471.21/221.11 MAYBE 471.21/221.12 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 471.21/221.12 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 471.21/221.12 471.21/221.12 471.21/221.12 Outermost Termination of the given OTRS could not be shown: 471.21/221.12 471.21/221.12 (0) OTRS 471.21/221.12 (1) Trivial-Transformation [SOUND, 0 ms] 471.21/221.12 (2) QTRS 471.21/221.12 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 471.21/221.12 (4) QDP 471.21/221.12 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (6) AND 471.21/221.12 (7) QDP 471.21/221.12 (8) UsableRulesProof [EQUIVALENT, 0 ms] 471.21/221.12 (9) QDP 471.21/221.12 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 471.21/221.12 (11) YES 471.21/221.12 (12) QDP 471.21/221.12 (13) TransformationProof [EQUIVALENT, 0 ms] 471.21/221.12 (14) QDP 471.21/221.12 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (16) QDP 471.21/221.12 (17) QDPOrderProof [EQUIVALENT, 9 ms] 471.21/221.12 (18) QDP 471.21/221.12 (19) QDPOrderProof [EQUIVALENT, 0 ms] 471.21/221.12 (20) QDP 471.21/221.12 (21) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] 471.21/221.12 (22) QTRS 471.21/221.12 (23) AAECC Innermost [EQUIVALENT, 0 ms] 471.21/221.12 (24) QTRS 471.21/221.12 (25) DependencyPairsProof [EQUIVALENT, 3 ms] 471.21/221.12 (26) QDP 471.21/221.12 (27) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (28) AND 471.21/221.12 (29) QDP 471.21/221.12 (30) UsableRulesProof [EQUIVALENT, 0 ms] 471.21/221.12 (31) QDP 471.21/221.12 (32) QReductionProof [EQUIVALENT, 0 ms] 471.21/221.12 (33) QDP 471.21/221.12 (34) QDPSizeChangeProof [EQUIVALENT, 0 ms] 471.21/221.12 (35) YES 471.21/221.12 (36) QDP 471.21/221.12 (37) UsableRulesProof [EQUIVALENT, 0 ms] 471.21/221.12 (38) QDP 471.21/221.12 (39) QReductionProof [EQUIVALENT, 0 ms] 471.21/221.12 (40) QDP 471.21/221.12 (41) TransformationProof [EQUIVALENT, 11 ms] 471.21/221.12 (42) QDP 471.21/221.12 (43) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (44) QDP 471.21/221.12 (45) TransformationProof [EQUIVALENT, 0 ms] 471.21/221.12 (46) QDP 471.21/221.12 (47) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (48) QDP 471.21/221.12 (49) TransformationProof [EQUIVALENT, 0 ms] 471.21/221.12 (50) QDP 471.21/221.12 (51) TransformationProof [EQUIVALENT, 0 ms] 471.21/221.12 (52) QDP 471.21/221.12 (53) TransformationProof [EQUIVALENT, 0 ms] 471.21/221.12 (54) QDP 471.21/221.12 (55) TransformationProof [EQUIVALENT, 0 ms] 471.21/221.12 (56) QDP 471.21/221.12 (57) TransformationProof [EQUIVALENT, 0 ms] 471.21/221.12 (58) QDP 471.21/221.12 (59) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (60) QDP 471.21/221.12 (61) TransformationProof [EQUIVALENT, 0 ms] 471.21/221.12 (62) QDP 471.21/221.12 (63) TransformationProof [EQUIVALENT, 0 ms] 471.21/221.12 (64) QDP 471.21/221.12 (65) TransformationProof [EQUIVALENT, 0 ms] 471.21/221.12 (66) QDP 471.21/221.12 (67) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (68) QDP 471.21/221.12 (69) TransformationProof [EQUIVALENT, 0 ms] 471.21/221.12 (70) QDP 471.21/221.12 (71) QDPOrderProof [EQUIVALENT, 27 ms] 471.21/221.12 (72) QDP 471.21/221.12 (73) QDPOrderProof [EQUIVALENT, 0 ms] 471.21/221.12 (74) QDP 471.21/221.12 (75) MNOCProof [EQUIVALENT, 0 ms] 471.21/221.12 (76) QDP 471.21/221.12 (77) SplitQDPProof [EQUIVALENT, 0 ms] 471.21/221.12 (78) AND 471.21/221.12 (79) QDP 471.21/221.12 (80) SemLabProof [SOUND, 0 ms] 471.21/221.12 (81) QDP 471.21/221.12 (82) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (83) QDP 471.21/221.12 (84) UsableRulesReductionPairsProof [EQUIVALENT, 7 ms] 471.21/221.12 (85) QDP 471.21/221.12 (86) MRRProof [EQUIVALENT, 6 ms] 471.21/221.12 (87) QDP 471.21/221.12 (88) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (89) QDP 471.21/221.12 (90) MRRProof [EQUIVALENT, 0 ms] 471.21/221.12 (91) QDP 471.21/221.12 (92) MRRProof [EQUIVALENT, 6 ms] 471.21/221.12 (93) QDP 471.21/221.12 (94) PisEmptyProof [SOUND, 0 ms] 471.21/221.12 (95) TRUE 471.21/221.12 (96) QDP 471.21/221.12 (97) SplitQDPProof [EQUIVALENT, 0 ms] 471.21/221.12 (98) AND 471.21/221.12 (99) QDP 471.21/221.12 (100) SemLabProof [SOUND, 0 ms] 471.21/221.12 (101) QDP 471.21/221.12 (102) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (103) QDP 471.21/221.12 (104) MRRProof [EQUIVALENT, 12 ms] 471.21/221.12 (105) QDP 471.21/221.12 (106) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (107) QDP 471.21/221.12 (108) MRRProof [EQUIVALENT, 0 ms] 471.21/221.12 (109) QDP 471.21/221.12 (110) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (111) QDP 471.21/221.12 (112) PisEmptyProof [SOUND, 0 ms] 471.21/221.12 (113) TRUE 471.21/221.12 (114) QDP 471.21/221.12 (115) SplitQDPProof [EQUIVALENT, 0 ms] 471.21/221.12 (116) AND 471.21/221.12 (117) QDP 471.21/221.12 (118) SemLabProof [SOUND, 0 ms] 471.21/221.12 (119) QDP 471.21/221.12 (120) MRRProof [EQUIVALENT, 0 ms] 471.21/221.12 (121) QDP 471.21/221.12 (122) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (123) QDP 471.21/221.12 (124) MRRProof [EQUIVALENT, 0 ms] 471.21/221.12 (125) QDP 471.21/221.12 (126) MRRProof [EQUIVALENT, 10 ms] 471.21/221.12 (127) QDP 471.21/221.12 (128) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (129) QDP 471.21/221.12 (130) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 471.21/221.12 (131) QDP 471.21/221.12 (132) PisEmptyProof [SOUND, 0 ms] 471.21/221.12 (133) TRUE 471.21/221.12 (134) QDP 471.21/221.12 (135) QReductionProof [EQUIVALENT, 0 ms] 471.21/221.12 (136) QDP 471.21/221.12 (137) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms] 471.21/221.12 (138) QTRS 471.21/221.12 (139) DependencyPairsProof [EQUIVALENT, 0 ms] 471.21/221.12 (140) QDP 471.21/221.12 (141) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (142) AND 471.21/221.12 (143) QDP 471.21/221.12 (144) UsableRulesProof [EQUIVALENT, 0 ms] 471.21/221.12 (145) QDP 471.21/221.12 (146) QReductionProof [EQUIVALENT, 0 ms] 471.21/221.12 (147) QDP 471.21/221.12 (148) MRRProof [EQUIVALENT, 0 ms] 471.21/221.12 (149) QDP 471.21/221.12 (150) UsableRulesReductionPairsProof [EQUIVALENT, 5 ms] 471.21/221.12 (151) QDP 471.21/221.12 (152) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (153) TRUE 471.21/221.12 (154) QDP 471.21/221.12 (155) UsableRulesProof [EQUIVALENT, 0 ms] 471.21/221.12 (156) QDP 471.21/221.12 (157) QReductionProof [EQUIVALENT, 0 ms] 471.21/221.12 (158) QDP 471.21/221.12 (159) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 471.21/221.12 (160) QDP 471.21/221.12 (161) UsableRulesReductionPairsProof [EQUIVALENT, 2 ms] 471.21/221.12 (162) QDP 471.21/221.12 (163) DependencyGraphProof [EQUIVALENT, 0 ms] 471.21/221.12 (164) TRUE 471.21/221.12 (165) QDP 471.21/221.12 (166) UsableRulesProof [EQUIVALENT, 0 ms] 471.21/221.12 (167) QDP 471.21/221.12 (168) QReductionProof [EQUIVALENT, 0 ms] 471.21/221.12 (169) QDP 471.21/221.12 (170) TransformationProof [EQUIVALENT, 0 ms] 471.21/221.12 (171) QDP 471.21/221.12 (172) UsableRulesProof [EQUIVALENT, 0 ms] 471.21/221.12 (173) QDP 471.21/221.12 (174) QReductionProof [EQUIVALENT, 0 ms] 471.21/221.12 (175) QDP 471.21/221.12 471.21/221.12 471.21/221.12 ---------------------------------------- 471.21/221.12 471.21/221.12 (0) 471.21/221.12 Obligation: 471.21/221.12 Term rewrite system R: 471.21/221.12 The TRS R consists of the following rules: 471.21/221.12 471.21/221.12 f(f(f(x))) -> c 471.21/221.12 b(u(x)) -> b(f(x)) 471.21/221.12 f(g(x)) -> f(f(x)) 471.21/221.12 f(b(x)) -> u(g(b(x))) 471.21/221.12 f(u(x)) -> u(g(x)) 471.21/221.12 471.21/221.12 471.21/221.12 471.21/221.12 Outermost Strategy. 471.21/221.12 471.21/221.12 ---------------------------------------- 471.21/221.12 471.21/221.12 (1) Trivial-Transformation (SOUND) 471.21/221.12 We applied the Trivial transformation to transform the outermost TRS to a standard TRS. 471.21/221.12 ---------------------------------------- 471.21/221.12 471.21/221.12 (2) 471.21/221.12 Obligation: 471.21/221.12 Q restricted rewrite system: 471.21/221.12 The TRS R consists of the following rules: 471.21/221.12 471.21/221.12 f(f(f(x))) -> c 471.21/221.12 b(u(x)) -> b(f(x)) 471.21/221.12 f(g(x)) -> f(f(x)) 471.21/221.12 f(b(x)) -> u(g(b(x))) 471.21/221.12 f(u(x)) -> u(g(x)) 471.21/221.12 471.21/221.12 Q is empty. 471.21/221.12 471.21/221.12 ---------------------------------------- 471.21/221.12 471.21/221.12 (3) DependencyPairsProof (EQUIVALENT) 471.21/221.12 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (4) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 B(u(x)) -> B(f(x)) 471.21/221.13 B(u(x)) -> F(x) 471.21/221.13 F(g(x)) -> F(f(x)) 471.21/221.13 F(g(x)) -> F(x) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 f(f(f(x))) -> c 471.21/221.13 b(u(x)) -> b(f(x)) 471.21/221.13 f(g(x)) -> f(f(x)) 471.21/221.13 f(b(x)) -> u(g(b(x))) 471.21/221.13 f(u(x)) -> u(g(x)) 471.21/221.13 471.21/221.13 Q is empty. 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (5) DependencyGraphProof (EQUIVALENT) 471.21/221.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (6) 471.21/221.13 Complex Obligation (AND) 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (7) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 F(g(x)) -> F(x) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 f(f(f(x))) -> c 471.21/221.13 b(u(x)) -> b(f(x)) 471.21/221.13 f(g(x)) -> f(f(x)) 471.21/221.13 f(b(x)) -> u(g(b(x))) 471.21/221.13 f(u(x)) -> u(g(x)) 471.21/221.13 471.21/221.13 Q is empty. 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (8) UsableRulesProof (EQUIVALENT) 471.21/221.13 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. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (9) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 F(g(x)) -> F(x) 471.21/221.13 471.21/221.13 R is empty. 471.21/221.13 Q is empty. 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (10) QDPSizeChangeProof (EQUIVALENT) 471.21/221.13 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. 471.21/221.13 471.21/221.13 From the DPs we obtained the following set of size-change graphs: 471.21/221.13 *F(g(x)) -> F(x) 471.21/221.13 The graph contains the following edges 1 > 1 471.21/221.13 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (11) 471.21/221.13 YES 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (12) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 B(u(x)) -> B(f(x)) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 f(f(f(x))) -> c 471.21/221.13 b(u(x)) -> b(f(x)) 471.21/221.13 f(g(x)) -> f(f(x)) 471.21/221.13 f(b(x)) -> u(g(b(x))) 471.21/221.13 f(u(x)) -> u(g(x)) 471.21/221.13 471.21/221.13 Q is empty. 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (13) TransformationProof (EQUIVALENT) 471.21/221.13 By narrowing [LPAR04] the rule B(u(x)) -> B(f(x)) at position [0] we obtained the following new rules [LPAR04]: 471.21/221.13 471.21/221.13 (B(u(f(f(x0)))) -> B(c),B(u(f(f(x0)))) -> B(c)) 471.21/221.13 (B(u(g(x0))) -> B(f(f(x0))),B(u(g(x0))) -> B(f(f(x0)))) 471.21/221.13 (B(u(b(x0))) -> B(u(g(b(x0)))),B(u(b(x0))) -> B(u(g(b(x0))))) 471.21/221.13 (B(u(u(x0))) -> B(u(g(x0))),B(u(u(x0))) -> B(u(g(x0)))) 471.21/221.13 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (14) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 B(u(f(f(x0)))) -> B(c) 471.21/221.13 B(u(g(x0))) -> B(f(f(x0))) 471.21/221.13 B(u(b(x0))) -> B(u(g(b(x0)))) 471.21/221.13 B(u(u(x0))) -> B(u(g(x0))) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 f(f(f(x))) -> c 471.21/221.13 b(u(x)) -> b(f(x)) 471.21/221.13 f(g(x)) -> f(f(x)) 471.21/221.13 f(b(x)) -> u(g(b(x))) 471.21/221.13 f(u(x)) -> u(g(x)) 471.21/221.13 471.21/221.13 Q is empty. 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (15) DependencyGraphProof (EQUIVALENT) 471.21/221.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (16) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 B(u(g(x0))) -> B(f(f(x0))) 471.21/221.13 B(u(b(x0))) -> B(u(g(b(x0)))) 471.21/221.13 B(u(u(x0))) -> B(u(g(x0))) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 f(f(f(x))) -> c 471.21/221.13 b(u(x)) -> b(f(x)) 471.21/221.13 f(g(x)) -> f(f(x)) 471.21/221.13 f(b(x)) -> u(g(b(x))) 471.21/221.13 f(u(x)) -> u(g(x)) 471.21/221.13 471.21/221.13 Q is empty. 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (17) QDPOrderProof (EQUIVALENT) 471.21/221.13 We use the reduction pair processor [LPAR04,JAR06]. 471.21/221.13 471.21/221.13 471.21/221.13 The following pairs can be oriented strictly and are deleted. 471.21/221.13 471.21/221.13 B(u(b(x0))) -> B(u(g(b(x0)))) 471.21/221.13 The remaining pairs can at least be oriented weakly. 471.21/221.13 Used ordering: Polynomial interpretation [POLO]: 471.21/221.13 471.21/221.13 POL(B(x_1)) = x_1 471.21/221.13 POL(b(x_1)) = 1 + x_1 471.21/221.13 POL(c) = 0 471.21/221.13 POL(f(x_1)) = 0 471.21/221.13 POL(g(x_1)) = 0 471.21/221.13 POL(u(x_1)) = x_1 471.21/221.13 471.21/221.13 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 471.21/221.13 471.21/221.13 f(f(f(x))) -> c 471.21/221.13 f(g(x)) -> f(f(x)) 471.21/221.13 f(b(x)) -> u(g(b(x))) 471.21/221.13 f(u(x)) -> u(g(x)) 471.21/221.13 b(u(x)) -> b(f(x)) 471.21/221.13 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (18) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 B(u(g(x0))) -> B(f(f(x0))) 471.21/221.13 B(u(u(x0))) -> B(u(g(x0))) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 f(f(f(x))) -> c 471.21/221.13 b(u(x)) -> b(f(x)) 471.21/221.13 f(g(x)) -> f(f(x)) 471.21/221.13 f(b(x)) -> u(g(b(x))) 471.21/221.13 f(u(x)) -> u(g(x)) 471.21/221.13 471.21/221.13 Q is empty. 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (19) QDPOrderProof (EQUIVALENT) 471.21/221.13 We use the reduction pair processor [LPAR04,JAR06]. 471.21/221.13 471.21/221.13 471.21/221.13 The following pairs can be oriented strictly and are deleted. 471.21/221.13 471.21/221.13 B(u(u(x0))) -> B(u(g(x0))) 471.21/221.13 The remaining pairs can at least be oriented weakly. 471.21/221.13 Used ordering: Matrix interpretation [MATRO]: 471.21/221.13 471.21/221.13 Non-tuple symbols: 471.21/221.13 <<< 471.21/221.13 M( b_1(x_1) ) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 471.21/221.13 >>> 471.21/221.13 471.21/221.13 <<< 471.21/221.13 M( c ) = [[1], [0]] 471.21/221.13 >>> 471.21/221.13 471.21/221.13 <<< 471.21/221.13 M( f_1(x_1) ) = [[1], [0]] + [[1, 1], [0, 0]] * x_1 471.21/221.13 >>> 471.21/221.13 471.21/221.13 <<< 471.21/221.13 M( g_1(x_1) ) = [[0], [1]] + [[0, 0], [1, 1]] * x_1 471.21/221.13 >>> 471.21/221.13 471.21/221.13 <<< 471.21/221.13 M( u_1(x_1) ) = [[1], [0]] + [[0, 0], [1, 0]] * x_1 471.21/221.13 >>> 471.21/221.13 471.21/221.13 Tuple symbols: 471.21/221.13 <<< 471.21/221.13 M( B_1(x_1) ) = [[0]] + [[0, 1]] * x_1 471.21/221.13 >>> 471.21/221.13 471.21/221.13 471.21/221.13 471.21/221.13 Matrix type: 471.21/221.13 471.21/221.13 We used a basic matrix type which is not further parametrizeable. 471.21/221.13 471.21/221.13 471.21/221.13 471.21/221.13 471.21/221.13 471.21/221.13 As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. 471.21/221.13 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 471.21/221.13 471.21/221.13 f(f(f(x))) -> c 471.21/221.13 f(g(x)) -> f(f(x)) 471.21/221.13 f(b(x)) -> u(g(b(x))) 471.21/221.13 f(u(x)) -> u(g(x)) 471.21/221.13 b(u(x)) -> b(f(x)) 471.21/221.13 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (20) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 B(u(g(x0))) -> B(f(f(x0))) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 f(f(f(x))) -> c 471.21/221.13 b(u(x)) -> b(f(x)) 471.21/221.13 f(g(x)) -> f(f(x)) 471.21/221.13 f(b(x)) -> u(g(b(x))) 471.21/221.13 f(u(x)) -> u(g(x)) 471.21/221.13 471.21/221.13 Q is empty. 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (21) Raffelsieper-Zantema-Transformation (SOUND) 471.21/221.13 We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (22) 471.21/221.13 Obligation: 471.21/221.13 Q restricted rewrite system: 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 down(f(f(f(x)))) -> up(c) 471.21/221.13 down(b(u(x))) -> up(b(f(x))) 471.21/221.13 down(f(g(x))) -> up(f(f(x))) 471.21/221.13 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.13 down(f(u(x))) -> up(u(g(x))) 471.21/221.13 top(up(x)) -> top(down(x)) 471.21/221.13 down(u(y2)) -> u_flat(down(y2)) 471.21/221.13 down(g(y3)) -> g_flat(down(y3)) 471.21/221.13 down(f(c)) -> f_flat(down(c)) 471.21/221.13 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.13 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.13 down(b(c)) -> b_flat(down(c)) 471.21/221.13 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.13 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.13 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.13 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.13 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.13 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.13 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.13 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.13 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.13 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.13 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.13 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.13 471.21/221.13 Q is empty. 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (23) AAECC Innermost (EQUIVALENT) 471.21/221.13 We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 471.21/221.13 down(u(y2)) -> u_flat(down(y2)) 471.21/221.13 down(g(y3)) -> g_flat(down(y3)) 471.21/221.13 down(f(c)) -> f_flat(down(c)) 471.21/221.13 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.13 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.13 down(b(c)) -> b_flat(down(c)) 471.21/221.13 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.13 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.13 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.13 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.13 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.13 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.13 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.13 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.13 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.13 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.13 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.13 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.13 down(f(f(f(x)))) -> up(c) 471.21/221.13 down(b(u(x))) -> up(b(f(x))) 471.21/221.13 down(f(g(x))) -> up(f(f(x))) 471.21/221.13 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.13 down(f(u(x))) -> up(u(g(x))) 471.21/221.13 471.21/221.13 The TRS R 2 is 471.21/221.13 top(up(x)) -> top(down(x)) 471.21/221.13 471.21/221.13 The signature Sigma is {top_1} 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (24) 471.21/221.13 Obligation: 471.21/221.13 Q restricted rewrite system: 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 down(f(f(f(x)))) -> up(c) 471.21/221.13 down(b(u(x))) -> up(b(f(x))) 471.21/221.13 down(f(g(x))) -> up(f(f(x))) 471.21/221.13 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.13 down(f(u(x))) -> up(u(g(x))) 471.21/221.13 top(up(x)) -> top(down(x)) 471.21/221.13 down(u(y2)) -> u_flat(down(y2)) 471.21/221.13 down(g(y3)) -> g_flat(down(y3)) 471.21/221.13 down(f(c)) -> f_flat(down(c)) 471.21/221.13 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.13 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.13 down(b(c)) -> b_flat(down(c)) 471.21/221.13 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.13 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.13 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.13 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.13 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.13 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.13 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.13 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.13 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.13 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.13 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.13 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.13 471.21/221.13 The set Q consists of the following terms: 471.21/221.13 471.21/221.13 down(f(f(f(x0)))) 471.21/221.13 down(b(u(x0))) 471.21/221.13 down(f(g(x0))) 471.21/221.13 down(f(b(x0))) 471.21/221.13 down(f(u(x0))) 471.21/221.13 top(up(x0)) 471.21/221.13 down(u(x0)) 471.21/221.13 down(g(x0)) 471.21/221.13 down(f(c)) 471.21/221.13 down(f(fresh_constant)) 471.21/221.13 down(b(f(x0))) 471.21/221.13 down(b(c)) 471.21/221.13 down(b(b(x0))) 471.21/221.13 down(b(g(x0))) 471.21/221.13 down(b(fresh_constant)) 471.21/221.13 down(f(f(c))) 471.21/221.13 down(f(f(b(x0)))) 471.21/221.13 down(f(f(u(x0)))) 471.21/221.13 down(f(f(g(x0)))) 471.21/221.13 down(f(f(fresh_constant))) 471.21/221.13 f_flat(up(x0)) 471.21/221.13 b_flat(up(x0)) 471.21/221.13 u_flat(up(x0)) 471.21/221.13 g_flat(up(x0)) 471.21/221.13 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (25) DependencyPairsProof (EQUIVALENT) 471.21/221.13 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (26) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 TOP(up(x)) -> TOP(down(x)) 471.21/221.13 TOP(up(x)) -> DOWN(x) 471.21/221.13 DOWN(u(y2)) -> U_FLAT(down(y2)) 471.21/221.13 DOWN(u(y2)) -> DOWN(y2) 471.21/221.13 DOWN(g(y3)) -> G_FLAT(down(y3)) 471.21/221.13 DOWN(g(y3)) -> DOWN(y3) 471.21/221.13 DOWN(f(c)) -> F_FLAT(down(c)) 471.21/221.13 DOWN(f(c)) -> DOWN(c) 471.21/221.13 DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant)) 471.21/221.13 DOWN(f(fresh_constant)) -> DOWN(fresh_constant) 471.21/221.13 DOWN(b(f(y10))) -> B_FLAT(down(f(y10))) 471.21/221.13 DOWN(b(f(y10))) -> DOWN(f(y10)) 471.21/221.13 DOWN(b(c)) -> B_FLAT(down(c)) 471.21/221.13 DOWN(b(c)) -> DOWN(c) 471.21/221.13 DOWN(b(b(y11))) -> B_FLAT(down(b(y11))) 471.21/221.13 DOWN(b(b(y11))) -> DOWN(b(y11)) 471.21/221.13 DOWN(b(g(y13))) -> B_FLAT(down(g(y13))) 471.21/221.13 DOWN(b(g(y13))) -> DOWN(g(y13)) 471.21/221.13 DOWN(b(fresh_constant)) -> B_FLAT(down(fresh_constant)) 471.21/221.13 DOWN(b(fresh_constant)) -> DOWN(fresh_constant) 471.21/221.13 DOWN(f(f(c))) -> F_FLAT(down(f(c))) 471.21/221.13 DOWN(f(f(c))) -> DOWN(f(c)) 471.21/221.13 DOWN(f(f(b(y16)))) -> F_FLAT(down(f(b(y16)))) 471.21/221.13 DOWN(f(f(b(y16)))) -> DOWN(f(b(y16))) 471.21/221.13 DOWN(f(f(u(y17)))) -> F_FLAT(down(f(u(y17)))) 471.21/221.13 DOWN(f(f(u(y17)))) -> DOWN(f(u(y17))) 471.21/221.13 DOWN(f(f(g(y18)))) -> F_FLAT(down(f(g(y18)))) 471.21/221.13 DOWN(f(f(g(y18)))) -> DOWN(f(g(y18))) 471.21/221.13 DOWN(f(f(fresh_constant))) -> F_FLAT(down(f(fresh_constant))) 471.21/221.13 DOWN(f(f(fresh_constant))) -> DOWN(f(fresh_constant)) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 down(f(f(f(x)))) -> up(c) 471.21/221.13 down(b(u(x))) -> up(b(f(x))) 471.21/221.13 down(f(g(x))) -> up(f(f(x))) 471.21/221.13 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.13 down(f(u(x))) -> up(u(g(x))) 471.21/221.13 top(up(x)) -> top(down(x)) 471.21/221.13 down(u(y2)) -> u_flat(down(y2)) 471.21/221.13 down(g(y3)) -> g_flat(down(y3)) 471.21/221.13 down(f(c)) -> f_flat(down(c)) 471.21/221.13 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.13 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.13 down(b(c)) -> b_flat(down(c)) 471.21/221.13 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.13 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.13 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.13 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.13 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.13 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.13 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.13 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.13 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.13 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.13 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.13 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.13 471.21/221.13 The set Q consists of the following terms: 471.21/221.13 471.21/221.13 down(f(f(f(x0)))) 471.21/221.13 down(b(u(x0))) 471.21/221.13 down(f(g(x0))) 471.21/221.13 down(f(b(x0))) 471.21/221.13 down(f(u(x0))) 471.21/221.13 top(up(x0)) 471.21/221.13 down(u(x0)) 471.21/221.13 down(g(x0)) 471.21/221.13 down(f(c)) 471.21/221.13 down(f(fresh_constant)) 471.21/221.13 down(b(f(x0))) 471.21/221.13 down(b(c)) 471.21/221.13 down(b(b(x0))) 471.21/221.13 down(b(g(x0))) 471.21/221.13 down(b(fresh_constant)) 471.21/221.13 down(f(f(c))) 471.21/221.13 down(f(f(b(x0)))) 471.21/221.13 down(f(f(u(x0)))) 471.21/221.13 down(f(f(g(x0)))) 471.21/221.13 down(f(f(fresh_constant))) 471.21/221.13 f_flat(up(x0)) 471.21/221.13 b_flat(up(x0)) 471.21/221.13 u_flat(up(x0)) 471.21/221.13 g_flat(up(x0)) 471.21/221.13 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (27) DependencyGraphProof (EQUIVALENT) 471.21/221.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 25 less nodes. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (28) 471.21/221.13 Complex Obligation (AND) 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (29) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 DOWN(g(y3)) -> DOWN(y3) 471.21/221.13 DOWN(u(y2)) -> DOWN(y2) 471.21/221.13 DOWN(b(b(y11))) -> DOWN(b(y11)) 471.21/221.13 DOWN(b(g(y13))) -> DOWN(g(y13)) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 down(f(f(f(x)))) -> up(c) 471.21/221.13 down(b(u(x))) -> up(b(f(x))) 471.21/221.13 down(f(g(x))) -> up(f(f(x))) 471.21/221.13 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.13 down(f(u(x))) -> up(u(g(x))) 471.21/221.13 top(up(x)) -> top(down(x)) 471.21/221.13 down(u(y2)) -> u_flat(down(y2)) 471.21/221.13 down(g(y3)) -> g_flat(down(y3)) 471.21/221.13 down(f(c)) -> f_flat(down(c)) 471.21/221.13 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.13 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.13 down(b(c)) -> b_flat(down(c)) 471.21/221.13 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.13 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.13 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.13 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.13 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.13 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.13 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.13 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.13 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.13 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.13 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.13 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.13 471.21/221.13 The set Q consists of the following terms: 471.21/221.13 471.21/221.13 down(f(f(f(x0)))) 471.21/221.13 down(b(u(x0))) 471.21/221.13 down(f(g(x0))) 471.21/221.13 down(f(b(x0))) 471.21/221.13 down(f(u(x0))) 471.21/221.13 top(up(x0)) 471.21/221.13 down(u(x0)) 471.21/221.13 down(g(x0)) 471.21/221.13 down(f(c)) 471.21/221.13 down(f(fresh_constant)) 471.21/221.13 down(b(f(x0))) 471.21/221.13 down(b(c)) 471.21/221.13 down(b(b(x0))) 471.21/221.13 down(b(g(x0))) 471.21/221.13 down(b(fresh_constant)) 471.21/221.13 down(f(f(c))) 471.21/221.13 down(f(f(b(x0)))) 471.21/221.13 down(f(f(u(x0)))) 471.21/221.13 down(f(f(g(x0)))) 471.21/221.13 down(f(f(fresh_constant))) 471.21/221.13 f_flat(up(x0)) 471.21/221.13 b_flat(up(x0)) 471.21/221.13 u_flat(up(x0)) 471.21/221.13 g_flat(up(x0)) 471.21/221.13 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (30) UsableRulesProof (EQUIVALENT) 471.21/221.13 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. