274.75/150.94 YES 274.75/150.95 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 274.75/150.95 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 274.75/150.95 274.75/150.95 274.75/150.95 Outermost Termination of the given OTRS could be proven: 274.75/150.95 274.75/150.95 (0) OTRS 274.75/150.95 (1) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] 274.75/150.95 (2) QTRS 274.75/150.95 (3) QTRSRRRProof [EQUIVALENT, 100 ms] 274.75/150.95 (4) QTRS 274.75/150.95 (5) AAECC Innermost [EQUIVALENT, 13 ms] 274.75/150.95 (6) QTRS 274.75/150.95 (7) DependencyPairsProof [EQUIVALENT, 0 ms] 274.75/150.95 (8) QDP 274.75/150.95 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 274.75/150.95 (10) AND 274.75/150.95 (11) QDP 274.75/150.95 (12) UsableRulesProof [EQUIVALENT, 0 ms] 274.75/150.95 (13) QDP 274.75/150.95 (14) QReductionProof [EQUIVALENT, 0 ms] 274.75/150.95 (15) QDP 274.75/150.95 (16) QDPSizeChangeProof [EQUIVALENT, 0 ms] 274.75/150.95 (17) YES 274.75/150.95 (18) QDP 274.75/150.95 (19) UsableRulesProof [EQUIVALENT, 0 ms] 274.75/150.95 (20) QDP 274.75/150.95 (21) QReductionProof [EQUIVALENT, 0 ms] 274.75/150.95 (22) QDP 274.75/150.95 (23) TransformationProof [EQUIVALENT, 0 ms] 274.75/150.95 (24) QDP 274.75/150.95 (25) DependencyGraphProof [EQUIVALENT, 0 ms] 274.75/150.95 (26) QDP 274.75/150.95 (27) UsableRulesProof [EQUIVALENT, 0 ms] 274.75/150.95 (28) QDP 274.75/150.95 (29) TransformationProof [EQUIVALENT, 0 ms] 274.75/150.95 (30) QDP 274.75/150.95 (31) TransformationProof [EQUIVALENT, 0 ms] 274.75/150.95 (32) QDP 274.75/150.95 (33) DependencyGraphProof [EQUIVALENT, 0 ms] 274.75/150.95 (34) QDP 274.75/150.95 (35) TransformationProof [EQUIVALENT, 0 ms] 274.75/150.95 (36) QDP 274.75/150.95 (37) DependencyGraphProof [EQUIVALENT, 0 ms] 274.75/150.95 (38) QDP 274.75/150.95 (39) TransformationProof [EQUIVALENT, 0 ms] 274.75/150.95 (40) QDP 274.75/150.95 (41) TransformationProof [EQUIVALENT, 0 ms] 274.75/150.95 (42) QDP 274.75/150.95 (43) DependencyGraphProof [EQUIVALENT, 0 ms] 274.75/150.95 (44) QDP 274.75/150.95 (45) TransformationProof [EQUIVALENT, 0 ms] 274.75/150.95 (46) QDP 274.75/150.95 (47) DependencyGraphProof [EQUIVALENT, 0 ms] 274.75/150.95 (48) QDP 274.75/150.95 (49) QDPOrderProof [EQUIVALENT, 1626 ms] 274.75/150.95 (50) QDP 274.75/150.95 (51) SplitQDPProof [EQUIVALENT, 0 ms] 274.75/150.95 (52) AND 274.75/150.95 (53) QDP 274.75/150.95 (54) SemLabProof [SOUND, 0 ms] 274.75/150.95 (55) QDP 274.75/150.95 (56) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 274.75/150.95 (57) QDP 274.75/150.95 (58) MRRProof [EQUIVALENT, 0 ms] 274.75/150.95 (59) QDP 274.75/150.95 (60) DependencyGraphProof [EQUIVALENT, 0 ms] 274.75/150.95 (61) QDP 274.75/150.95 (62) MRRProof [EQUIVALENT, 5 ms] 274.75/150.95 (63) QDP 274.75/150.95 (64) QDPOrderProof [EQUIVALENT, 0 ms] 274.75/150.95 (65) QDP 274.75/150.95 (66) QDPOrderProof [EQUIVALENT, 8 ms] 274.75/150.95 (67) QDP 274.75/150.95 (68) PisEmptyProof [SOUND, 0 ms] 274.75/150.95 (69) TRUE 274.75/150.95 (70) QDP 274.75/150.95 (71) SplitQDPProof [EQUIVALENT, 0 ms] 274.75/150.95 (72) AND 274.75/150.95 (73) QDP 274.75/150.95 (74) SemLabProof [SOUND, 0 ms] 274.75/150.95 (75) QDP 274.75/150.95 (76) UsableRulesReductionPairsProof [EQUIVALENT, 6 ms] 274.75/150.95 (77) QDP 274.75/150.95 (78) MRRProof [EQUIVALENT, 0 ms] 274.75/150.95 (79) QDP 274.75/150.95 (80) DependencyGraphProof [EQUIVALENT, 0 ms] 274.75/150.95 (81) QDP 274.75/150.95 (82) QDPOrderProof [EQUIVALENT, 8 ms] 274.75/150.95 (83) QDP 274.75/150.95 (84) QDPOrderProof [EQUIVALENT, 0 ms] 274.75/150.95 (85) QDP 274.75/150.95 (86) PisEmptyProof [SOUND, 0 ms] 274.75/150.95 (87) TRUE 274.75/150.95 (88) QDP 274.75/150.95 (89) SplitQDPProof [EQUIVALENT, 0 ms] 274.75/150.95 (90) AND 274.75/150.95 (91) QDP 274.75/150.95 (92) SemLabProof [SOUND, 0 ms] 274.75/150.95 (93) QDP 274.75/150.95 (94) MRRProof [EQUIVALENT, 0 ms] 274.75/150.95 (95) QDP 274.75/150.95 (96) DependencyGraphProof [EQUIVALENT, 0 ms] 274.75/150.95 (97) QDP 274.75/150.95 (98) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 274.75/150.95 (99) QDP 274.75/150.95 (100) MRRProof [EQUIVALENT, 0 ms] 274.75/150.95 (101) QDP 274.75/150.95 (102) PisEmptyProof [SOUND, 0 ms] 274.75/150.95 (103) TRUE 274.75/150.95 (104) QDP 274.75/150.95 (105) SplitQDPProof [EQUIVALENT, 0 ms] 274.75/150.95 (106) AND 274.75/150.95 (107) QDP 274.75/150.95 (108) SemLabProof [SOUND, 0 ms] 274.75/150.95 (109) QDP 274.75/150.95 (110) MRRProof [EQUIVALENT, 0 ms] 274.75/150.95 (111) QDP 274.75/150.95 (112) DependencyGraphProof [EQUIVALENT, 0 ms] 274.75/150.95 (113) QDP 274.75/150.95 (114) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 274.75/150.95 (115) QDP 274.75/150.95 (116) MRRProof [EQUIVALENT, 0 ms] 274.75/150.95 (117) QDP 274.75/150.95 (118) PisEmptyProof [SOUND, 0 ms] 274.75/150.95 (119) TRUE 274.75/150.95 (120) QDP 274.75/150.95 (121) SemLabProof [SOUND, 4526 ms] 274.75/150.95 (122) QDP 274.75/150.95 (123) DependencyGraphProof [EQUIVALENT, 0 ms] 274.75/150.95 (124) AND 274.75/150.95 (125) QDP 274.75/150.95 (126) MRRProof [EQUIVALENT, 0 ms] 274.75/150.95 (127) QDP 274.75/150.95 (128) MRRProof [EQUIVALENT, 0 ms] 274.75/150.95 (129) QDP 274.75/150.95 (130) MRRProof [EQUIVALENT, 0 ms] 274.75/150.95 (131) QDP 274.75/150.95 (132) PisEmptyProof [EQUIVALENT, 0 ms] 274.75/150.95 (133) YES 274.75/150.95 (134) QDP 274.75/150.95 (135) MRRProof [EQUIVALENT, 0 ms] 274.75/150.95 (136) QDP 274.75/150.95 (137) MRRProof [EQUIVALENT, 4 ms] 274.75/150.95 (138) QDP 274.75/150.95 (139) MRRProof [EQUIVALENT, 0 ms] 274.75/150.95 (140) QDP 274.75/150.95 (141) PisEmptyProof [EQUIVALENT, 0 ms] 274.75/150.95 (142) YES 274.75/150.95 274.75/150.95 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (0) 274.75/150.95 Obligation: 274.75/150.95 Term rewrite system R: 274.75/150.95 The TRS R consists of the following rules: 274.75/150.95 274.75/150.95 g(b) -> g(f(f(f(f(f(b)))))) 274.75/150.95 f(f(b)) -> b 274.75/150.95 f(b) -> b 274.75/150.95 g(f(b)) -> c 274.75/150.95 274.75/150.95 274.75/150.95 274.75/150.95 Outermost Strategy. 274.75/150.95 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (1) Raffelsieper-Zantema-Transformation (SOUND) 274.75/150.95 We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (2) 274.75/150.95 Obligation: 274.75/150.95 Q restricted rewrite system: 274.75/150.95 The TRS R consists of the following rules: 274.75/150.95 274.75/150.95 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.95 down(f(f(b))) -> up(b) 274.75/150.95 down(f(b)) -> up(b) 274.75/150.95 down(g(f(b))) -> up(c) 274.75/150.95 top(up(x)) -> top(down(x)) 274.75/150.95 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.95 down(g(c)) -> g_flat(down(c)) 274.75/150.95 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.95 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.95 down(f(c)) -> f_flat(down(c)) 274.75/150.95 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.95 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.95 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.95 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.95 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.95 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.95 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.95 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.95 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.95 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.95 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.95 274.75/150.95 Q is empty. 