.0(u.0(x0)) 361.56/136.61 down.0(u.1(x0)) 361.56/136.61 down.0(a.0(c.)) 361.56/136.61 down.0(a.0(b.0(x0))) 361.56/136.61 down.0(a.0(b.1(x0))) 361.56/136.61 down.0(a.0(d.0(x0))) 361.56/136.61 down.0(a.0(d.1(x0))) 361.56/136.61 down.0(a.1(fresh_constant.)) 361.56/136.61 down.0(b.0(a.0(x0))) 361.56/136.61 down.0(b.0(a.1(x0))) 361.56/136.61 down.0(b.0(c.)) 361.56/136.61 down.0(b.0(b.0(x0))) 361.56/136.61 down.0(b.0(b.1(x0))) 361.56/136.61 down.0(b.0(d.0(x0))) 361.56/136.61 down.0(b.0(d.1(x0))) 361.56/136.61 down.0(b.1(fresh_constant.)) 361.56/136.61 down.0(d.0(c.)) 361.56/136.61 down.0(d.0(u.0(x0))) 361.56/136.61 down.0(d.0(u.1(x0))) 361.56/136.61 down.0(d.0(d.0(x0))) 361.56/136.61 down.0(d.0(d.1(x0))) 361.56/136.61 down.0(d.1(fresh_constant.)) 361.56/136.61 down.0(a.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(x0)))) 361.56/136.61 down.0(a.0(a.0(b.1(x0)))) 361.56/136.61 down.0(a.0(a.0(u.0(x0)))) 361.56/136.61 down.0(a.0(a.0(u.1(x0)))) 361.56/136.61 down.0(a.0(a.0(d.0(x0)))) 361.56/136.61 down.0(a.0(a.0(d.1(x0)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) 361.56/136.61 a_flat.0(up.0(x0)) 361.56/136.61 a_flat.0(up.1(x0)) 361.56/136.61 b_flat.0(up.0(x0)) 361.56/136.61 b_flat.0(up.1(x0)) 361.56/136.61 u_flat.0(up.0(x0)) 361.56/136.61 u_flat.0(up.1(x0)) 361.56/136.61 d_flat.0(up.0(x0)) 361.56/136.61 d_flat.0(up.1(x0)) 361.56/136.61 361.56/136.61 We have to consider all minimal (P,Q,R)-chains. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (127) DependencyGraphProof (EQUIVALENT) 361.56/136.61 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (128) 361.56/136.61 Obligation: 361.56/136.61 Q DP problem: 361.56/136.61 The TRS P consists of the following rules: 361.56/136.61 361.56/136.61 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.61 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.61 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0)))) 361.56/136.61 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.61 TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0)))) 361.56/136.61 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 361.56/136.61 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(a.0(b.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.1(x0))))) 361.56/136.61 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.61 TOP.0(up.0(a.0(a.0(d.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.1(x0))))) 361.56/136.61 361.56/136.61 The TRS R consists of the following rules: 361.56/136.61 361.56/136.61 down.0(a.0(a.0(a.0(x)))) -> up.0(c.) 361.56/136.61 down.0(a.0(a.0(a.1(x)))) -> up.0(c.) 361.56/136.61 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.61 down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x))) 361.56/136.61 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.61 down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x))) 361.56/136.61 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.61 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.61 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.61 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.61 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.61 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 361.56/136.61 down.0(a.0(c.)) -> a_flat.0(down.0(c.)) 361.56/136.61 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.61 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.61 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.61 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.61 down.0(a.1(fresh_constant.)) -> a_flat.0(down.1(fresh_constant.)) 361.56/136.61 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.61 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.61 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 361.56/136.61 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.61 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.61 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.61 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.61 down.0(b.1(fresh_constant.)) -> b_flat.0(down.1(fresh_constant.)) 361.56/136.61 down.0(d.0(c.)) -> d_flat.0(down.0(c.)) 361.56/136.61 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.61 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.61 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.61 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.61 down.0(d.1(fresh_constant.)) -> d_flat.0(down.1(fresh_constant.)) 361.56/136.61 down.0(a.0(a.0(c.))) -> a_flat.0(down.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.61 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.61 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.61 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.61 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.61 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) -> a_flat.0(down.0(a.1(fresh_constant.))) 361.56/136.61 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.61 a_flat.0(up.1(x_1)) -> up.0(a.1(x_1)) 361.56/136.61 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.61 d_flat.0(up.1(x_1)) -> up.0(d.1(x_1)) 361.56/136.61 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.61 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 361.56/136.61 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.61 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 361.56/136.61 361.56/136.61 The set Q consists of the following terms: 361.56/136.61 361.56/136.61 down.0(a.0(a.0(a.0(x0)))) 361.56/136.61 down.0(a.0(a.0(a.1(x0)))) 361.56/136.61 down.0(b.0(u.0(x0))) 361.56/136.61 down.0(b.0(u.1(x0))) 361.56/136.61 down.0(d.0(a.0(x0))) 361.56/136.61 down.0(d.0(a.1(x0))) 361.56/136.61 down.0(d.0(b.0(x0))) 361.56/136.61 down.0(d.0(b.1(x0))) 361.56/136.61 down.0(a.0(u.0(x0))) 361.56/136.61 down.0(a.0(u.1(x0))) 361.56/136.61 down.0(u.0(x0)) 361.56/136.61 down.0(u.1(x0)) 361.56/136.61 down.0(a.0(c.)) 361.56/136.61 down.0(a.0(b.0(x0))) 361.56/136.61 down.0(a.0(b.1(x0))) 361.56/136.61 down.0(a.0(d.0(x0))) 361.56/136.61 down.0(a.0(d.1(x0))) 361.56/136.61 down.0(a.1(fresh_constant.)) 361.56/136.61 down.0(b.0(a.0(x0))) 361.56/136.61 down.0(b.0(a.1(x0))) 361.56/136.61 down.0(b.0(c.)) 361.56/136.61 down.0(b.0(b.0(x0))) 361.56/136.61 down.0(b.0(b.1(x0))) 361.56/136.61 down.0(b.0(d.0(x0))) 361.56/136.61 down.0(b.0(d.1(x0))) 361.56/136.61 down.0(b.1(fresh_constant.)) 361.56/136.61 down.0(d.0(c.)) 361.56/136.61 down.0(d.0(u.0(x0))) 361.56/136.61 down.0(d.0(u.1(x0))) 361.56/136.61 down.0(d.0(d.0(x0))) 361.56/136.61 down.0(d.0(d.1(x0))) 361.56/136.61 down.0(d.1(fresh_constant.)) 361.56/136.61 down.0(a.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(x0)))) 361.56/136.61 down.0(a.0(a.0(b.1(x0)))) 361.56/136.61 down.0(a.0(a.0(u.0(x0)))) 361.56/136.61 down.0(a.0(a.0(u.1(x0)))) 361.56/136.61 down.0(a.0(a.0(d.0(x0)))) 361.56/136.61 down.0(a.0(a.0(d.1(x0)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) 361.56/136.61 a_flat.0(up.0(x0)) 361.56/136.61 a_flat.0(up.1(x0)) 361.56/136.61 b_flat.0(up.0(x0)) 361.56/136.61 b_flat.0(up.1(x0)) 361.56/136.61 u_flat.0(up.0(x0)) 361.56/136.61 u_flat.0(up.1(x0)) 361.56/136.61 d_flat.0(up.0(x0)) 361.56/136.61 d_flat.0(up.1(x0)) 361.56/136.61 361.56/136.61 We have to consider all minimal (P,Q,R)-chains. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (129) UsableRulesReductionPairsProof (EQUIVALENT) 361.56/136.61 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. 361.56/136.61 361.56/136.61 No dependency pairs are removed. 361.56/136.61 361.56/136.61 The following rules are removed from R: 361.56/136.61 361.56/136.61 a_flat.0(up.1(x_1)) -> up.0(a.1(x_1)) 361.56/136.61 d_flat.0(up.1(x_1)) -> up.0(d.1(x_1)) 361.56/136.61 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 361.56/136.61 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 361.56/136.61 Used ordering: POLO with Polynomial interpretation [POLO]: 361.56/136.61 361.56/136.61 POL(TOP.0(x_1)) = x_1 361.56/136.61 POL(a.0(x_1)) = x_1 361.56/136.61 POL(a.1(x_1)) = x_1 361.56/136.61 POL(a_flat.0(x_1)) = x_1 361.56/136.61 POL(b.0(x_1)) = x_1 361.56/136.61 POL(b.1(x_1)) = x_1 361.56/136.61 POL(b_flat.0(x_1)) = x_1 361.56/136.61 POL(c.) = 0 361.56/136.61 POL(d.0(x_1)) = 1 + x_1 361.56/136.61 POL(d.1(x_1)) = 1 + x_1 361.56/136.61 POL(d_flat.0(x_1)) = 1 + x_1 361.56/136.61 POL(down.0(x_1)) = 1 + x_1 361.56/136.61 POL(down.1(x_1)) = 1 + x_1 361.56/136.61 POL(fresh_constant.) = 0 361.56/136.61 POL(u.0(x_1)) = 1 + x_1 361.56/136.61 POL(u.1(x_1)) = 1 + x_1 361.56/136.61 POL(u_flat.0(x_1)) = 1 + x_1 361.56/136.61 POL(up.0(x_1)) = 1 + x_1 361.56/136.61 POL(up.1(x_1)) = 1 + x_1 361.56/136.61 361.56/136.61 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (130) 361.56/136.61 Obligation: 361.56/136.61 Q DP problem: 361.56/136.61 The TRS P consists of the following rules: 361.56/136.61 361.56/136.61 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.61 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.61 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0)))) 361.56/136.61 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.61 TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0)))) 361.56/136.61 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 361.56/136.61 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(a.0(b.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.1(x0))))) 361.56/136.61 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.61 TOP.0(up.0(a.0(a.0(d.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.1(x0))))) 361.56/136.61 361.56/136.61 The TRS R consists of the following rules: 361.56/136.61 361.56/136.61 down.0(d.1(fresh_constant.)) -> d_flat.0(down.1(fresh_constant.)) 361.56/136.61 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.61 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.61 down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x))) 361.56/136.61 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.61 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.61 down.0(d.0(c.)) -> d_flat.0(down.0(c.)) 361.56/136.61 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.61 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.61 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.61 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.61 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.61 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 361.56/136.61 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.61 down.0(a.0(a.0(a.0(x)))) -> up.0(c.) 361.56/136.61 down.0(a.0(a.0(a.1(x)))) -> up.0(c.) 361.56/136.61 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.61 down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x))) 361.56/136.61 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.61 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.61 down.0(a.0(c.)) -> a_flat.0(down.0(c.)) 361.56/136.61 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.61 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.61 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.61 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.61 down.0(a.1(fresh_constant.)) -> a_flat.0(down.1(fresh_constant.)) 