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (31) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 DOWN(g(y3)) -> DOWN(y3) 471.21/221.13 DOWN(u(y2)) -> DOWN(y2) 471.21/221.13 DOWN(b(b(y11))) -> DOWN(b(y11)) 471.21/221.13 DOWN(b(g(y13))) -> DOWN(g(y13)) 471.21/221.13 471.21/221.13 R is empty. 471.21/221.13 The set Q consists of the following terms: 471.21/221.13 471.21/221.13 down(f(f(f(x0)))) 471.21/221.13 down(b(u(x0))) 471.21/221.13 down(f(g(x0))) 471.21/221.13 down(f(b(x0))) 471.21/221.13 down(f(u(x0))) 471.21/221.13 top(up(x0)) 471.21/221.13 down(u(x0)) 471.21/221.13 down(g(x0)) 471.21/221.13 down(f(c)) 471.21/221.13 down(f(fresh_constant)) 471.21/221.13 down(b(f(x0))) 471.21/221.13 down(b(c)) 471.21/221.13 down(b(b(x0))) 471.21/221.13 down(b(g(x0))) 471.21/221.13 down(b(fresh_constant)) 471.21/221.13 down(f(f(c))) 471.21/221.13 down(f(f(b(x0)))) 471.21/221.13 down(f(f(u(x0)))) 471.21/221.13 down(f(f(g(x0)))) 471.21/221.13 down(f(f(fresh_constant))) 471.21/221.13 f_flat(up(x0)) 471.21/221.13 b_flat(up(x0)) 471.21/221.13 u_flat(up(x0)) 471.21/221.13 g_flat(up(x0)) 471.21/221.13 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (32) QReductionProof (EQUIVALENT) 471.21/221.13 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 471.21/221.13 471.21/221.13 down(f(f(f(x0)))) 471.21/221.13 down(b(u(x0))) 471.21/221.13 down(f(g(x0))) 471.21/221.13 down(f(b(x0))) 471.21/221.13 down(f(u(x0))) 471.21/221.13 top(up(x0)) 471.21/221.13 down(u(x0)) 471.21/221.13 down(g(x0)) 471.21/221.13 down(f(c)) 471.21/221.13 down(f(fresh_constant)) 471.21/221.13 down(b(f(x0))) 471.21/221.13 down(b(c)) 471.21/221.13 down(b(b(x0))) 471.21/221.13 down(b(g(x0))) 471.21/221.13 down(b(fresh_constant)) 471.21/221.13 down(f(f(c))) 471.21/221.13 down(f(f(b(x0)))) 471.21/221.13 down(f(f(u(x0)))) 471.21/221.13 down(f(f(g(x0)))) 471.21/221.13 down(f(f(fresh_constant))) 471.21/221.13 f_flat(up(x0)) 471.21/221.13 b_flat(up(x0)) 471.21/221.13 u_flat(up(x0)) 471.21/221.13 g_flat(up(x0)) 471.21/221.13 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (33) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 DOWN(g(y3)) -> DOWN(y3) 471.21/221.13 DOWN(u(y2)) -> DOWN(y2) 471.21/221.13 DOWN(b(b(y11))) -> DOWN(b(y11)) 471.21/221.13 DOWN(b(g(y13))) -> DOWN(g(y13)) 471.21/221.13 471.21/221.13 R is empty. 471.21/221.13 Q is empty. 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (34) QDPSizeChangeProof (EQUIVALENT) 471.21/221.13 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. 471.21/221.13 471.21/221.13 From the DPs we obtained the following set of size-change graphs: 471.21/221.13 *DOWN(b(b(y11))) -> DOWN(b(y11)) 471.21/221.13 The graph contains the following edges 1 > 1 471.21/221.13 471.21/221.13 471.21/221.13 *DOWN(b(g(y13))) -> DOWN(g(y13)) 471.21/221.13 The graph contains the following edges 1 > 1 471.21/221.13 471.21/221.13 471.21/221.13 *DOWN(g(y3)) -> DOWN(y3) 471.21/221.13 The graph contains the following edges 1 > 1 471.21/221.13 471.21/221.13 471.21/221.13 *DOWN(u(y2)) -> DOWN(y2) 471.21/221.13 The graph contains the following edges 1 > 1 471.21/221.13 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (35) 471.21/221.13 YES 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (36) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 TOP(up(x)) -> TOP(down(x)) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 down(f(f(f(x)))) -> up(c) 471.21/221.13 down(b(u(x))) -> up(b(f(x))) 471.21/221.13 down(f(g(x))) -> up(f(f(x))) 471.21/221.13 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.13 down(f(u(x))) -> up(u(g(x))) 471.21/221.13 top(up(x)) -> top(down(x)) 471.21/221.13 down(u(y2)) -> u_flat(down(y2)) 471.21/221.13 down(g(y3)) -> g_flat(down(y3)) 471.21/221.13 down(f(c)) -> f_flat(down(c)) 471.21/221.13 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.13 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.13 down(b(c)) -> b_flat(down(c)) 471.21/221.13 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.13 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.13 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.13 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.13 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.13 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.13 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.13 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.13 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.13 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.13 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.13 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.13 471.21/221.13 The set Q consists of the following terms: 471.21/221.13 471.21/221.13 down(f(f(f(x0)))) 471.21/221.13 down(b(u(x0))) 471.21/221.13 down(f(g(x0))) 471.21/221.13 down(f(b(x0))) 471.21/221.13 down(f(u(x0))) 471.21/221.13 top(up(x0)) 471.21/221.13 down(u(x0)) 471.21/221.13 down(g(x0)) 471.21/221.13 down(f(c)) 471.21/221.13 down(f(fresh_constant)) 471.21/221.13 down(b(f(x0))) 471.21/221.13 down(b(c)) 471.21/221.13 down(b(b(x0))) 471.21/221.13 down(b(g(x0))) 471.21/221.13 down(b(fresh_constant)) 471.21/221.13 down(f(f(c))) 471.21/221.13 down(f(f(b(x0)))) 471.21/221.13 down(f(f(u(x0)))) 471.21/221.13 down(f(f(g(x0)))) 471.21/221.13 down(f(f(fresh_constant))) 471.21/221.13 f_flat(up(x0)) 471.21/221.13 b_flat(up(x0)) 471.21/221.13 u_flat(up(x0)) 471.21/221.13 g_flat(up(x0)) 471.21/221.13 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (37) UsableRulesProof (EQUIVALENT) 471.21/221.13 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. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (38) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 TOP(up(x)) -> TOP(down(x)) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 down(f(f(f(x)))) -> up(c) 471.21/221.13 down(b(u(x))) -> up(b(f(x))) 471.21/221.13 down(f(g(x))) -> up(f(f(x))) 471.21/221.13 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.13 down(f(u(x))) -> up(u(g(x))) 471.21/221.13 down(u(y2)) -> u_flat(down(y2)) 471.21/221.13 down(g(y3)) -> g_flat(down(y3)) 471.21/221.13 down(f(c)) -> f_flat(down(c)) 471.21/221.13 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.13 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.13 down(b(c)) -> b_flat(down(c)) 471.21/221.13 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.13 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.13 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.13 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.13 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.13 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.13 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.13 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.13 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.13 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.13 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.13 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.13 471.21/221.13 The set Q consists of the following terms: 471.21/221.13 471.21/221.13 down(f(f(f(x0)))) 471.21/221.13 down(b(u(x0))) 471.21/221.13 down(f(g(x0))) 471.21/221.13 down(f(b(x0))) 471.21/221.13 down(f(u(x0))) 471.21/221.13 top(up(x0)) 471.21/221.13 down(u(x0)) 471.21/221.13 down(g(x0)) 471.21/221.13 down(f(c)) 471.21/221.13 down(f(fresh_constant)) 471.21/221.13 down(b(f(x0))) 471.21/221.13 down(b(c)) 471.21/221.13 down(b(b(x0))) 471.21/221.13 down(b(g(x0))) 471.21/221.13 down(b(fresh_constant)) 471.21/221.13 down(f(f(c))) 471.21/221.13 down(f(f(b(x0)))) 471.21/221.13 down(f(f(u(x0)))) 471.21/221.13 down(f(f(g(x0)))) 471.21/221.13 down(f(f(fresh_constant))) 471.21/221.13 f_flat(up(x0)) 471.21/221.13 b_flat(up(x0)) 471.21/221.13 u_flat(up(x0)) 471.21/221.13 g_flat(up(x0)) 471.21/221.13 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (39) QReductionProof (EQUIVALENT) 471.21/221.13 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 471.21/221.13 471.21/221.13 top(up(x0)) 471.21/221.13 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (40) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 TOP(up(x)) -> TOP(down(x)) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 down(f(f(f(x)))) -> up(c) 471.21/221.13 down(b(u(x))) -> up(b(f(x))) 471.21/221.13 down(f(g(x))) -> up(f(f(x))) 471.21/221.13 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.13 down(f(u(x))) -> up(u(g(x))) 471.21/221.13 down(u(y2)) -> u_flat(down(y2)) 471.21/221.13 down(g(y3)) -> g_flat(down(y3)) 471.21/221.13 down(f(c)) -> f_flat(down(c)) 471.21/221.13 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.13 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.13 down(b(c)) -> b_flat(down(c)) 471.21/221.13 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.13 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.13 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.13 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.13 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.13 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.13 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.13 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.13 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.13 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.13 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.13 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.13 471.21/221.13 The set Q consists of the following terms: 471.21/221.13 471.21/221.13 down(f(f(f(x0)))) 471.21/221.13 down(b(u(x0))) 471.21/221.13 down(f(g(x0))) 471.21/221.13 down(f(b(x0))) 471.21/221.13 down(f(u(x0))) 471.21/221.13 down(u(x0)) 471.21/221.13 down(g(x0)) 471.21/221.13 down(f(c)) 471.21/221.13 down(f(fresh_constant)) 471.21/221.13 down(b(f(x0))) 471.21/221.13 down(b(c)) 471.21/221.13 down(b(b(x0))) 471.21/221.13 down(b(g(x0))) 471.21/221.13 down(b(fresh_constant)) 471.21/221.13 down(f(f(c))) 471.21/221.13 down(f(f(b(x0)))) 471.21/221.13 down(f(f(u(x0)))) 471.21/221.13 down(f(f(g(x0)))) 471.21/221.13 down(f(f(fresh_constant))) 471.21/221.13 f_flat(up(x0)) 471.21/221.13 b_flat(up(x0)) 471.21/221.13 u_flat(up(x0)) 471.21/221.13 g_flat(up(x0)) 471.21/221.13 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (41) TransformationProof (EQUIVALENT) 471.21/221.13 By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]: 471.21/221.13 471.21/221.13 (TOP(up(f(f(f(x0))))) -> TOP(up(c)),TOP(up(f(f(f(x0))))) -> TOP(up(c))) 471.21/221.13 (TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))),TOP(up(b(u(x0)))) -> TOP(up(b(f(x0))))) 471.21/221.13 (TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))),TOP(up(f(g(x0)))) -> TOP(up(f(f(x0))))) 471.21/221.13 (TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))),TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0)))))) 471.21/221.13 (TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))),TOP(up(f(u(x0)))) -> TOP(up(u(g(x0))))) 471.21/221.13 (TOP(up(u(x0))) -> TOP(u_flat(down(x0))),TOP(up(u(x0))) -> TOP(u_flat(down(x0)))) 471.21/221.13 (TOP(up(g(x0))) -> TOP(g_flat(down(x0))),TOP(up(g(x0))) -> TOP(g_flat(down(x0)))) 471.21/221.13 (TOP(up(f(c))) -> TOP(f_flat(down(c))),TOP(up(f(c))) -> TOP(f_flat(down(c)))) 471.21/221.13 (TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))),TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant)))) 471.21/221.13 (TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))),TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0))))) 471.21/221.13 (TOP(up(b(c))) -> TOP(b_flat(down(c))),TOP(up(b(c))) -> TOP(b_flat(down(c)))) 471.21/221.13 (TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))),TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))) 471.21/221.13 (TOP(up(b(g(x0)))) -> TOP(b_flat(down(g(x0)))),TOP(up(b(g(x0)))) -> TOP(b_flat(down(g(x0))))) 471.21/221.13 (TOP(up(b(fresh_constant))) -> TOP(b_flat(down(fresh_constant))),TOP(up(b(fresh_constant))) -> TOP(b_flat(down(fresh_constant)))) 471.21/221.13 (TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))),TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c))))) 471.21/221.13 (TOP(up(f(f(b(x0))))) -> TOP(f_flat(down(f(b(x0))))),TOP(up(f(f(b(x0))))) -> TOP(f_flat(down(f(b(x0)))))) 471.21/221.13 (TOP(up(f(f(u(x0))))) -> TOP(f_flat(down(f(u(x0))))),TOP(up(f(f(u(x0))))) -> TOP(f_flat(down(f(u(x0)))))) 471.21/221.13 (TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))),TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))) 471.21/221.13 (TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))),TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))) 471.21/221.13 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (42) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 TOP(up(f(f(f(x0))))) -> TOP(up(c)) 471.21/221.13 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.13 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.13 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.13 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.13 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.13 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.13 TOP(up(f(c))) -> TOP(f_flat(down(c))) 471.21/221.13 TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))) 471.21/221.13 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.13 TOP(up(b(c))) -> TOP(b_flat(down(c))) 471.21/221.13 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.13 TOP(up(b(g(x0)))) -> TOP(b_flat(down(g(x0)))) 471.21/221.13 TOP(up(b(fresh_constant))) -> TOP(b_flat(down(fresh_constant))) 471.21/221.13 TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) 471.21/221.13 TOP(up(f(f(b(x0))))) -> TOP(f_flat(down(f(b(x0))))) 471.21/221.13 TOP(up(f(f(u(x0))))) -> TOP(f_flat(down(f(u(x0))))) 471.21/221.13 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 471.21/221.13 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 down(f(f(f(x)))) -> up(c) 471.21/221.13 down(b(u(x))) -> up(b(f(x))) 471.21/221.13 down(f(g(x))) -> up(f(f(x))) 471.21/221.13 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.13 down(f(u(x))) -> up(u(g(x))) 471.21/221.13 down(u(y2)) -> u_flat(down(y2)) 471.21/221.13 down(g(y3)) -> g_flat(down(y3)) 471.21/221.13 down(f(c)) -> f_flat(down(c)) 471.21/221.13 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.13 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.13 down(b(c)) -> b_flat(down(c)) 471.21/221.13 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.13 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.13 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.13 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.13 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.13 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.13 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.13 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.13 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.13 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.13 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.13 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.13 471.21/221.13 The set Q consists of the following terms: 471.21/221.13 471.21/221.13 down(f(f(f(x0)))) 471.21/221.13 down(b(u(x0))) 471.21/221.13 down(f(g(x0))) 471.21/221.13 down(f(b(x0))) 471.21/221.13 down(f(u(x0))) 471.21/221.13 down(u(x0)) 471.21/221.13 down(g(x0)) 471.21/221.13 down(f(c)) 471.21/221.13 down(f(fresh_constant)) 471.21/221.13 down(b(f(x0))) 471.21/221.13 down(b(c)) 471.21/221.13 down(b(b(x0))) 471.21/221.13 down(b(g(x0))) 471.21/221.13 down(b(fresh_constant)) 471.21/221.13 down(f(f(c))) 471.21/221.13 down(f(f(b(x0)))) 471.21/221.13 down(f(f(u(x0)))) 471.21/221.13 down(f(f(g(x0)))) 471.21/221.13 down(f(f(fresh_constant))) 471.21/221.13 f_flat(up(x0)) 471.21/221.13 b_flat(up(x0)) 471.21/221.13 u_flat(up(x0)) 471.21/221.13 g_flat(up(x0)) 471.21/221.13 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (43) DependencyGraphProof (EQUIVALENT) 471.21/221.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (44) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) 471.21/221.13 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.13 TOP(up(f(f(b(x0))))) -> TOP(f_flat(down(f(b(x0))))) 471.21/221.13 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.13 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.13 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.13 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.13 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.13 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.13 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.13 TOP(up(b(g(x0)))) -> TOP(b_flat(down(g(x0)))) 471.21/221.13 TOP(up(f(f(u(x0))))) -> TOP(f_flat(down(f(u(x0))))) 471.21/221.13 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 471.21/221.13 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 down(f(f(f(x)))) -> up(c) 471.21/221.13 down(b(u(x))) -> up(b(f(x))) 471.21/221.13 down(f(g(x))) -> up(f(f(x))) 471.21/221.13 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.13 down(f(u(x))) -> up(u(g(x))) 471.21/221.13 down(u(y2)) -> u_flat(down(y2)) 471.21/221.13 down(g(y3)) -> g_flat(down(y3)) 471.21/221.13 down(f(c)) -> f_flat(down(c)) 471.21/221.13 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.13 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.13 down(b(c)) -> b_flat(down(c)) 471.21/221.13 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.13 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.13 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.13 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.13 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.13 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.13 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.13 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.13 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.13 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.13 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.13 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.13 471.21/221.13 The set Q consists of the following terms: 471.21/221.13 471.21/221.13 down(f(f(f(x0)))) 471.21/221.13 down(b(u(x0))) 471.21/221.13 down(f(g(x0))) 471.21/221.13 down(f(b(x0))) 471.21/221.13 down(f(u(x0))) 471.21/221.13 down(u(x0)) 471.21/221.13 down(g(x0)) 471.21/221.13 down(f(c)) 471.21/221.13 down(f(fresh_constant)) 471.21/221.13 down(b(f(x0))) 471.21/221.13 down(b(c)) 471.21/221.13 down(b(b(x0))) 471.21/221.13 down(b(g(x0))) 471.21/221.13 down(b(fresh_constant)) 471.21/221.13 down(f(f(c))) 471.21/221.13 down(f(f(b(x0)))) 471.21/221.13 down(f(f(u(x0)))) 471.21/221.13 down(f(f(g(x0)))) 471.21/221.13 down(f(f(fresh_constant))) 471.21/221.13 f_flat(up(x0)) 471.21/221.13 b_flat(up(x0)) 471.21/221.13 u_flat(up(x0)) 471.21/221.13 g_flat(up(x0)) 471.21/221.13 471.21/221.13 We have to consider all minimal (P,Q,R)-chains. 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (45) TransformationProof (EQUIVALENT) 471.21/221.13 By rewriting [LPAR04] the rule TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) at position [0,0] we obtained the following new rules [LPAR04]: 471.21/221.13 471.21/221.13 (TOP(up(f(f(c)))) -> TOP(f_flat(f_flat(down(c)))),TOP(up(f(f(c)))) -> TOP(f_flat(f_flat(down(c))))) 471.21/221.13 471.21/221.13 471.21/221.13 ---------------------------------------- 471.21/221.13 471.21/221.13 (46) 471.21/221.13 Obligation: 471.21/221.13 Q DP problem: 471.21/221.13 The TRS P consists of the following rules: 471.21/221.13 471.21/221.13 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.13 TOP(up(f(f(b(x0))))) -> TOP(f_flat(down(f(b(x0))))) 471.21/221.13 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.13 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.13 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.13 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.13 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.13 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.13 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.13 TOP(up(b(g(x0)))) -> TOP(b_flat(down(g(x0)))) 471.21/221.13 TOP(up(f(f(u(x0))))) -> TOP(f_flat(down(f(u(x0))))) 471.21/221.13 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 471.21/221.13 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 471.21/221.13 TOP(up(f(f(c)))) -> TOP(f_flat(f_flat(down(c)))) 471.21/221.13 471.21/221.13 The TRS R consists of the following rules: 471.21/221.13 471.21/221.13 down(f(f(f(x)))) -> up(c) 471.21/221.13 down(b(u(x))) -> up(b(f(x))) 471.21/221.13 down(f(g(x))) -> up(f(f(x))) 471.21/221.13 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.13 down(f(u(x))) -> up(u(g(x))) 471.21/221.13 down(u(y2)) -> u_flat(down(y2)) 471.21/221.13 down(g(y3)) -> g_flat(down(y3)) 471.21/221.13 down(f(c)) -> f_flat(down(c)) 471.21/221.13 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.13 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.13 down(b(c)) -> b_flat(down(c)) 471.21/221.13 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.13 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.13 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.13 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.13 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.13 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.13 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.13 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (47) DependencyGraphProof (EQUIVALENT) 471.21/221.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (48) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(f_flat(down(f(b(x0))))) 471.21/221.14 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(f_flat(down(f(u(x0))))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(down(g(x0)))) 471.21/221.14 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 471.21/221.14 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (49) TransformationProof (EQUIVALENT) 471.21/221.14 By rewriting [LPAR04] the rule TOP(up(f(f(b(x0))))) -> TOP(f_flat(down(f(b(x0))))) at position [0,0] we obtained the following new rules [LPAR04]: 471.21/221.14 471.21/221.14 (TOP(up(f(f(b(x0))))) -> TOP(f_flat(up(u(g(b(x0)))))),TOP(up(f(f(b(x0))))) -> TOP(f_flat(up(u(g(b(x0))))))) 471.21/221.14 471.21/221.14 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (50) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(f_flat(down(f(u(x0))))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(down(g(x0)))) 471.21/221.14 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 471.21/221.14 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(f_flat(up(u(g(b(x0)))))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (51) TransformationProof (EQUIVALENT) 471.21/221.14 By rewriting [LPAR04] the rule TOP(up(f(f(u(x0))))) -> TOP(f_flat(down(f(u(x0))))) at position [0,0] we obtained the following new rules [LPAR04]: 471.21/221.14 471.21/221.14 (TOP(up(f(f(u(x0))))) -> TOP(f_flat(up(u(g(x0))))),TOP(up(f(f(u(x0))))) -> TOP(f_flat(up(u(g(x0)))))) 471.21/221.14 471.21/221.14 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (52) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(down(g(x0)))) 471.21/221.14 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 471.21/221.14 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(f_flat(up(u(g(b(x0)))))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(f_flat(up(u(g(x0))))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (53) TransformationProof (EQUIVALENT) 471.21/221.14 By rewriting [LPAR04] the rule TOP(up(b(g(x0)))) -> TOP(b_flat(down(g(x0)))) at position [0,0] we obtained the following new rules [LPAR04]: 471.21/221.14 471.21/221.14 (TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))),TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0))))) 471.21/221.14 471.21/221.14 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (54) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 471.21/221.14 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(f_flat(up(u(g(b(x0)))))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(f_flat(up(u(g(x0))))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (55) TransformationProof (EQUIVALENT) 471.21/221.14 By rewriting [LPAR04] the rule TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) at position [0,0] we obtained the following new rules [LPAR04]: 471.21/221.14 471.21/221.14 (TOP(up(f(f(g(x0))))) -> TOP(f_flat(up(f(f(x0))))),TOP(up(f(f(g(x0))))) -> TOP(f_flat(up(f(f(x0)))))) 471.21/221.14 471.21/221.14 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (56) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(f_flat(up(u(g(b(x0)))))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(f_flat(up(u(g(x0))))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.14 TOP(up(f(f(g(x0))))) -> TOP(f_flat(up(f(f(x0))))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (57) TransformationProof (EQUIVALENT) 471.21/221.14 By rewriting [LPAR04] the rule TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) at position [0,0] we obtained the following new rules [LPAR04]: 471.21/221.14 471.21/221.14 (TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant)))),TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant))))) 471.21/221.14 471.21/221.14 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (58) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(f_flat(up(u(g(b(x0)))))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(f_flat(up(u(g(x0))))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.14 TOP(up(f(f(g(x0))))) -> TOP(f_flat(up(f(f(x0))))) 471.21/221.14 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant)))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (59) DependencyGraphProof (EQUIVALENT) 471.21/221.