274.75/150.95 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (3) QTRSRRRProof (EQUIVALENT) 274.75/150.95 Used ordering: 274.75/150.95 Polynomial interpretation [POLO]: 274.75/150.95 274.75/150.95 POL(b) = 0 274.75/150.95 POL(c) = 0 274.75/150.95 POL(down(x_1)) = 2*x_1 274.75/150.95 POL(f(x_1)) = x_1 274.75/150.95 POL(f_flat(x_1)) = x_1 274.75/150.95 POL(fresh_constant) = 0 274.75/150.95 POL(g(x_1)) = 1 + x_1 274.75/150.95 POL(g_flat(x_1)) = 2 + x_1 274.75/150.95 POL(top(x_1)) = 2*x_1 274.75/150.95 POL(up(x_1)) = 2*x_1 274.75/150.95 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 274.75/150.95 274.75/150.95 down(g(f(b))) -> up(c) 274.75/150.95 274.75/150.95 274.75/150.95 274.75/150.95 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (4) 274.75/150.95 Obligation: 274.75/150.95 Q restricted rewrite system: 274.75/150.95 The TRS R consists of the following rules: 274.75/150.95 274.75/150.95 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.95 down(f(f(b))) -> up(b) 274.75/150.95 down(f(b)) -> up(b) 274.75/150.95 top(up(x)) -> top(down(x)) 274.75/150.95 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.95 down(g(c)) -> g_flat(down(c)) 274.75/150.95 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.95 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.95 down(f(c)) -> f_flat(down(c)) 274.75/150.95 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.95 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.95 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.95 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.95 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.95 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.95 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.95 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.95 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.95 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.95 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.95 274.75/150.95 Q is empty. 274.75/150.95 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (5) AAECC Innermost (EQUIVALENT) 274.75/150.95 We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 274.75/150.95 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.95 down(g(c)) -> g_flat(down(c)) 274.75/150.95 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.95 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.95 down(f(c)) -> f_flat(down(c)) 274.75/150.95 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.95 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.95 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.95 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.95 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.95 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.95 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.95 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.95 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.95 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.95 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.95 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.95 down(f(f(b))) -> up(b) 274.75/150.95 down(f(b)) -> up(b) 274.75/150.95 274.75/150.95 The TRS R 2 is 274.75/150.95 top(up(x)) -> top(down(x)) 274.75/150.95 274.75/150.95 The signature Sigma is {top_1} 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (6) 274.75/150.95 Obligation: 274.75/150.95 Q restricted rewrite system: 274.75/150.95 The TRS R consists of the following rules: 274.75/150.95 274.75/150.95 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.95 down(f(f(b))) -> up(b) 274.75/150.95 down(f(b)) -> up(b) 274.75/150.95 top(up(x)) -> top(down(x)) 274.75/150.95 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.95 down(g(c)) -> g_flat(down(c)) 274.75/150.95 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.95 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.95 down(f(c)) -> f_flat(down(c)) 