361.56/136.61 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.61 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.61 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 361.56/136.61 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.61 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.61 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.61 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.61 down.0(b.1(fresh_constant.)) -> b_flat.0(down.1(fresh_constant.)) 361.56/136.61 down.0(a.0(a.0(c.))) -> a_flat.0(down.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.61 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.61 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.61 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.61 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.61 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) -> a_flat.0(down.0(a.1(fresh_constant.))) 361.56/136.61 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.61 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.61 361.56/136.61 The set Q consists of the following terms: 361.56/136.61 361.56/136.61 down.0(a.0(a.0(a.0(x0)))) 361.56/136.61 down.0(a.0(a.0(a.1(x0)))) 361.56/136.61 down.0(b.0(u.0(x0))) 361.56/136.61 down.0(b.0(u.1(x0))) 361.56/136.61 down.0(d.0(a.0(x0))) 361.56/136.61 down.0(d.0(a.1(x0))) 361.56/136.61 down.0(d.0(b.0(x0))) 361.56/136.61 down.0(d.0(b.1(x0))) 361.56/136.61 down.0(a.0(u.0(x0))) 361.56/136.61 down.0(a.0(u.1(x0))) 361.56/136.61 down.0(u.0(x0)) 361.56/136.61 down.0(u.1(x0)) 361.56/136.61 down.0(a.0(c.)) 361.56/136.61 down.0(a.0(b.0(x0))) 361.56/136.61 down.0(a.0(b.1(x0))) 361.56/136.61 down.0(a.0(d.0(x0))) 361.56/136.61 down.0(a.0(d.1(x0))) 361.56/136.61 down.0(a.1(fresh_constant.)) 361.56/136.61 down.0(b.0(a.0(x0))) 361.56/136.61 down.0(b.0(a.1(x0))) 361.56/136.61 down.0(b.0(c.)) 361.56/136.61 down.0(b.0(b.0(x0))) 361.56/136.61 down.0(b.0(b.1(x0))) 361.56/136.61 down.0(b.0(d.0(x0))) 361.56/136.61 down.0(b.0(d.1(x0))) 361.56/136.61 down.0(b.1(fresh_constant.)) 361.56/136.61 down.0(d.0(c.)) 361.56/136.61 down.0(d.0(u.0(x0))) 361.56/136.61 down.0(d.0(u.1(x0))) 361.56/136.61 down.0(d.0(d.0(x0))) 361.56/136.61 down.0(d.0(d.1(x0))) 361.56/136.61 down.0(d.1(fresh_constant.)) 361.56/136.61 down.0(a.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(x0)))) 361.56/136.61 down.0(a.0(a.0(b.1(x0)))) 361.56/136.61 down.0(a.0(a.0(u.0(x0)))) 361.56/136.61 down.0(a.0(a.0(u.1(x0)))) 361.56/136.61 down.0(a.0(a.0(d.0(x0)))) 361.56/136.61 down.0(a.0(a.0(d.1(x0)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) 361.56/136.61 a_flat.0(up.0(x0)) 361.56/136.61 a_flat.0(up.1(x0)) 361.56/136.61 b_flat.0(up.0(x0)) 361.56/136.61 b_flat.0(up.1(x0)) 361.56/136.61 u_flat.0(up.0(x0)) 361.56/136.61 u_flat.0(up.1(x0)) 361.56/136.61 d_flat.0(up.0(x0)) 361.56/136.61 d_flat.0(up.1(x0)) 361.56/136.61 361.56/136.61 We have to consider all minimal (P,Q,R)-chains. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (131) MRRProof (EQUIVALENT) 361.56/136.61 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. 361.56/136.61 361.56/136.61 361.56/136.61 Strictly oriented rules of the TRS R: 361.56/136.61 361.56/136.61 down.0(d.1(fresh_constant.)) -> d_flat.0(down.1(fresh_constant.)) 361.56/136.61 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 361.56/136.61 down.0(a.1(fresh_constant.)) -> a_flat.0(down.1(fresh_constant.)) 361.56/136.61 down.0(b.1(fresh_constant.)) -> b_flat.0(down.1(fresh_constant.)) 361.56/136.61 361.56/136.61 Used ordering: Polynomial interpretation [POLO]: 361.56/136.61 361.56/136.61 POL(TOP.0(x_1)) = x_1 361.56/136.61 POL(a.0(x_1)) = x_1 361.56/136.61 POL(a.1(x_1)) = x_1 361.56/136.61 POL(a_flat.0(x_1)) = x_1 361.56/136.61 POL(b.0(x_1)) = 1 + x_1 361.56/136.61 POL(b.1(x_1)) = 1 + x_1 361.56/136.61 POL(b_flat.0(x_1)) = 1 + x_1 361.56/136.61 POL(c.) = 0 361.56/136.61 POL(d.0(x_1)) = x_1 361.56/136.61 POL(d.1(x_1)) = x_1 361.56/136.61 POL(d_flat.0(x_1)) = x_1 361.56/136.61 POL(down.0(x_1)) = 1 + x_1 361.56/136.61 POL(down.1(x_1)) = x_1 361.56/136.61 POL(fresh_constant.) = 0 361.56/136.61 POL(u.0(x_1)) = x_1 361.56/136.61 POL(u.1(x_1)) = x_1 361.56/136.61 POL(u_flat.0(x_1)) = x_1 361.56/136.61 POL(up.0(x_1)) = 1 + x_1 361.56/136.61 361.56/136.61 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (132) 361.56/136.61 Obligation: 361.56/136.61 Q DP problem: 361.56/136.61 The TRS P consists of the following rules: 361.56/136.61 361.56/136.61 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.61 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.61 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0)))) 361.56/136.61 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.61 TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0)))) 361.56/136.61 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 361.56/136.61 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0)))) 361.56/136.61 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(a.0(b.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.1(x0))))) 361.56/136.61 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.61 TOP.0(up.0(a.0(a.0(d.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.1(x0))))) 361.56/136.61 361.56/136.61 The TRS R consists of the following rules: 361.56/136.61 361.56/136.61 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.61 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.61 down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x))) 361.56/136.61 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.61 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.61 down.0(d.0(c.)) -> d_flat.0(down.0(c.)) 361.56/136.61 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.61 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.61 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.61 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.61 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.61 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.61 down.0(a.0(a.0(a.0(x)))) -> up.0(c.) 361.56/136.61 down.0(a.0(a.0(a.1(x)))) -> up.0(c.) 361.56/136.61 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.61 down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x))) 361.56/136.61 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.61 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.61 down.0(a.0(c.)) -> a_flat.0(down.0(c.)) 361.56/136.61 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.61 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.61 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.61 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.61 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.61 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.61 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 361.56/136.61 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.61 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.61 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.61 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.61 down.0(a.0(a.0(c.))) -> a_flat.0(down.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.61 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.61 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.61 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.61 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.61 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) -> a_flat.0(down.0(a.1(fresh_constant.))) 361.56/136.61 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.61 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.61 361.56/136.61 The set Q consists of the following terms: 361.56/136.61 361.56/136.61 down.0(a.0(a.0(a.0(x0)))) 361.56/136.61 down.0(a.0(a.0(a.1(x0)))) 361.56/136.61 down.0(b.0(u.0(x0))) 361.56/136.61 down.0(b.0(u.1(x0))) 361.56/136.61 down.0(d.0(a.0(x0))) 361.56/136.61 down.0(d.0(a.1(x0))) 361.56/136.61 down.0(d.0(b.0(x0))) 361.56/136.61 down.0(d.0(b.1(x0))) 361.56/136.61 down.0(a.0(u.0(x0))) 361.56/136.61 down.0(a.0(u.1(x0))) 361.56/136.61 down.0(u.0(x0)) 361.56/136.61 down.0(u.1(x0)) 361.56/136.61 down.0(a.0(c.)) 361.56/136.61 down.0(a.0(b.0(x0))) 361.56/136.61 down.0(a.0(b.1(x0))) 361.56/136.61 down.0(a.0(d.0(x0))) 361.56/136.61 down.0(a.0(d.1(x0))) 361.56/136.61 down.0(a.1(fresh_constant.)) 361.56/136.61 down.0(b.0(a.0(x0))) 361.56/136.61 down.0(b.0(a.1(x0))) 361.56/136.61 down.0(b.0(c.)) 361.56/136.61 down.0(b.0(b.0(x0))) 361.56/136.61 down.0(b.0(b.1(x0))) 361.56/136.61 down.0(b.0(d.0(x0))) 361.56/136.61 down.0(b.0(d.1(x0))) 361.56/136.61 down.0(b.1(fresh_constant.)) 361.56/136.61 down.0(d.0(c.)) 361.56/136.61 down.0(d.0(u.0(x0))) 361.56/136.61 down.0(d.0(u.1(x0))) 361.56/136.61 down.0(d.0(d.0(x0))) 361.56/136.61 down.0(d.0(d.1(x0))) 361.56/136.61 down.0(d.1(fresh_constant.)) 361.56/136.61 down.0(a.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(x0)))) 361.56/136.61 down.0(a.0(a.0(b.1(x0)))) 361.56/136.61 down.0(a.0(a.0(u.0(x0)))) 361.56/136.61 down.0(a.0(a.0(u.1(x0)))) 361.56/136.61 down.0(a.0(a.0(d.0(x0)))) 361.56/136.61 down.0(a.0(a.0(d.1(x0)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) 361.56/136.61 a_flat.0(up.0(x0)) 361.56/136.61 a_flat.0(up.1(x0)) 361.56/136.61 b_flat.0(up.0(x0)) 361.56/136.61 b_flat.0(up.1(x0)) 361.56/136.61 u_flat.0(up.0(x0)) 361.56/136.61 u_flat.0(up.1(x0)) 361.56/136.61 d_flat.0(up.0(x0)) 361.56/136.61 d_flat.0(up.1(x0)) 361.56/136.61 361.56/136.61 We have to consider all minimal (P,Q,R)-chains. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (133) DependencyGraphProof (EQUIVALENT) 361.56/136.61 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (134) 361.56/136.61 Obligation: 361.56/136.61 Q DP problem: 361.56/136.61 The TRS P consists of the following rules: 361.56/136.61 361.56/136.61 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.61 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.61 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.61 361.56/136.61 The TRS R consists of the following rules: 361.56/136.61 361.56/136.61 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.61 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.61 down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x))) 361.56/136.61 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.61 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.61 down.0(d.0(c.)) -> d_flat.0(down.0(c.)) 361.56/136.61 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.61 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.61 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.61 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.61 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.61 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.61 down.0(a.0(a.0(a.0(x)))) -> up.0(c.) 361.56/136.61 down.0(a.0(a.0(a.1(x)))) -> up.0(c.) 361.56/136.61 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.61 down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x))) 361.56/136.61 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.61 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.61 down.0(a.0(c.)) -> a_flat.0(down.0(c.)) 361.56/136.61 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.61 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.61 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.61 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.61 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.61 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.61 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 361.56/136.61 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.61 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.61 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.61 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.61 down.0(a.0(a.0(c.))) -> a_flat.0(down.