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (60) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(f_flat(up(u(g(b(x0)))))) 471.21/221.14 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(f_flat(up(u(g(x0))))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.14 TOP(up(f(f(g(x0))))) -> TOP(f_flat(up(f(f(x0))))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (61) TransformationProof (EQUIVALENT) 471.21/221.14 By rewriting [LPAR04] the rule TOP(up(f(f(b(x0))))) -> TOP(f_flat(up(u(g(b(x0)))))) at position [0] we obtained the following new rules [LPAR04]: 471.21/221.14 471.21/221.14 (TOP(up(f(f(b(x0))))) -> TOP(up(f(u(g(b(x0)))))),TOP(up(f(f(b(x0))))) -> TOP(up(f(u(g(b(x0))))))) 471.21/221.14 471.21/221.14 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (62) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(f_flat(up(u(g(x0))))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.14 TOP(up(f(f(g(x0))))) -> TOP(f_flat(up(f(f(x0))))) 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(up(f(u(g(b(x0)))))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (63) TransformationProof (EQUIVALENT) 471.21/221.14 By rewriting [LPAR04] the rule TOP(up(f(f(u(x0))))) -> TOP(f_flat(up(u(g(x0))))) at position [0] we obtained the following new rules [LPAR04]: 471.21/221.14 471.21/221.14 (TOP(up(f(f(u(x0))))) -> TOP(up(f(u(g(x0))))),TOP(up(f(f(u(x0))))) -> TOP(up(f(u(g(x0)))))) 471.21/221.14 471.21/221.14 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (64) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.14 TOP(up(f(f(g(x0))))) -> TOP(f_flat(up(f(f(x0))))) 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(up(f(u(g(b(x0)))))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(up(f(u(g(x0))))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (65) TransformationProof (EQUIVALENT) 471.21/221.14 By rewriting [LPAR04] the rule TOP(up(f(f(g(x0))))) -> TOP(f_flat(up(f(f(x0))))) at position [0] we obtained the following new rules [LPAR04]: 471.21/221.14 471.21/221.14 (TOP(up(f(f(g(x0))))) -> TOP(up(f(f(f(x0))))),TOP(up(f(f(g(x0))))) -> TOP(up(f(f(f(x0)))))) 471.21/221.14 471.21/221.14 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (66) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(up(f(u(g(b(x0)))))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(up(f(u(g(x0))))) 471.21/221.14 TOP(up(f(f(g(x0))))) -> TOP(up(f(f(f(x0))))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (67) DependencyGraphProof (EQUIVALENT) 471.21/221.14 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (68) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(up(f(u(g(b(x0)))))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(up(f(u(g(x0))))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (69) TransformationProof (EQUIVALENT) 471.21/221.14 By forward instantiating [JAR06] the rule TOP(up(f(g(x0)))) -> TOP(up(f(f(x0)))) we obtained the following new rules [LPAR04]: 471.21/221.14 471.21/221.14 (TOP(up(f(g(b(y_0))))) -> TOP(up(f(f(b(y_0))))),TOP(up(f(g(b(y_0))))) -> TOP(up(f(f(b(y_0)))))) 471.21/221.14 (TOP(up(f(g(u(y_0))))) -> TOP(up(f(f(u(y_0))))),TOP(up(f(g(u(y_0))))) -> TOP(up(f(f(u(y_0)))))) 471.21/221.14 471.21/221.14 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (70) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(up(f(u(g(b(x0)))))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(up(f(u(g(x0))))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.14 TOP(up(f(g(b(y_0))))) -> TOP(up(f(f(b(y_0))))) 471.21/221.14 TOP(up(f(g(u(y_0))))) -> TOP(up(f(f(u(y_0))))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (71) QDPOrderProof (EQUIVALENT) 471.21/221.14 We use the reduction pair processor [LPAR04,JAR06]. 471.21/221.14 471.21/221.14 471.21/221.14 The following pairs can be oriented strictly and are deleted. 471.21/221.14 471.21/221.14 TOP(up(f(g(b(y_0))))) -> TOP(up(f(f(b(y_0))))) 471.21/221.14 TOP(up(f(g(u(y_0))))) -> TOP(up(f(f(u(y_0))))) 471.21/221.14 The remaining pairs can at least be oriented weakly. 471.21/221.14 Used ordering: Polynomial interpretation [POLO]: 471.21/221.14 471.21/221.14 POL(TOP(x_1)) = x_1 471.21/221.14 POL(b(x_1)) = 0 471.21/221.14 POL(b_flat(x_1)) = 0 471.21/221.14 POL(c) = 0 471.21/221.14 POL(down(x_1)) = 0 471.21/221.14 POL(f(x_1)) = x_1 471.21/221.14 POL(f_flat(x_1)) = 0 471.21/221.14 POL(fresh_constant) = 0 471.21/221.14 POL(g(x_1)) = 1 471.21/221.14 POL(g_flat(x_1)) = 1 471.21/221.14 POL(u(x_1)) = 0 471.21/221.14 POL(u_flat(x_1)) = 0 471.21/221.14 POL(up(x_1)) = x_1 471.21/221.14 471.21/221.14 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 471.21/221.14 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 471.21/221.14 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (72) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(up(f(u(g(b(x0)))))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(up(f(u(g(x0))))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (73) QDPOrderProof (EQUIVALENT) 471.21/221.14 We use the reduction pair processor [LPAR04,JAR06]. 471.21/221.14 471.21/221.14 471.21/221.14 The following pairs can be oriented strictly and are deleted. 471.21/221.14 471.21/221.14 TOP(up(f(f(b(x0))))) -> TOP(up(f(u(g(b(x0)))))) 471.21/221.14 TOP(up(f(u(x0)))) -> TOP(up(u(g(x0)))) 471.21/221.14 TOP(up(f(f(u(x0))))) -> TOP(up(f(u(g(x0))))) 471.21/221.14 TOP(up(f(b(x0)))) -> TOP(up(u(g(b(x0))))) 471.21/221.14 The remaining pairs can at least be oriented weakly. 471.21/221.14 Used ordering: Polynomial interpretation [POLO]: 471.21/221.14 471.21/221.14 POL(TOP(x_1)) = x_1 471.21/221.14 POL(b(x_1)) = 0 471.21/221.14 POL(b_flat(x_1)) = 0 471.21/221.14 POL(c) = 0 471.21/221.14 POL(down(x_1)) = 0 471.21/221.14 POL(f(x_1)) = 1 + x_1 471.21/221.14 POL(f_flat(x_1)) = 0 471.21/221.14 POL(fresh_constant) = 0 471.21/221.14 POL(g(x_1)) = 0 471.21/221.14 POL(g_flat(x_1)) = 0 471.21/221.14 POL(u(x_1)) = 0 471.21/221.14 POL(u_flat(x_1)) = 0 471.21/221.14 POL(up(x_1)) = x_1 471.21/221.14 471.21/221.14 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 471.21/221.14 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 471.21/221.14 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (74) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (75) MNOCProof (EQUIVALENT) 471.21/221.14 We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (76) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 Q is empty. 471.21/221.14 We have to consider all (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (77) SplitQDPProof (EQUIVALENT) 471.21/221.14 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 471.21/221.14 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (78) 471.21/221.14 Complex Obligation (AND) 471.21/221.14 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (79) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.14 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.14 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.14 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.14 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.14 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down(f(f(f(x)))) -> up(c) 471.21/221.14 down(b(u(x))) -> up(b(f(x))) 471.21/221.14 down(f(g(x))) -> up(f(f(x))) 471.21/221.14 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.14 down(f(u(x))) -> up(u(g(x))) 471.21/221.14 down(u(y2)) -> u_flat(down(y2)) 471.21/221.14 down(g(y3)) -> g_flat(down(y3)) 471.21/221.14 down(f(c)) -> f_flat(down(c)) 471.21/221.14 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 471.21/221.14 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.14 down(b(c)) -> b_flat(down(c)) 471.21/221.14 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.14 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.14 down(b(fresh_constant)) -> b_flat(down(fresh_constant)) 471.21/221.14 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.14 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.14 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.14 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.14 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.14 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.14 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.14 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.14 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down(f(f(f(x0)))) 471.21/221.14 down(b(u(x0))) 471.21/221.14 down(f(g(x0))) 471.21/221.14 down(f(b(x0))) 471.21/221.14 down(f(u(x0))) 471.21/221.14 down(u(x0)) 471.21/221.14 down(g(x0)) 471.21/221.14 down(f(c)) 471.21/221.14 down(f(fresh_constant)) 471.21/221.14 down(b(f(x0))) 471.21/221.14 down(b(c)) 471.21/221.14 down(b(b(x0))) 471.21/221.14 down(b(g(x0))) 471.21/221.14 down(b(fresh_constant)) 471.21/221.14 down(f(f(c))) 471.21/221.14 down(f(f(b(x0)))) 471.21/221.14 down(f(f(u(x0)))) 471.21/221.14 down(f(f(g(x0)))) 471.21/221.14 down(f(f(fresh_constant))) 471.21/221.14 f_flat(up(x0)) 471.21/221.14 b_flat(up(x0)) 471.21/221.14 u_flat(up(x0)) 471.21/221.14 g_flat(up(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (80) SemLabProof (SOUND) 471.21/221.14 We found the following model for the rules of the TRSs R and P. 471.21/221.14 Interpretation over the domain with elements from 0 to 1. 471.21/221.14 c: 0 471.21/221.14 TOP: 0 471.21/221.14 u: 0 471.21/221.14 g: 0 471.21/221.14 b: 0 471.21/221.14 down: 0 471.21/221.14 f: 0 471.21/221.14 fresh_constant: 1 471.21/221.14 up: 0 471.21/221.14 u_flat: 0 471.21/221.14 f_flat: 0 471.21/221.14 b_flat: 0 471.21/221.14 g_flat: 0 471.21/221.14 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 471.21/221.14 ---------------------------------------- 471.21/221.14 471.21/221.14 (81) 471.21/221.14 Obligation: 471.21/221.14 Q DP problem: 471.21/221.14 The TRS P consists of the following rules: 471.21/221.14 471.21/221.14 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.21/221.14 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.21/221.14 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(f.1(x0)))) 471.21/221.14 TOP.0(up.0(u.1(x0))) -> TOP.0(u_flat.0(down.1(x0))) 471.21/221.14 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.21/221.14 TOP.0(up.0(g.1(x0))) -> TOP.0(g_flat.0(down.1(x0))) 471.21/221.14 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.21/221.14 TOP.0(up.0(b.0(f.1(x0)))) -> TOP.0(b_flat.0(down.0(f.1(x0)))) 471.21/221.14 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.21/221.14 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.21/221.14 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.21/221.14 TOP.0(up.0(b.0(g.1(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.1(x0)))) 471.21/221.14 471.21/221.14 The TRS R consists of the following rules: 471.21/221.14 471.21/221.14 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.21/221.14 down.0(f.0(f.0(f.1(x)))) -> up.0(c.) 471.21/221.14 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.21/221.14 down.0(b.0(u.1(x))) -> up.0(b.0(f.1(x))) 471.21/221.14 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.21/221.14 down.0(f.0(g.1(x))) -> up.0(f.0(f.1(x))) 471.21/221.14 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.21/221.14 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.21/221.14 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.21/221.14 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.21/221.14 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.21/221.14 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 471.21/221.14 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.21/221.14 down.0(g.1(y3)) -> g_flat.0(down.1(y3)) 471.21/221.14 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 471.21/221.14 down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.)) 471.21/221.14 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.21/221.14 down.0(b.0(f.1(y10))) -> b_flat.0(down.0(f.1(y10))) 471.21/221.14 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 471.21/221.14 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.21/221.14 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.21/221.14 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.21/221.14 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.21/221.14 down.0(b.1(fresh_constant.)) -> b_flat.0(down.1(fresh_constant.)) 471.21/221.14 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.21/221.14 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.21/221.14 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.21/221.14 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.21/221.14 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.21/221.14 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.21/221.14 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.21/221.14 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 471.21/221.14 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.21/221.14 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 471.21/221.14 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.21/221.14 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 471.21/221.14 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.21/221.14 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.21/221.14 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.21/221.14 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.21/221.14 471.21/221.14 The set Q consists of the following terms: 471.21/221.14 471.21/221.14 down.0(f.0(f.0(f.0(x0)))) 471.21/221.14 down.0(f.0(f.0(f.1(x0)))) 471.21/221.14 down.0(b.0(u.0(x0))) 471.21/221.14 down.0(b.0(u.1(x0))) 471.21/221.14 down.0(f.0(g.0(x0))) 471.21/221.14 down.0(f.0(g.1(x0))) 471.21/221.14 down.0(f.0(b.0(x0))) 471.21/221.14 down.0(f.0(b.1(x0))) 471.21/221.14 down.0(f.0(u.0(x0))) 471.21/221.14 down.0(f.0(u.1(x0))) 471.21/221.14 down.0(u.0(x0)) 471.21/221.14 down.0(u.1(x0)) 471.21/221.14 down.0(g.0(x0)) 471.21/221.14 down.0(g.1(x0)) 471.21/221.14 down.0(f.0(c.)) 471.21/221.14 down.0(f.1(fresh_constant.)) 471.21/221.14 down.0(b.0(f.0(x0))) 471.21/221.14 down.0(b.0(f.1(x0))) 471.21/221.14 down.0(b.0(c.)) 471.21/221.14 down.0(b.0(b.0(x0))) 471.21/221.14 down.0(b.0(b.1(x0))) 471.21/221.14 down.0(b.0(g.0(x0))) 471.21/221.14 down.0(b.0(g.1(x0))) 471.21/221.14 down.0(b.1(fresh_constant.)) 471.21/221.14 down.0(f.0(f.0(c.))) 471.21/221.14 down.0(f.0(f.0(b.0(x0)))) 471.21/221.14 down.0(f.0(f.0(b.1(x0)))) 471.21/221.14 down.0(f.0(f.0(u.0(x0)))) 471.21/221.14 down.0(f.0(f.0(u.1(x0)))) 471.21/221.14 down.0(f.0(f.0(g.0(x0)))) 471.21/221.14 down.0(f.0(f.0(g.1(x0)))) 471.21/221.14 down.0(f.0(f.1(fresh_constant.))) 471.21/221.14 f_flat.0(up.0(x0)) 471.21/221.14 f_flat.0(up.1(x0)) 471.21/221.14 b_flat.0(up.0(x0)) 471.21/221.14 b_flat.0(up.1(x0)) 471.21/221.14 u_flat.0(up.0(x0)) 471.21/221.14 u_flat.0(up.1(x0)) 471.21/221.14 g_flat.0(up.0(x0)) 471.21/221.14 g_flat.0(up.1(x0)) 471.21/221.14 471.21/221.14 We have to consider all minimal (P,Q,R)-chains. 471.21/221.14 ---------------------------------------- 471.21/221.15 471.21/221.15 (82) DependencyGraphProof (EQUIVALENT) 471.21/221.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 471.21/221.15 ---------------------------------------- 471.21/221.15 471.21/221.15 (83) 471.21/221.15 Obligation: 471.21/221.15 Q DP problem: 471.21/221.15 The TRS P consists of the following rules: 471.21/221.15 471.21/221.15 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.21/221.15 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.21/221.15 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(f.1(x0)))) 471.21/221.15 TOP.0(up.0(b.0(f.1(x0)))) -> TOP.0(b_flat.0(down.0(f.1(x0)))) 471.21/221.15 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.21/221.15 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.21/221.15 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.21/221.15 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.21/221.15 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.21/221.15 471.21/221.15 The TRS R consists of the following rules: 471.21/221.15 471.21/221.15 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.21/221.15 down.0(f.0(f.0(f.1(x)))) -> up.0(c.) 471.21/221.15 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.21/221.15 down.0(b.0(u.1(x))) -> up.0(b.0(f.1(x))) 471.21/221.15 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.21/221.15 down.0(f.0(g.1(x))) -> up.0(f.0(f.1(x))) 471.21/221.15 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.21/221.15 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.21/221.15 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.21/221.15 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.21/221.15 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.21/221.15 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 471.21/221.15 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.21/221.15 down.0(g.1(y3)) -> g_flat.0(down.1(y3)) 471.21/221.15 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 471.21/221.15 down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.)) 471.21/221.15 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.21/221.15 down.0(b.0(f.1(y10))) -> b_flat.0(down.0(f.1(y10))) 471.21/221.15 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 471.21/221.15 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.21/221.15 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.21/221.15 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.21/221.15 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.21/221.15 down.0(b.1(fresh_constant.)) -> b_flat.0(down.1(fresh_constant.)) 471.21/221.15 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.21/221.15 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.21/221.15 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.21/221.15 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.21/221.15 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.21/221.15 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.21/221.15 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.21/221.15 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 471.21/221.15 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.21/221.15 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 471.21/221.15 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.21/221.15 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 471.21/221.15 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.21/221.15 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.21/221.15 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.21/221.15 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.21/221.15 471.21/221.15 The set Q consists of the following terms: 471.21/221.15 471.21/221.15 down.0(f.0(f.0(f.0(x0)))) 471.21/221.15 down.0(f.0(f.0(f.1(x0)))) 471.21/221.15 down.0(b.0(u.0(x0))) 471.21/221.15 down.0(b.0(u.1(x0))) 471.21/221.15 down.0(f.0(g.0(x0))) 471.21/221.15 down.0(f.0(g.1(x0))) 471.21/221.15 down.0(f.0(b.0(x0))) 471.21/221.15 down.0(f.0(b.1(x0))) 471.21/221.15 down.0(f.0(u.0(x0))) 471.21/221.15 down.0(f.0(u.1(x0))) 471.21/221.15 down.0(u.0(x0)) 471.21/221.15 down.0(u.1(x0)) 471.21/221.15 down.0(g.0(x0)) 471.21/221.15 down.0(g.1(x0)) 471.21/221.15 down.0(f.0(c.)) 471.21/221.15 down.0(f.1(fresh_constant.)) 471.21/221.15 down.0(b.0(f.0(x0))) 471.21/221.15 down.0(b.0(f.1(x0))) 471.21/221.15 down.0(b.0(c.)) 471.21/221.15 down.0(b.0(b.0(x0))) 471.21/221.15 down.0(b.0(b.1(x0))) 471.21/221.15 down.0(b.0(g.0(x0))) 471.21/221.15 down.0(b.0(g.1(x0))) 471.21/221.15 down.0(b.1(fresh_constant.)) 471.21/221.15 down.0(f.0(f.0(c.))) 471.21/221.15 down.0(f.0(f.0(b.0(x0)))) 471.21/221.15 down.0(f.0(f.0(b.1(x0)))) 471.21/221.15 down.0(f.0(f.0(u.0(x0)))) 471.21/221.15 down.0(f.0(f.0(u.1(x0)))) 471.21/221.15 down.0(f.0(f.0(g.0(x0)))) 471.21/221.15 down.0(f.0(f.0(g.1(x0)))) 471.21/221.15 down.0(f.0(f.1(fresh_constant.))) 471.21/221.15 f_flat.0(up.0(x0)) 471.21/221.15 f_flat.0(up.1(x0)) 471.21/221.15 b_flat.0(up.0(x0)) 471.21/221.15 b_flat.0(up.1(x0)) 471.21/221.15 u_flat.0(up.0(x0)) 471.21/221.15 u_flat.0(up.1(x0)) 471.21/221.15 g_flat.0(up.0(x0)) 471.21/221.15 g_flat.0(up.1(x0)) 471.21/221.15 471.21/221.15 We have to consider all minimal (P,Q,R)-chains. 471.21/221.15 ---------------------------------------- 471.21/221.15 471.21/221.15 (84) UsableRulesReductionPairsProof (EQUIVALENT) 471.21/221.15 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. 471.21/221.15 471.21/221.15 No dependency pairs are removed. 471.21/221.15 471.21/221.15 The following rules are removed from R: 471.21/221.15 471.21/221.15 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 471.21/221.15 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 471.21/221.15 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.21/221.15 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.21/221.15 Used ordering: POLO with Polynomial interpretation [POLO]: 471.21/221.15 471.21/221.15 POL(TOP.0(x_1)) = x_1 471.21/221.15 POL(b.0(x_1)) = 1 + x_1 471.21/221.15 POL(b.1(x_1)) = 1 + x_1 471.21/221.15 POL(b_flat.0(x_1)) = 1 + x_1 471.21/221.15 POL(c.) = 0 471.21/221.15 POL(down.0(x_1)) = 1 + x_1 471.21/221.15 POL(down.1(x_1)) = 1 + x_1 471.21/221.15 POL(f.0(x_1)) = x_1 471.21/221.15 POL(f.1(x_1)) = x_1 471.21/221.15 POL(f_flat.0(x_1)) = x_1 471.21/221.15 POL(fresh_constant.) = 0 471.21/221.15 POL(g.0(x_1)) = x_1 471.21/221.15 POL(g.1(x_1)) = x_1 471.21/221.15 POL(g_flat.0(x_1)) = x_1 471.21/221.15 POL(u.0(x_1)) = x_1 471.21/221.15 POL(u.1(x_1)) = x_1 471.21/221.15 POL(u_flat.0(x_1)) = x_1 471.21/221.15 POL(up.0(x_1)) = 1 + x_1 471.21/221.15 POL(up.1(x_1)) = 1 + x_1 471.21/221.15 471.21/221.15 471.21/221.15 ---------------------------------------- 471.21/221.15 471.21/221.15 (85) 471.21/221.15 Obligation: 471.21/221.15 Q DP problem: 471.21/221.15 The TRS P consists of the following rules: 471.21/221.15 471.21/221.15 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.21/221.15 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.21/221.15 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(f.1(x0)))) 471.21/221.15 TOP.0(up.0(b.0(f.1(x0)))) -> TOP.0(b_flat.0(down.0(f.1(x0)))) 471.21/221.15 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.21/221.15 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.21/221.15 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.21/221.15 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.21/221.15 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.21/221.15 471.21/221.15 The TRS R consists of the following rules: 471.21/221.15 471.21/221.15 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.21/221.15 down.0(f.0(f.0(f.1(x)))) -> up.0(c.) 471.21/221.15 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.21/221.15 down.0(b.0(u.1(x))) -> up.0(b.0(f.1(x))) 471.21/221.15 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.21/221.15 down.0(f.0(g.1(x))) -> up.0(f.0(f.1(x))) 471.21/221.15 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.21/221.15 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.21/221.15 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.21/221.15 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.21/221.15 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.21/221.15 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 471.21/221.15 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.21/221.15 down.0(g.1(y3)) -> g_flat.0(down.1(y3)) 471.21/221.15 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 471.21/221.15 down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.)) 471.21/221.15 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.21/221.15 down.0(b.0(f.1(y10))) -> b_flat.0(down.0(f.1(y10))) 471.21/221.15 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 471.21/221.15 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.21/221.15 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.21/221.15 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.21/221.15 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.21/221.15 down.0(b.1(fresh_constant.)) -> b_flat.0(down.1(fresh_constant.)) 471.21/221.15 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.21/221.15 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.21/221.15 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.21/221.15 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.21/221.15 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.21/221.15 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.21/221.15 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.21/221.15 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 471.21/221.15 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.21/221.15 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.21/221.15 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.21/221.15 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.21/221.15 471.21/221.15 The set Q consists of the following terms: 471.21/221.15 471.21/221.15 down.0(f.0(f.0(f.0(x0)))) 471.21/221.15 down.0(f.0(f.0(f.1(x0)))) 471.21/221.15 down.0(b.0(u.0(x0))) 471.21/221.15 down.0(b.0(u.1(x0))) 471.21/221.15 down.0(f.0(g.0(x0))) 471.21/221.15 down.0(f.0(g.1(x0))) 471.21/221.15 down.0(f.0(b.0(x0))) 471.21/221.15 down.0(f.0(b.1(x0))) 471.21/221.15 down.0(f.0(u.0(x0))) 471.21/221.15 down.0(f.0(u.1(x0))) 471.21/221.15 down.0(u.0(x0)) 471.21/221.15 down.0(u.1(x0)) 471.21/221.15 down.0(g.0(x0)) 471.21/221.15 down.0(g.1(x0)) 471.21/221.15 down.0(f.0(c.)) 471.21/221.15 down.0(f.1(fresh_constant.)) 471.21/221.15 down.0(b.0(f.0(x0))) 471.21/221.15 down.0(b.0(f.1(x0))) 471.21/221.15 down.0(b.0(c.)) 471.21/221.15 down.0(b.0(b.0(x0))) 471.21/221.15 down.0(b.0(b.1(x0))) 471.21/221.15 down.0(b.0(g.0(x0))) 471.21/221.15 down.0(b.0(g.1(x0))) 471.21/221.15 down.0(b.1(fresh_constant.)) 471.21/221.15 down.0(f.0(f.0(c.))) 471.21/221.15 down.0(f.0(f.0(b.0(x0)))) 471.21/221.15 down.0(f.0(f.0(b.1(x0)))) 471.21/221.15 down.0(f.0(f.0(u.0(x0)))) 471.21/221.15 down.0(f.0(f.0(u.1(x0)))) 471.21/221.15 down.0(f.0(f.0(g.0(x0)))) 471.21/221.15 down.0(f.0(f.0(g.1(x0)))) 471.21/221.15 down.0(f.0(f.1(fresh_constant.))) 471.21/221.15 f_flat.0(up.0(x0)) 471.21/221.15 f_flat.0(up.1(x0)) 471.21/221.15 b_flat.0(up.0(x0)) 471.21/221.15 b_flat.0(up.1(x0)) 471.21/221.15 u_flat.0(up.0(x0)) 471.21/221.15 u_flat.0(up.1(x0)) 471.21/221.15 g_flat.0(up.0(x0)) 471.21/221.15 g_flat.0(up.1(x0)) 471.21/221.15 471.21/221.15 We have to consider all minimal (P,Q,R)-chains. 471.21/221.15 ---------------------------------------- 471.21/221.15 471.21/221.15 (86) MRRProof (EQUIVALENT) 471.21/221.15 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. 471.21/221.15 471.21/221.15 471.21/221.15 Strictly oriented rules of the TRS R: 471.21/221.15 471.21/221.15 down.0(f.0(f.0(f.1(x)))) -> up.0(c.) 471.21/221.15 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 471.21/221.15 down.0(g.1(y3)) -> g_flat.0(down.1(y3)) 471.21/221.15 down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.)) 471.21/221.15 down.0(b.1(fresh_constant.)) -> b_flat.0(down.1(fresh_constant.)) 