274.75/150.95 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.95 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.95 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.95 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.95 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.95 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.95 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.95 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.95 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.95 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.95 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.95 274.75/150.95 The set Q consists of the following terms: 274.75/150.95 274.75/150.95 down(g(b)) 274.75/150.95 down(f(f(b))) 274.75/150.95 down(f(b)) 274.75/150.95 top(up(x0)) 274.75/150.95 down(g(g(x0))) 274.75/150.95 down(g(c)) 274.75/150.95 down(g(fresh_constant)) 274.75/150.95 down(f(g(x0))) 274.75/150.95 down(f(c)) 274.75/150.95 down(f(fresh_constant)) 274.75/150.95 down(g(f(g(x0)))) 274.75/150.95 down(g(f(f(x0)))) 274.75/150.95 down(g(f(c))) 274.75/150.95 down(g(f(fresh_constant))) 274.75/150.95 down(f(f(g(x0)))) 274.75/150.95 down(f(f(f(x0)))) 274.75/150.95 down(f(f(c))) 274.75/150.95 down(f(f(fresh_constant))) 274.75/150.95 g_flat(up(x0)) 274.75/150.95 f_flat(up(x0)) 274.75/150.95 274.75/150.95 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (7) DependencyPairsProof (EQUIVALENT) 274.75/150.95 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (8) 274.75/150.95 Obligation: 274.75/150.95 Q DP problem: 274.75/150.95 The TRS P consists of the following rules: 274.75/150.95 274.75/150.95 TOP(up(x)) -> TOP(down(x)) 274.75/150.95 TOP(up(x)) -> DOWN(x) 274.75/150.95 DOWN(g(g(y3))) -> G_FLAT(down(g(y3))) 274.75/150.95 DOWN(g(g(y3))) -> DOWN(g(y3)) 274.75/150.95 DOWN(g(c)) -> G_FLAT(down(c)) 274.75/150.95 DOWN(g(c)) -> DOWN(c) 274.75/150.95 DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant)) 274.75/150.95 DOWN(g(fresh_constant)) -> DOWN(fresh_constant) 274.75/150.95 DOWN(f(g(y6))) -> F_FLAT(down(g(y6))) 274.75/150.95 DOWN(f(g(y6))) -> DOWN(g(y6)) 274.75/150.95 DOWN(f(c)) -> F_FLAT(down(c)) 274.75/150.95 DOWN(f(c)) -> DOWN(c) 274.75/150.95 DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant)) 274.75/150.95 DOWN(f(fresh_constant)) -> DOWN(fresh_constant) 274.75/150.95 DOWN(g(f(g(y9)))) -> G_FLAT(down(f(g(y9)))) 274.75/150.95 DOWN(g(f(g(y9)))) -> DOWN(f(g(y9))) 274.75/150.95 DOWN(g(f(f(y10)))) -> G_FLAT(down(f(f(y10)))) 274.75/150.95 DOWN(g(f(f(y10)))) -> DOWN(f(f(y10))) 274.75/150.95 DOWN(g(f(c))) -> G_FLAT(down(f(c))) 274.75/150.95 DOWN(g(f(c))) -> DOWN(f(c)) 274.75/150.95 DOWN(g(f(fresh_constant))) -> G_FLAT(down(f(fresh_constant))) 274.75/150.95 DOWN(g(f(fresh_constant))) -> DOWN(f(fresh_constant)) 274.75/150.95 DOWN(f(f(g(y12)))) -> F_FLAT(down(f(g(y12)))) 274.75/150.95 DOWN(f(f(g(y12)))) -> DOWN(f(g(y12))) 274.75/150.95 DOWN(f(f(f(y13)))) -> F_FLAT(down(f(f(y13)))) 274.75/150.95 DOWN(f(f(f(y13)))) -> DOWN(f(f(y13))) 274.75/150.95 DOWN(f(f(c))) -> F_FLAT(down(f(c))) 274.75/150.95 DOWN(f(f(c))) -> DOWN(f(c)) 274.75/150.95 DOWN(f(f(fresh_constant))) -> F_FLAT(down(f(fresh_constant))) 274.75/150.95 DOWN(f(f(fresh_constant))) -> DOWN(f(fresh_constant)) 274.75/150.95 274.75/150.95 The TRS R consists of the following rules: 274.75/150.95 274.75/150.95 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.95 down(f(f(b))) -> up(b) 274.75/150.95 down(f(b)) -> up(b) 274.75/150.95 top(up(x)) -> top(down(x)) 274.75/150.95 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.95 down(g(c)) -> g_flat(down(c)) 274.75/150.95 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.95 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.95 down(f(c)) -> f_flat(down(c)) 274.75/150.95 