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.61 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.61 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.61 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.61 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.61 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) -> a_flat.0(down.0(a.1(fresh_constant.))) 361.56/136.61 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.61 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.61 361.56/136.61 The set Q consists of the following terms: 361.56/136.61 361.56/136.61 down.0(a.0(a.0(a.0(x0)))) 361.56/136.61 down.0(a.0(a.0(a.1(x0)))) 361.56/136.61 down.0(b.0(u.0(x0))) 361.56/136.61 down.0(b.0(u.1(x0))) 361.56/136.61 down.0(d.0(a.0(x0))) 361.56/136.61 down.0(d.0(a.1(x0))) 361.56/136.61 down.0(d.0(b.0(x0))) 361.56/136.61 down.0(d.0(b.1(x0))) 361.56/136.61 down.0(a.0(u.0(x0))) 361.56/136.61 down.0(a.0(u.1(x0))) 361.56/136.61 down.0(u.0(x0)) 361.56/136.61 down.0(u.1(x0)) 361.56/136.61 down.0(a.0(c.)) 361.56/136.61 down.0(a.0(b.0(x0))) 361.56/136.61 down.0(a.0(b.1(x0))) 361.56/136.61 down.0(a.0(d.0(x0))) 361.56/136.61 down.0(a.0(d.1(x0))) 361.56/136.61 down.0(a.1(fresh_constant.)) 361.56/136.61 down.0(b.0(a.0(x0))) 361.56/136.61 down.0(b.0(a.1(x0))) 361.56/136.61 down.0(b.0(c.)) 361.56/136.61 down.0(b.0(b.0(x0))) 361.56/136.61 down.0(b.0(b.1(x0))) 361.56/136.61 down.0(b.0(d.0(x0))) 361.56/136.61 down.0(b.0(d.1(x0))) 361.56/136.61 down.0(b.1(fresh_constant.)) 361.56/136.61 down.0(d.0(c.)) 361.56/136.61 down.0(d.0(u.0(x0))) 361.56/136.61 down.0(d.0(u.1(x0))) 361.56/136.61 down.0(d.0(d.0(x0))) 361.56/136.61 down.0(d.0(d.1(x0))) 361.56/136.61 down.0(d.1(fresh_constant.)) 361.56/136.61 down.0(a.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(x0)))) 361.56/136.61 down.0(a.0(a.0(b.1(x0)))) 361.56/136.61 down.0(a.0(a.0(u.0(x0)))) 361.56/136.61 down.0(a.0(a.0(u.1(x0)))) 361.56/136.61 down.0(a.0(a.0(d.0(x0)))) 361.56/136.61 down.0(a.0(a.0(d.1(x0)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) 361.56/136.61 a_flat.0(up.0(x0)) 361.56/136.61 a_flat.0(up.1(x0)) 361.56/136.61 b_flat.0(up.0(x0)) 361.56/136.61 b_flat.0(up.1(x0)) 361.56/136.61 u_flat.0(up.0(x0)) 361.56/136.61 u_flat.0(up.1(x0)) 361.56/136.61 d_flat.0(up.0(x0)) 361.56/136.61 d_flat.0(up.1(x0)) 361.56/136.61 361.56/136.61 We have to consider all minimal (P,Q,R)-chains. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (135) MRRProof (EQUIVALENT) 361.56/136.61 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. 361.56/136.61 361.56/136.61 361.56/136.61 Strictly oriented rules of the TRS R: 361.56/136.61 361.56/136.61 down.0(a.0(a.0(a.1(x)))) -> up.0(c.) 361.56/136.61 361.56/136.61 Used ordering: Polynomial interpretation [POLO]: 361.56/136.61 361.56/136.61 POL(TOP.0(x_1)) = x_1 361.56/136.61 POL(a.0(x_1)) = x_1 361.56/136.61 POL(a.1(x_1)) = 1 + x_1 361.56/136.61 POL(a_flat.0(x_1)) = x_1 361.56/136.61 POL(b.0(x_1)) = x_1 361.56/136.61 POL(b.1(x_1)) = x_1 361.56/136.61 POL(b_flat.0(x_1)) = x_1 361.56/136.61 POL(c.) = 0 361.56/136.61 POL(d.0(x_1)) = x_1 361.56/136.61 POL(d.1(x_1)) = 1 + x_1 361.56/136.61 POL(d_flat.0(x_1)) = x_1 361.56/136.61 POL(down.0(x_1)) = x_1 361.56/136.61 POL(fresh_constant.) = 0 361.56/136.61 POL(u.0(x_1)) = x_1 361.56/136.61 POL(u.1(x_1)) = 1 + x_1 361.56/136.61 POL(u_flat.0(x_1)) = x_1 361.56/136.61 POL(up.0(x_1)) = x_1 361.56/136.61 361.56/136.61 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (136) 361.56/136.61 Obligation: 361.56/136.61 Q DP problem: 361.56/136.61 The TRS P consists of the following rules: 361.56/136.61 361.56/136.61 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.61 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.61 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.61 361.56/136.61 The TRS R consists of the following rules: 361.56/136.61 361.56/136.61 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.61 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.61 down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x))) 361.56/136.61 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.61 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.61 down.0(d.0(c.)) -> d_flat.0(down.0(c.)) 361.56/136.61 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.61 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.61 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.61 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.61 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.61 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.61 down.0(a.0(a.0(a.0(x)))) -> up.0(c.) 361.56/136.61 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.61 down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x))) 361.56/136.61 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.61 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.61 down.0(a.0(c.)) -> a_flat.0(down.0(c.)) 361.56/136.61 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.61 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.61 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.61 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.61 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.61 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.61 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 361.56/136.61 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.61 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.61 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.61 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.61 down.0(a.0(a.0(c.))) -> a_flat.0(down.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.61 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.61 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.61 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.61 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.61 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) -> a_flat.0(down.0(a.1(fresh_constant.))) 361.56/136.61 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.61 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.61 361.56/136.61 The set Q consists of the following terms: 361.56/136.61 361.56/136.61 down.0(a.0(a.0(a.0(x0)))) 361.56/136.61 down.0(a.0(a.0(a.1(x0)))) 361.56/136.61 down.0(b.0(u.0(x0))) 361.56/136.61 down.0(b.0(u.1(x0))) 361.56/136.61 down.0(d.0(a.0(x0))) 361.56/136.61 down.0(d.0(a.1(x0))) 361.56/136.61 down.0(d.0(b.0(x0))) 361.56/136.61 down.0(d.0(b.1(x0))) 361.56/136.61 down.0(a.0(u.0(x0))) 361.56/136.61 down.0(a.0(u.1(x0))) 361.56/136.61 down.0(u.0(x0)) 361.56/136.61 down.0(u.1(x0)) 361.56/136.61 down.0(a.0(c.)) 361.56/136.61 down.0(a.0(b.0(x0))) 361.56/136.61 down.0(a.0(b.1(x0))) 361.56/136.61 down.0(a.0(d.0(x0))) 361.56/136.61 down.0(a.0(d.1(x0))) 361.56/136.61 down.0(a.1(fresh_constant.)) 361.56/136.61 down.0(b.0(a.0(x0))) 361.56/136.61 down.0(b.0(a.1(x0))) 361.56/136.61 down.0(b.0(c.)) 361.56/136.61 down.0(b.0(b.0(x0))) 361.56/136.61 down.0(b.0(b.1(x0))) 361.56/136.61 down.0(b.0(d.0(x0))) 361.56/136.61 down.0(b.0(d.1(x0))) 361.56/136.61 down.0(b.1(fresh_constant.)) 361.56/136.61 down.0(d.0(c.)) 361.56/136.61 down.0(d.0(u.0(x0))) 361.56/136.61 down.0(d.0(u.1(x0))) 361.56/136.61 down.0(d.0(d.0(x0))) 361.56/136.61 down.0(d.0(d.1(x0))) 361.56/136.61 down.0(d.1(fresh_constant.)) 361.56/136.61 down.0(a.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(x0)))) 361.56/136.61 down.0(a.0(a.0(b.1(x0)))) 361.56/136.61 down.0(a.0(a.0(u.0(x0)))) 361.56/136.61 down.0(a.0(a.0(u.1(x0)))) 361.56/136.61 down.0(a.0(a.0(d.0(x0)))) 361.56/136.61 down.0(a.0(a.0(d.1(x0)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) 361.56/136.61 a_flat.0(up.0(x0)) 361.56/136.61 a_flat.0(up.1(x0)) 361.56/136.61 b_flat.0(up.0(x0)) 361.56/136.61 b_flat.0(up.1(x0)) 361.56/136.61 u_flat.0(up.0(x0)) 361.56/136.61 u_flat.0(up.1(x0)) 361.56/136.61 d_flat.0(up.0(x0)) 361.56/136.61 d_flat.0(up.1(x0)) 361.56/136.61 361.56/136.61 We have to consider all minimal (P,Q,R)-chains. 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (137) MRRProof (EQUIVALENT) 361.56/136.61 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. 361.56/136.61 361.56/136.61 361.56/136.61 Strictly oriented rules of the TRS R: 361.56/136.61 361.56/136.61 down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x))) 361.56/136.61 down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x))) 361.56/136.61 361.56/136.61 Used ordering: Polynomial interpretation [POLO]: 361.56/136.61 361.56/136.61 POL(TOP.0(x_1)) = x_1 361.56/136.61 POL(a.0(x_1)) = x_1 361.56/136.61 POL(a.1(x_1)) = 1 + x_1 361.56/136.61 POL(a_flat.0(x_1)) = x_1 361.56/136.61 POL(b.0(x_1)) = x_1 361.56/136.61 POL(b.1(x_1)) = x_1 361.56/136.61 POL(b_flat.0(x_1)) = x_1 361.56/136.61 POL(c.) = 0 361.56/136.61 POL(d.0(x_1)) = x_1 361.56/136.61 POL(d.1(x_1)) = x_1 361.56/136.61 POL(d_flat.0(x_1)) = x_1 361.56/136.61 POL(down.0(x_1)) = x_1 361.56/136.61 POL(fresh_constant.) = 0 361.56/136.61 POL(u.0(x_1)) = x_1 361.56/136.61 POL(u.1(x_1)) = 1 + x_1 361.56/136.61 POL(u_flat.0(x_1)) = x_1 361.56/136.61 POL(up.0(x_1)) = x_1 361.56/136.61 361.56/136.61 361.56/136.61 ---------------------------------------- 361.56/136.61 361.56/136.61 (138) 361.56/136.61 Obligation: 361.56/136.61 Q DP problem: 361.56/136.61 The TRS P consists of the following rules: 361.56/136.61 361.56/136.61 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.61 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.61 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.61 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.61 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.61 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.61 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.61 361.56/136.61 The TRS R consists of the following rules: 361.56/136.61 361.56/136.61 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.61 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.61 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.61 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.61 down.0(d.0(c.)) -> d_flat.0(down.0(c.)) 361.56/136.61 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.61 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.61 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.61 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.61 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.61 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.61 down.0(a.0(a.0(a.0(x)))) -> up.0(c.) 361.56/136.61 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.61 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.61 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.61 down.0(a.0(c.)) -> a_flat.0(down.0(c.)) 361.56/136.61 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.61 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.61 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.61 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.61 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.61 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.61 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 361.56/136.61 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.61 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.61 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.61 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.61 down.0(a.0(a.0(c.))) -> a_flat.0(down.