471.21/221.15 471.21/221.15 Used ordering: Polynomial interpretation [POLO]: 471.21/221.15 471.21/221.15 POL(TOP.0(x_1)) = x_1 471.21/221.15 POL(b.0(x_1)) = x_1 471.21/221.15 POL(b.1(x_1)) = 1 + x_1 471.21/221.15 POL(b_flat.0(x_1)) = x_1 471.21/221.15 POL(c.) = 0 471.21/221.15 POL(down.0(x_1)) = x_1 471.21/221.15 POL(down.1(x_1)) = x_1 471.21/221.15 POL(f.0(x_1)) = x_1 471.21/221.15 POL(f.1(x_1)) = 1 + x_1 471.21/221.15 POL(f_flat.0(x_1)) = x_1 471.21/221.15 POL(fresh_constant.) = 0 471.21/221.15 POL(g.0(x_1)) = x_1 471.21/221.15 POL(g.1(x_1)) = 1 + x_1 471.21/221.15 POL(g_flat.0(x_1)) = x_1 471.21/221.15 POL(u.0(x_1)) = x_1 471.21/221.15 POL(u.1(x_1)) = 1 + x_1 471.21/221.15 POL(u_flat.0(x_1)) = x_1 471.21/221.15 POL(up.0(x_1)) = x_1 471.21/221.15 471.21/221.15 471.21/221.15 ---------------------------------------- 471.21/221.15 471.21/221.15 (87) 471.21/221.15 Obligation: 471.21/221.15 Q DP problem: 471.21/221.15 The TRS P consists of the following rules: 471.21/221.15 471.21/221.15 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.21/221.15 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.21/221.15 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(f.1(x0)))) 471.21/221.15 TOP.0(up.0(b.0(f.1(x0)))) -> TOP.0(b_flat.0(down.0(f.1(x0)))) 471.21/221.15 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.21/221.15 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.21/221.15 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.21/221.15 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.21/221.15 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.21/221.15 471.21/221.15 The TRS R consists of the following rules: 471.21/221.15 471.21/221.15 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.21/221.15 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.21/221.15 down.0(b.0(u.1(x))) -> up.0(b.0(f.1(x))) 471.21/221.15 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.21/221.15 down.0(f.0(g.1(x))) -> up.0(f.0(f.1(x))) 471.21/221.15 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.21/221.15 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.21/221.15 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.21/221.15 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.21/221.15 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.21/221.15 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.21/221.15 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 471.21/221.15 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.21/221.15 down.0(b.0(f.1(y10))) -> b_flat.0(down.0(f.1(y10))) 471.21/221.15 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 471.21/221.15 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.21/221.15 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.21/221.15 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.21/221.15 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.21/221.15 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.21/221.15 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.21/221.15 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.21/221.15 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.21/221.15 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.21/221.15 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.21/221.15 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.21/221.15 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 471.21/221.15 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.21/221.15 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.21/221.15 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.21/221.15 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.21/221.15 471.21/221.15 The set Q consists of the following terms: 471.21/221.15 471.21/221.15 down.0(f.0(f.0(f.0(x0)))) 471.21/221.15 down.0(f.0(f.0(f.1(x0)))) 471.21/221.15 down.0(b.0(u.0(x0))) 471.21/221.15 down.0(b.0(u.1(x0))) 471.21/221.15 down.0(f.0(g.0(x0))) 471.21/221.15 down.0(f.0(g.1(x0))) 471.21/221.15 down.0(f.0(b.0(x0))) 471.21/221.15 down.0(f.0(b.1(x0))) 471.21/221.15 down.0(f.0(u.0(x0))) 471.21/221.15 down.0(f.0(u.1(x0))) 471.21/221.15 down.0(u.0(x0)) 471.21/221.15 down.0(u.1(x0)) 471.21/221.15 down.0(g.0(x0)) 471.21/221.15 down.0(g.1(x0)) 471.21/221.15 down.0(f.0(c.)) 471.21/221.15 down.0(f.1(fresh_constant.)) 471.21/221.15 down.0(b.0(f.0(x0))) 471.21/221.15 down.0(b.0(f.1(x0))) 471.21/221.15 down.0(b.0(c.)) 471.21/221.15 down.0(b.0(b.0(x0))) 471.21/221.15 down.0(b.0(b.1(x0))) 471.21/221.15 down.0(b.0(g.0(x0))) 471.21/221.15 down.0(b.0(g.1(x0))) 471.21/221.15 down.0(b.1(fresh_constant.)) 471.21/221.15 down.0(f.0(f.0(c.))) 471.21/221.15 down.0(f.0(f.0(b.0(x0)))) 471.21/221.15 down.0(f.0(f.0(b.1(x0)))) 471.21/221.15 down.0(f.0(f.0(u.0(x0)))) 471.21/221.15 down.0(f.0(f.0(u.1(x0)))) 471.21/221.15 down.0(f.0(f.0(g.0(x0)))) 471.21/221.15 down.0(f.0(f.0(g.1(x0)))) 471.21/221.15 down.0(f.0(f.1(fresh_constant.))) 471.21/221.15 f_flat.0(up.0(x0)) 471.21/221.15 f_flat.0(up.1(x0)) 471.21/221.15 b_flat.0(up.0(x0)) 471.21/221.15 b_flat.0(up.1(x0)) 471.21/221.15 u_flat.0(up.0(x0)) 471.21/221.15 u_flat.0(up.1(x0)) 471.21/221.15 g_flat.0(up.0(x0)) 471.21/221.15 g_flat.0(up.1(x0)) 471.21/221.15 471.21/221.15 We have to consider all minimal (P,Q,R)-chains. 471.21/221.15 ---------------------------------------- 471.21/221.15 471.21/221.15 (88) DependencyGraphProof (EQUIVALENT) 471.21/221.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 471.21/221.15 ---------------------------------------- 471.21/221.15 471.21/221.15 (89) 471.21/221.15 Obligation: 471.21/221.15 Q DP problem: 471.21/221.15 The TRS P consists of the following rules: 471.21/221.15 471.21/221.15 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.21/221.15 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.21/221.15 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.21/221.15 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.21/221.15 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.21/221.15 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.21/221.15 471.21/221.15 The TRS R consists of the following rules: 471.21/221.15 471.21/221.15 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.21/221.15 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.21/221.15 down.0(b.0(u.1(x))) -> up.0(b.0(f.1(x))) 471.21/221.15 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.21/221.15 down.0(f.0(g.1(x))) -> up.0(f.0(f.1(x))) 471.21/221.15 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.21/221.15 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.21/221.15 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.21/221.15 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.21/221.15 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.21/221.15 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.21/221.15 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 471.21/221.15 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.21/221.15 down.0(b.0(f.1(y10))) -> b_flat.0(down.0(f.1(y10))) 471.21/221.15 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 471.21/221.15 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.21/221.15 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.21/221.15 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.21/221.15 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.21/221.15 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.21/221.15 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.21/221.15 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.21/221.15 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.21/221.15 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.21/221.15 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.21/221.15 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.21/221.15 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 471.21/221.15 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.21/221.15 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.21/221.15 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.21/221.15 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.21/221.15 471.21/221.15 The set Q consists of the following terms: 471.21/221.15 471.21/221.15 down.0(f.0(f.0(f.0(x0)))) 471.21/221.15 down.0(f.0(f.0(f.1(x0)))) 471.21/221.15 down.0(b.0(u.0(x0))) 471.21/221.15 down.0(b.0(u.1(x0))) 471.21/221.15 down.0(f.0(g.0(x0))) 471.21/221.15 down.0(f.0(g.1(x0))) 471.21/221.15 down.0(f.0(b.0(x0))) 471.21/221.15 down.0(f.0(b.1(x0))) 471.21/221.15 down.0(f.0(u.0(x0))) 471.21/221.15 down.0(f.0(u.1(x0))) 471.21/221.15 down.0(u.0(x0)) 471.21/221.15 down.0(u.1(x0)) 471.21/221.15 down.0(g.0(x0)) 471.21/221.15 down.0(g.1(x0)) 471.21/221.15 down.0(f.0(c.)) 471.21/221.15 down.0(f.1(fresh_constant.)) 471.21/221.15 down.0(b.0(f.0(x0))) 471.21/221.15 down.0(b.0(f.1(x0))) 471.21/221.15 down.0(b.0(c.)) 471.21/221.15 down.0(b.0(b.0(x0))) 471.21/221.15 down.0(b.0(b.1(x0))) 471.21/221.15 down.0(b.0(g.0(x0))) 471.21/221.15 down.0(b.0(g.1(x0))) 471.21/221.15 down.0(b.1(fresh_constant.)) 471.21/221.15 down.0(f.0(f.0(c.))) 471.21/221.15 down.0(f.0(f.0(b.0(x0)))) 471.21/221.15 down.0(f.0(f.0(b.1(x0)))) 471.21/221.15 down.0(f.0(f.0(u.0(x0)))) 471.21/221.15 down.0(f.0(f.0(u.1(x0)))) 471.21/221.15 down.0(f.0(f.0(g.0(x0)))) 471.21/221.15 down.0(f.0(f.0(g.1(x0)))) 471.21/221.15 down.0(f.0(f.1(fresh_constant.))) 471.21/221.15 f_flat.0(up.0(x0)) 471.21/221.15 f_flat.0(up.1(x0)) 471.21/221.15 b_flat.0(up.0(x0)) 471.21/221.15 b_flat.0(up.1(x0)) 471.21/221.15 u_flat.0(up.0(x0)) 471.21/221.15 u_flat.0(up.1(x0)) 471.21/221.15 g_flat.0(up.0(x0)) 471.21/221.15 g_flat.0(up.1(x0)) 471.21/221.15 471.21/221.15 We have to consider all minimal (P,Q,R)-chains. 471.21/221.15 ---------------------------------------- 471.21/221.15 471.21/221.15 (90) MRRProof (EQUIVALENT) 471.21/221.15 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. 471.21/221.15 471.21/221.15 471.21/221.15 Strictly oriented rules of the TRS R: 471.21/221.15 471.21/221.15 down.0(b.0(u.1(x))) -> up.0(b.0(f.1(x))) 471.21/221.15 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.21/221.15 471.21/221.15 Used ordering: Polynomial interpretation [POLO]: 471.21/221.15 471.21/221.15 POL(TOP.0(x_1)) = x_1 471.21/221.15 POL(b.0(x_1)) = x_1 471.21/221.15 POL(b.1(x_1)) = x_1 471.21/221.15 POL(b_flat.0(x_1)) = x_1 471.21/221.15 POL(c.) = 0 471.21/221.15 POL(down.0(x_1)) = x_1 471.21/221.15 POL(f.0(x_1)) = x_1 471.21/221.15 POL(f.1(x_1)) = x_1 471.21/221.15 POL(f_flat.0(x_1)) = x_1 471.21/221.15 POL(fresh_constant.) = 0 471.21/221.15 POL(g.0(x_1)) = x_1 471.21/221.15 POL(g.1(x_1)) = x_1 471.21/221.15 POL(g_flat.0(x_1)) = x_1 471.21/221.15 POL(u.0(x_1)) = x_1 471.21/221.15 POL(u.1(x_1)) = 1 + x_1 471.21/221.15 POL(u_flat.0(x_1)) = x_1 471.21/221.15 POL(up.0(x_1)) = x_1 471.21/221.15 471.21/221.15 471.21/221.15 ---------------------------------------- 471.21/221.15 471.21/221.15 (91) 471.21/221.15 Obligation: 471.21/221.15 Q DP problem: 471.21/221.15 The TRS P consists of the following rules: 471.21/221.15 471.21/221.15 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.21/221.15 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.21/221.15 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.21/221.15 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.21/221.15 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.21/221.15 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.21/221.15 471.21/221.15 The TRS R consists of the following rules: 471.21/221.15 471.21/221.15 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.21/221.15 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.21/221.15 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.21/221.15 down.0(f.0(g.1(x))) -> up.0(f.0(f.1(x))) 471.21/221.15 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.21/221.15 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.21/221.15 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.21/221.15 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.21/221.15 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.21/221.15 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 471.21/221.15 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.21/221.15 down.0(b.0(f.1(y10))) -> b_flat.0(down.0(f.1(y10))) 471.21/221.15 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 471.21/221.15 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.21/221.15 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.21/221.15 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.21/221.15 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.21/221.15 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.21/221.15 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.21/221.15 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.21/221.15 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.21/221.15 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.21/221.15 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.21/221.15 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.21/221.15 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 471.21/221.15 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.21/221.15 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.21/221.15 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.21/221.15 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.21/221.15 471.21/221.15 The set Q consists of the following terms: 471.21/221.15 471.21/221.15 down.0(f.0(f.0(f.0(x0)))) 471.21/221.15 down.0(f.0(f.0(f.1(x0)))) 471.21/221.15 down.0(b.0(u.0(x0))) 471.21/221.15 down.0(b.0(u.1(x0))) 471.21/221.15 down.0(f.0(g.0(x0))) 471.21/221.15 down.0(f.0(g.1(x0))) 471.21/221.15 down.0(f.0(b.0(x0))) 471.21/221.15 down.0(f.0(b.1(x0))) 471.21/221.15 down.0(f.0(u.0(x0))) 471.21/221.15 down.0(f.0(u.1(x0))) 471.21/221.15 down.0(u.0(x0)) 471.21/221.15 down.0(u.1(x0)) 471.21/221.15 down.0(g.0(x0)) 471.21/221.15 down.0(g.1(x0)) 471.21/221.15 down.0(f.0(c.)) 471.21/221.15 down.0(f.1(fresh_constant.)) 471.21/221.15 down.0(b.0(f.0(x0))) 471.21/221.15 down.0(b.0(f.1(x0))) 471.21/221.15 down.0(b.0(c.)) 471.21/221.15 down.0(b.0(b.0(x0))) 471.21/221.15 down.0(b.0(b.1(x0))) 471.21/221.15 down.0(b.0(g.0(x0))) 471.21/221.15 down.0(b.0(g.1(x0))) 471.21/221.15 down.0(b.1(fresh_constant.)) 471.21/221.15 down.0(f.0(f.0(c.))) 471.21/221.15 down.0(f.0(f.0(b.0(x0)))) 471.21/221.15 down.0(f.0(f.0(b.1(x0)))) 471.21/221.15 down.0(f.0(f.0(u.0(x0)))) 471.21/221.15 down.0(f.0(f.0(u.1(x0)))) 471.21/221.15 down.0(f.0(f.0(g.0(x0)))) 471.21/221.15 down.0(f.0(f.0(g.1(x0)))) 471.21/221.15 down.0(f.0(f.1(fresh_constant.))) 471.21/221.15 f_flat.0(up.0(x0)) 471.21/221.15 f_flat.0(up.1(x0)) 471.21/221.15 b_flat.0(up.0(x0)) 471.21/221.15 b_flat.0(up.1(x0)) 471.21/221.15 u_flat.0(up.0(x0)) 471.21/221.15 u_flat.0(up.1(x0)) 471.21/221.15 g_flat.0(up.0(x0)) 471.21/221.15 g_flat.0(up.1(x0)) 471.21/221.15 471.21/221.15 We have to consider all minimal (P,Q,R)-chains. 471.21/221.15 ---------------------------------------- 471.21/221.15 471.21/221.15 (92) MRRProof (EQUIVALENT) 471.21/221.15 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. 471.21/221.15 471.21/221.15 471.21/221.15 Strictly oriented rules of the TRS R: 471.21/221.15 471.21/221.15 down.0(f.0(g.1(x))) -> up.0(f.0(f.1(x))) 471.21/221.15 471.21/221.15 Used ordering: Polynomial interpretation [POLO]: 471.21/221.15 471.21/221.15 POL(TOP.0(x_1)) = x_1 471.21/221.15 POL(b.0(x_1)) = x_1 471.21/221.15 POL(b.1(x_1)) = x_1 471.21/221.15 POL(b_flat.0(x_1)) = x_1 471.21/221.15 POL(c.) = 0 471.21/221.15 POL(down.0(x_1)) = x_1 471.21/221.15 POL(f.0(x_1)) = x_1 471.21/221.15 POL(f.1(x_1)) = x_1 471.21/221.15 POL(f_flat.0(x_1)) = x_1 471.21/221.15 POL(fresh_constant.) = 0 471.21/221.15 POL(g.0(x_1)) = x_1 471.21/221.15 POL(g.1(x_1)) = 1 + x_1 471.21/221.15 POL(g_flat.0(x_1)) = x_1 471.21/221.15 POL(u.0(x_1)) = x_1 471.21/221.15 POL(u.1(x_1)) = x_1 471.21/221.15 POL(u_flat.0(x_1)) = x_1 471.21/221.15 POL(up.0(x_1)) = x_1 471.21/221.15 471.21/221.15 471.21/221.15 ---------------------------------------- 471.21/221.15 471.21/221.15 (93) 471.21/221.15 Obligation: 471.21/221.15 Q DP problem: 471.21/221.15 The TRS P consists of the following rules: 471.21/221.15 471.21/221.15 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.21/221.15 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.21/221.15 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.21/221.15 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.21/221.15 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.21/221.15 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.21/221.15 471.21/221.15 The TRS R consists of the following rules: 471.21/221.15 471.21/221.15 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.21/221.15 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.21/221.15 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.21/221.15 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.21/221.15 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.21/221.15 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.21/221.15 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.21/221.15 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.21/221.15 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 471.21/221.15 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.21/221.15 down.0(b.0(f.1(y10))) -> b_flat.0(down.0(f.1(y10))) 471.21/221.15 down.0(b.0(c.)) -> b_flat.0(down.0(c.)) 471.21/221.15 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.21/221.15 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.21/221.15 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.21/221.15 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.21/221.15 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.21/221.15 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.21/221.15 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.21/221.15 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.21/221.15 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.21/221.15 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.21/221.15 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.21/221.15 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 471.21/221.15 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.21/221.15 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.21/221.15 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.21/221.15 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.21/221.15 471.21/221.15 The set Q consists of the following terms: 471.21/221.15 471.21/221.15 down.0(f.0(f.0(f.0(x0)))) 471.21/221.15 down.0(f.0(f.0(f.1(x0)))) 471.21/221.15 down.0(b.0(u.0(x0))) 471.21/221.15 down.0(b.0(u.1(x0))) 471.21/221.15 down.0(f.0(g.0(x0))) 471.21/221.15 down.0(f.0(g.1(x0))) 471.21/221.15 down.0(f.0(b.0(x0))) 471.21/221.15 down.0(f.0(b.1(x0))) 471.21/221.15 down.0(f.0(u.0(x0))) 471.21/221.15 down.0(f.0(u.1(x0))) 471.21/221.15 down.0(u.0(x0)) 471.21/221.15 down.0(u.1(x0)) 471.21/221.15 down.0(g.0(x0)) 471.21/221.15 down.0(g.1(x0)) 471.21/221.15 down.0(f.0(c.)) 471.21/221.15 down.0(f.1(fresh_constant.)) 471.21/221.15 down.0(b.0(f.0(x0))) 471.21/221.15 down.0(b.0(f.1(x0))) 471.21/221.15 down.0(b.0(c.)) 471.21/221.15 down.0(b.0(b.0(x0))) 471.21/221.15 down.0(b.0(b.1(x0))) 471.21/221.15 down.0(b.0(g.0(x0))) 471.21/221.15 down.0(b.0(g.1(x0))) 471.21/221.15 down.0(b.1(fresh_constant.)) 471.21/221.15 down.0(f.0(f.0(c.))) 471.21/221.15 down.0(f.0(f.0(b.0(x0)))) 471.21/221.15 down.0(f.0(f.0(b.1(x0)))) 471.21/221.15 down.0(f.0(f.0(u.0(x0)))) 471.21/221.15 down.0(f.0(f.0(u.1(x0)))) 471.21/221.15 down.0(f.0(f.0(g.0(x0)))) 471.21/221.15 down.0(f.0(f.0(g.1(x0)))) 471.21/221.15 down.0(f.0(f.1(fresh_constant.))) 471.21/221.15 f_flat.0(up.0(x0)) 471.21/221.15 f_flat.0(up.1(x0)) 471.21/221.15 b_flat.0(up.0(x0)) 471.21/221.15 b_flat.0(up.1(x0)) 471.21/221.15 u_flat.0(up.0(x0)) 471.21/221.15 u_flat.0(up.1(x0)) 471.21/221.15 g_flat.0(up.0(x0)) 471.21/221.15 g_flat.0(up.1(x0)) 471.21/221.15 471.21/221.15 We have to consider all minimal (P,Q,R)-chains. 471.21/221.15 ---------------------------------------- 471.21/221.15 471.21/221.15 (94) PisEmptyProof (SOUND) 471.21/221.16 The TRS P is empty. Hence, there is no (P,Q,R) chain. 471.21/221.16 ---------------------------------------- 471.21/221.16 471.21/221.16 (95) 471.21/221.16 TRUE 471.21/221.16 471.21/221.16 ---------------------------------------- 471.21/221.16 471.21/221.16 (96) 471.21/221.16 Obligation: 471.21/221.16 Q DP problem: 471.21/221.16 The TRS P consists of the following rules: 471.21/221.16 471.21/221.16 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.16 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.16 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.16 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.16 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.16 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.16 471.21/221.16 The TRS R consists of the following rules: 471.21/221.16 471.21/221.16 down(f(f(f(x)))) -> up(c) 471.21/221.16 down(b(u(x))) -> up(b(f(x))) 471.21/221.16 down(f(g(x))) -> up(f(f(x))) 471.21/221.16 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.16 down(f(u(x))) -> up(u(g(x))) 471.21/221.16 down(u(y2)) -> u_flat(down(y2)) 471.21/221.16 down(g(y3)) -> g_flat(down(y3)) 471.21/221.16 down(f(c)) -> f_flat(down(c)) 471.21/221.16 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.16 down(b(c)) -> b_flat(down(c)) 471.21/221.16 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.16 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.16 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.16 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.16 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.16 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.16 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.16 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.16 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.16 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.16 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.16 471.21/221.16 The set Q consists of the following terms: 471.21/221.16 471.21/221.16 down(f(f(f(x0)))) 471.21/221.16 down(b(u(x0))) 471.21/221.16 down(f(g(x0))) 471.21/221.16 down(f(b(x0))) 471.21/221.16 down(f(u(x0))) 471.21/221.16 down(u(x0)) 471.21/221.16 down(g(x0)) 471.21/221.16 down(f(c)) 471.21/221.16 down(f(fresh_constant)) 471.21/221.16 down(b(f(x0))) 471.21/221.16 down(b(c)) 471.21/221.16 down(b(b(x0))) 471.21/221.16 down(b(g(x0))) 471.21/221.16 down(b(fresh_constant)) 471.21/221.16 down(f(f(c))) 471.21/221.16 down(f(f(b(x0)))) 471.21/221.16 down(f(f(u(x0)))) 471.21/221.16 down(f(f(g(x0)))) 471.21/221.16 down(f(f(fresh_constant))) 471.21/221.16 f_flat(up(x0)) 471.21/221.16 b_flat(up(x0)) 471.21/221.16 u_flat(up(x0)) 471.21/221.16 g_flat(up(x0)) 471.21/221.16 471.21/221.16 We have to consider all minimal (P,Q,R)-chains. 471.21/221.16 ---------------------------------------- 471.21/221.16 471.21/221.16 (97) SplitQDPProof (EQUIVALENT) 471.21/221.16 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 471.21/221.16 471.21/221.16 ---------------------------------------- 471.21/221.16 471.21/221.16 (98) 471.21/221.16 Complex Obligation (AND) 471.21/221.16 471.21/221.16 ---------------------------------------- 471.21/221.16 471.21/221.16 (99) 471.21/221.16 Obligation: 471.21/221.16 Q DP problem: 471.21/221.16 The TRS P consists of the following rules: 471.21/221.16 471.21/221.16 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.21/221.16 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.21/221.16 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.21/221.16 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.21/221.16 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.21/221.16 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.21/221.16 471.21/221.16 The TRS R consists of the following rules: 471.21/221.16 471.21/221.16 down(f(f(f(x)))) -> up(c) 471.21/221.16 down(b(u(x))) -> up(b(f(x))) 471.21/221.16 down(f(g(x))) -> up(f(f(x))) 471.21/221.16 down(f(b(x))) -> up(u(g(b(x)))) 471.21/221.16 down(f(u(x))) -> up(u(g(x))) 471.21/221.16 down(u(y2)) -> u_flat(down(y2)) 471.21/221.16 down(g(y3)) -> g_flat(down(y3)) 471.21/221.16 down(f(c)) -> f_flat(down(c)) 471.21/221.16 down(b(f(y10))) -> b_flat(down(f(y10))) 471.21/221.16 down(b(c)) -> b_flat(down(c)) 471.21/221.16 down(b(b(y11))) -> b_flat(down(b(y11))) 471.21/221.16 down(b(g(y13))) -> b_flat(down(g(y13))) 471.21/221.16 down(f(f(c))) -> f_flat(down(f(c))) 471.21/221.16 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.21/221.16 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.21/221.16 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.21/221.16 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.21/221.16 g_flat(up(x_1)) -> up(g(x_1)) 471.21/221.16 b_flat(up(x_1)) -> up(b(x_1)) 471.21/221.16 f_flat(up(x_1)) -> up(f(x_1)) 471.21/221.16 u_flat(up(x_1)) -> up(u(x_1)) 471.21/221.16 471.21/221.16 The set Q consists of the following terms: 471.21/221.16 471.21/221.16 down(f(f(f(x0)))) 471.21/221.16 down(b(u(x0))) 471.21/221.16 down(f(g(x0))) 471.21/221.16 down(f(b(x0))) 471.21/221.16 down(f(u(x0))) 471.21/221.16 down(u(x0)) 471.21/221.16 down(g(x0)) 471.21/221.16 down(f(c)) 471.21/221.16 down(f(fresh_constant)) 471.21/221.16 down(b(f(x0))) 471.21/221.16 down(b(c)) 471.21/221.16 down(b(b(x0))) 471.21/221.16 down(b(g(x0))) 471.21/221.16 down(b(fresh_constant)) 471.21/221.16 down(f(f(c))) 471.21/221.16 down(f(f(b(x0)))) 471.21/221.16 down(f(f(u(x0)))) 471.21/221.16 down(f(f(g(x0)))) 471.21/221.16 down(f(f(fresh_constant))) 471.21/221.16 f_flat(up(x0)) 471.21/221.16 b_flat(up(x0)) 471.21/221.16 u_flat(up(x0)) 471.21/221.16 g_flat(up(x0)) 471.21/221.16 471.21/221.16 We have to consider all minimal (P,Q,R)-chains. 471.21/221.16 ---------------------------------------- 471.21/221.16 471.21/221.16 (100) SemLabProof (SOUND) 471.21/221.16 We found the following model for the rules of the TRSs R and P. 471.21/221.16 Interpretation over the domain with elements from 0 to 1. 