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.95 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.95 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.95 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.95 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.95 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.95 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.95 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.95 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.95 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.95 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.95 274.75/150.95 The set Q consists of the following terms: 274.75/150.95 274.75/150.95 down(g(b)) 274.75/150.95 down(f(f(b))) 274.75/150.95 down(f(b)) 274.75/150.95 top(up(x0)) 274.75/150.95 down(g(g(x0))) 274.75/150.95 down(g(c)) 274.75/150.95 down(g(fresh_constant)) 274.75/150.95 down(f(g(x0))) 274.75/150.95 down(f(c)) 274.75/150.95 down(f(fresh_constant)) 274.75/150.95 down(g(f(g(x0)))) 274.75/150.95 down(g(f(f(x0)))) 274.75/150.95 down(g(f(c))) 274.75/150.95 down(g(f(fresh_constant))) 274.75/150.95 down(f(f(g(x0)))) 274.75/150.95 down(f(f(f(x0)))) 274.75/150.95 down(f(f(c))) 274.75/150.95 down(f(f(fresh_constant))) 274.75/150.95 g_flat(up(x0)) 274.75/150.95 f_flat(up(x0)) 274.75/150.95 274.75/150.95 We have to consider all minimal (P,Q,R)-chains. 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (9) DependencyGraphProof (EQUIVALENT) 274.75/150.95 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 23 less nodes. 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (10) 274.75/150.95 Complex Obligation (AND) 274.75/150.95 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (11) 274.75/150.95 Obligation: 274.75/150.95 Q DP problem: 274.75/150.95 The TRS P consists of the following rules: 274.75/150.95 274.75/150.95 DOWN(g(f(g(y9)))) -> DOWN(f(g(y9))) 274.75/150.95 DOWN(f(g(y6))) -> DOWN(g(y6)) 274.75/150.95 DOWN(g(g(y3))) -> DOWN(g(y3)) 274.75/150.95 DOWN(g(f(f(y10)))) -> DOWN(f(f(y10))) 274.75/150.95 DOWN(f(f(g(y12)))) -> DOWN(f(g(y12))) 274.75/150.95 DOWN(f(f(f(y13)))) -> DOWN(f(f(y13))) 274.75/150.95 274.75/150.95 The TRS R consists of the following rules: 274.75/150.95 274.75/150.95 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.95 down(f(f(b))) -> up(b) 274.75/150.95 down(f(b)) -> up(b) 274.75/150.95 top(up(x)) -> top(down(x)) 274.75/150.95 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.95 down(g(c)) -> g_flat(down(c)) 274.75/150.95 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.95 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.95 down(f(c)) -> f_flat(down(c)) 274.75/150.95 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.95 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.95 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.95 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.95 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.95 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.95 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.95 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.95 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.95 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.95 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.95 274.75/150.95 The set Q consists of the following terms: 274.75/150.95 274.75/150.95 down(g(b)) 274.75/150.95 down(f(f(b))) 274.75/150.95 down(f(b)) 274.75/150.95 top(up(x0)) 274.75/150.95 down(g(g(x0))) 274.75/150.95 down(g(c)) 274.75/150.95 down(g(fresh_constant)) 274.75/150.95 down(f(g(x0))) 274.75/150.95 down(f(c)) 274.75/150.95 down(f(fresh_constant)) 274.75/150.95 down(g(f(g(x0)))) 274.75/150.95 down(g(f(f(x0)))) 274.75/150.95 down(g(f(c))) 274.75/150.95 down(g(f(fresh_constant))) 274.75/150.95 down(f(f(g(x0)))) 274.75/150.95 down(f(f(f(x0)))) 274.75/150.95 down(f(f(c))) 274.75/150.95 down(f(f(fresh_constant))) 274.75/150.95 g_flat(up(x0)) 274.75/150.95 f_flat(up(x0)) 274.75/150.95 274.75/150.95 We have to consider all minimal (P,Q,R)-chains. 