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.61 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.61 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.61 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.61 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.61 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) -> a_flat.0(down.0(a.1(fresh_constant.))) 361.56/136.61 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.61 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.61 361.56/136.61 The set Q consists of the following terms: 361.56/136.61 361.56/136.61 down.0(a.0(a.0(a.0(x0)))) 361.56/136.61 down.0(a.0(a.0(a.1(x0)))) 361.56/136.61 down.0(b.0(u.0(x0))) 361.56/136.61 down.0(b.0(u.1(x0))) 361.56/136.61 down.0(d.0(a.0(x0))) 361.56/136.61 down.0(d.0(a.1(x0))) 361.56/136.61 down.0(d.0(b.0(x0))) 361.56/136.61 down.0(d.0(b.1(x0))) 361.56/136.61 down.0(a.0(u.0(x0))) 361.56/136.61 down.0(a.0(u.1(x0))) 361.56/136.61 down.0(u.0(x0)) 361.56/136.61 down.0(u.1(x0)) 361.56/136.61 down.0(a.0(c.)) 361.56/136.61 down.0(a.0(b.0(x0))) 361.56/136.61 down.0(a.0(b.1(x0))) 361.56/136.61 down.0(a.0(d.0(x0))) 361.56/136.61 down.0(a.0(d.1(x0))) 361.56/136.61 down.0(a.1(fresh_constant.)) 361.56/136.61 down.0(b.0(a.0(x0))) 361.56/136.61 down.0(b.0(a.1(x0))) 361.56/136.61 down.0(b.0(c.)) 361.56/136.61 down.0(b.0(b.0(x0))) 361.56/136.61 down.0(b.0(b.1(x0))) 361.56/136.61 down.0(b.0(d.0(x0))) 361.56/136.61 down.0(b.0(d.1(x0))) 361.56/136.61 down.0(b.1(fresh_constant.)) 361.56/136.61 down.0(d.0(c.)) 361.56/136.61 down.0(d.0(u.0(x0))) 361.56/136.61 down.0(d.0(u.1(x0))) 361.56/136.61 down.0(d.0(d.0(x0))) 361.56/136.61 down.0(d.0(d.1(x0))) 361.56/136.61 down.0(d.1(fresh_constant.)) 361.56/136.61 down.0(a.0(a.0(c.))) 361.56/136.61 down.0(a.0(a.0(b.0(x0)))) 361.56/136.61 down.0(a.0(a.0(b.1(x0)))) 361.56/136.61 down.0(a.0(a.0(u.0(x0)))) 361.56/136.61 down.0(a.0(a.0(u.1(x0)))) 361.56/136.61 down.0(a.0(a.0(d.0(x0)))) 361.56/136.61 down.0(a.0(a.0(d.1(x0)))) 361.56/136.61 down.0(a.0(a.1(fresh_constant.))) 361.56/136.61 a_flat.0(up.0(x0)) 361.56/136.61 a_flat.0(up.1(x0)) 361.56/136.61 b_flat.0(up.0(x0)) 361.56/136.61 b_flat.0(up.1(x0)) 361.56/136.61 u_flat.0(up.0(x0)) 361.56/136.61 u_flat.0(up.1(x0)) 361.56/136.61 d_flat.0(up.0(x0)) 361.56/136.61 d_flat.0(up.1(x0)) 361.56/136.61 361.56/136.61 We have to consider all minimal (P,Q,R)-chains. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (139) MRRProof (EQUIVALENT) 361.56/136.62 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. 361.56/136.62 361.56/136.62 361.56/136.62 Strictly oriented rules of the TRS R: 361.56/136.62 361.56/136.62 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.62 361.56/136.62 Used ordering: Polynomial interpretation [POLO]: 361.56/136.62 361.56/136.62 POL(TOP.0(x_1)) = x_1 361.56/136.62 POL(a.0(x_1)) = x_1 361.56/136.62 POL(a.1(x_1)) = x_1 361.56/136.62 POL(a_flat.0(x_1)) = x_1 361.56/136.62 POL(b.0(x_1)) = x_1 361.56/136.62 POL(b.1(x_1)) = x_1 361.56/136.62 POL(b_flat.0(x_1)) = x_1 361.56/136.62 POL(c.) = 0 361.56/136.62 POL(d.0(x_1)) = x_1 361.56/136.62 POL(d.1(x_1)) = x_1 361.56/136.62 POL(d_flat.0(x_1)) = x_1 361.56/136.62 POL(down.0(x_1)) = x_1 361.56/136.62 POL(fresh_constant.) = 0 361.56/136.62 POL(u.0(x_1)) = x_1 361.56/136.62 POL(u.1(x_1)) = 1 + x_1 361.56/136.62 POL(u_flat.0(x_1)) = x_1 361.56/136.62 POL(up.0(x_1)) = x_1 361.56/136.62 361.56/136.62 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (140) 361.56/136.62 Obligation: 361.56/136.62 Q DP problem: 361.56/136.62 The TRS P consists of the following rules: 361.56/136.62 361.56/136.62 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.62 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.62 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.62 361.56/136.62 The TRS R consists of the following rules: 361.56/136.62 361.56/136.62 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.62 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.62 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.62 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.62 down.0(d.0(c.)) -> d_flat.0(down.0(c.)) 361.56/136.62 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.62 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.62 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.62 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.62 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.62 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.62 down.0(a.0(a.0(a.0(x)))) -> up.0(c.) 361.56/136.62 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.62 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.62 down.0(a.0(c.)) -> a_flat.0(down.0(c.)) 361.56/136.62 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.62 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.62 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.62 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.62 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.62 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.62 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 361.56/136.62 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.62 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.62 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.62 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.62 down.0(a.0(a.0(c.))) -> a_flat.0(down.0(a.0(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.62 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.62 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.62 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.62 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.62 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.62 down.0(a.0(a.1(fresh_constant.))) -> a_flat.0(down.0(a.1(fresh_constant.))) 361.56/136.62 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.62 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.62 361.56/136.62 The set Q consists of the following terms: 361.56/136.62 361.56/136.62 down.0(a.0(a.0(a.0(x0)))) 361.56/136.62 down.0(a.0(a.0(a.1(x0)))) 361.56/136.62 down.0(b.0(u.0(x0))) 361.56/136.62 down.0(b.0(u.1(x0))) 361.56/136.62 down.0(d.0(a.0(x0))) 361.56/136.62 down.0(d.0(a.1(x0))) 361.56/136.62 down.0(d.0(b.0(x0))) 361.56/136.62 down.0(d.0(b.1(x0))) 361.56/136.62 down.0(a.0(u.0(x0))) 361.56/136.62 down.0(a.0(u.1(x0))) 361.56/136.62 down.0(u.0(x0)) 361.56/136.62 down.0(u.1(x0)) 361.56/136.62 down.0(a.0(c.)) 361.56/136.62 down.0(a.0(b.0(x0))) 361.56/136.62 down.0(a.0(b.1(x0))) 361.56/136.62 down.0(a.0(d.0(x0))) 361.56/136.62 down.0(a.0(d.1(x0))) 361.56/136.62 down.0(a.1(fresh_constant.)) 361.56/136.62 down.0(b.0(a.0(x0))) 361.56/136.62 down.0(b.0(a.1(x0))) 361.56/136.62 down.0(b.0(c.)) 361.56/136.62 down.0(b.0(b.0(x0))) 361.56/136.62 down.0(b.0(b.1(x0))) 361.56/136.62 down.0(b.0(d.0(x0))) 361.56/136.62 down.0(b.0(d.1(x0))) 361.56/136.62 down.0(b.1(fresh_constant.)) 361.56/136.62 down.0(d.0(c.)) 361.56/136.62 down.0(d.0(u.0(x0))) 361.56/136.62 down.0(d.0(u.1(x0))) 361.56/136.62 down.0(d.0(d.0(x0))) 361.56/136.62 down.0(d.0(d.1(x0))) 361.56/136.62 down.0(d.1(fresh_constant.)) 361.56/136.62 down.0(a.0(a.0(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(x0)))) 361.56/136.62 down.0(a.0(a.0(b.1(x0)))) 361.56/136.62 down.0(a.0(a.0(u.0(x0)))) 361.56/136.62 down.0(a.0(a.0(u.1(x0)))) 361.56/136.62 down.0(a.0(a.0(d.0(x0)))) 361.56/136.62 down.0(a.0(a.0(d.1(x0)))) 361.56/136.62 down.0(a.0(a.1(fresh_constant.))) 361.56/136.62 a_flat.0(up.0(x0)) 361.56/136.62 a_flat.0(up.1(x0)) 361.56/136.62 b_flat.0(up.0(x0)) 361.56/136.62 b_flat.0(up.1(x0)) 361.56/136.62 u_flat.0(up.0(x0)) 361.56/136.62 u_flat.0(up.1(x0)) 361.56/136.62 d_flat.0(up.0(x0)) 361.56/136.62 d_flat.0(up.1(x0)) 361.56/136.62 361.56/136.62 We have to consider all minimal (P,Q,R)-chains. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (141) PisEmptyProof (SOUND) 361.56/136.62 The TRS P is empty. Hence, there is no (P,Q,R) chain. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (142) 361.56/136.62 TRUE 361.56/136.62 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (143) 361.56/136.62 Obligation: 361.56/136.62 Q DP problem: 361.56/136.62 The TRS P consists of the following rules: 361.56/136.62 361.56/136.62 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.62 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.62 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.62 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.62 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.62 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.62 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.62 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.62 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.62 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.62 TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0))))) 361.56/136.62 361.56/136.62 The TRS R consists of the following rules: 361.56/136.62 361.56/136.62 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.62 down(d(a(x))) -> up(a(d(x))) 361.56/136.62 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.62 down(d(c)) -> d_flat(down(c)) 361.56/136.62 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.62 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.62 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.62 down(u(y2)) -> u_flat(down(y2)) 361.56/136.62 down(a(a(a(x)))) -> up(c) 361.56/136.62 down(b(u(x))) -> up(b(d(x))) 361.56/136.62 down(a(u(x))) -> up(u(a(x))) 361.56/136.62 down(a(c)) -> a_flat(down(c)) 361.56/136.62 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.62 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.62 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.62 down(b(c)) -> b_flat(down(c)) 361.56/136.62 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.62 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.62 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.62 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.62 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.62 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.62 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.62 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.62 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.62 361.56/136.62 The set Q consists of the following terms: 361.56/136.62 361.56/136.62 down(a(a(a(x0)))) 361.56/136.62 down(b(u(x0))) 361.56/136.62 down(d(a(x0))) 361.56/136.62 down(d(b(x0))) 361.56/136.62 down(a(u(x0))) 361.56/136.62 down(u(x0)) 361.56/136.62 down(a(c)) 361.56/136.62 down(a(b(x0))) 361.56/136.62 down(a(d(x0))) 361.56/136.62 down(a(fresh_constant)) 361.56/136.62 down(b(a(x0))) 361.56/136.62 down(b(c)) 361.56/136.62 down(b(b(x0))) 361.56/136.62 down(b(d(x0))) 361.56/136.62 down(b(fresh_constant)) 361.56/136.62 down(d(c)) 361.56/136.62 down(d(u(x0))) 361.56/136.62 down(d(d(x0))) 361.56/136.62 down(d(fresh_constant)) 361.56/136.62 down(a(a(c))) 361.56/136.62 down(a(a(b(x0)))) 361.56/136.62 down(a(a(u(x0)))) 361.56/136.62 down(a(a(d(x0)))) 361.56/136.62 down(a(a(fresh_constant))) 361.56/136.62 a_flat(up(x0)) 361.56/136.62 b_flat(up(x0)) 361.56/136.62 u_flat(up(x0)) 361.56/136.62 d_flat(up(x0)) 361.56/136.62 361.56/136.62 We have to consider all minimal (P,Q,R)-chains. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (144) SplitQDPProof (EQUIVALENT) 361.56/136.62 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 361.56/136.62 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (145) 361.56/136.62 Complex Obligation (AND) 361.56/136.62 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (146) 361.56/136.62 Obligation: 361.56/136.62 Q DP problem: 361.56/136.62 The TRS P consists of the following rules: 361.56/136.62 361.56/136.62 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.62 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.62 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.62 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.62 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.62 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.62 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.62 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.62 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.62 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.62 TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0))))) 361.56/136.62 361.56/136.62 The TRS R consists of the following rules: 361.56/136.62 361.56/136.62 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.62 down(d(a(x))) -> up(a(d(x))) 361.56/136.62 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.62 down(d(c)) -> d_flat(down(c)) 361.56/136.62 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.62 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.62 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.62 down(u(y2)) -> u_flat(down(y2)) 361.56/136.62 down(a(a(a(x)))) -> up(c) 361.56/136.62 down(b(u(x))) -> up(b(d(x))) 361.56/136.62 down(a(u(x))) -> up(u(a(x))) 361.56/136.62 down(a(c)) -> a_flat(down(c)) 361.56/136.62 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.62 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.62 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.62 down(b(c)) -> b_flat(down(c)) 361.56/136.62 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.62 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.62 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.62 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.62 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.62 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.62 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.62 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.62 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.62 361.56/136.62 The set Q consists of the following terms: 361.56/136.62 361.56/136.62 down(a(a(a(x0)))) 361.56/136.62 down(b(u(x0))) 361.56/136.62 down(d(a(x0))) 361.56/136.62 down(d(b(x0))) 361.56/136.62 down(a(u(x0))) 361.56/136.62 down(u(x0)) 361.56/136.62 down(a(c)) 361.56/136.62 down(a(b(x0))) 361.56/136.62 down(a(d(x0))) 361.56/136.62 down(a(fresh_constant)) 361.56/136.62 down(b(a(x0))) 361.56/136.62 down(b(c)) 361.56/136.62 down(b(b(x0))) 361.56/136.62 down(b(d(x0))) 361.56/136.62 down(b(fresh_constant)) 361.56/136.62 down(d(c)) 361.56/136.62 down(d(u(x0))) 361.56/136.62 down(d(d(x0))) 361.56/136.62 down(d(fresh_constant)) 361.56/136.62 down(a(a(c))) 361.56/136.62 down(a(a(b(x0)))) 361.56/136.62 down(a(a(u(x0)))) 361.56/136.62 down(a(a(d(x0)))) 361.56/136.62 down(a(a(fresh_constant))) 361.56/136.62 a_flat(up(x0)) 361.56/136.62 b_flat(up(x0)) 361.56/136.62 u_flat(up(x0)) 361.56/136.62 d_flat(up(x0)) 361.56/136.62 361.56/136.62 We have to consider all minimal (P,Q,R)-chains. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (147) SemLabProof (SOUND) 361.56/136.62 We found the following model for the rules of the TRSs R and P. 361.56/136.62 Interpretation over the domain with elements from 0 to 1. 361.56/136.62 a: 0 361.56/136.62 c: 1 361.56/136.62 TOP: 0 361.56/136.62 u: 0 361.56/136.62 b: 0 361.56/136.62 d: 0 361.56/136.62 down: 0 361.56/136.62 d_flat: 0 361.56/136.62 fresh_constant: 0 361.56/136.62 up: 0 361.56/136.62 u_flat: 0 361.56/136.62 b_flat: 0 361.56/136.62 a_flat: 0 361.56/136.62 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (148) 361.56/136.62 Obligation: 361.56/136.62 Q DP problem: 361.56/136.62 The TRS P consists of the following rules: 361.56/136.62 361.56/136.62 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0)))) 361.56/136.62 TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0)))) 361.56/136.62 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.62 TOP.0(up.0(u.1(x0))) -> TOP.0(u_flat.0(down.1(x0))) 361.56/136.62 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0)))) 361.56/136.62 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0)))) 361.56/136.62 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0)))) 361.56/136.62 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 361.56/136.62 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0)))) 361.56/136.62 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.62 TOP.0(up.0(d.0(u.1(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.1(x0)))) 361.56/136.62 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.62 TOP.0(up.0(a.0(a.0(b.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.1(x0))))) 361.56/136.62 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.62 TOP.0(up.0(a.0(a.0(d.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.1(x0))))) 361.56/136.62 361.56/136.62 The TRS R consists of the following rules: 361.56/136.62 361.56/136.62 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.62 a_flat.0(up.1(x_1)) -> up.0(a.1(x_1)) 361.56/136.62 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.62 down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x))) 361.56/136.62 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.62 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.62 down.0(d.1(c.)) -> d_flat.0(down.1(c.)) 361.56/136.62 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.62 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.62 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.62 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.62 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.62 d_flat.0(up.1(x_1)) -> up.0(d.1(x_1)) 361.56/136.62 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.62 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 361.56/136.62 down.0(a.0(a.0(a.0(x)))) -> up.1(c.) 361.56/136.62 down.0(a.0(a.0(a.1(x)))) -> up.1(c.) 361.56/136.62 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.62 down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x))) 361.56/136.62 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.62 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.62 down.0(a.1(c.)) -> a_flat.0(down.1(c.)) 361.56/136.62 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.62 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.62 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.62 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.62 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.62 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.62 down.0(b.1(c.)) -> b_flat.0(down.1(c.)) 361.56/136.62 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.62 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.62 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.62 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.62 down.0(a.0(a.1(c.))) -> a_flat.0(down.0(a.1(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.62 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.62 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.62 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.62 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.62 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.62 down.0(a.0(a.0(fresh_constant.))) -> a_flat.0(down.0(a.0(fresh_constant.))) 361.56/136.62 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.62 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 361.56/136.62 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.62 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 361.56/136.62 361.56/136.62 The set Q consists of the following terms: 361.56/136.62 361.56/136.62 down.0(a.0(a.0(a.0(x0)))) 361.56/136.62 down.0(a.0(a.0(a.1(x0)))) 361.56/136.62 down.0(b.0(u.0(x0))) 361.56/136.62 down.0(b.0(u.1(x0))) 361.56/136.62 down.0(d.0(a.0(x0))) 361.56/136.62 down.0(d.0(a.1(x0))) 361.56/136.62 down.0(d.0(b.0(x0))) 361.56/136.62 down.0(d.0(b.1(x0))) 361.56/136.62 down.0(a.0(u.0(x0))) 361.56/136.62 down.0(a.0(u.1(x0))) 361.56/136.62 down.0(u.0(x0)) 361.56/136.62 down.0(u.1(x0)) 361.56/136.62 down.0(a.1(c.)) 361.56/136.62 down.0(a.0(b.0(x0))) 361.56/136.62 down.0(a.0(b.1(x0))) 361.56/136.62 down.0(a.0(d.0(x0))) 361.56/136.62 down.0(a.0(d.1(x0))) 361.56/136.62 down.0(a.0(fresh_constant.)) 361.56/136.62 down.0(b.0(a.0(x0))) 361.56/136.62 down.0(b.0(a.1(x0))) 361.56/136.62 down.0(b.1(c.)) 361.56/136.62 down.0(b.0(b.0(x0))) 361.56/136.62 down.0(b.0(b.1(x0))) 361.56/136.62 down.0(b.0(d.0(x0))) 361.56/136.62 down.0(b.0(d.1(x0))) 361.56/136.62 down.0(b.0(fresh_constant.)) 361.56/136.62 down.0(d.1(c.)) 361.56/136.62 down.0(d.0(u.0(x0))) 361.56/136.62 down.0(d.0(u.1(x0))) 361.56/136.62 down.0(d.0(d.0(x0))) 361.56/136.62 down.0(d.0(d.1(x0))) 361.56/136.62 down.0(d.0(fresh_constant.)) 361.56/136.62 down.0(a.0(a.1(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(x0)))) 361.56/136.62 down.0(a.0(a.0(b.1(x0)))) 361.56/136.62 down.0(a.0(a.0(u.0(x0)))) 361.56/136.62 down.0(a.0(a.0(u.1(x0)))) 361.56/136.62 down.0(a.0(a.0(d.0(x0)))) 361.56/136.62 down.0(a.0(a.0(d.1(x0)))) 361.56/136.62 down.0(a.0(a.0(fresh_constant.))) 361.56/136.62 a_flat.0(up.0(x0)) 361.56/136.62 a_flat.0(up.1(x0)) 361.56/136.62 b_flat.0(up.0(x0)) 361.56/136.62 b_flat.0(up.1(x0)) 361.56/136.62 u_flat.0(up.0(x0)) 361.56/136.62 u_flat.0(up.1(x0)) 361.56/136.62 d_flat.0(up.0(x0)) 361.56/136.62 d_flat.0(up.1(x0)) 361.56/136.62 361.56/136.62 We have to consider all minimal (P,Q,R)-chains. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (149) DependencyGraphProof (EQUIVALENT) 361.56/136.62 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (150) 361.56/136.62 Obligation: 361.56/136.62 Q DP problem: 361.56/136.62 The TRS P consists of the following rules: 361.56/136.62 361.56/136.62 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0)))) 361.56/136.62 TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0)))) 361.56/136.62 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.62 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.62 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0)))) 361.56/136.62 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0)))) 361.56/136.62 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.62 TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0)))) 361.56/136.62 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 361.56/136.62 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0)))) 361.56/136.62 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(a.0(b.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.1(x0))))) 361.56/136.62 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.62 TOP.0(up.0(a.0(a.0(d.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.1(x0))))) 361.56/136.62 361.56/136.62 The TRS R consists of the following rules: 361.56/136.62 361.56/136.62 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.62 a_flat.0(up.1(x_1)) -> up.0(a.1(x_1)) 361.56/136.62 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.62 down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x))) 361.56/136.62 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.62 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.62 down.0(d.1(c.)) -> d_flat.0(down.1(c.)) 361.56/136.62 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.62 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.62 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.62 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.62 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.62 d_flat.0(up.1(x_1)) -> up.0(d.1(x_1)) 361.56/136.62 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.62 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 361.56/136.62 down.0(a.0(a.0(a.0(x)))) -> up.1(c.) 361.56/136.62 down.0(a.0(a.0(a.1(x)))) -> up.1(c.) 361.56/136.62 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.62 down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x))) 361.56/136.62 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.62 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.62 down.0(a.1(c.)) -> a_flat.0(down.1(c.)) 361.56/136.62 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.62 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.62 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.62 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.62 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.62 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.62 down.0(b.1(c.)) -> b_flat.0(down.1(c.)) 361.56/136.62 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.62 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.62 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.62 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.62 down.0(a.0(a.1(c.))) -> a_flat.0(down.0(a.1(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.62 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.62 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.62 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.62 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.62 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.62 down.0(a.0(a.0(fresh_constant.))) -> a_flat.0(down.0(a.0(fresh_constant.))) 361.56/136.62 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.62 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 361.56/136.62 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.62 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 361.56/136.62 361.56/136.62 The set Q consists of the following terms: 361.56/136.62 361.56/136.62 down.0(a.0(a.0(a.0(x0)))) 361.56/136.62 down.0(a.0(a.0(a.1(x0)))) 361.56/136.62 down.0(b.0(u.0(x0))) 361.56/136.62 down.0(b.0(u.1(x0))) 361.56/136.62 down.0(d.0(a.0(x0))) 361.56/136.62 down.0(d.0(a.1(x0))) 361.56/136.62 down.0(d.0(b.0(x0))) 361.56/136.62 down.0(d.0(b.1(x0))) 361.56/136.62 down.0(a.0(u.0(x0))) 361.56/136.62 down.0(a.0(u.1(x0))) 361.56/136.62 down.0(u.0(x0)) 361.56/136.62 down.0(u.1(x0)) 361.56/136.62 down.0(a.1(c.)) 361.56/136.62 down.0(a.0(b.0(x0))) 361.56/136.62 down.0(a.0(b.1(x0))) 361.56/136.62 down.0(a.0(d.0(x0))) 361.56/136.62 down.0(a.0(d.1(x0))) 361.56/136.62 down.0(a.0(fresh_constant.)) 361.56/136.62 down.0(b.0(a.0(x0))) 361.56/136.62 down.0(b.0(a.1(x0))) 361.56/136.62 down.0(b.1(c.)) 361.56/136.62 down.0(b.0(b.0(x0))) 361.56/136.62 down.0(b.0(b.1(x0))) 361.56/136.62 down.0(b.0(d.0(x0))) 361.56/136.62 down.0(b.0(d.1(x0))) 361.56/136.62 down.0(b.0(fresh_constant.)) 361.56/136.62 down.0(d.1(c.)) 361.56/136.62 down.0(d.0(u.0(x0))) 361.56/136.62 down.0(d.0(u.1(x0))) 361.56/136.62 down.0(d.0(d.0(x0))) 361.56/136.62 down.0(d.0(d.1(x0))) 361.56/136.62 down.0(d.0(fresh_constant.)) 361.56/136.62 down.0(a.0(a.1(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(x0)))) 361.56/136.62 down.0(a.0(a.0(b.1(x0)))) 361.56/136.62 down.0(a.0(a.0(u.0(x0)))) 361.56/136.62 down.0(a.0(a.0(u.1(x0)))) 361.56/136.62 down.0(a.0(a.0(d.0(x0)))) 361.56/136.62 down.0(a.0(a.0(d.1(x0)))) 361.56/136.62 down.0(a.0(a.0(fresh_constant.))) 361.56/136.62 a_flat.0(up.0(x0)) 361.56/136.62 a_flat.0(up.1(x0)) 361.56/136.62 b_flat.0(up.0(x0)) 361.56/136.62 b_flat.0(up.1(x0)) 361.56/136.62 u_flat.0(up.0(x0)) 361.56/136.62 u_flat.0(up.1(x0)) 361.56/136.62 d_flat.0(up.0(x0)) 361.56/136.62 d_flat.0(up.1(x0)) 361.56/136.62 361.56/136.62 We have to consider all minimal (P,Q,R)-chains. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (151) MRRProof (EQUIVALENT) 361.56/136.62 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. 361.56/136.62 361.56/136.62 361.56/136.62 Strictly oriented rules of the TRS R: 361.56/136.62 361.56/136.62 down.0(d.1(c.)) -> d_flat.0(down.1(c.)) 361.56/136.62 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 361.56/136.62 down.0(a.1(c.)) -> a_flat.0(down.1(c.)) 361.56/136.62 down.0(b.1(c.)) -> b_flat.0(down.1(c.)) 361.56/136.62 361.56/136.62 Used ordering: Polynomial interpretation [POLO]: 361.56/136.62 361.56/136.62 POL(TOP.0(x_1)) = x_1 361.56/136.62 POL(a.0(x_1)) = x_1 361.56/136.62 POL(a.1(x_1)) = x_1 361.56/136.62 POL(a_flat.0(x_1)) = x_1 361.56/136.62 POL(b.0(x_1)) = x_1 361.56/136.62 POL(b.1(x_1)) = x_1 361.56/136.62 POL(b_flat.0(x_1)) = x_1 361.56/136.62 POL(c.) = 0 361.56/136.62 POL(d.0(x_1)) = x_1 361.56/136.62 POL(d.1(x_1)) = x_1 361.56/136.62 POL(d_flat.0(x_1)) = x_1 361.56/136.62 POL(down.0(x_1)) = 1 + x_1 361.56/136.62 POL(down.1(x_1)) = x_1 361.56/136.62 POL(fresh_constant.) = 0 361.56/136.62 POL(u.0(x_1)) = x_1 361.56/136.62 POL(u.1(x_1)) = x_1 361.56/136.62 POL(u_flat.0(x_1)) = x_1 361.56/136.62 POL(up.0(x_1)) = 1 + x_1 361.56/136.62 POL(up.1(x_1)) = 1 + x_1 361.56/136.62 361.56/136.62 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (152) 361.56/136.62 Obligation: 361.56/136.62 Q DP problem: 361.56/136.62 The TRS P consists of the following rules: 361.56/136.62 361.56/136.62 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0)))) 361.56/136.62 TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0)))) 361.56/136.62 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.62 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.62 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0)))) 361.56/136.62 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0)))) 361.56/136.62 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.62 TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0)))) 361.56/136.62 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 361.56/136.62 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0)))) 361.56/136.62 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(a.0(b.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.1(x0))))) 361.56/136.62 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.62 TOP.0(up.0(a.0(a.0(d.1(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.1(x0))))) 361.56/136.62 361.56/136.62 The TRS R consists of the following rules: 361.56/136.62 361.56/136.62 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.62 a_flat.0(up.1(x_1)) -> up.0(a.1(x_1)) 361.56/136.62 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.62 down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x))) 361.56/136.62 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.62 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.62 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.62 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.62 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.62 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.62 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.62 d_flat.0(up.1(x_1)) -> up.0(d.1(x_1)) 361.56/136.62 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.62 down.0(a.0(a.0(a.0(x)))) -> up.1(c.) 361.56/136.62 down.0(a.0(a.0(a.1(x)))) -> up.1(c.) 361.56/136.62 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.62 down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x))) 361.56/136.62 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.62 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.62 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.62 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.62 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.62 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.62 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.62 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.62 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.62 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.62 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.62 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.62 down.0(a.0(a.1(c.))) -> a_flat.0(down.0(a.1(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.62 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.62 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.62 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.62 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.62 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.62 down.0(a.0(a.0(fresh_constant.))) -> a_flat.0(down.0(a.0(fresh_constant.))) 361.56/136.62 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.62 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 361.56/136.62 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.62 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 361.56/136.62 361.56/136.62 The set Q consists of the following terms: 361.56/136.62 361.56/136.62 down.0(a.0(a.0(a.0(x0)))) 361.56/136.62 down.0(a.0(a.0(a.1(x0)))) 361.56/136.62 down.0(b.0(u.0(x0))) 361.56/136.62 down.0(b.0(u.1(x0))) 361.56/136.62 down.0(d.0(a.0(x0))) 361.56/136.62 down.0(d.0(a.1(x0))) 361.56/136.62 down.0(d.0(b.0(x0))) 361.56/136.62 down.0(d.0(b.1(x0))) 361.56/136.62 down.0(a.0(u.0(x0))) 361.56/136.62 down.0(a.0(u.1(x0))) 361.56/136.62 down.0(u.0(x0)) 361.56/136.62 down.0(u.1(x0)) 361.56/136.62 down.0(a.1(c.)) 361.56/136.62 down.0(a.0(b.0(x0))) 361.56/136.62 down.0(a.0(b.1(x0))) 361.56/136.62 down.0(a.0(d.0(x0))) 361.56/136.62 down.0(a.0(d.1(x0))) 361.56/136.62 down.0(a.0(fresh_constant.)) 