471.21/221.16 c: 1 471.21/221.16 TOP: 0 471.21/221.16 u: 0 471.21/221.16 g: 0 471.21/221.16 b: 0 471.21/221.16 down: 0 471.21/221.16 f: 0 471.21/221.16 fresh_constant: 0 471.21/221.16 up: 0 471.21/221.16 u_flat: 0 471.21/221.16 f_flat: 0 471.21/221.16 b_flat: 0 471.21/221.16 g_flat: 0 471.21/221.16 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 471.21/221.16 ---------------------------------------- 471.21/221.16 471.21/221.16 (101) 471.21/221.16 Obligation: 471.21/221.16 Q DP problem: 471.21/221.16 The TRS P consists of the following rules: 471.21/221.16 471.21/221.16 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.21/221.16 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.21/221.16 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(f.1(x0)))) 471.21/221.16 TOP.0(up.0(b.0(f.1(x0)))) -> TOP.0(b_flat.0(down.0(f.1(x0)))) 471.21/221.16 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.21/221.16 TOP.0(up.0(u.1(x0))) -> TOP.0(u_flat.0(down.1(x0))) 471.21/221.16 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.21/221.16 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.21/221.16 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.21/221.16 TOP.0(up.0(b.0(g.1(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.1(x0)))) 471.21/221.16 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.21/221.16 TOP.0(up.0(g.1(x0))) -> TOP.0(g_flat.0(down.1(x0))) 471.21/221.16 471.21/221.16 The TRS R consists of the following rules: 471.21/221.16 471.21/221.16 down.0(f.0(f.0(f.0(x)))) -> up.1(c.) 471.21/221.16 down.0(f.0(f.0(f.1(x)))) -> up.1(c.) 471.21/221.16 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.21/221.16 down.0(b.0(u.1(x))) -> up.0(b.0(f.1(x))) 471.21/221.16 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.21/221.16 down.0(f.0(g.1(x))) -> up.0(f.0(f.1(x))) 471.21/221.16 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.21/221.16 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.21/221.16 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.21/221.16 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.21/221.16 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.21/221.16 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 471.21/221.16 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.21/221.16 down.0(g.1(y3)) -> g_flat.0(down.1(y3)) 471.21/221.16 down.0(f.1(c.)) -> f_flat.0(down.1(c.)) 471.21/221.16 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.21/221.16 down.0(b.0(f.1(y10))) -> b_flat.0(down.0(f.1(y10))) 471.21/221.16 down.0(b.1(c.)) -> b_flat.0(down.1(c.)) 471.21/221.16 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.21/221.16 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.21/221.16 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.21/221.16 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.21/221.16 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 471.21/221.16 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.21/221.16 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.21/221.16 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.21/221.16 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.21/221.16 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.21/221.16 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.21/221.16 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 471.21/221.16 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.21/221.16 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.21/221.16 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.21/221.16 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 471.21/221.16 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.21/221.16 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 471.21/221.16 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.21/221.16 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.21/221.16 471.21/221.16 The set Q consists of the following terms: 471.21/221.16 471.21/221.16 down.0(f.0(f.0(f.0(x0)))) 471.21/221.16 down.0(f.0(f.0(f.1(x0)))) 471.21/221.16 down.0(b.0(u.0(x0))) 471.21/221.16 down.0(b.0(u.1(x0))) 471.21/221.16 down.0(f.0(g.0(x0))) 471.21/221.16 down.0(f.0(g.1(x0))) 471.21/221.16 down.0(f.0(b.0(x0))) 471.21/221.16 down.0(f.0(b.1(x0))) 471.21/221.16 down.0(f.0(u.0(x0))) 471.21/221.16 down.0(f.0(u.1(x0))) 471.21/221.16 down.0(u.0(x0)) 471.21/221.16 down.0(u.1(x0)) 471.21/221.16 down.0(g.0(x0)) 471.21/221.16 down.0(g.1(x0)) 471.21/221.16 down.0(f.1(c.)) 471.21/221.16 down.0(f.0(fresh_constant.)) 471.21/221.16 down.0(b.0(f.0(x0))) 471.21/221.16 down.0(b.0(f.1(x0))) 471.21/221.16 down.0(b.1(c.)) 471.21/221.16 down.0(b.0(b.0(x0))) 471.21/221.16 down.0(b.0(b.1(x0))) 471.21/221.16 down.0(b.0(g.0(x0))) 471.21/221.16 down.0(b.0(g.1(x0))) 471.21/221.16 down.0(b.0(fresh_constant.)) 471.21/221.16 down.0(f.0(f.1(c.))) 471.21/221.16 down.0(f.0(f.0(b.0(x0)))) 471.21/221.16 down.0(f.0(f.0(b.1(x0)))) 471.21/221.16 down.0(f.0(f.0(u.0(x0)))) 471.21/221.16 down.0(f.0(f.0(u.1(x0)))) 471.21/221.16 down.0(f.0(f.0(g.0(x0)))) 471.21/221.16 down.0(f.0(f.0(g.1(x0)))) 471.21/221.16 down.0(f.0(f.0(fresh_constant.))) 471.21/221.16 f_flat.0(up.0(x0)) 471.21/221.16 f_flat.0(up.1(x0)) 471.21/221.16 b_flat.0(up.0(x0)) 471.21/221.16 b_flat.0(up.1(x0)) 471.21/221.16 u_flat.0(up.0(x0)) 471.21/221.16 u_flat.0(up.1(x0)) 471.21/221.16 g_flat.0(up.0(x0)) 471.21/221.16 g_flat.0(up.1(x0)) 471.21/221.16 471.21/221.16 We have to consider all minimal (P,Q,R)-chains. 471.21/221.16 ---------------------------------------- 471.21/221.16 471.21/221.16 (102) DependencyGraphProof (EQUIVALENT) 471.21/221.16 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 471.21/221.16 ---------------------------------------- 471.21/221.16 471.21/221.16 (103) 471.21/221.16 Obligation: 471.21/221.16 Q DP problem: 471.21/221.16 The TRS P consists of the following rules: 471.21/221.16 471.21/221.16 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.21/221.16 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.21/221.16 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(f.1(x0)))) 471.21/221.16 TOP.0(up.0(b.0(f.1(x0)))) -> TOP.0(b_flat.0(down.0(f.1(x0)))) 471.21/221.16 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.21/221.16 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.21/221.16 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.21/221.16 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.21/221.16 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.21/221.16 471.21/221.16 The TRS R consists of the following rules: 471.21/221.16 471.21/221.16 down.0(f.0(f.0(f.0(x)))) -> up.1(c.) 471.21/221.16 down.0(f.0(f.0(f.1(x)))) -> up.1(c.) 471.21/221.16 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.21/221.16 down.0(b.0(u.1(x))) -> up.0(b.0(f.1(x))) 471.21/221.16 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.21/221.16 down.0(f.0(g.1(x))) -> up.0(f.0(f.1(x))) 471.21/221.16 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.21/221.16 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.21/221.16 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.21/221.16 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.21/221.16 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.21/221.16 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 471.21/221.16 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.21/221.16 down.0(g.1(y3)) -> g_flat.0(down.1(y3)) 471.21/221.16 down.0(f.1(c.)) -> f_flat.0(down.1(c.)) 471.21/221.16 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.21/221.16 down.0(b.0(f.1(y10))) -> b_flat.0(down.0(f.1(y10))) 471.21/221.16 down.0(b.1(c.)) -> b_flat.0(down.1(c.)) 471.21/221.16 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.21/221.16 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.21/221.16 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.21/221.16 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.21/221.16 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 471.21/221.16 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.21/221.16 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.21/221.16 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.21/221.16 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.21/221.16 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.21/221.16 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.21/221.16 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 471.21/221.16 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.21/221.16 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.21/221.16 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.21/221.16 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 471.21/221.16 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.21/221.16 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 471.21/221.16 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.21/221.16 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.21/221.16 471.21/221.16 The set Q consists of the following terms: 471.21/221.16 471.21/221.16 down.0(f.0(f.0(f.0(x0)))) 471.21/221.16 down.0(f.0(f.0(f.1(x0)))) 471.21/221.16 down.0(b.0(u.0(x0))) 471.21/221.16 down.0(b.0(u.1(x0))) 471.21/221.16 down.0(f.0(g.0(x0))) 471.21/221.16 down.0(f.0(g.1(x0))) 471.21/221.16 down.0(f.0(b.0(x0))) 471.21/221.16 down.0(f.0(b.1(x0))) 471.21/221.16 down.0(f.0(u.0(x0))) 471.21/221.16 down.0(f.0(u.1(x0))) 471.21/221.16 down.0(u.0(x0)) 471.21/221.16 down.0(u.1(x0)) 471.21/221.16 down.0(g.0(x0)) 471.21/221.16 down.0(g.1(x0)) 471.21/221.16 down.0(f.1(c.)) 471.21/221.16 down.0(f.0(fresh_constant.)) 471.21/221.16 down.0(b.0(f.0(x0))) 471.21/221.16 down.0(b.0(f.1(x0))) 471.21/221.16 down.0(b.1(c.)) 471.21/221.16 down.0(b.0(b.0(x0))) 471.21/221.16 down.0(b.0(b.1(x0))) 471.21/221.16 down.0(b.0(g.0(x0))) 471.21/221.16 down.0(b.0(g.1(x0))) 471.21/221.16 down.0(b.0(fresh_constant.)) 471.21/221.16 down.0(f.0(f.1(c.))) 471.21/221.16 down.0(f.0(f.0(b.0(x0)))) 471.21/221.16 down.0(f.0(f.0(b.1(x0)))) 471.21/221.16 down.0(f.0(f.0(u.0(x0)))) 471.21/221.16 down.0(f.0(f.0(u.1(x0)))) 471.21/221.16 down.0(f.0(f.0(g.0(x0)))) 471.21/221.16 down.0(f.0(f.0(g.1(x0)))) 471.21/221.16 down.0(f.0(f.0(fresh_constant.))) 471.21/221.16 f_flat.0(up.0(x0)) 471.21/221.16 f_flat.0(up.1(x0)) 471.21/221.16 b_flat.0(up.0(x0)) 471.21/221.16 b_flat.0(up.1(x0)) 471.21/221.16 u_flat.0(up.0(x0)) 471.21/221.16 u_flat.0(up.1(x0)) 471.21/221.16 g_flat.0(up.0(x0)) 471.21/221.16 g_flat.0(up.1(x0)) 471.21/221.16 471.21/221.16 We have to consider all minimal (P,Q,R)-chains. 471.21/221.16 ---------------------------------------- 471.21/221.16 471.21/221.16 (104) MRRProof (EQUIVALENT) 471.21/221.16 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. 471.21/221.16 471.21/221.16 471.21/221.16 Strictly oriented rules of the TRS R: 471.21/221.16 471.21/221.16 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 471.21/221.16 down.0(g.1(y3)) -> g_flat.0(down.1(y3)) 471.21/221.16 down.0(f.1(c.)) -> f_flat.0(down.1(c.)) 471.21/221.16 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 471.21/221.16 471.21/221.16 Used ordering: Polynomial interpretation [POLO]: 471.21/221.16 471.21/221.16 POL(TOP.0(x_1)) = x_1 471.21/221.16 POL(b.0(x_1)) = 1 + x_1 471.21/221.16 POL(b.1(x_1)) = x_1 471.21/221.16 POL(b_flat.0(x_1)) = 1 + x_1 471.21/221.16 POL(c.) = 0 471.21/221.16 POL(down.0(x_1)) = 1 + x_1 471.21/221.16 POL(down.1(x_1)) = x_1 471.21/221.16 POL(f.0(x_1)) = x_1 471.21/221.16 POL(f.1(x_1)) = x_1 471.21/221.16 POL(f_flat.0(x_1)) = x_1 471.21/221.16 POL(fresh_constant.) = 0 471.21/221.16 POL(g.0(x_1)) = x_1 471.21/221.16 POL(g.1(x_1)) = x_1 471.21/221.16 POL(g_flat.0(x_1)) = x_1 471.21/221.16 POL(u.0(x_1)) = x_1 471.21/221.16 POL(u.1(x_1)) = x_1 471.21/221.16 POL(u_flat.0(x_1)) = x_1 471.21/221.16 POL(up.0(x_1)) = 1 + x_1 471.21/221.16 POL(up.1(x_1)) = 1 + x_1 471.21/221.16 471.21/221.16 471.21/221.16 ---------------------------------------- 471.21/221.16 471.21/221.16 (105) 471.21/221.16 Obligation: 471.21/221.16 Q DP problem: 471.21/221.16 The TRS P consists of the following rules: 471.21/221.16 471.21/221.16 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.21/221.16 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.21/221.16 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(f.1(x0)))) 471.21/221.16 TOP.0(up.0(b.0(f.1(x0)))) -> TOP.0(b_flat.0(down.0(f.1(x0)))) 471.21/221.16 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.21/221.16 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.21/221.16 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.21/221.16 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.21/221.16 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.21/221.16 471.21/221.16 The TRS R consists of the following rules: 471.21/221.16 471.21/221.16 down.0(f.0(f.0(f.0(x)))) -> up.1(c.) 471.21/221.16 down.0(f.0(f.0(f.1(x)))) -> up.1(c.) 471.21/221.16 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.21/221.16 down.0(b.0(u.1(x))) -> up.0(b.0(f.1(x))) 471.21/221.16 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.21/221.16 down.0(f.0(g.1(x))) -> up.0(f.0(f.1(x))) 471.21/221.16 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.21/221.16 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.21/221.16 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.21/221.16 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.21/221.16 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.21/221.16 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.21/221.16 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.21/221.16 down.0(b.0(f.1(y10))) -> b_flat.0(down.0(f.1(y10))) 471.21/221.16 down.0(b.1(c.)) -> b_flat.0(down.1(c.)) 471.21/221.16 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.21/221.16 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.21/221.16 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.21/221.16 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.21/221.16 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 471.21/221.16 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.21/221.16 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.21/221.16 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.21/221.16 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.21/221.16 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.21/221.16 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.21/221.16 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 471.21/221.16 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.21/221.16 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.21/221.16 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.21/221.16 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.21/221.16 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 471.21/221.16 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.21/221.16 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.21/221.16 471.21/221.16 The set Q consists of the following terms: 471.21/221.16 471.21/221.16 down.0(f.0(f.0(f.0(x0)))) 471.21/221.16 down.0(f.0(f.0(f.1(x0)))) 471.21/221.16 down.0(b.0(u.0(x0))) 471.21/221.16 down.0(b.0(u.1(x0))) 471.21/221.16 down.0(f.0(g.0(x0))) 471.21/221.16 down.0(f.0(g.1(x0))) 471.21/221.16 down.0(f.0(b.0(x0))) 471.21/221.16 down.0(f.0(b.1(x0))) 471.21/221.16 down.0(f.0(u.0(x0))) 471.21/221.16 down.0(f.0(u.1(x0))) 471.21/221.16 down.0(u.0(x0)) 471.21/221.16 down.0(u.1(x0)) 471.21/221.16 down.0(g.0(x0)) 471.21/221.16 down.0(g.1(x0)) 471.21/221.16 down.0(f.1(c.)) 471.21/221.16 down.0(f.0(fresh_constant.)) 471.21/221.16 down.0(b.0(f.0(x0))) 471.21/221.16 down.0(b.0(f.1(x0))) 471.21/221.16 down.0(b.1(c.)) 471.21/221.16 down.0(b.0(b.0(x0))) 471.21/221.16 down.0(b.0(b.1(x0))) 471.21/221.16 down.0(b.0(g.0(x0))) 471.21/221.16 down.0(b.0(g.1(x0))) 471.21/221.16 down.0(b.0(fresh_constant.)) 471.21/221.16 down.0(f.0(f.1(c.))) 471.21/221.16 down.0(f.0(f.0(b.0(x0)))) 471.21/221.16 down.0(f.0(f.0(b.1(x0)))) 471.21/221.16 down.0(f.0(f.0(u.0(x0)))) 471.21/221.16 down.0(f.0(f.0(u.1(x0)))) 471.21/221.16 down.0(f.0(f.0(g.0(x0)))) 471.21/221.16 down.0(f.0(f.0(g.1(x0)))) 471.21/221.16 down.0(f.0(f.0(fresh_constant.))) 471.21/221.16 f_flat.0(up.0(x0)) 471.21/221.16 f_flat.0(up.1(x0)) 471.34/221.16 b_flat.0(up.0(x0)) 471.34/221.16 b_flat.0(up.1(x0)) 471.34/221.16 u_flat.0(up.0(x0)) 471.34/221.16 u_flat.0(up.1(x0)) 471.34/221.16 g_flat.0(up.0(x0)) 471.34/221.16 g_flat.0(up.1(x0)) 471.34/221.16 471.34/221.16 We have to consider all minimal (P,Q,R)-chains. 471.34/221.16 ---------------------------------------- 471.34/221.16 471.34/221.16 (106) DependencyGraphProof (EQUIVALENT) 471.34/221.16 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 471.34/221.16 ---------------------------------------- 471.34/221.16 471.34/221.16 (107) 471.34/221.16 Obligation: 471.34/221.16 Q DP problem: 471.34/221.16 The TRS P consists of the following rules: 471.34/221.16 471.34/221.16 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.34/221.16 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.34/221.16 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.34/221.16 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.34/221.16 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.34/221.16 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.34/221.16 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.34/221.16 471.34/221.16 The TRS R consists of the following rules: 471.34/221.16 471.34/221.16 down.0(f.0(f.0(f.0(x)))) -> up.1(c.) 471.34/221.16 down.0(f.0(f.0(f.1(x)))) -> up.1(c.) 471.34/221.16 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.34/221.16 down.0(b.0(u.1(x))) -> up.0(b.0(f.1(x))) 471.34/221.16 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.34/221.16 down.0(f.0(g.1(x))) -> up.0(f.0(f.1(x))) 471.34/221.16 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.34/221.16 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.34/221.16 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.34/221.16 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.34/221.16 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.34/221.16 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.34/221.16 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.34/221.16 down.0(b.0(f.1(y10))) -> b_flat.0(down.0(f.1(y10))) 471.34/221.16 down.0(b.1(c.)) -> b_flat.0(down.1(c.)) 471.34/221.16 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.34/221.16 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.34/221.16 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.34/221.16 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.34/221.16 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 471.34/221.16 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.34/221.16 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.34/221.16 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.34/221.16 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.34/221.16 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.34/221.16 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.34/221.16 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 471.34/221.16 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.34/221.16 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.34/221.16 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.34/221.16 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.34/221.16 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 471.34/221.16 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.34/221.16 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.34/221.16 471.34/221.16 The set Q consists of the following terms: 471.34/221.16 471.34/221.16 down.0(f.0(f.0(f.0(x0)))) 471.34/221.16 down.0(f.0(f.0(f.1(x0)))) 471.34/221.16 down.0(b.0(u.0(x0))) 471.34/221.16 down.0(b.0(u.1(x0))) 471.34/221.16 down.0(f.0(g.0(x0))) 471.34/221.16 down.0(f.0(g.1(x0))) 471.34/221.16 down.0(f.0(b.0(x0))) 471.34/221.16 down.0(f.0(b.1(x0))) 471.34/221.16 down.0(f.0(u.0(x0))) 471.34/221.16 down.0(f.0(u.1(x0))) 471.34/221.16 down.0(u.0(x0)) 471.34/221.16 down.0(u.1(x0)) 471.34/221.16 down.0(g.0(x0)) 471.34/221.16 down.0(g.1(x0)) 471.34/221.16 down.0(f.1(c.)) 471.34/221.16 down.0(f.0(fresh_constant.)) 471.34/221.16 down.0(b.0(f.0(x0))) 471.34/221.16 down.0(b.0(f.1(x0))) 471.34/221.16 down.0(b.1(c.)) 471.34/221.16 down.0(b.0(b.0(x0))) 471.34/221.16 down.0(b.0(b.1(x0))) 471.34/221.16 down.0(b.0(g.0(x0))) 471.34/221.16 down.0(b.0(g.1(x0))) 471.34/221.16 down.0(b.0(fresh_constant.)) 471.34/221.16 down.0(f.0(f.1(c.))) 471.34/221.16 down.0(f.0(f.0(b.0(x0)))) 471.34/221.16 down.0(f.0(f.0(b.1(x0)))) 471.34/221.16 down.0(f.0(f.0(u.0(x0)))) 471.34/221.16 down.0(f.0(f.0(u.1(x0)))) 471.34/221.16 down.0(f.0(f.0(g.0(x0)))) 471.34/221.16 down.0(f.0(f.0(g.1(x0)))) 471.34/221.16 down.0(f.0(f.0(fresh_constant.))) 471.34/221.16 f_flat.0(up.0(x0)) 471.34/221.16 f_flat.0(up.1(x0)) 471.34/221.16 b_flat.0(up.0(x0)) 471.34/221.16 b_flat.0(up.1(x0)) 471.34/221.16 u_flat.0(up.0(x0)) 471.34/221.16 u_flat.0(up.1(x0)) 471.34/221.16 g_flat.0(up.0(x0)) 471.34/221.16 g_flat.0(up.1(x0)) 471.34/221.16 471.34/221.16 We have to consider all minimal (P,Q,R)-chains. 471.34/221.16 ---------------------------------------- 471.34/221.16 471.34/221.16 (108) MRRProof (EQUIVALENT) 471.34/221.16 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. 471.34/221.16 471.34/221.16 471.34/221.16 Strictly oriented rules of the TRS R: 471.34/221.16 471.34/221.16 down.0(b.1(c.)) -> b_flat.0(down.1(c.)) 471.34/221.16 471.34/221.16 Used ordering: Polynomial interpretation [POLO]: 471.34/221.16 471.34/221.16 POL(TOP.0(x_1)) = x_1 471.34/221.16 POL(b.0(x_1)) = x_1 471.34/221.16 POL(b.1(x_1)) = x_1 471.34/221.16 POL(b_flat.0(x_1)) = x_1 471.34/221.16 POL(c.) = 0 471.34/221.16 POL(down.0(x_1)) = 1 + x_1 471.34/221.16 POL(down.1(x_1)) = x_1 471.34/221.16 POL(f.0(x_1)) = x_1 471.34/221.16 POL(f.1(x_1)) = x_1 471.34/221.16 POL(f_flat.0(x_1)) = x_1 471.34/221.16 POL(fresh_constant.) = 0 471.34/221.16 POL(g.0(x_1)) = x_1 471.34/221.16 POL(g.1(x_1)) = x_1 471.34/221.16 POL(g_flat.0(x_1)) = x_1 471.34/221.16 POL(u.0(x_1)) = x_1 471.34/221.16 POL(u.1(x_1)) = x_1 471.34/221.16 POL(u_flat.0(x_1)) = x_1 471.34/221.16 POL(up.0(x_1)) = 1 + x_1 471.34/221.16 POL(up.1(x_1)) = 1 + x_1 471.34/221.16 471.34/221.16 471.34/221.16 ---------------------------------------- 471.34/221.16 471.34/221.16 (109) 471.34/221.16 Obligation: 471.34/221.16 Q DP problem: 471.34/221.16 The TRS P consists of the following rules: 471.34/221.16 471.34/221.16 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.34/221.16 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.34/221.16 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.34/221.16 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.34/221.16 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.34/221.16 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.34/221.16 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.34/221.16 471.34/221.16 The TRS R consists of the following rules: 471.34/221.16 471.34/221.16 down.0(f.0(f.0(f.0(x)))) -> up.1(c.) 471.34/221.16 down.0(f.0(f.0(f.1(x)))) -> up.1(c.) 471.34/221.16 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.34/221.16 down.0(b.0(u.1(x))) -> up.0(b.0(f.1(x))) 471.34/221.16 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.34/221.16 down.0(f.0(g.1(x))) -> up.0(f.0(f.1(x))) 471.34/221.16 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.34/221.16 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.34/221.16 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.34/221.16 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.34/221.16 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.34/221.16 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.34/221.16 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.34/221.16 down.0(b.0(f.1(y10))) -> b_flat.0(down.0(f.1(y10))) 471.34/221.16 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.34/221.16 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.34/221.16 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.34/221.16 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.34/221.16 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 471.34/221.16 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.34/221.16 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.34/221.16 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.34/221.16 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.34/221.16 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.34/221.16 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.34/221.16 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 471.34/221.16 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.34/221.16 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.34/221.16 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.34/221.16 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.34/221.16 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 471.34/221.16 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.34/221.16 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.34/221.16 471.34/221.16 The set Q consists of the following terms: 471.34/221.16 471.34/221.16 down.0(f.0(f.0(f.0(x0)))) 471.34/221.16 down.0(f.0(f.0(f.1(x0)))) 471.34/221.16 down.0(b.0(u.0(x0))) 471.34/221.16 down.0(b.0(u.1(x0))) 471.34/221.16 down.0(f.0(g.0(x0))) 471.34/221.16 down.0(f.0(g.1(x0))) 471.34/221.16 down.0(f.0(b.0(x0))) 471.34/221.16 down.0(f.0(b.1(x0))) 471.34/221.16 down.0(f.0(u.0(x0))) 471.34/221.16 down.0(f.0(u.1(x0))) 471.34/221.16 down.0(u.0(x0)) 471.34/221.16 down.0(u.1(x0)) 471.34/221.16 down.0(g.0(x0)) 471.34/221.16 down.0(g.1(x0)) 471.34/221.16 down.0(f.1(c.)) 471.34/221.16 down.0(f.0(fresh_constant.)) 471.34/221.16 down.0(b.0(f.0(x0))) 471.34/221.16 down.0(b.0(f.1(x0))) 471.34/221.16 down.0(b.1(c.)) 471.34/221.16 down.0(b.0(b.0(x0))) 471.34/221.16 down.0(b.0(b.1(x0))) 471.34/221.16 down.0(b.0(g.0(x0))) 471.34/221.16 down.0(b.0(g.1(x0))) 471.34/221.16 down.0(b.0(fresh_constant.)) 471.34/221.16 down.0(f.0(f.1(c.))) 471.34/221.16 down.0(f.0(f.0(b.0(x0)))) 471.34/221.16 down.0(f.0(f.0(b.1(x0)))) 471.34/221.16 down.0(f.0(f.0(u.0(x0)))) 471.34/221.16 down.0(f.0(f.0(u.1(x0)))) 471.34/221.16 down.0(f.0(f.0(g.0(x0)))) 471.34/221.16 down.0(f.0(f.0(g.1(x0)))) 471.34/221.16 down.0(f.0(f.0(fresh_constant.))) 471.34/221.16 f_flat.0(up.0(x0)) 471.34/221.16 f_flat.0(up.1(x0)) 471.34/221.16 b_flat.0(up.0(x0)) 471.34/221.16 b_flat.0(up.1(x0)) 471.34/221.16 u_flat.0(up.0(x0)) 471.34/221.16 u_flat.0(up.1(x0)) 471.34/221.16 g_flat.0(up.0(x0)) 471.34/221.16 g_flat.0(up.1(x0)) 471.34/221.16 471.34/221.16 We have to consider all minimal (P,Q,R)-chains. 