361.56/136.62 down.0(b.0(a.0(x0))) 361.56/136.62 down.0(b.0(a.1(x0))) 361.56/136.62 down.0(b.1(c.)) 361.56/136.62 down.0(b.0(b.0(x0))) 361.56/136.62 down.0(b.0(b.1(x0))) 361.56/136.62 down.0(b.0(d.0(x0))) 361.56/136.62 down.0(b.0(d.1(x0))) 361.56/136.62 down.0(b.0(fresh_constant.)) 361.56/136.62 down.0(d.1(c.)) 361.56/136.62 down.0(d.0(u.0(x0))) 361.56/136.62 down.0(d.0(u.1(x0))) 361.56/136.62 down.0(d.0(d.0(x0))) 361.56/136.62 down.0(d.0(d.1(x0))) 361.56/136.62 down.0(d.0(fresh_constant.)) 361.56/136.62 down.0(a.0(a.1(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(x0)))) 361.56/136.62 down.0(a.0(a.0(b.1(x0)))) 361.56/136.62 down.0(a.0(a.0(u.0(x0)))) 361.56/136.62 down.0(a.0(a.0(u.1(x0)))) 361.56/136.62 down.0(a.0(a.0(d.0(x0)))) 361.56/136.62 down.0(a.0(a.0(d.1(x0)))) 361.56/136.62 down.0(a.0(a.0(fresh_constant.))) 361.56/136.62 a_flat.0(up.0(x0)) 361.56/136.62 a_flat.0(up.1(x0)) 361.56/136.62 b_flat.0(up.0(x0)) 361.56/136.62 b_flat.0(up.1(x0)) 361.56/136.62 u_flat.0(up.0(x0)) 361.56/136.62 u_flat.0(up.1(x0)) 361.56/136.62 d_flat.0(up.0(x0)) 361.56/136.62 d_flat.0(up.1(x0)) 361.56/136.62 361.56/136.62 We have to consider all minimal (P,Q,R)-chains. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (153) DependencyGraphProof (EQUIVALENT) 361.56/136.62 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (154) 361.56/136.62 Obligation: 361.56/136.62 Q DP problem: 361.56/136.62 The TRS P consists of the following rules: 361.56/136.62 361.56/136.62 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.62 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.62 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.62 361.56/136.62 The TRS R consists of the following rules: 361.56/136.62 361.56/136.62 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.62 a_flat.0(up.1(x_1)) -> up.0(a.1(x_1)) 361.56/136.62 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.62 down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x))) 361.56/136.62 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.62 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.62 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.62 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.62 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.62 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.62 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.62 d_flat.0(up.1(x_1)) -> up.0(d.1(x_1)) 361.56/136.62 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.62 down.0(a.0(a.0(a.0(x)))) -> up.1(c.) 361.56/136.62 down.0(a.0(a.0(a.1(x)))) -> up.1(c.) 361.56/136.62 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.62 down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x))) 361.56/136.62 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.62 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.62 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.62 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.62 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.62 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.62 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.62 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.62 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.62 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.62 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.62 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.62 down.0(a.0(a.1(c.))) -> a_flat.0(down.0(a.1(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.62 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.62 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.62 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.62 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.62 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.62 down.0(a.0(a.0(fresh_constant.))) -> a_flat.0(down.0(a.0(fresh_constant.))) 361.56/136.62 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.62 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 361.56/136.62 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.62 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 361.56/136.62 361.56/136.62 The set Q consists of the following terms: 361.56/136.62 361.56/136.62 down.0(a.0(a.0(a.0(x0)))) 361.56/136.62 down.0(a.0(a.0(a.1(x0)))) 361.56/136.62 down.0(b.0(u.0(x0))) 361.56/136.62 down.0(b.0(u.1(x0))) 361.56/136.62 down.0(d.0(a.0(x0))) 361.56/136.62 down.0(d.0(a.1(x0))) 361.56/136.62 down.0(d.0(b.0(x0))) 361.56/136.62 down.0(d.0(b.1(x0))) 361.56/136.62 down.0(a.0(u.0(x0))) 361.56/136.62 down.0(a.0(u.1(x0))) 361.56/136.62 down.0(u.0(x0)) 361.56/136.62 down.0(u.1(x0)) 361.56/136.62 down.0(a.1(c.)) 361.56/136.62 down.0(a.0(b.0(x0))) 361.56/136.62 down.0(a.0(b.1(x0))) 361.56/136.62 down.0(a.0(d.0(x0))) 361.56/136.62 down.0(a.0(d.1(x0))) 361.56/136.62 down.0(a.0(fresh_constant.)) 361.56/136.62 down.0(b.0(a.0(x0))) 361.56/136.62 down.0(b.0(a.1(x0))) 361.56/136.62 down.0(b.1(c.)) 361.56/136.62 down.0(b.0(b.0(x0))) 361.56/136.62 down.0(b.0(b.1(x0))) 361.56/136.62 down.0(b.0(d.0(x0))) 361.56/136.62 down.0(b.0(d.1(x0))) 361.56/136.62 down.0(b.0(fresh_constant.)) 361.56/136.62 down.0(d.1(c.)) 361.56/136.62 down.0(d.0(u.0(x0))) 361.56/136.62 down.0(d.0(u.1(x0))) 361.56/136.62 down.0(d.0(d.0(x0))) 361.56/136.62 down.0(d.0(d.1(x0))) 361.56/136.62 down.0(d.0(fresh_constant.)) 361.56/136.62 down.0(a.0(a.1(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(x0)))) 361.56/136.62 down.0(a.0(a.0(b.1(x0)))) 361.56/136.62 down.0(a.0(a.0(u.0(x0)))) 361.56/136.62 down.0(a.0(a.0(u.1(x0)))) 361.56/136.62 down.0(a.0(a.0(d.0(x0)))) 361.56/136.62 down.0(a.0(a.0(d.1(x0)))) 361.56/136.62 down.0(a.0(a.0(fresh_constant.))) 361.56/136.62 a_flat.0(up.0(x0)) 361.56/136.62 a_flat.0(up.1(x0)) 361.56/136.62 b_flat.0(up.0(x0)) 361.56/136.62 b_flat.0(up.1(x0)) 361.56/136.62 u_flat.0(up.0(x0)) 361.56/136.62 u_flat.0(up.1(x0)) 361.56/136.62 d_flat.0(up.0(x0)) 361.56/136.62 d_flat.0(up.1(x0)) 361.56/136.62 361.56/136.62 We have to consider all minimal (P,Q,R)-chains. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (155) MRRProof (EQUIVALENT) 361.56/136.62 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. 361.56/136.62 361.56/136.62 361.56/136.62 Strictly oriented rules of the TRS R: 361.56/136.62 361.56/136.62 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 361.56/136.62 361.56/136.62 Used ordering: Polynomial interpretation [POLO]: 361.56/136.62 361.56/136.62 POL(TOP.0(x_1)) = x_1 361.56/136.62 POL(a.0(x_1)) = x_1 361.56/136.62 POL(a.1(x_1)) = x_1 361.56/136.62 POL(a_flat.0(x_1)) = x_1 361.56/136.62 POL(b.0(x_1)) = 1 + x_1 361.56/136.62 POL(b.1(x_1)) = x_1 361.56/136.62 POL(b_flat.0(x_1)) = 1 + x_1 361.56/136.62 POL(c.) = 0 361.56/136.62 POL(d.0(x_1)) = x_1 361.56/136.62 POL(d.1(x_1)) = x_1 361.56/136.62 POL(d_flat.0(x_1)) = x_1 361.56/136.62 POL(down.0(x_1)) = x_1 361.56/136.62 POL(fresh_constant.) = 0 361.56/136.62 POL(u.0(x_1)) = x_1 361.56/136.62 POL(u.1(x_1)) = x_1 361.56/136.62 POL(u_flat.0(x_1)) = x_1 361.56/136.62 POL(up.0(x_1)) = x_1 361.56/136.62 POL(up.1(x_1)) = x_1 361.56/136.62 361.56/136.62 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (156) 361.56/136.62 Obligation: 361.56/136.62 Q DP problem: 361.56/136.62 The TRS P consists of the following rules: 361.56/136.62 361.56/136.62 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.62 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.62 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.62 361.56/136.62 The TRS R consists of the following rules: 361.56/136.62 361.56/136.62 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.62 a_flat.0(up.1(x_1)) -> up.0(a.1(x_1)) 361.56/136.62 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.62 down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x))) 361.56/136.62 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.62 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.62 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.62 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.62 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.62 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.62 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.62 d_flat.0(up.1(x_1)) -> up.0(d.1(x_1)) 361.56/136.62 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.62 down.0(a.0(a.0(a.0(x)))) -> up.1(c.) 361.56/136.62 down.0(a.0(a.0(a.1(x)))) -> up.1(c.) 361.56/136.62 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.62 down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x))) 361.56/136.62 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.62 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.62 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.62 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.62 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.62 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.62 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.62 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.62 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.62 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.62 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.62 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.62 down.0(a.0(a.1(c.))) -> a_flat.0(down.0(a.1(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.62 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.62 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.62 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.62 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.62 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.62 down.0(a.0(a.0(fresh_constant.))) -> a_flat.0(down.0(a.0(fresh_constant.))) 361.56/136.62 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.62 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 361.56/136.62 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.62 361.56/136.62 The set Q consists of the following terms: 361.56/136.62 361.56/136.62 down.0(a.0(a.0(a.0(x0)))) 361.56/136.62 down.0(a.0(a.0(a.1(x0)))) 361.56/136.62 down.0(b.0(u.0(x0))) 361.56/136.62 down.0(b.0(u.1(x0))) 361.56/136.62 down.0(d.0(a.0(x0))) 361.56/136.62 down.0(d.0(a.1(x0))) 361.56/136.62 down.0(d.0(b.0(x0))) 361.56/136.62 down.0(d.0(b.1(x0))) 361.56/136.62 down.0(a.0(u.0(x0))) 361.56/136.62 down.0(a.0(u.1(x0))) 361.56/136.62 down.0(u.0(x0)) 361.56/136.62 down.0(u.1(x0)) 361.56/136.62 down.0(a.1(c.)) 361.56/136.62 down.0(a.0(b.0(x0))) 361.56/136.62 down.0(a.0(b.1(x0))) 361.56/136.62 down.0(a.0(d.0(x0))) 361.56/136.62 down.0(a.0(d.1(x0))) 361.56/136.62 down.0(a.0(fresh_constant.)) 361.56/136.62 down.0(b.0(a.0(x0))) 361.56/136.62 down.0(b.0(a.1(x0))) 361.56/136.62 down.0(b.1(c.)) 361.56/136.62 down.0(b.0(b.0(x0))) 361.56/136.62 down.0(b.0(b.1(x0))) 361.56/136.62 down.0(b.0(d.0(x0))) 361.56/136.62 down.0(b.0(d.1(x0))) 361.56/136.62 down.0(b.0(fresh_constant.)) 361.56/136.62 down.0(d.1(c.)) 361.56/136.62 down.0(d.0(u.0(x0))) 361.56/136.62 down.0(d.0(u.1(x0))) 361.56/136.62 down.0(d.0(d.0(x0))) 361.56/136.62 down.0(d.0(d.1(x0))) 361.56/136.62 down.0(d.0(fresh_constant.)) 361.56/136.62 down.0(a.0(a.1(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(x0)))) 361.56/136.62 down.0(a.0(a.0(b.1(x0)))) 361.56/136.62 down.0(a.0(a.0(u.0(x0)))) 361.56/136.62 down.0(a.0(a.0(u.1(x0)))) 361.56/136.62 down.0(a.0(a.0(d.0(x0)))) 361.56/136.62 down.0(a.0(a.0(d.1(x0)))) 361.56/136.62 down.0(a.0(a.0(fresh_constant.))) 361.56/136.62 a_flat.0(up.0(x0)) 361.56/136.62 a_flat.0(up.1(x0)) 361.56/136.62 b_flat.0(up.0(x0)) 361.56/136.62 b_flat.0(up.1(x0)) 361.56/136.62 u_flat.0(up.0(x0)) 361.56/136.62 u_flat.0(up.1(x0)) 361.56/136.62 d_flat.0(up.0(x0)) 361.56/136.62 d_flat.0(up.1(x0)) 361.56/136.62 361.56/136.62 We have to consider all minimal (P,Q,R)-chains. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (157) MRRProof (EQUIVALENT) 361.56/136.62 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. 361.56/136.62 361.56/136.62 361.56/136.62 Strictly oriented rules of the TRS R: 361.56/136.62 361.56/136.62 down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x))) 361.56/136.62 d_flat.0(up.1(x_1)) -> up.0(d.1(x_1)) 361.56/136.62 down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x))) 361.56/136.62 361.56/136.62 Used ordering: Polynomial interpretation [POLO]: 361.56/136.62 361.56/136.62 POL(TOP.0(x_1)) = x_1 361.56/136.62 POL(a.0(x_1)) = x_1 361.56/136.62 POL(a.1(x_1)) = x_1 361.56/136.62 POL(a_flat.0(x_1)) = x_1 361.56/136.62 POL(b.0(x_1)) = x_1 361.56/136.62 POL(b.1(x_1)) = x_1 361.56/136.62 POL(b_flat.0(x_1)) = x_1 361.56/136.62 POL(c.) = 0 361.56/136.62 POL(d.0(x_1)) = 1 + x_1 361.56/136.62 POL(d.1(x_1)) = x_1 361.56/136.62 POL(d_flat.0(x_1)) = 1 + x_1 361.56/136.62 POL(down.0(x_1)) = x_1 361.56/136.62 POL(fresh_constant.) = 0 361.56/136.62 POL(u.0(x_1)) = 1 + x_1 361.56/136.62 POL(u.1(x_1)) = 1 + x_1 361.56/136.62 POL(u_flat.0(x_1)) = 1 + x_1 361.56/136.62 POL(up.0(x_1)) = x_1 361.56/136.62 POL(up.1(x_1)) = x_1 361.56/136.62 361.56/136.62 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (158) 361.56/136.62 Obligation: 361.56/136.62 Q DP problem: 361.56/136.62 The TRS P consists of the following rules: 361.56/136.62 361.56/136.62 TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 361.56/136.62 TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0)))) 361.56/136.62 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 361.56/136.62 TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0)))) 361.56/136.62 TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0)))) 361.56/136.62 TOP.0(up.0(a.0(a.0(b.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(b.0(x0))))) 361.56/136.62 TOP.0(up.0(a.0(a.0(d.0(x0))))) -> TOP.0(a_flat.0(a_flat.0(down.0(d.0(x0))))) 361.56/136.62 361.56/136.62 The TRS R consists of the following rules: 361.56/136.62 361.56/136.62 a_flat.0(up.0(x_1)) -> up.0(a.0(x_1)) 361.56/136.62 a_flat.0(up.1(x_1)) -> up.0(a.1(x_1)) 361.56/136.62 down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x))) 361.56/136.62 down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x)))) 361.56/136.62 down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x)))) 361.56/136.62 down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17))) 361.56/136.62 down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17))) 361.56/136.62 down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18))) 361.56/136.62 down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18))) 361.56/136.62 d_flat.0(up.0(x_1)) -> up.0(d.0(x_1)) 361.56/136.62 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 361.56/136.62 down.0(a.0(a.0(a.0(x)))) -> up.1(c.) 361.56/136.62 down.0(a.0(a.0(a.1(x)))) -> up.1(c.) 361.56/136.62 down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x))) 361.56/136.62 down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x))) 361.56/136.62 down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x))) 361.56/136.62 down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6))) 361.56/136.62 down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6))) 361.56/136.62 down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8))) 361.56/136.62 down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8))) 361.56/136.62 down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10))) 361.56/136.62 down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10))) 361.56/136.62 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 361.56/136.62 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 361.56/136.62 down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13))) 361.56/136.62 down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13))) 361.56/136.62 down.0(a.0(a.1(c.))) -> a_flat.0(down.0(a.1(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(y21)))) -> a_flat.0(down.0(a.0(b.0(y21)))) 361.56/136.62 down.0(a.0(a.0(b.1(y21)))) -> a_flat.0(down.0(a.0(b.1(y21)))) 361.56/136.62 down.0(a.0(a.0(u.0(y22)))) -> a_flat.0(down.0(a.0(u.0(y22)))) 361.56/136.62 down.0(a.0(a.0(u.1(y22)))) -> a_flat.0(down.0(a.0(u.1(y22)))) 361.56/136.62 down.0(a.0(a.0(d.0(y23)))) -> a_flat.0(down.0(a.0(d.0(y23)))) 361.56/136.62 down.0(a.0(a.0(d.1(y23)))) -> a_flat.0(down.0(a.0(d.1(y23)))) 361.56/136.62 down.0(a.0(a.0(fresh_constant.))) -> a_flat.0(down.0(a.0(fresh_constant.))) 361.56/136.62 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 361.56/136.62 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 361.56/136.62 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 361.56/136.62 361.56/136.62 The set Q consists of the following terms: 361.56/136.62 361.56/136.62 down.0(a.0(a.0(a.0(x0)))) 361.56/136.62 down.0(a.0(a.0(a.1(x0)))) 361.56/136.62 down.0(b.0(u.0(x0))) 361.56/136.62 down.0(b.0(u.1(x0))) 361.56/136.62 down.0(d.0(a.0(x0))) 361.56/136.62 down.0(d.0(a.1(x0))) 361.56/136.62 down.0(d.0(b.0(x0))) 361.56/136.62 down.0(d.0(b.1(x0))) 361.56/136.62 down.0(a.0(u.0(x0))) 361.56/136.62 down.0(a.0(u.1(x0))) 361.56/136.62 down.0(u.0(x0)) 361.56/136.62 down.0(u.1(x0)) 361.56/136.62 down.0(a.1(c.)) 361.56/136.62 down.0(a.0(b.0(x0))) 361.56/136.62 down.0(a.0(b.1(x0))) 361.56/136.62 down.0(a.0(d.0(x0))) 361.56/136.62 down.0(a.0(d.1(x0))) 361.56/136.62 down.0(a.0(fresh_constant.)) 361.56/136.62 down.0(b.0(a.0(x0))) 361.56/136.62 down.0(b.0(a.1(x0))) 361.56/136.62 down.0(b.1(c.)) 361.56/136.62 down.0(b.0(b.0(x0))) 361.56/136.62 down.0(b.0(b.1(x0))) 361.56/136.62 down.0(b.0(d.0(x0))) 361.56/136.62 down.0(b.0(d.1(x0))) 361.56/136.62 down.0(b.0(fresh_constant.)) 361.56/136.62 down.0(d.1(c.)) 361.56/136.62 down.0(d.0(u.0(x0))) 361.56/136.62 down.0(d.0(u.1(x0))) 361.56/136.62 down.0(d.0(d.0(x0))) 361.56/136.62 down.0(d.0(d.1(x0))) 361.56/136.62 down.0(d.0(fresh_constant.)) 361.56/136.62 down.0(a.0(a.1(c.))) 361.56/136.62 down.0(a.0(a.0(b.0(x0)))) 361.56/136.62 down.0(a.0(a.0(b.1(x0)))) 361.56/136.62 down.0(a.0(a.0(u.0(x0)))) 361.56/136.62 down.0(a.0(a.0(u.1(x0)))) 361.56/136.62 down.0(a.0(a.0(d.0(x0)))) 361.56/136.62 down.0(a.0(a.0(d.1(x0)))) 361.56/136.62 down.0(a.0(a.0(fresh_constant.))) 361.56/136.62 a_flat.0(up.0(x0)) 361.56/136.62 a_flat.0(up.1(x0)) 361.56/136.62 b_flat.0(up.0(x0)) 361.56/136.62 b_flat.0(up.1(x0)) 361.56/136.62 u_flat.0(up.0(x0)) 361.56/136.62 u_flat.0(up.1(x0)) 361.56/136.62 d_flat.0(up.0(x0)) 361.56/136.62 d_flat.0(up.1(x0)) 361.56/136.62 361.56/136.62 We have to consider all minimal (P,Q,R)-chains. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (159) PisEmptyProof (SOUND) 361.56/136.62 The TRS P is empty. Hence, there is no (P,Q,R) chain. 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (160) 361.56/136.62 TRUE 361.56/136.62 361.56/136.62 ---------------------------------------- 361.56/136.62 361.56/136.62 (161) 361.56/136.62 Obligation: 361.56/136.62 Q DP problem: 361.56/136.62 The TRS P consists of the following rules: 361.56/136.62 361.56/136.62 TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))) 361.56/136.62 TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))) 361.56/136.62 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 361.56/136.62 TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))) 361.56/136.62 TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))) 361.56/136.62 TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))) 361.56/136.62 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 361.56/136.62 TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))) 361.56/136.62 TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))) 361.56/136.62 TOP(up(a(a(b(x0))))) -> TOP(a_flat(a_flat(down(b(x0))))) 361.56/136.62 TOP(up(a(a(d(x0))))) -> TOP(a_flat(a_flat(down(d(x0))))) 361.56/136.62 361.56/136.62 The TRS R consists of the following rules: 361.56/136.62 361.56/136.62 a_flat(up(x_1)) -> up(a(x_1)) 361.56/136.62 down(d(a(x))) -> up(a(d(x))) 361.56/136.62 down(d(b(x))) -> up(u(a(b(x)))) 361.56/136.62 down(d(u(y17))) -> d_flat(down(u(y17))) 361.56/136.62 down(d(d(y18))) -> d_flat(down(d(y18))) 361.56/136.62 d_flat(up(x_1)) -> up(d(x_1)) 361.56/136.62 down(u(y2)) -> u_flat(down(y2)) 361.56/136.62 down(a(a(a(x)))) -> up(c) 361.56/136.62 down(b(u(x))) -> up(b(d(x))) 361.56/136.62 down(a(u(x))) -> up(u(a(x))) 361.56/136.62 down(a(b(y6))) -> a_flat(down(b(y6))) 361.56/136.62 down(a(d(y8))) -> a_flat(down(d(y8))) 361.56/136.62 down(b(a(y10))) -> b_flat(down(a(y10))) 361.56/136.62 down(b(b(y11))) -> b_flat(down(b(y11))) 361.56/136.62 down(b(d(y13))) -> b_flat(down(d(y13))) 361.56/136.62 down(a(a(c))) -> a_flat(down(a(c))) 361.56/136.62 down(a(a(b(y21)))) -> a_flat(down(a(b(y21)))) 361.56/136.62 down(a(a(u(y22)))) -> a_flat(down(a(u(y22)))) 361.56/136.62 down(a(a(d(y23)))) -> a_flat(down(a(d(y23)))) 361.56/136.62 down(a(a(fresh_constant))) -> a_flat(down(a(fresh_constant))) 361.56/136.62 u_flat(up(x_1)) -> up(u(x_1)) 361.56/136.62 b_flat(up(x_1)) -> up(b(x_1)) 361.56/136.62 361.56/136.62 The set Q consists of the following terms: 361.56/136.62 361.56/136.62 down(a(a(a(x0)))) 361.56/136.62 down(b(u(x0))) 361.56/136.62 down(d(a(x0))) 361.56/136.62 down(d(b(x0))) 361.56/136.62 down(a(u(x0))) 361.56/136.62 down(u(x0)) 361.56/136.62 down(a(c)) 361.56/136.62 down(a(b(x0))) 361.56/136.62 down(a(d(x0))) 361.56/136.62 down(a(fresh_constant)) 361.56/136.62 down(b(a(x0))) 361.56/136.62 down(b(c)) 361.56/136.62 down(b(b(x0))) 361.56/136.62 down(b(d(x0))) 361.56/136.62 down(b(fresh_constant)) 361.56/136.62 down(d(c)) 361.56/136.62 down(d(u(x0))) 361.56/136.62 down(d(d(x0))) 361.56/136.62 down(d(fresh_constant)) 361.56/136.62 down(a(a(c))) 361.56/136.62 down(a(a(b(x0)))) 361.56/136.62 down(a(a(u(x0)))) 361.56/136.62 down(a(a(d(x0)))) 361.56/136.62 down(a(a(fresh_constant))) 361.56/136.62 a_flat(up(x0)) 361.56/136.62 b_flat(up(x0)) 361.56/136.62 u_flat(up(x0)) 361.56/136.62 d_flat(up(x0)) 361.56/136.62 361.56/136.62 We have to consider all minimal (P,Q,R)-chains. 361.73/136.70 EOF