471.34/221.16 ---------------------------------------- 471.34/221.16 471.34/221.16 (110) DependencyGraphProof (EQUIVALENT) 471.34/221.16 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 471.34/221.16 ---------------------------------------- 471.34/221.16 471.34/221.16 (111) 471.34/221.16 Obligation: 471.34/221.16 Q DP problem: 471.34/221.16 The TRS P consists of the following rules: 471.34/221.16 471.34/221.16 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.34/221.16 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.34/221.16 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.34/221.16 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.34/221.16 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.34/221.16 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.34/221.16 471.34/221.16 The TRS R consists of the following rules: 471.34/221.16 471.34/221.16 down.0(f.0(f.0(f.0(x)))) -> up.1(c.) 471.34/221.16 down.0(f.0(f.0(f.1(x)))) -> up.1(c.) 471.34/221.16 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.34/221.16 down.0(b.0(u.1(x))) -> up.0(b.0(f.1(x))) 471.34/221.16 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.34/221.16 down.0(f.0(g.1(x))) -> up.0(f.0(f.1(x))) 471.34/221.16 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.34/221.16 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.34/221.16 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.34/221.16 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.34/221.16 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.34/221.16 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.34/221.16 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.34/221.16 down.0(b.0(f.1(y10))) -> b_flat.0(down.0(f.1(y10))) 471.34/221.16 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.34/221.16 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.34/221.16 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.34/221.16 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.34/221.16 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 471.34/221.16 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.34/221.16 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.34/221.16 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.34/221.16 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.34/221.16 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.34/221.16 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.34/221.16 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 471.34/221.16 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.34/221.16 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.34/221.16 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.34/221.16 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.34/221.16 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 471.34/221.16 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.34/221.16 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.34/221.16 471.34/221.16 The set Q consists of the following terms: 471.34/221.16 471.34/221.16 down.0(f.0(f.0(f.0(x0)))) 471.34/221.16 down.0(f.0(f.0(f.1(x0)))) 471.34/221.16 down.0(b.0(u.0(x0))) 471.34/221.16 down.0(b.0(u.1(x0))) 471.34/221.16 down.0(f.0(g.0(x0))) 471.34/221.16 down.0(f.0(g.1(x0))) 471.34/221.16 down.0(f.0(b.0(x0))) 471.34/221.16 down.0(f.0(b.1(x0))) 471.34/221.16 down.0(f.0(u.0(x0))) 471.34/221.16 down.0(f.0(u.1(x0))) 471.34/221.16 down.0(u.0(x0)) 471.34/221.16 down.0(u.1(x0)) 471.34/221.16 down.0(g.0(x0)) 471.34/221.16 down.0(g.1(x0)) 471.34/221.16 down.0(f.1(c.)) 471.34/221.16 down.0(f.0(fresh_constant.)) 471.34/221.16 down.0(b.0(f.0(x0))) 471.34/221.16 down.0(b.0(f.1(x0))) 471.34/221.16 down.0(b.1(c.)) 471.34/221.16 down.0(b.0(b.0(x0))) 471.34/221.16 down.0(b.0(b.1(x0))) 471.34/221.16 down.0(b.0(g.0(x0))) 471.34/221.16 down.0(b.0(g.1(x0))) 471.34/221.16 down.0(b.0(fresh_constant.)) 471.34/221.16 down.0(f.0(f.1(c.))) 471.34/221.16 down.0(f.0(f.0(b.0(x0)))) 471.34/221.16 down.0(f.0(f.0(b.1(x0)))) 471.34/221.16 down.0(f.0(f.0(u.0(x0)))) 471.34/221.16 down.0(f.0(f.0(u.1(x0)))) 471.34/221.16 down.0(f.0(f.0(g.0(x0)))) 471.34/221.16 down.0(f.0(f.0(g.1(x0)))) 471.34/221.16 down.0(f.0(f.0(fresh_constant.))) 471.34/221.16 f_flat.0(up.0(x0)) 471.34/221.16 f_flat.0(up.1(x0)) 471.34/221.16 b_flat.0(up.0(x0)) 471.34/221.16 b_flat.0(up.1(x0)) 471.34/221.16 u_flat.0(up.0(x0)) 471.34/221.16 u_flat.0(up.1(x0)) 471.34/221.16 g_flat.0(up.0(x0)) 471.34/221.16 g_flat.0(up.1(x0)) 471.34/221.16 471.34/221.16 We have to consider all minimal (P,Q,R)-chains. 471.34/221.16 ---------------------------------------- 471.34/221.16 471.34/221.16 (112) PisEmptyProof (SOUND) 471.34/221.16 The TRS P is empty. Hence, there is no (P,Q,R) chain. 471.34/221.16 ---------------------------------------- 471.34/221.16 471.34/221.16 (113) 471.34/221.16 TRUE 471.34/221.16 471.34/221.16 ---------------------------------------- 471.34/221.16 471.34/221.16 (114) 471.34/221.16 Obligation: 471.34/221.16 Q DP problem: 471.34/221.16 The TRS P consists of the following rules: 471.34/221.16 471.34/221.16 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.34/221.16 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.34/221.16 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.34/221.16 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.34/221.16 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.34/221.16 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.34/221.16 471.34/221.16 The TRS R consists of the following rules: 471.34/221.16 471.34/221.16 down(f(f(f(x)))) -> up(c) 471.34/221.16 down(b(u(x))) -> up(b(f(x))) 471.34/221.16 down(f(g(x))) -> up(f(f(x))) 471.34/221.16 down(f(b(x))) -> up(u(g(b(x)))) 471.34/221.16 down(f(u(x))) -> up(u(g(x))) 471.34/221.16 down(u(y2)) -> u_flat(down(y2)) 471.34/221.16 down(g(y3)) -> g_flat(down(y3)) 471.34/221.16 down(b(f(y10))) -> b_flat(down(f(y10))) 471.34/221.16 down(b(b(y11))) -> b_flat(down(b(y11))) 471.34/221.16 down(b(g(y13))) -> b_flat(down(g(y13))) 471.34/221.16 down(f(f(c))) -> f_flat(down(f(c))) 471.34/221.16 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.34/221.16 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.34/221.16 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.34/221.16 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.34/221.16 g_flat(up(x_1)) -> up(g(x_1)) 471.34/221.16 b_flat(up(x_1)) -> up(b(x_1)) 471.34/221.16 f_flat(up(x_1)) -> up(f(x_1)) 471.34/221.16 u_flat(up(x_1)) -> up(u(x_1)) 471.34/221.16 471.34/221.16 The set Q consists of the following terms: 471.34/221.16 471.34/221.16 down(f(f(f(x0)))) 471.34/221.16 down(b(u(x0))) 471.34/221.16 down(f(g(x0))) 471.34/221.16 down(f(b(x0))) 471.34/221.16 down(f(u(x0))) 471.34/221.16 down(u(x0)) 471.34/221.16 down(g(x0)) 471.34/221.16 down(f(c)) 471.34/221.16 down(f(fresh_constant)) 471.34/221.16 down(b(f(x0))) 471.34/221.16 down(b(c)) 471.34/221.16 down(b(b(x0))) 471.34/221.16 down(b(g(x0))) 471.34/221.16 down(b(fresh_constant)) 471.34/221.16 down(f(f(c))) 471.34/221.16 down(f(f(b(x0)))) 471.34/221.16 down(f(f(u(x0)))) 471.34/221.16 down(f(f(g(x0)))) 471.34/221.16 down(f(f(fresh_constant))) 471.34/221.16 f_flat(up(x0)) 471.34/221.16 b_flat(up(x0)) 471.34/221.16 u_flat(up(x0)) 471.34/221.16 g_flat(up(x0)) 471.34/221.16 471.34/221.16 We have to consider all minimal (P,Q,R)-chains. 471.34/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (115) SplitQDPProof (EQUIVALENT) 471.35/221.16 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 471.35/221.16 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (116) 471.35/221.16 Complex Obligation (AND) 471.35/221.16 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (117) 471.35/221.16 Obligation: 471.35/221.16 Q DP problem: 471.35/221.16 The TRS P consists of the following rules: 471.35/221.16 471.35/221.16 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.35/221.16 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.35/221.16 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.35/221.16 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.35/221.16 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.35/221.16 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.35/221.16 471.35/221.16 The TRS R consists of the following rules: 471.35/221.16 471.35/221.16 down(f(f(f(x)))) -> up(c) 471.35/221.16 down(b(u(x))) -> up(b(f(x))) 471.35/221.16 down(f(g(x))) -> up(f(f(x))) 471.35/221.16 down(f(b(x))) -> up(u(g(b(x)))) 471.35/221.16 down(f(u(x))) -> up(u(g(x))) 471.35/221.16 down(u(y2)) -> u_flat(down(y2)) 471.35/221.16 down(g(y3)) -> g_flat(down(y3)) 471.35/221.16 down(b(f(y10))) -> b_flat(down(f(y10))) 471.35/221.16 down(b(b(y11))) -> b_flat(down(b(y11))) 471.35/221.16 down(b(g(y13))) -> b_flat(down(g(y13))) 471.35/221.16 down(f(f(c))) -> f_flat(down(f(c))) 471.35/221.16 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.35/221.16 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.35/221.16 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.35/221.16 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 471.35/221.16 g_flat(up(x_1)) -> up(g(x_1)) 471.35/221.16 b_flat(up(x_1)) -> up(b(x_1)) 471.35/221.16 f_flat(up(x_1)) -> up(f(x_1)) 471.35/221.16 u_flat(up(x_1)) -> up(u(x_1)) 471.35/221.16 471.35/221.16 The set Q consists of the following terms: 471.35/221.16 471.35/221.16 down(f(f(f(x0)))) 471.35/221.16 down(b(u(x0))) 471.35/221.16 down(f(g(x0))) 471.35/221.16 down(f(b(x0))) 471.35/221.16 down(f(u(x0))) 471.35/221.16 down(u(x0)) 471.35/221.16 down(g(x0)) 471.35/221.16 down(f(c)) 471.35/221.16 down(f(fresh_constant)) 471.35/221.16 down(b(f(x0))) 471.35/221.16 down(b(c)) 471.35/221.16 down(b(b(x0))) 471.35/221.16 down(b(g(x0))) 471.35/221.16 down(b(fresh_constant)) 471.35/221.16 down(f(f(c))) 471.35/221.16 down(f(f(b(x0)))) 471.35/221.16 down(f(f(u(x0)))) 471.35/221.16 down(f(f(g(x0)))) 471.35/221.16 down(f(f(fresh_constant))) 471.35/221.16 f_flat(up(x0)) 471.35/221.16 b_flat(up(x0)) 471.35/221.16 u_flat(up(x0)) 471.35/221.16 g_flat(up(x0)) 471.35/221.16 471.35/221.16 We have to consider all minimal (P,Q,R)-chains. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (118) SemLabProof (SOUND) 471.35/221.16 We found the following model for the rules of the TRSs R and P. 471.35/221.16 Interpretation over the domain with elements from 0 to 1. 471.35/221.16 c: 0 471.35/221.16 TOP: 0 471.35/221.16 u: 0 471.35/221.16 g: 0 471.35/221.16 b: 0 471.35/221.16 down: 0 471.35/221.16 f: x0 471.35/221.16 fresh_constant: 1 471.35/221.16 up: 0 471.35/221.16 u_flat: 0 471.35/221.16 b_flat: 0 471.35/221.16 f_flat: 0 471.35/221.16 g_flat: 0 471.35/221.16 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (119) 471.35/221.16 Obligation: 471.35/221.16 Q DP problem: 471.35/221.16 The TRS P consists of the following rules: 471.35/221.16 471.35/221.16 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(b.1(f.1(x0)))) -> TOP.0(b_flat.0(down.1(f.1(x0)))) 471.35/221.16 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.1(f.1(x0)))) 471.35/221.16 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.35/221.16 TOP.0(up.0(u.1(x0))) -> TOP.0(u_flat.0(down.1(x0))) 471.35/221.16 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.35/221.16 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(g.1(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.1(x0)))) 471.35/221.16 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.35/221.16 TOP.0(up.0(g.1(x0))) -> TOP.0(g_flat.0(down.1(x0))) 471.35/221.16 471.35/221.16 The TRS R consists of the following rules: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.35/221.16 down.1(f.1(f.1(f.1(x)))) -> up.0(c.) 471.35/221.16 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.35/221.16 down.0(b.0(u.1(x))) -> up.0(b.1(f.1(x))) 471.35/221.16 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.35/221.16 down.0(f.0(g.1(x))) -> up.1(f.1(f.1(x))) 471.35/221.16 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.35/221.16 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.35/221.16 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.35/221.16 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.35/221.16 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.35/221.16 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 471.35/221.16 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.35/221.16 down.0(g.1(y3)) -> g_flat.0(down.1(y3)) 471.35/221.16 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.35/221.16 down.0(b.1(f.1(y10))) -> b_flat.0(down.1(f.1(y10))) 471.35/221.16 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.35/221.16 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.35/221.16 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.35/221.16 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.35/221.16 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.35/221.16 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.35/221.16 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.35/221.16 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.35/221.16 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.35/221.16 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.))) 471.35/221.16 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.35/221.16 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.35/221.16 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.35/221.16 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 471.35/221.16 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.35/221.16 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 471.35/221.16 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.35/221.16 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.35/221.16 471.35/221.16 The set Q consists of the following terms: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x0)))) 471.35/221.16 down.1(f.1(f.1(f.1(x0)))) 471.35/221.16 down.0(b.0(u.0(x0))) 471.35/221.16 down.0(b.0(u.1(x0))) 471.35/221.16 down.0(f.0(g.0(x0))) 471.35/221.16 down.0(f.0(g.1(x0))) 471.35/221.16 down.0(f.0(b.0(x0))) 471.35/221.16 down.0(f.0(b.1(x0))) 471.35/221.16 down.0(f.0(u.0(x0))) 471.35/221.16 down.0(f.0(u.1(x0))) 471.35/221.16 down.0(u.0(x0)) 471.35/221.16 down.0(u.1(x0)) 471.35/221.16 down.0(g.0(x0)) 471.35/221.16 down.0(g.1(x0)) 471.35/221.16 down.0(f.0(c.)) 471.35/221.16 down.1(f.1(fresh_constant.)) 471.35/221.16 down.0(b.0(f.0(x0))) 471.35/221.16 down.0(b.1(f.1(x0))) 471.35/221.16 down.0(b.0(c.)) 471.35/221.16 down.0(b.0(b.0(x0))) 471.35/221.16 down.0(b.0(b.1(x0))) 471.35/221.16 down.0(b.0(g.0(x0))) 471.35/221.16 down.0(b.0(g.1(x0))) 471.35/221.16 down.0(b.1(fresh_constant.)) 471.35/221.16 down.0(f.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(x0)))) 471.35/221.16 down.0(f.0(f.0(b.1(x0)))) 471.35/221.16 down.0(f.0(f.0(u.0(x0)))) 471.35/221.16 down.0(f.0(f.0(u.1(x0)))) 471.35/221.16 down.0(f.0(f.0(g.0(x0)))) 471.35/221.16 down.0(f.0(f.0(g.1(x0)))) 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) 471.35/221.16 f_flat.0(up.0(x0)) 471.35/221.16 f_flat.0(up.1(x0)) 471.35/221.16 b_flat.0(up.0(x0)) 471.35/221.16 b_flat.0(up.1(x0)) 471.35/221.16 u_flat.0(up.0(x0)) 471.35/221.16 u_flat.0(up.1(x0)) 471.35/221.16 g_flat.0(up.0(x0)) 471.35/221.16 g_flat.0(up.1(x0)) 471.35/221.16 471.35/221.16 We have to consider all minimal (P,Q,R)-chains. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (120) MRRProof (EQUIVALENT) 471.35/221.16 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. 471.35/221.16 471.35/221.16 Strictly oriented dependency pairs: 471.35/221.16 471.35/221.16 TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.1(f.1(x0)))) 471.35/221.16 TOP.0(up.0(u.1(x0))) -> TOP.0(u_flat.0(down.1(x0))) 471.35/221.16 TOP.0(up.0(b.0(g.1(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.1(x0)))) 471.35/221.16 TOP.0(up.0(g.1(x0))) -> TOP.0(g_flat.0(down.1(x0))) 471.35/221.16 471.35/221.16 Strictly oriented rules of the TRS R: 471.35/221.16 471.35/221.16 down.0(b.0(u.1(x))) -> up.0(b.1(f.1(x))) 471.35/221.16 down.0(u.1(y2)) -> u_flat.0(down.1(y2)) 471.35/221.16 down.0(g.1(y3)) -> g_flat.0(down.1(y3)) 471.35/221.16 b_flat.0(up.1(x_1)) -> up.0(b.1(x_1)) 471.35/221.16 471.35/221.16 Used ordering: Polynomial interpretation [POLO]: 471.35/221.16 471.35/221.16 POL(TOP.0(x_1)) = x_1 471.35/221.16 POL(b.0(x_1)) = x_1 471.35/221.16 POL(b.1(x_1)) = x_1 471.35/221.16 POL(b_flat.0(x_1)) = x_1 471.35/221.16 POL(c.) = 0 471.35/221.16 POL(down.0(x_1)) = x_1 471.35/221.16 POL(down.1(x_1)) = x_1 471.35/221.16 POL(f.0(x_1)) = x_1 471.35/221.16 POL(f.1(x_1)) = x_1 471.35/221.16 POL(f_flat.0(x_1)) = x_1 471.35/221.16 POL(fresh_constant.) = 0 471.35/221.16 POL(g.0(x_1)) = x_1 471.35/221.16 POL(g.1(x_1)) = 1 + x_1 471.35/221.16 POL(g_flat.0(x_1)) = x_1 471.35/221.16 POL(u.0(x_1)) = x_1 471.35/221.16 POL(u.1(x_1)) = 1 + x_1 471.35/221.16 POL(u_flat.0(x_1)) = x_1 471.35/221.16 POL(up.0(x_1)) = x_1 471.35/221.16 POL(up.1(x_1)) = 1 + x_1 471.35/221.16 471.35/221.16 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (121) 471.35/221.16 Obligation: 471.35/221.16 Q DP problem: 471.35/221.16 The TRS P consists of the following rules: 471.35/221.16 471.35/221.16 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(b.1(f.1(x0)))) -> TOP.0(b_flat.0(down.1(f.1(x0)))) 471.35/221.16 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.35/221.16 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.35/221.16 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.35/221.16 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.35/221.16 471.35/221.16 The TRS R consists of the following rules: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.35/221.16 down.1(f.1(f.1(f.1(x)))) -> up.0(c.) 471.35/221.16 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.35/221.16 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.35/221.16 down.0(f.0(g.1(x))) -> up.1(f.1(f.1(x))) 471.35/221.16 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.35/221.16 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.35/221.16 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.35/221.16 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.35/221.16 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.35/221.16 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.35/221.16 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.35/221.16 down.0(b.1(f.1(y10))) -> b_flat.0(down.1(f.1(y10))) 471.35/221.16 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.35/221.16 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.35/221.16 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.35/221.16 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.35/221.16 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.35/221.16 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.35/221.16 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.35/221.16 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.35/221.16 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.35/221.16 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.))) 471.35/221.16 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.35/221.16 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.35/221.16 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.35/221.16 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.35/221.16 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 471.35/221.16 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.35/221.16 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.35/221.16 471.35/221.16 The set Q consists of the following terms: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x0)))) 471.35/221.16 down.1(f.1(f.1(f.1(x0)))) 471.35/221.16 down.0(b.0(u.0(x0))) 471.35/221.16 down.0(b.0(u.1(x0))) 471.35/221.16 down.0(f.0(g.0(x0))) 471.35/221.16 down.0(f.0(g.1(x0))) 471.35/221.16 down.0(f.0(b.0(x0))) 471.35/221.16 down.0(f.0(b.1(x0))) 471.35/221.16 down.0(f.0(u.0(x0))) 471.35/221.16 down.0(f.0(u.1(x0))) 471.35/221.16 down.0(u.0(x0)) 471.35/221.16 down.0(u.1(x0)) 471.35/221.16 down.0(g.0(x0)) 471.35/221.16 down.0(g.1(x0)) 471.35/221.16 down.0(f.0(c.)) 471.35/221.16 down.1(f.1(fresh_constant.)) 471.35/221.16 down.0(b.0(f.0(x0))) 471.35/221.16 down.0(b.1(f.1(x0))) 471.35/221.16 down.0(b.0(c.)) 471.35/221.16 down.0(b.0(b.0(x0))) 471.35/221.16 down.0(b.0(b.1(x0))) 471.35/221.16 down.0(b.0(g.0(x0))) 471.35/221.16 down.0(b.0(g.1(x0))) 471.35/221.16 down.0(b.1(fresh_constant.)) 471.35/221.16 down.0(f.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(x0)))) 471.35/221.16 down.0(f.0(f.0(b.1(x0)))) 471.35/221.16 down.0(f.0(f.0(u.0(x0)))) 471.35/221.16 down.0(f.0(f.0(u.1(x0)))) 471.35/221.16 down.0(f.0(f.0(g.0(x0)))) 471.35/221.16 down.0(f.0(f.0(g.1(x0)))) 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) 471.35/221.16 f_flat.0(up.0(x0)) 471.35/221.16 f_flat.0(up.1(x0)) 471.35/221.16 b_flat.0(up.0(x0)) 471.35/221.16 b_flat.0(up.1(x0)) 471.35/221.16 u_flat.0(up.0(x0)) 471.35/221.16 u_flat.0(up.1(x0)) 471.35/221.16 g_flat.0(up.0(x0)) 471.35/221.16 g_flat.0(up.1(x0)) 471.35/221.16 471.35/221.16 We have to consider all minimal (P,Q,R)-chains. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (122) DependencyGraphProof (EQUIVALENT) 471.35/221.16 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (123) 471.35/221.16 Obligation: 471.35/221.16 Q DP problem: 471.35/221.16 The TRS P consists of the following rules: 471.35/221.16 471.35/221.16 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.35/221.16 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.35/221.16 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.35/221.16 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.35/221.16 471.35/221.16 The TRS R consists of the following rules: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.35/221.16 down.1(f.1(f.1(f.1(x)))) -> up.0(c.) 471.35/221.16 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.35/221.16 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.35/221.16 down.0(f.0(g.1(x))) -> up.1(f.1(f.1(x))) 471.35/221.16 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.35/221.16 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.35/221.16 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.35/221.16 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.35/221.16 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.35/221.16 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.35/221.16 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.35/221.16 down.0(b.1(f.1(y10))) -> b_flat.0(down.1(f.1(y10))) 471.35/221.16 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.35/221.16 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.35/221.16 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.35/221.16 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.35/221.16 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.35/221.16 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.35/221.16 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.35/221.16 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.35/221.16 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.35/221.16 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.))) 471.35/221.16 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.35/221.16 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.35/221.16 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.35/221.16 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.35/221.16 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 471.35/221.16 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.35/221.16 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.35/221.16 471.35/221.16 The set Q consists of the following terms: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x0)))) 471.35/221.16 down.1(f.1(f.1(f.1(x0)))) 471.35/221.16 down.0(b.0(u.0(x0))) 471.35/221.16 down.0(b.0(u.1(x0))) 471.35/221.16 down.0(f.0(g.0(x0))) 471.35/221.16 down.0(f.0(g.1(x0))) 471.35/221.16 down.0(f.0(b.0(x0))) 471.35/221.16 down.0(f.0(b.1(x0))) 471.35/221.16 down.0(f.0(u.0(x0))) 471.35/221.16 down.0(f.0(u.1(x0))) 471.35/221.16 down.0(u.0(x0)) 471.35/221.16 down.0(u.1(x0)) 471.35/221.16 down.0(g.0(x0)) 471.35/221.16 down.0(g.1(x0)) 471.35/221.16 down.0(f.0(c.)) 471.35/221.16 down.1(f.1(fresh_constant.)) 471.35/221.16 down.0(b.0(f.0(x0))) 471.35/221.16 down.0(b.1(f.1(x0))) 471.35/221.16 down.0(b.0(c.)) 471.35/221.16 down.0(b.0(b.0(x0))) 471.35/221.16 down.0(b.0(b.1(x0))) 471.35/221.16 down.0(b.0(g.0(x0))) 471.35/221.16 down.0(b.0(g.1(x0))) 471.35/221.16 down.0(b.1(fresh_constant.)) 471.35/221.16 down.0(f.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(x0)))) 471.35/221.16 down.0(f.0(f.0(b.1(x0)))) 471.35/221.16 down.0(f.0(f.0(u.0(x0)))) 471.35/221.16 down.0(f.0(f.0(u.1(x0)))) 471.35/221.16 down.0(f.0(f.0(g.0(x0)))) 471.35/221.16 down.0(f.0(f.0(g.1(x0)))) 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) 471.35/221.16 f_flat.0(up.0(x0)) 471.35/221.16 f_flat.0(up.1(x0)) 471.35/221.16 b_flat.0(up.0(x0)) 471.35/221.16 b_flat.0(up.1(x0)) 471.35/221.16 u_flat.0(up.0(x0)) 471.35/221.16 u_flat.0(up.1(x0)) 471.35/221.16 g_flat.0(up.0(x0)) 471.35/221.16 g_flat.0(up.1(x0)) 471.35/221.16 471.35/221.16 We have to consider all minimal (P,Q,R)-chains. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (124) MRRProof (EQUIVALENT) 471.35/221.16 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. 471.35/221.16 471.35/221.16 471.35/221.16 Strictly oriented rules of the TRS R: 471.35/221.16 471.35/221.16 down.1(f.1(f.1(f.1(x)))) -> up.0(c.) 471.35/221.16 471.35/221.16 Used ordering: Polynomial interpretation [POLO]: 471.35/221.16 471.35/221.16 POL(TOP.0(x_1)) = x_1 471.35/221.16 POL(b.0(x_1)) = x_1 471.35/221.16 POL(b.1(x_1)) = 1 + x_1 471.35/221.16 POL(b_flat.0(x_1)) = x_1 471.35/221.16 POL(c.) = 0 471.35/221.16 POL(down.0(x_1)) = x_1 471.35/221.16 POL(down.1(x_1)) = 1 + x_1 471.35/221.16 POL(f.0(x_1)) = x_1 471.35/221.16 POL(f.1(x_1)) = x_1 471.35/221.16 POL(f_flat.0(x_1)) = x_1 471.35/221.16 POL(fresh_constant.) = 0 471.35/221.16 POL(g.0(x_1)) = x_1 471.35/221.16 POL(g.1(x_1)) = x_1 471.35/221.16 POL(g_flat.0(x_1)) = x_1 471.35/221.16 POL(u.0(x_1)) = x_1 471.35/221.16 POL(u.1(x_1)) = x_1 471.35/221.16 POL(u_flat.0(x_1)) = x_1 471.35/221.16 POL(up.0(x_1)) = x_1 471.35/221.16 POL(up.1(x_1)) = x_1 471.35/221.16 471.35/221.16 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (125) 471.35/221.16 Obligation: 471.35/221.16 Q DP problem: 471.35/221.16 The TRS P consists of the following rules: 471.35/221.16 471.35/221.16 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.35/221.16 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.35/221.16 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.35/221.16 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.35/221.16 471.35/221.16 The TRS R consists of the following rules: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.35/221.16 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.35/221.16 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.35/221.16 down.0(f.0(g.1(x))) -> up.1(f.1(f.1(x))) 471.35/221.16 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.35/221.16 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.35/221.16 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.35/221.16 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.35/221.16 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.35/221.16 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.35/221.16 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.35/221.16 down.0(b.1(f.1(y10))) -> b_flat.0(down.1(f.1(y10))) 471.35/221.16 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.35/221.16 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.35/221.16 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.35/221.16 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.35/221.16 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.35/221.16 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.35/221.16 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.35/221.16 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.35/221.16 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.35/221.16 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.))) 471.35/221.16 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.35/221.16 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.35/221.16 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.35/221.16 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.35/221.16 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 471.35/221.16 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.35/221.16 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.35/221.16 471.35/221.16 The set Q consists of the following terms: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x0)))) 471.35/221.16 down.1(f.1(f.1(f.1(x0)))) 471.35/221.16 down.0(b.0(u.0(x0))) 471.35/221.16 down.0(b.0(u.1(x0))) 471.35/221.16 down.0(f.0(g.0(x0))) 471.35/221.16 down.0(f.0(g.1(x0))) 471.35/221.16 down.0(f.0(b.0(x0))) 471.35/221.16 down.0(f.0(b.1(x0))) 471.35/221.16 down.0(f.0(u.0(x0))) 471.35/221.16 down.0(f.0(u.1(x0))) 471.35/221.16 down.0(u.0(x0)) 471.35/221.16 down.0(u.1(x0)) 471.35/221.16 down.0(g.0(x0)) 471.35/221.16 down.0(g.1(x0)) 471.35/221.16 down.0(f.0(c.)) 471.35/221.16 down.1(f.1(fresh_constant.)) 471.35/221.16 down.0(b.0(f.0(x0))) 471.35/221.16 down.0(b.1(f.1(x0))) 471.35/221.16 down.0(b.0(c.)) 471.35/221.16 down.0(b.0(b.0(x0))) 471.35/221.16 down.0(b.0(b.1(x0))) 471.35/221.16 down.0(b.0(g.0(x0))) 471.35/221.16 down.0(b.0(g.1(x0))) 471.35/221.16 down.0(b.1(fresh_constant.)) 471.35/221.16 down.0(f.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(x0)))) 471.35/221.16 down.0(f.0(f.0(b.1(x0)))) 471.35/221.16 down.0(f.0(f.0(u.0(x0)))) 471.35/221.16 down.0(f.0(f.0(u.1(x0)))) 471.35/221.16 down.0(f.0(f.0(g.0(x0)))) 471.35/221.16 down.0(f.0(f.0(g.1(x0)))) 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) 471.35/221.16 f_flat.0(up.0(x0)) 471.35/221.16 f_flat.0(up.1(x0)) 471.35/221.16 b_flat.0(up.0(x0)) 471.35/221.16 b_flat.0(up.1(x0)) 471.35/221.16 u_flat.0(up.0(x0)) 471.35/221.16 u_flat.0(up.1(x0)) 471.35/221.16 g_flat.0(up.0(x0)) 471.35/221.16 g_flat.0(up.1(x0)) 471.35/221.16 471.35/221.16 We have to consider all minimal (P,Q,R)-chains. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (126) MRRProof (EQUIVALENT) 471.35/221.16 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. 471.35/221.16 471.35/221.16 471.35/221.16 Strictly oriented rules of the TRS R: 471.35/221.16 471.35/221.16 down.0(b.1(f.1(y10))) -> b_flat.0(down.1(f.1(y10))) 471.35/221.16 471.35/221.16 Used ordering: Polynomial interpretation [POLO]: 471.35/221.16 471.35/221.16 POL(TOP.0(x_1)) = x_1 471.35/221.16 POL(b.0(x_1)) = x_1 471.35/221.16 POL(b.1(x_1)) = 1 + x_1 471.35/221.16 POL(b_flat.0(x_1)) = x_1 471.35/221.16 POL(c.) = 0 471.35/221.16 POL(down.0(x_1)) = x_1 471.35/221.16 POL(down.1(x_1)) = x_1 471.35/221.16 POL(f.0(x_1)) = x_1 471.35/221.16 POL(f.1(x_1)) = x_1 471.35/221.16 POL(f_flat.0(x_1)) = x_1 471.35/221.16 POL(fresh_constant.) = 0 471.35/221.16 POL(g.0(x_1)) = x_1 471.35/221.16 POL(g.1(x_1)) = x_1 471.35/221.16 POL(g_flat.0(x_1)) = x_1 471.35/221.16 POL(u.0(x_1)) = x_1 471.35/221.16 POL(u.1(x_1)) = x_1 471.35/221.16 POL(u_flat.0(x_1)) = x_1 471.35/221.16 POL(up.0(x_1)) = x_1 471.35/221.16 POL(up.1(x_1)) = x_1 471.35/221.16 471.35/221.16 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (127) 471.35/221.16 Obligation: 471.35/221.16 Q DP problem: 471.35/221.16 The TRS P consists of the following rules: 471.35/221.16 471.35/221.16 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.35/221.16 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.35/221.16 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0)))) 471.35/221.16 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.35/221.16 471.35/221.16 The TRS R consists of the following rules: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.35/221.16 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.35/221.16 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.35/221.16 down.0(f.0(g.1(x))) -> up.1(f.1(f.1(x))) 471.35/221.16 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.35/221.16 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.35/221.16 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.35/221.16 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.35/221.16 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.35/221.16 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.35/221.16 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.35/221.16 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.35/221.16 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.35/221.16 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.35/221.16 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.35/221.16 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.35/221.16 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.35/221.16 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.35/221.16 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.35/221.16 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.35/221.16 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.))) 471.35/221.16 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.35/221.16 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.35/221.16 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.35/221.16 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.35/221.16 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 471.35/221.16 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.35/221.16 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.35/221.16 471.35/221.16 The set Q consists of the following terms: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x0)))) 471.35/221.16 down.1(f.1(f.1(f.1(x0)))) 471.35/221.16 down.0(b.0(u.0(x0))) 471.35/221.16 down.0(b.0(u.1(x0))) 471.35/221.16 down.0(f.0(g.0(x0))) 471.35/221.16 down.0(f.0(g.1(x0))) 471.35/221.16 down.0(f.0(b.0(x0))) 471.35/221.16 down.0(f.0(b.1(x0))) 471.35/221.16 down.0(f.0(u.0(x0))) 471.35/221.16 down.0(f.0(u.1(x0))) 471.35/221.16 down.0(u.0(x0)) 471.35/221.16 down.0(u.1(x0)) 471.35/221.16 down.0(g.0(x0)) 471.35/221.16 down.0(g.1(x0)) 471.35/221.16 down.0(f.0(c.)) 471.35/221.16 down.1(f.1(fresh_constant.)) 471.35/221.16 down.0(b.0(f.0(x0))) 471.35/221.16 down.0(b.1(f.1(x0))) 471.35/221.16 down.0(b.0(c.)) 471.35/221.16 down.0(b.0(b.0(x0))) 471.35/221.16 down.0(b.0(b.1(x0))) 471.35/221.16 down.0(b.0(g.0(x0))) 471.35/221.16 down.0(b.0(g.1(x0))) 471.35/221.16 down.0(b.1(fresh_constant.)) 471.35/221.16 down.0(f.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(x0)))) 471.35/221.16 down.0(f.0(f.0(b.1(x0)))) 471.35/221.16 down.0(f.0(f.0(u.0(x0)))) 471.35/221.16 down.0(f.0(f.0(u.1(x0)))) 471.35/221.16 down.0(f.0(f.0(g.0(x0)))) 471.35/221.16 down.0(f.0(f.0(g.1(x0)))) 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) 471.35/221.16 f_flat.0(up.0(x0)) 471.35/221.16 f_flat.0(up.1(x0)) 471.35/221.16 b_flat.0(up.0(x0)) 471.35/221.16 b_flat.0(up.1(x0)) 471.35/221.16 u_flat.0(up.0(x0)) 471.35/221.16 u_flat.0(up.1(x0)) 471.35/221.16 g_flat.0(up.0(x0)) 471.35/221.16 g_flat.0(up.1(x0)) 471.35/221.16 471.35/221.16 We have to consider all minimal (P,Q,R)-chains. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (128) DependencyGraphProof (EQUIVALENT) 471.35/221.16 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (129) 471.35/221.16 Obligation: 471.35/221.16 Q DP problem: 471.35/221.16 The TRS P consists of the following rules: 471.35/221.16 471.35/221.16 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.35/221.16 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.35/221.16 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.35/221.16 471.35/221.16 The TRS R consists of the following rules: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.35/221.16 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.35/221.16 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.35/221.16 down.0(f.0(g.1(x))) -> up.1(f.1(f.1(x))) 471.35/221.16 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.35/221.16 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.35/221.16 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.35/221.16 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.35/221.16 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.35/221.16 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.35/221.16 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.35/221.16 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.35/221.16 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.35/221.16 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.35/221.16 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.35/221.16 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.35/221.16 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.35/221.16 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.35/221.16 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.35/221.16 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.35/221.16 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.))) 471.35/221.16 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.35/221.16 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.35/221.16 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.35/221.16 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.35/221.16 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 471.35/221.16 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.35/221.16 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.35/221.16 471.35/221.16 The set Q consists of the following terms: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x0)))) 471.35/221.16 down.1(f.1(f.1(f.1(x0)))) 471.35/221.16 down.0(b.0(u.0(x0))) 471.35/221.16 down.0(b.0(u.1(x0))) 471.35/221.16 down.0(f.0(g.0(x0))) 471.35/221.16 down.0(f.0(g.1(x0))) 471.35/221.16 down.0(f.0(b.0(x0))) 471.35/221.16 down.0(f.0(b.1(x0))) 471.35/221.16 down.0(f.0(u.0(x0))) 471.35/221.16 down.0(f.0(u.1(x0))) 471.35/221.16 down.0(u.0(x0)) 471.35/221.16 down.0(u.1(x0)) 471.35/221.16 down.0(g.0(x0)) 471.35/221.16 down.0(g.1(x0)) 471.35/221.16 down.0(f.0(c.)) 471.35/221.16 down.1(f.1(fresh_constant.)) 471.35/221.16 down.0(b.0(f.0(x0))) 471.35/221.16 down.0(b.1(f.1(x0))) 471.35/221.16 down.0(b.0(c.)) 471.35/221.16 down.0(b.0(b.0(x0))) 471.35/221.16 down.0(b.0(b.1(x0))) 471.35/221.16 down.0(b.0(g.0(x0))) 471.35/221.16 down.0(b.0(g.1(x0))) 471.35/221.16 down.0(b.1(fresh_constant.)) 471.35/221.16 down.0(f.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(x0)))) 471.35/221.16 down.0(f.0(f.0(b.1(x0)))) 471.35/221.16 down.0(f.0(f.0(u.0(x0)))) 471.35/221.16 down.0(f.0(f.0(u.1(x0)))) 471.35/221.16 down.0(f.0(f.0(g.0(x0)))) 471.35/221.16 down.0(f.0(f.0(g.1(x0)))) 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) 471.35/221.16 f_flat.0(up.0(x0)) 471.35/221.16 f_flat.0(up.1(x0)) 471.35/221.16 b_flat.0(up.0(x0)) 471.35/221.16 b_flat.0(up.1(x0)) 471.35/221.16 u_flat.0(up.0(x0)) 471.35/221.16 u_flat.0(up.1(x0)) 471.35/221.16 g_flat.0(up.0(x0)) 471.35/221.16 g_flat.0(up.1(x0)) 471.35/221.16 471.35/221.16 We have to consider all minimal (P,Q,R)-chains. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (130) UsableRulesReductionPairsProof (EQUIVALENT) 471.35/221.16 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. 471.35/221.16 471.35/221.16 No dependency pairs are removed. 471.35/221.16 471.35/221.16 The following rules are removed from R: 471.35/221.16 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.))) 471.35/221.16 Used ordering: POLO with Polynomial interpretation [POLO]: 471.35/221.16 471.35/221.16 POL(TOP.0(x_1)) = x_1 471.35/221.16 POL(b.0(x_1)) = x_1 471.35/221.16 POL(b.1(x_1)) = x_1 471.35/221.16 POL(b_flat.0(x_1)) = x_1 471.35/221.16 POL(c.) = 0 471.35/221.16 POL(down.0(x_1)) = 1 + x_1 471.35/221.16 POL(f.0(x_1)) = x_1 471.35/221.16 POL(f.1(x_1)) = x_1 471.35/221.16 POL(f_flat.0(x_1)) = x_1 471.35/221.16 POL(g.0(x_1)) = x_1 471.35/221.16 POL(g.1(x_1)) = x_1 471.35/221.16 POL(g_flat.0(x_1)) = x_1 471.35/221.16 POL(u.0(x_1)) = x_1 471.35/221.16 POL(u.1(x_1)) = x_1 471.35/221.16 POL(u_flat.0(x_1)) = x_1 471.35/221.16 POL(up.0(x_1)) = 1 + x_1 471.35/221.16 POL(up.1(x_1)) = 1 + x_1 471.35/221.16 471.35/221.16 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (131) 471.35/221.16 Obligation: 471.35/221.16 Q DP problem: 471.35/221.16 The TRS P consists of the following rules: 471.35/221.16 471.35/221.16 TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(f.0(x0)))) -> TOP.0(b_flat.0(down.0(f.0(x0)))) 471.35/221.16 TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0))) 471.35/221.16 TOP.0(up.0(g.0(x0))) -> TOP.0(g_flat.0(down.0(x0))) 471.35/221.16 TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0)))) 471.35/221.16 TOP.0(up.0(b.0(g.0(x0)))) -> TOP.0(b_flat.0(g_flat.0(down.0(x0)))) 471.35/221.16 471.35/221.16 The TRS R consists of the following rules: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x)))) -> up.0(c.) 471.35/221.16 down.0(b.0(u.0(x))) -> up.0(b.0(f.0(x))) 471.35/221.16 down.0(f.0(g.0(x))) -> up.0(f.0(f.0(x))) 471.35/221.16 down.0(f.0(g.1(x))) -> up.1(f.1(f.1(x))) 471.35/221.16 down.0(f.0(b.0(x))) -> up.0(u.0(g.0(b.0(x)))) 471.35/221.16 down.0(f.0(b.1(x))) -> up.0(u.0(g.0(b.1(x)))) 471.35/221.16 down.0(f.0(u.0(x))) -> up.0(u.0(g.0(x))) 471.35/221.16 down.0(f.0(u.1(x))) -> up.0(u.0(g.1(x))) 471.35/221.16 down.0(u.0(y2)) -> u_flat.0(down.0(y2)) 471.35/221.16 down.0(g.0(y3)) -> g_flat.0(down.0(y3)) 471.35/221.16 down.0(b.0(f.0(y10))) -> b_flat.0(down.0(f.0(y10))) 471.35/221.16 down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11))) 471.35/221.16 down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11))) 471.35/221.16 down.0(b.0(g.0(y13))) -> b_flat.0(down.0(g.0(y13))) 471.35/221.16 down.0(b.0(g.1(y13))) -> b_flat.0(down.0(g.1(y13))) 471.35/221.16 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(y16)))) -> f_flat.0(down.0(f.0(b.0(y16)))) 471.35/221.16 down.0(f.0(f.0(b.1(y16)))) -> f_flat.0(down.0(f.0(b.1(y16)))) 471.35/221.16 down.0(f.0(f.0(u.0(y17)))) -> f_flat.0(down.0(f.0(u.0(y17)))) 471.35/221.16 down.0(f.0(f.0(u.1(y17)))) -> f_flat.0(down.0(f.0(u.1(y17)))) 471.35/221.16 down.0(f.0(f.0(g.0(y18)))) -> f_flat.0(down.0(f.0(g.0(y18)))) 471.35/221.16 down.0(f.0(f.0(g.1(y18)))) -> f_flat.0(down.0(f.0(g.1(y18)))) 471.35/221.16 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 471.35/221.16 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 471.35/221.16 b_flat.0(up.0(x_1)) -> up.0(b.0(x_1)) 471.35/221.16 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 471.35/221.16 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 471.35/221.16 u_flat.0(up.0(x_1)) -> up.0(u.0(x_1)) 471.35/221.16 u_flat.0(up.1(x_1)) -> up.0(u.1(x_1)) 471.35/221.16 471.35/221.16 The set Q consists of the following terms: 471.35/221.16 471.35/221.16 down.0(f.0(f.0(f.0(x0)))) 471.35/221.16 down.1(f.1(f.1(f.1(x0)))) 471.35/221.16 down.0(b.0(u.0(x0))) 471.35/221.16 down.0(b.0(u.1(x0))) 471.35/221.16 down.0(f.0(g.0(x0))) 471.35/221.16 down.0(f.0(g.1(x0))) 471.35/221.16 down.0(f.0(b.0(x0))) 471.35/221.16 down.0(f.0(b.1(x0))) 471.35/221.16 down.0(f.0(u.0(x0))) 471.35/221.16 down.0(f.0(u.1(x0))) 471.35/221.16 down.0(u.0(x0)) 471.35/221.16 down.0(u.1(x0)) 471.35/221.16 down.0(g.0(x0)) 471.35/221.16 down.0(g.1(x0)) 471.35/221.16 down.0(f.0(c.)) 471.35/221.16 down.1(f.1(fresh_constant.)) 471.35/221.16 down.0(b.0(f.0(x0))) 471.35/221.16 down.0(b.1(f.1(x0))) 471.35/221.16 down.0(b.0(c.)) 471.35/221.16 down.0(b.0(b.0(x0))) 471.35/221.16 down.0(b.0(b.1(x0))) 471.35/221.16 down.0(b.0(g.0(x0))) 471.35/221.16 down.0(b.0(g.1(x0))) 471.35/221.16 down.0(b.1(fresh_constant.)) 471.35/221.16 down.0(f.0(f.0(c.))) 471.35/221.16 down.0(f.0(f.0(b.0(x0)))) 471.35/221.16 down.0(f.0(f.0(b.1(x0)))) 471.35/221.16 down.0(f.0(f.0(u.0(x0)))) 471.35/221.16 down.0(f.0(f.0(u.1(x0)))) 471.35/221.16 down.0(f.0(f.0(g.0(x0)))) 471.35/221.16 down.0(f.0(f.0(g.1(x0)))) 471.35/221.16 down.1(f.1(f.1(fresh_constant.))) 471.35/221.16 f_flat.0(up.0(x0)) 471.35/221.16 f_flat.0(up.1(x0)) 471.35/221.16 b_flat.0(up.0(x0)) 471.35/221.16 b_flat.0(up.1(x0)) 471.35/221.16 u_flat.0(up.0(x0)) 471.35/221.16 u_flat.0(up.1(x0)) 471.35/221.16 g_flat.0(up.0(x0)) 471.35/221.16 g_flat.0(up.1(x0)) 471.35/221.16 471.35/221.16 We have to consider all minimal (P,Q,R)-chains. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (132) PisEmptyProof (SOUND) 471.35/221.16 The TRS P is empty. Hence, there is no (P,Q,R) chain. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (133) 471.35/221.16 TRUE 471.35/221.16 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (134) 471.35/221.16 Obligation: 471.35/221.16 Q DP problem: 471.35/221.16 The TRS P consists of the following rules: 471.35/221.16 471.35/221.16 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.35/221.16 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.35/221.16 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.35/221.16 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.35/221.16 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.35/221.16 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.35/221.16 471.35/221.16 The TRS R consists of the following rules: 471.35/221.16 471.35/221.16 down(f(f(f(x)))) -> up(c) 471.35/221.16 down(b(u(x))) -> up(b(f(x))) 471.35/221.16 down(f(g(x))) -> up(f(f(x))) 471.35/221.16 down(f(b(x))) -> up(u(g(b(x)))) 471.35/221.16 down(f(u(x))) -> up(u(g(x))) 471.35/221.16 down(u(y2)) -> u_flat(down(y2)) 471.35/221.16 down(g(y3)) -> g_flat(down(y3)) 471.35/221.16 down(b(f(y10))) -> b_flat(down(f(y10))) 471.35/221.16 down(b(b(y11))) -> b_flat(down(b(y11))) 471.35/221.16 down(b(g(y13))) -> b_flat(down(g(y13))) 471.35/221.16 down(f(f(c))) -> f_flat(down(f(c))) 471.35/221.16 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.35/221.16 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.35/221.16 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.35/221.16 g_flat(up(x_1)) -> up(g(x_1)) 471.35/221.16 b_flat(up(x_1)) -> up(b(x_1)) 471.35/221.16 f_flat(up(x_1)) -> up(f(x_1)) 471.35/221.16 u_flat(up(x_1)) -> up(u(x_1)) 471.35/221.16 471.35/221.16 The set Q consists of the following terms: 471.35/221.16 471.35/221.16 down(f(f(f(x0)))) 471.35/221.16 down(b(u(x0))) 471.35/221.16 down(f(g(x0))) 471.35/221.16 down(f(b(x0))) 471.35/221.16 down(f(u(x0))) 471.35/221.16 down(u(x0)) 471.35/221.16 down(g(x0)) 471.35/221.16 down(f(c)) 471.35/221.16 down(f(fresh_constant)) 471.35/221.16 down(b(f(x0))) 471.35/221.16 down(b(c)) 471.35/221.16 down(b(b(x0))) 471.35/221.16 down(b(g(x0))) 471.35/221.16 down(b(fresh_constant)) 471.35/221.16 down(f(f(c))) 471.35/221.16 down(f(f(b(x0)))) 471.35/221.16 down(f(f(u(x0)))) 471.35/221.16 down(f(f(g(x0)))) 471.35/221.16 down(f(f(fresh_constant))) 471.35/221.16 f_flat(up(x0)) 471.35/221.16 b_flat(up(x0)) 471.35/221.16 u_flat(up(x0)) 471.35/221.16 g_flat(up(x0)) 471.35/221.16 471.35/221.16 We have to consider all minimal (P,Q,R)-chains. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (135) QReductionProof (EQUIVALENT) 471.35/221.16 We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN]. 471.35/221.16 471.35/221.16 down(f(fresh_constant)) 471.35/221.16 down(b(fresh_constant)) 471.35/221.16 down(f(f(fresh_constant))) 471.35/221.16 471.35/221.16 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (136) 471.35/221.16 Obligation: 471.35/221.16 Q DP problem: 471.35/221.16 The TRS P consists of the following rules: 471.35/221.16 471.35/221.16 TOP(up(b(u(x0)))) -> TOP(up(b(f(x0)))) 471.35/221.16 TOP(up(b(f(x0)))) -> TOP(b_flat(down(f(x0)))) 471.35/221.16 TOP(up(u(x0))) -> TOP(u_flat(down(x0))) 471.35/221.16 TOP(up(g(x0))) -> TOP(g_flat(down(x0))) 471.35/221.16 TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))) 471.35/221.16 TOP(up(b(g(x0)))) -> TOP(b_flat(g_flat(down(x0)))) 471.35/221.16 471.35/221.16 The TRS R consists of the following rules: 471.35/221.16 471.35/221.16 down(f(f(f(x)))) -> up(c) 471.35/221.16 down(b(u(x))) -> up(b(f(x))) 471.35/221.16 down(f(g(x))) -> up(f(f(x))) 471.35/221.16 down(f(b(x))) -> up(u(g(b(x)))) 471.35/221.16 down(f(u(x))) -> up(u(g(x))) 471.35/221.16 down(u(y2)) -> u_flat(down(y2)) 471.35/221.16 down(g(y3)) -> g_flat(down(y3)) 471.35/221.16 down(b(f(y10))) -> b_flat(down(f(y10))) 471.35/221.16 down(b(b(y11))) -> b_flat(down(b(y11))) 471.35/221.16 down(b(g(y13))) -> b_flat(down(g(y13))) 471.35/221.16 down(f(f(c))) -> f_flat(down(f(c))) 471.35/221.16 down(f(f(b(y16)))) -> f_flat(down(f(b(y16)))) 471.35/221.16 down(f(f(u(y17)))) -> f_flat(down(f(u(y17)))) 471.35/221.16 down(f(f(g(y18)))) -> f_flat(down(f(g(y18)))) 471.35/221.16 g_flat(up(x_1)) -> up(g(x_1)) 471.35/221.16 b_flat(up(x_1)) -> up(b(x_1)) 471.35/221.16 f_flat(up(x_1)) -> up(f(x_1)) 471.35/221.16 u_flat(up(x_1)) -> up(u(x_1)) 471.35/221.16 471.35/221.16 The set Q consists of the following terms: 471.35/221.16 471.35/221.16 down(f(f(f(x0)))) 471.35/221.16 down(b(u(x0))) 471.35/221.16 down(f(g(x0))) 471.35/221.16 down(f(b(x0))) 471.35/221.16 down(f(u(x0))) 471.35/221.16 down(u(x0)) 471.35/221.16 down(g(x0)) 471.35/221.16 down(f(c)) 471.35/221.16 down(b(f(x0))) 471.35/221.16 down(b(c)) 471.35/221.16 down(b(b(x0))) 471.35/221.16 down(b(g(x0))) 471.35/221.16 down(f(f(c))) 471.35/221.16 down(f(f(b(x0)))) 471.35/221.16 down(f(f(u(x0)))) 471.35/221.16 down(f(f(g(x0)))) 471.35/221.16 f_flat(up(x0)) 471.35/221.16 b_flat(up(x0)) 471.35/221.16 u_flat(up(x0)) 471.35/221.16 g_flat(up(x0)) 471.35/221.16 471.35/221.16 We have to consider all (P,Q,R)-chains. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (137) Thiemann-SpecialC-Transformation (EQUIVALENT) 471.35/221.16 We applied the Thiemann-SpecialC transformation to transform the outermost TRS to an innermost TRS. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (138) 471.35/221.16 Obligation: 471.35/221.16 Q restricted rewrite system: 471.35/221.16 The TRS R consists of the following rules: 471.35/221.16 471.35/221.16 top(go_up(x)) -> top(reduce(x)) 471.35/221.16 reduce(f(x_1)) -> check_f(redex_f(x_1)) 471.35/221.16 reduce(b(x_1)) -> check_b(redex_b(x_1)) 471.35/221.16 redex_f(f(f(x))) -> result_f(c) 471.35/221.16 redex_b(u(x)) -> result_b(b(f(x))) 471.35/221.16 redex_f(g(x)) -> result_f(f(f(x))) 471.35/221.16 redex_f(b(x)) -> result_f(u(g(b(x)))) 471.35/221.16 redex_f(u(x)) -> result_f(u(g(x))) 471.35/221.16 check_f(result_f(x)) -> go_up(x) 471.35/221.16 check_b(result_b(x)) -> go_up(x) 471.35/221.16 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 471.35/221.16 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 471.35/221.16 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 471.35/221.16 reduce(g(x_1)) -> in_g_1(reduce(x_1)) 471.35/221.16 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 471.35/221.16 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 471.35/221.16 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 471.35/221.16 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 471.35/221.16 471.35/221.16 The set Q consists of the following terms: 471.35/221.16 471.35/221.16 top(go_up(x0)) 471.35/221.16 reduce(f(x0)) 471.35/221.16 reduce(b(x0)) 471.35/221.16 redex_f(f(f(x0))) 471.35/221.16 redex_b(u(x0)) 471.35/221.16 redex_f(g(x0)) 471.35/221.16 redex_f(b(x0)) 471.35/221.16 redex_f(u(x0)) 471.35/221.16 check_f(result_f(x0)) 471.35/221.16 check_b(result_b(x0)) 471.35/221.16 check_f(redex_f(x0)) 471.35/221.16 check_b(redex_b(x0)) 471.35/221.16 reduce(u(x0)) 471.35/221.16 reduce(g(x0)) 471.35/221.16 in_f_1(go_up(x0)) 471.35/221.16 in_b_1(go_up(x0)) 471.35/221.16 in_u_1(go_up(x0)) 471.35/221.16 in_g_1(go_up(x0)) 471.35/221.16 471.35/221.16 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (139) DependencyPairsProof (EQUIVALENT) 471.35/221.16 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (140) 471.35/221.16 Obligation: 471.35/221.16 Q DP problem: 471.35/221.16 The TRS P consists of the following rules: 471.35/221.16 471.35/221.16 TOP(go_up(x)) -> TOP(reduce(x)) 471.35/221.16 TOP(go_up(x)) -> REDUCE(x) 471.35/221.16 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 471.35/221.16 REDUCE(f(x_1)) -> REDEX_F(x_1) 471.35/221.16 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 471.35/221.16 REDUCE(b(x_1)) -> REDEX_B(x_1) 471.35/221.16 CHECK_F(redex_f(x_1)) -> IN_F_1(reduce(x_1)) 471.35/221.16 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 471.35/221.16 CHECK_B(redex_b(x_1)) -> IN_B_1(reduce(x_1)) 471.35/221.16 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 471.35/221.16 REDUCE(u(x_1)) -> IN_U_1(reduce(x_1)) 471.35/221.16 REDUCE(u(x_1)) -> REDUCE(x_1) 471.35/221.16 REDUCE(g(x_1)) -> IN_G_1(reduce(x_1)) 471.35/221.16 REDUCE(g(x_1)) -> REDUCE(x_1) 471.35/221.16 471.35/221.16 The TRS R consists of the following rules: 471.35/221.16 471.35/221.16 top(go_up(x)) -> top(reduce(x)) 471.35/221.16 reduce(f(x_1)) -> check_f(redex_f(x_1)) 471.35/221.16 reduce(b(x_1)) -> check_b(redex_b(x_1)) 471.35/221.16 redex_f(f(f(x))) -> result_f(c) 471.35/221.16 redex_b(u(x)) -> result_b(b(f(x))) 471.35/221.16 redex_f(g(x)) -> result_f(f(f(x))) 471.35/221.16 redex_f(b(x)) -> result_f(u(g(b(x)))) 471.35/221.16 redex_f(u(x)) -> result_f(u(g(x))) 471.35/221.16 check_f(result_f(x)) -> go_up(x) 471.35/221.16 check_b(result_b(x)) -> go_up(x) 471.35/221.16 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 471.35/221.16 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 471.35/221.16 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 471.35/221.16 reduce(g(x_1)) -> in_g_1(reduce(x_1)) 471.35/221.16 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 471.35/221.16 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 471.35/221.16 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 471.35/221.16 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 471.35/221.16 471.35/221.16 The set Q consists of the following terms: 471.35/221.16 471.35/221.16 top(go_up(x0)) 471.35/221.16 reduce(f(x0)) 471.35/221.16 reduce(b(x0)) 471.35/221.16 redex_f(f(f(x0))) 471.35/221.16 redex_b(u(x0)) 471.35/221.16 redex_f(g(x0)) 471.35/221.16 redex_f(b(x0)) 471.35/221.16 redex_f(u(x0)) 471.35/221.16 check_f(result_f(x0)) 471.35/221.16 check_b(result_b(x0)) 471.35/221.16 check_f(redex_f(x0)) 471.35/221.16 check_b(redex_b(x0)) 471.35/221.16 reduce(u(x0)) 471.35/221.16 reduce(g(x0)) 471.35/221.16 in_f_1(go_up(x0)) 471.35/221.16 in_b_1(go_up(x0)) 471.35/221.16 in_u_1(go_up(x0)) 471.35/221.16 in_g_1(go_up(x0)) 471.35/221.16 471.35/221.16 We have to consider all minimal (P,Q,R)-chains. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (141) DependencyGraphProof (EQUIVALENT) 471.35/221.16 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 7 less nodes. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (142) 471.35/221.16 Complex Obligation (AND) 471.35/221.16 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (143) 471.35/221.16 Obligation: 471.35/221.16 Q DP problem: 471.35/221.16 The TRS P consists of the following rules: 471.35/221.16 471.35/221.16 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 471.35/221.16 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 471.35/221.16 471.35/221.16 The TRS R consists of the following rules: 471.35/221.16 471.35/221.16 top(go_up(x)) -> top(reduce(x)) 471.35/221.16 reduce(f(x_1)) -> check_f(redex_f(x_1)) 471.35/221.16 reduce(b(x_1)) -> check_b(redex_b(x_1)) 471.35/221.16 redex_f(f(f(x))) -> result_f(c) 471.35/221.16 redex_b(u(x)) -> result_b(b(f(x))) 471.35/221.16 redex_f(g(x)) -> result_f(f(f(x))) 471.35/221.16 redex_f(b(x)) -> result_f(u(g(b(x)))) 471.35/221.16 redex_f(u(x)) -> result_f(u(g(x))) 471.35/221.16 check_f(result_f(x)) -> go_up(x) 471.35/221.16 check_b(result_b(x)) -> go_up(x) 471.35/221.16 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 471.35/221.16 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 471.35/221.16 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 471.35/221.16 reduce(g(x_1)) -> in_g_1(reduce(x_1)) 471.35/221.16 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 471.35/221.16 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 471.35/221.16 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 471.35/221.16 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 471.35/221.16 471.35/221.16 The set Q consists of the following terms: 471.35/221.16 471.35/221.16 top(go_up(x0)) 471.35/221.16 reduce(f(x0)) 471.35/221.16 reduce(b(x0)) 471.35/221.16 redex_f(f(f(x0))) 471.35/221.16 redex_b(u(x0)) 471.35/221.16 redex_f(g(x0)) 471.35/221.16 redex_f(b(x0)) 471.35/221.16 redex_f(u(x0)) 471.35/221.16 check_f(result_f(x0)) 471.35/221.16 check_b(result_b(x0)) 471.35/221.16 check_f(redex_f(x0)) 471.35/221.16 check_b(redex_b(x0)) 471.35/221.16 reduce(u(x0)) 471.35/221.16 reduce(g(x0)) 471.35/221.16 in_f_1(go_up(x0)) 471.35/221.16 in_b_1(go_up(x0)) 471.35/221.16 in_u_1(go_up(x0)) 471.35/221.16 in_g_1(go_up(x0)) 471.35/221.16 471.35/221.16 We have to consider all minimal (P,Q,R)-chains. 471.35/221.16 ---------------------------------------- 471.35/221.16 471.35/221.16 (144) UsableRulesProof (EQUIVALENT) 471.35/221.16 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. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (145) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 471.35/221.17 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 471.35/221.17 471.35/221.17 The TRS R consists of the following rules: 471.35/221.17 471.35/221.17 redex_f(f(f(x))) -> result_f(c) 471.35/221.17 redex_f(g(x)) -> result_f(f(f(x))) 471.35/221.17 redex_f(b(x)) -> result_f(u(g(b(x)))) 471.35/221.17 redex_f(u(x)) -> result_f(u(g(x))) 471.35/221.17 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 top(go_up(x0)) 471.35/221.17 reduce(f(x0)) 471.35/221.17 reduce(b(x0)) 471.35/221.17 redex_f(f(f(x0))) 471.35/221.17 redex_b(u(x0)) 471.35/221.17 redex_f(g(x0)) 471.35/221.17 redex_f(b(x0)) 471.35/221.17 redex_f(u(x0)) 471.35/221.17 check_f(result_f(x0)) 471.35/221.17 check_b(result_b(x0)) 471.35/221.17 check_f(redex_f(x0)) 471.35/221.17 check_b(redex_b(x0)) 471.35/221.17 reduce(u(x0)) 471.35/221.17 reduce(g(x0)) 471.35/221.17 in_f_1(go_up(x0)) 471.35/221.17 in_b_1(go_up(x0)) 471.35/221.17 in_u_1(go_up(x0)) 471.35/221.17 in_g_1(go_up(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (146) QReductionProof (EQUIVALENT) 471.35/221.17 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 471.35/221.17 471.35/221.17 top(go_up(x0)) 471.35/221.17 reduce(f(x0)) 471.35/221.17 reduce(b(x0)) 471.35/221.17 redex_b(u(x0)) 471.35/221.17 check_f(result_f(x0)) 471.35/221.17 check_b(result_b(x0)) 471.35/221.17 check_f(redex_f(x0)) 471.35/221.17 check_b(redex_b(x0)) 471.35/221.17 reduce(u(x0)) 471.35/221.17 reduce(g(x0)) 471.35/221.17 in_f_1(go_up(x0)) 471.35/221.17 in_b_1(go_up(x0)) 471.35/221.17 in_u_1(go_up(x0)) 471.35/221.17 in_g_1(go_up(x0)) 471.35/221.17 471.35/221.17 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (147) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 471.35/221.17 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 471.35/221.17 471.35/221.17 The TRS R consists of the following rules: 471.35/221.17 471.35/221.17 redex_f(f(f(x))) -> result_f(c) 471.35/221.17 redex_f(g(x)) -> result_f(f(f(x))) 471.35/221.17 redex_f(b(x)) -> result_f(u(g(b(x)))) 471.35/221.17 redex_f(u(x)) -> result_f(u(g(x))) 471.35/221.17 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 redex_f(f(f(x0))) 471.35/221.17 redex_f(g(x0)) 471.35/221.17 redex_f(b(x0)) 471.35/221.17 redex_f(u(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (148) MRRProof (EQUIVALENT) 471.35/221.17 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. 471.35/221.17 471.35/221.17 471.35/221.17 Strictly oriented rules of the TRS R: 471.35/221.17 471.35/221.17 redex_f(f(f(x))) -> result_f(c) 471.35/221.17 redex_f(g(x)) -> result_f(f(f(x))) 471.35/221.17 redex_f(b(x)) -> result_f(u(g(b(x)))) 471.35/221.17 redex_f(u(x)) -> result_f(u(g(x))) 471.35/221.17 471.35/221.17 Used ordering: Polynomial interpretation [POLO]: 471.35/221.17 471.35/221.17 POL(CHECK_F(x_1)) = 1 + x_1 471.35/221.17 POL(REDUCE(x_1)) = 2 + 2*x_1 471.35/221.17 POL(b(x_1)) = x_1 471.35/221.17 POL(c) = 0 471.35/221.17 POL(f(x_1)) = x_1 471.35/221.17 POL(g(x_1)) = 2*x_1 471.35/221.17 POL(redex_f(x_1)) = 1 + 2*x_1 471.35/221.17 POL(result_f(x_1)) = x_1 471.35/221.17 POL(u(x_1)) = x_1 471.35/221.17 471.35/221.17 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (149) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 471.35/221.17 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 471.35/221.17 471.35/221.17 R is empty. 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 redex_f(f(f(x0))) 471.35/221.17 redex_f(g(x0)) 471.35/221.17 redex_f(b(x0)) 471.35/221.17 redex_f(u(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (150) UsableRulesReductionPairsProof (EQUIVALENT) 471.35/221.17 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. 471.35/221.17 471.35/221.17 The following dependency pairs can be deleted: 471.35/221.17 471.35/221.17 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 471.35/221.17 No rules are removed from R. 471.35/221.17 471.35/221.17 Used ordering: POLO with Polynomial interpretation [POLO]: 471.35/221.17 471.35/221.17 POL(CHECK_F(x_1)) = 2*x_1 471.35/221.17 POL(REDUCE(x_1)) = 2*x_1 471.35/221.17 POL(f(x_1)) = 2*x_1 471.35/221.17 POL(redex_f(x_1)) = x_1 471.35/221.17 471.35/221.17 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (151) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 471.35/221.17 471.35/221.17 R is empty. 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 redex_f(f(f(x0))) 471.35/221.17 redex_f(g(x0)) 471.35/221.17 redex_f(b(x0)) 471.35/221.17 redex_f(u(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (152) DependencyGraphProof (EQUIVALENT) 471.35/221.17 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (153) 471.35/221.17 TRUE 471.35/221.17 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (154) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 471.35/221.17 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 471.35/221.17 REDUCE(g(x_1)) -> REDUCE(x_1) 471.35/221.17 REDUCE(u(x_1)) -> REDUCE(x_1) 471.35/221.17 471.35/221.17 The TRS R consists of the following rules: 471.35/221.17 471.35/221.17 top(go_up(x)) -> top(reduce(x)) 471.35/221.17 reduce(f(x_1)) -> check_f(redex_f(x_1)) 471.35/221.17 reduce(b(x_1)) -> check_b(redex_b(x_1)) 471.35/221.17 redex_f(f(f(x))) -> result_f(c) 471.35/221.17 redex_b(u(x)) -> result_b(b(f(x))) 471.35/221.17 redex_f(g(x)) -> result_f(f(f(x))) 471.35/221.17 redex_f(b(x)) -> result_f(u(g(b(x)))) 471.35/221.17 redex_f(u(x)) -> result_f(u(g(x))) 471.35/221.17 check_f(result_f(x)) -> go_up(x) 471.35/221.17 check_b(result_b(x)) -> go_up(x) 471.35/221.17 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 471.35/221.17 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 471.35/221.17 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 471.35/221.17 reduce(g(x_1)) -> in_g_1(reduce(x_1)) 471.35/221.17 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 471.35/221.17 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 471.35/221.17 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 471.35/221.17 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 471.35/221.17 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 top(go_up(x0)) 471.35/221.17 reduce(f(x0)) 471.35/221.17 reduce(b(x0)) 471.35/221.17 redex_f(f(f(x0))) 471.35/221.17 redex_b(u(x0)) 471.35/221.17 redex_f(g(x0)) 471.35/221.17 redex_f(b(x0)) 471.35/221.17 redex_f(u(x0)) 471.35/221.17 check_f(result_f(x0)) 471.35/221.17 check_b(result_b(x0)) 471.35/221.17 check_f(redex_f(x0)) 471.35/221.17 check_b(redex_b(x0)) 471.35/221.17 reduce(u(x0)) 471.35/221.17 reduce(g(x0)) 471.35/221.17 in_f_1(go_up(x0)) 471.35/221.17 in_b_1(go_up(x0)) 471.35/221.17 in_u_1(go_up(x0)) 471.35/221.17 in_g_1(go_up(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (155) UsableRulesProof (EQUIVALENT) 471.35/221.17 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. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (156) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 471.35/221.17 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 471.35/221.17 REDUCE(g(x_1)) -> REDUCE(x_1) 471.35/221.17 REDUCE(u(x_1)) -> REDUCE(x_1) 471.35/221.17 471.35/221.17 The TRS R consists of the following rules: 471.35/221.17 471.35/221.17 redex_b(u(x)) -> result_b(b(f(x))) 471.35/221.17 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 top(go_up(x0)) 471.35/221.17 reduce(f(x0)) 471.35/221.17 reduce(b(x0)) 471.35/221.17 redex_f(f(f(x0))) 471.35/221.17 redex_b(u(x0)) 471.35/221.17 redex_f(g(x0)) 471.35/221.17 redex_f(b(x0)) 471.35/221.17 redex_f(u(x0)) 471.35/221.17 check_f(result_f(x0)) 471.35/221.17 check_b(result_b(x0)) 471.35/221.17 check_f(redex_f(x0)) 471.35/221.17 check_b(redex_b(x0)) 471.35/221.17 reduce(u(x0)) 471.35/221.17 reduce(g(x0)) 471.35/221.17 in_f_1(go_up(x0)) 471.35/221.17 in_b_1(go_up(x0)) 471.35/221.17 in_u_1(go_up(x0)) 471.35/221.17 in_g_1(go_up(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (157) QReductionProof (EQUIVALENT) 471.35/221.17 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 471.35/221.17 471.35/221.17 top(go_up(x0)) 471.35/221.17 reduce(f(x0)) 471.35/221.17 reduce(b(x0)) 471.35/221.17 redex_f(f(f(x0))) 471.35/221.17 redex_f(g(x0)) 471.35/221.17 redex_f(b(x0)) 471.35/221.17 redex_f(u(x0)) 471.35/221.17 check_f(result_f(x0)) 471.35/221.17 check_b(result_b(x0)) 471.35/221.17 check_f(redex_f(x0)) 471.35/221.17 check_b(redex_b(x0)) 471.35/221.17 reduce(u(x0)) 471.35/221.17 reduce(g(x0)) 471.35/221.17 in_f_1(go_up(x0)) 471.35/221.17 in_b_1(go_up(x0)) 471.35/221.17 in_u_1(go_up(x0)) 471.35/221.17 in_g_1(go_up(x0)) 471.35/221.17 471.35/221.17 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (158) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 471.35/221.17 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 471.35/221.17 REDUCE(g(x_1)) -> REDUCE(x_1) 471.35/221.17 REDUCE(u(x_1)) -> REDUCE(x_1) 471.35/221.17 471.35/221.17 The TRS R consists of the following rules: 471.35/221.17 471.35/221.17 redex_b(u(x)) -> result_b(b(f(x))) 471.35/221.17 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 redex_b(u(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (159) UsableRulesReductionPairsProof (EQUIVALENT) 471.35/221.17 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. 471.35/221.17 471.35/221.17 The following dependency pairs can be deleted: 471.35/221.17 471.35/221.17 REDUCE(g(x_1)) -> REDUCE(x_1) 471.35/221.17 REDUCE(u(x_1)) -> REDUCE(x_1) 471.35/221.17 The following rules are removed from R: 471.35/221.17 471.35/221.17 redex_b(u(x)) -> result_b(b(f(x))) 471.35/221.17 Used ordering: POLO with Polynomial interpretation [POLO]: 471.35/221.17 471.35/221.17 POL(CHECK_B(x_1)) = x_1 471.35/221.17 POL(REDUCE(x_1)) = 2*x_1 471.35/221.17 POL(b(x_1)) = 2*x_1 471.35/221.17 POL(f(x_1)) = 2*x_1 471.35/221.17 POL(g(x_1)) = 2*x_1 471.35/221.17 POL(redex_b(x_1)) = 2*x_1 471.35/221.17 POL(result_b(x_1)) = x_1 471.35/221.17 POL(u(x_1)) = 2*x_1 471.35/221.17 471.35/221.17 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (160) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 471.35/221.17 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 471.35/221.17 471.35/221.17 R is empty. 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 redex_b(u(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (161) UsableRulesReductionPairsProof (EQUIVALENT) 471.35/221.17 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. 471.35/221.17 471.35/221.17 The following dependency pairs can be deleted: 471.35/221.17 471.35/221.17 REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1)) 471.35/221.17 No rules are removed from R. 471.35/221.17 471.35/221.17 Used ordering: POLO with Polynomial interpretation [POLO]: 471.35/221.17 471.35/221.17 POL(CHECK_B(x_1)) = 2*x_1 471.35/221.17 POL(REDUCE(x_1)) = 2*x_1 471.35/221.17 POL(b(x_1)) = 2*x_1 471.35/221.17 POL(redex_b(x_1)) = x_1 471.35/221.17 471.35/221.17 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (162) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 CHECK_B(redex_b(x_1)) -> REDUCE(x_1) 471.35/221.17 471.35/221.17 R is empty. 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 redex_b(u(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (163) DependencyGraphProof (EQUIVALENT) 471.35/221.17 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (164) 471.35/221.17 TRUE 471.35/221.17 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (165) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 TOP(go_up(x)) -> TOP(reduce(x)) 471.35/221.17 471.35/221.17 The TRS R consists of the following rules: 471.35/221.17 471.35/221.17 top(go_up(x)) -> top(reduce(x)) 471.35/221.17 reduce(f(x_1)) -> check_f(redex_f(x_1)) 471.35/221.17 reduce(b(x_1)) -> check_b(redex_b(x_1)) 471.35/221.17 redex_f(f(f(x))) -> result_f(c) 471.35/221.17 redex_b(u(x)) -> result_b(b(f(x))) 471.35/221.17 redex_f(g(x)) -> result_f(f(f(x))) 471.35/221.17 redex_f(b(x)) -> result_f(u(g(b(x)))) 471.35/221.17 redex_f(u(x)) -> result_f(u(g(x))) 471.35/221.17 check_f(result_f(x)) -> go_up(x) 471.35/221.17 check_b(result_b(x)) -> go_up(x) 471.35/221.17 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 471.35/221.17 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 471.35/221.17 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 471.35/221.17 reduce(g(x_1)) -> in_g_1(reduce(x_1)) 471.35/221.17 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 471.35/221.17 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 471.35/221.17 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 471.35/221.17 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 471.35/221.17 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 top(go_up(x0)) 471.35/221.17 reduce(f(x0)) 471.35/221.17 reduce(b(x0)) 471.35/221.17 redex_f(f(f(x0))) 471.35/221.17 redex_b(u(x0)) 471.35/221.17 redex_f(g(x0)) 471.35/221.17 redex_f(b(x0)) 471.35/221.17 redex_f(u(x0)) 471.35/221.17 check_f(result_f(x0)) 471.35/221.17 check_b(result_b(x0)) 471.35/221.17 check_f(redex_f(x0)) 471.35/221.17 check_b(redex_b(x0)) 471.35/221.17 reduce(u(x0)) 471.35/221.17 reduce(g(x0)) 471.35/221.17 in_f_1(go_up(x0)) 471.35/221.17 in_b_1(go_up(x0)) 471.35/221.17 in_u_1(go_up(x0)) 471.35/221.17 in_g_1(go_up(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (166) UsableRulesProof (EQUIVALENT) 471.35/221.17 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. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (167) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 TOP(go_up(x)) -> TOP(reduce(x)) 471.35/221.17 471.35/221.17 The TRS R consists of the following rules: 471.35/221.17 471.35/221.17 reduce(f(x_1)) -> check_f(redex_f(x_1)) 471.35/221.17 reduce(b(x_1)) -> check_b(redex_b(x_1)) 471.35/221.17 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 471.35/221.17 reduce(g(x_1)) -> in_g_1(reduce(x_1)) 471.35/221.17 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 471.35/221.17 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 471.35/221.17 redex_b(u(x)) -> result_b(b(f(x))) 471.35/221.17 check_b(result_b(x)) -> go_up(x) 471.35/221.17 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 471.35/221.17 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 471.35/221.17 redex_f(f(f(x))) -> result_f(c) 471.35/221.17 redex_f(g(x)) -> result_f(f(f(x))) 471.35/221.17 redex_f(b(x)) -> result_f(u(g(b(x)))) 471.35/221.17 redex_f(u(x)) -> result_f(u(g(x))) 471.35/221.17 check_f(result_f(x)) -> go_up(x) 471.35/221.17 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 471.35/221.17 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 471.35/221.17 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 top(go_up(x0)) 471.35/221.17 reduce(f(x0)) 471.35/221.17 reduce(b(x0)) 471.35/221.17 redex_f(f(f(x0))) 471.35/221.17 redex_b(u(x0)) 471.35/221.17 redex_f(g(x0)) 471.35/221.17 redex_f(b(x0)) 471.35/221.17 redex_f(u(x0)) 471.35/221.17 check_f(result_f(x0)) 471.35/221.17 check_b(result_b(x0)) 471.35/221.17 check_f(redex_f(x0)) 471.35/221.17 check_b(redex_b(x0)) 471.35/221.17 reduce(u(x0)) 471.35/221.17 reduce(g(x0)) 471.35/221.17 in_f_1(go_up(x0)) 471.35/221.17 in_b_1(go_up(x0)) 471.35/221.17 in_u_1(go_up(x0)) 471.35/221.17 in_g_1(go_up(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (168) QReductionProof (EQUIVALENT) 471.35/221.17 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 471.35/221.17 471.35/221.17 top(go_up(x0)) 471.35/221.17 471.35/221.17 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (169) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 TOP(go_up(x)) -> TOP(reduce(x)) 471.35/221.17 471.35/221.17 The TRS R consists of the following rules: 471.35/221.17 471.35/221.17 reduce(f(x_1)) -> check_f(redex_f(x_1)) 471.35/221.17 reduce(b(x_1)) -> check_b(redex_b(x_1)) 471.35/221.17 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 471.35/221.17 reduce(g(x_1)) -> in_g_1(reduce(x_1)) 471.35/221.17 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 471.35/221.17 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 471.35/221.17 redex_b(u(x)) -> result_b(b(f(x))) 471.35/221.17 check_b(result_b(x)) -> go_up(x) 471.35/221.17 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 471.35/221.17 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 471.35/221.17 redex_f(f(f(x))) -> result_f(c) 471.35/221.17 redex_f(g(x)) -> result_f(f(f(x))) 471.35/221.17 redex_f(b(x)) -> result_f(u(g(b(x)))) 471.35/221.17 redex_f(u(x)) -> result_f(u(g(x))) 471.35/221.17 check_f(result_f(x)) -> go_up(x) 471.35/221.17 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 471.35/221.17 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 471.35/221.17 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 reduce(f(x0)) 471.35/221.17 reduce(b(x0)) 471.35/221.17 redex_f(f(f(x0))) 471.35/221.17 redex_b(u(x0)) 471.35/221.17 redex_f(g(x0)) 471.35/221.17 redex_f(b(x0)) 471.35/221.17 redex_f(u(x0)) 471.35/221.17 check_f(result_f(x0)) 471.35/221.17 check_b(result_b(x0)) 471.35/221.17 check_f(redex_f(x0)) 471.35/221.17 check_b(redex_b(x0)) 471.35/221.17 reduce(u(x0)) 471.35/221.17 reduce(g(x0)) 471.35/221.17 in_f_1(go_up(x0)) 471.35/221.17 in_b_1(go_up(x0)) 471.35/221.17 in_u_1(go_up(x0)) 471.35/221.17 in_g_1(go_up(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (170) TransformationProof (EQUIVALENT) 471.35/221.17 By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]: 471.35/221.17 471.35/221.17 (TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))),TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))) 471.35/221.17 (TOP(go_up(b(x0))) -> TOP(check_b(redex_b(x0))),TOP(go_up(b(x0))) -> TOP(check_b(redex_b(x0)))) 471.35/221.17 (TOP(go_up(u(x0))) -> TOP(in_u_1(reduce(x0))),TOP(go_up(u(x0))) -> TOP(in_u_1(reduce(x0)))) 471.35/221.17 (TOP(go_up(g(x0))) -> TOP(in_g_1(reduce(x0))),TOP(go_up(g(x0))) -> TOP(in_g_1(reduce(x0)))) 471.35/221.17 471.35/221.17 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (171) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 471.35/221.17 TOP(go_up(b(x0))) -> TOP(check_b(redex_b(x0))) 471.35/221.17 TOP(go_up(u(x0))) -> TOP(in_u_1(reduce(x0))) 471.35/221.17 TOP(go_up(g(x0))) -> TOP(in_g_1(reduce(x0))) 471.35/221.17 471.35/221.17 The TRS R consists of the following rules: 471.35/221.17 471.35/221.17 reduce(f(x_1)) -> check_f(redex_f(x_1)) 471.35/221.17 reduce(b(x_1)) -> check_b(redex_b(x_1)) 471.35/221.17 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 471.35/221.17 reduce(g(x_1)) -> in_g_1(reduce(x_1)) 471.35/221.17 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 471.35/221.17 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 471.35/221.17 redex_b(u(x)) -> result_b(b(f(x))) 471.35/221.17 check_b(result_b(x)) -> go_up(x) 471.35/221.17 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 471.35/221.17 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 471.35/221.17 redex_f(f(f(x))) -> result_f(c) 471.35/221.17 redex_f(g(x)) -> result_f(f(f(x))) 471.35/221.17 redex_f(b(x)) -> result_f(u(g(b(x)))) 471.35/221.17 redex_f(u(x)) -> result_f(u(g(x))) 471.35/221.17 check_f(result_f(x)) -> go_up(x) 471.35/221.17 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 471.35/221.17 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 471.35/221.17 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 reduce(f(x0)) 471.35/221.17 reduce(b(x0)) 471.35/221.17 redex_f(f(f(x0))) 471.35/221.17 redex_b(u(x0)) 471.35/221.17 redex_f(g(x0)) 471.35/221.17 redex_f(b(x0)) 471.35/221.17 redex_f(u(x0)) 471.35/221.17 check_f(result_f(x0)) 471.35/221.17 check_b(result_b(x0)) 471.35/221.17 check_f(redex_f(x0)) 471.35/221.17 check_b(redex_b(x0)) 471.35/221.17 reduce(u(x0)) 471.35/221.17 reduce(g(x0)) 471.35/221.17 in_f_1(go_up(x0)) 471.35/221.17 in_b_1(go_up(x0)) 471.35/221.17 in_u_1(go_up(x0)) 471.35/221.17 in_g_1(go_up(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (172) UsableRulesProof (EQUIVALENT) 471.35/221.17 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. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (173) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 TOP(go_up(x)) -> TOP(reduce(x)) 471.35/221.17 471.35/221.17 The TRS R consists of the following rules: 471.35/221.17 471.35/221.17 reduce(f(x_1)) -> check_f(redex_f(x_1)) 471.35/221.17 reduce(b(x_1)) -> check_b(redex_b(x_1)) 471.35/221.17 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 471.35/221.17 reduce(g(x_1)) -> in_g_1(reduce(x_1)) 471.35/221.17 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 471.35/221.17 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 471.35/221.17 redex_b(u(x)) -> result_b(b(f(x))) 471.35/221.17 check_b(result_b(x)) -> go_up(x) 471.35/221.17 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 471.35/221.17 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 471.35/221.17 redex_f(f(f(x))) -> result_f(c) 471.35/221.17 redex_f(g(x)) -> result_f(f(f(x))) 471.35/221.17 redex_f(b(x)) -> result_f(u(g(b(x)))) 471.35/221.17 redex_f(u(x)) -> result_f(u(g(x))) 471.35/221.17 check_f(result_f(x)) -> go_up(x) 471.35/221.17 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 471.35/221.17 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 471.35/221.17 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 top(go_up(x0)) 471.35/221.17 reduce(f(x0)) 471.35/221.17 reduce(b(x0)) 471.35/221.17 redex_f(f(f(x0))) 471.35/221.17 redex_b(u(x0)) 471.35/221.17 redex_f(g(x0)) 471.35/221.17 redex_f(b(x0)) 471.35/221.17 redex_f(u(x0)) 471.35/221.17 check_f(result_f(x0)) 471.35/221.17 check_b(result_b(x0)) 471.35/221.17 check_f(redex_f(x0)) 471.35/221.17 check_b(redex_b(x0)) 471.35/221.17 reduce(u(x0)) 471.35/221.17 reduce(g(x0)) 471.35/221.17 in_f_1(go_up(x0)) 471.35/221.17 in_b_1(go_up(x0)) 471.35/221.17 in_u_1(go_up(x0)) 471.35/221.17 in_g_1(go_up(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (174) QReductionProof (EQUIVALENT) 471.35/221.17 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 471.35/221.17 471.35/221.17 top(go_up(x0)) 471.35/221.17 471.35/221.17 471.35/221.17 ---------------------------------------- 471.35/221.17 471.35/221.17 (175) 471.35/221.17 Obligation: 471.35/221.17 Q DP problem: 471.35/221.17 The TRS P consists of the following rules: 471.35/221.17 471.35/221.17 TOP(go_up(x)) -> TOP(reduce(x)) 471.35/221.17 471.35/221.17 The TRS R consists of the following rules: 471.35/221.17 471.35/221.17 reduce(f(x_1)) -> check_f(redex_f(x_1)) 471.35/221.17 reduce(b(x_1)) -> check_b(redex_b(x_1)) 471.35/221.17 reduce(u(x_1)) -> in_u_1(reduce(x_1)) 471.35/221.17 reduce(g(x_1)) -> in_g_1(reduce(x_1)) 471.35/221.17 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 471.35/221.17 in_u_1(go_up(x_1)) -> go_up(u(x_1)) 471.35/221.17 redex_b(u(x)) -> result_b(b(f(x))) 471.35/221.17 check_b(result_b(x)) -> go_up(x) 471.35/221.17 check_b(redex_b(x_1)) -> in_b_1(reduce(x_1)) 471.35/221.17 in_b_1(go_up(x_1)) -> go_up(b(x_1)) 471.35/221.17 redex_f(f(f(x))) -> result_f(c) 471.35/221.17 redex_f(g(x)) -> result_f(f(f(x))) 471.35/221.17 redex_f(b(x)) -> result_f(u(g(b(x)))) 471.35/221.17 redex_f(u(x)) -> result_f(u(g(x))) 471.35/221.17 check_f(result_f(x)) -> go_up(x) 471.35/221.17 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 471.35/221.17 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 471.35/221.17 471.35/221.17 The set Q consists of the following terms: 471.35/221.17 471.35/221.17 reduce(f(x0)) 471.35/221.17 reduce(b(x0)) 471.35/221.17 redex_f(f(f(x0))) 471.35/221.17 redex_b(u(x0)) 471.35/221.17 redex_f(g(x0)) 471.35/221.17 redex_f(b(x0)) 471.35/221.17 redex_f(u(x0)) 471.35/221.17 check_f(result_f(x0)) 471.35/221.17 check_b(result_b(x0)) 471.35/221.17 check_f(redex_f(x0)) 471.35/221.17 check_b(redex_b(x0)) 471.35/221.17 reduce(u(x0)) 471.35/221.17 reduce(g(x0)) 471.35/221.17 in_f_1(go_up(x0)) 471.35/221.17 in_b_1(go_up(x0)) 471.35/221.17 in_u_1(go_up(x0)) 471.35/221.17 in_g_1(go_up(x0)) 471.35/221.17 471.35/221.17 We have to consider all minimal (P,Q,R)-chains. 471.49/221.26 EOF