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (12) UsableRulesProof (EQUIVALENT) 274.75/150.95 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. 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (13) 274.75/150.95 Obligation: 274.75/150.95 Q DP problem: 274.75/150.95 The TRS P consists of the following rules: 274.75/150.95 274.75/150.95 DOWN(g(f(g(y9)))) -> DOWN(f(g(y9))) 274.75/150.95 DOWN(f(g(y6))) -> DOWN(g(y6)) 274.75/150.95 DOWN(g(g(y3))) -> DOWN(g(y3)) 274.75/150.95 DOWN(g(f(f(y10)))) -> DOWN(f(f(y10))) 274.75/150.95 DOWN(f(f(g(y12)))) -> DOWN(f(g(y12))) 274.75/150.95 DOWN(f(f(f(y13)))) -> DOWN(f(f(y13))) 274.75/150.95 274.75/150.95 R is empty. 274.75/150.95 The set Q consists of the following terms: 274.75/150.95 274.75/150.95 down(g(b)) 274.75/150.95 down(f(f(b))) 274.75/150.95 down(f(b)) 274.75/150.95 top(up(x0)) 274.75/150.95 down(g(g(x0))) 274.75/150.95 down(g(c)) 274.75/150.95 down(g(fresh_constant)) 274.75/150.95 down(f(g(x0))) 274.75/150.95 down(f(c)) 274.75/150.95 down(f(fresh_constant)) 274.75/150.95 down(g(f(g(x0)))) 274.75/150.95 down(g(f(f(x0)))) 274.75/150.95 down(g(f(c))) 274.75/150.95 down(g(f(fresh_constant))) 274.75/150.95 down(f(f(g(x0)))) 274.75/150.95 down(f(f(f(x0)))) 274.75/150.95 down(f(f(c))) 274.75/150.95 down(f(f(fresh_constant))) 274.75/150.95 g_flat(up(x0)) 274.75/150.95 f_flat(up(x0)) 274.75/150.95 274.75/150.95 We have to consider all minimal (P,Q,R)-chains. 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (14) QReductionProof (EQUIVALENT) 274.75/150.95 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 274.75/150.95 274.75/150.95 down(g(b)) 274.75/150.95 down(f(f(b))) 274.75/150.95 down(f(b)) 274.75/150.95 top(up(x0)) 274.75/150.95 down(g(g(x0))) 274.75/150.95 down(g(c)) 274.75/150.95 down(g(fresh_constant)) 274.75/150.95 down(f(g(x0))) 274.75/150.95 down(f(c)) 274.75/150.95 down(f(fresh_constant)) 274.75/150.95 down(g(f(g(x0)))) 274.75/150.95 down(g(f(f(x0)))) 274.75/150.95 down(g(f(c))) 274.75/150.95 down(g(f(fresh_constant))) 274.75/150.95 down(f(f(g(x0)))) 274.75/150.95 down(f(f(f(x0)))) 274.75/150.95 down(f(f(c))) 274.75/150.95 down(f(f(fresh_constant))) 274.75/150.95 g_flat(up(x0)) 274.75/150.95 f_flat(up(x0)) 274.75/150.95 274.75/150.95 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (15) 274.75/150.95 Obligation: 274.75/150.95 Q DP problem: 274.75/150.95 The TRS P consists of the following rules: 274.75/150.95 274.75/150.95 DOWN(g(f(g(y9)))) -> DOWN(f(g(y9))) 274.75/150.95 DOWN(f(g(y6))) -> DOWN(g(y6)) 274.75/150.95 DOWN(g(g(y3))) -> DOWN(g(y3)) 274.75/150.95 DOWN(g(f(f(y10)))) -> DOWN(f(f(y10))) 274.75/150.95 DOWN(f(f(g(y12)))) -> DOWN(f(g(y12))) 274.75/150.95 DOWN(f(f(f(y13)))) -> DOWN(f(f(y13))) 274.75/150.95 274.75/150.95 R is empty. 274.75/150.95 Q is empty. 274.75/150.95 We have to consider all minimal (P,Q,R)-chains. 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (16) QDPSizeChangeProof (EQUIVALENT) 274.75/150.95 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. 274.75/150.95 274.75/150.95 From the DPs we obtained the following set of size-change graphs: 274.75/150.95 *DOWN(f(g(y6))) -> DOWN(g(y6)) 274.75/150.95 The graph contains the following edges 1 > 1 274.75/150.95 274.75/150.95 274.75/150.95 *DOWN(g(g(y3))) -> DOWN(g(y3)) 274.75/150.95 The graph contains the following edges 1 > 1 274.75/150.95 274.75/150.95 274.75/150.95 *DOWN(g(f(g(y9)))) -> DOWN(f(g(y9))) 274.75/150.95 The graph contains the following edges 1 > 1 274.75/150.95 274.75/150.95 274.75/150.95 *DOWN(g(f(f(y10)))) -> DOWN(f(f(y10))) 274.75/150.95 The graph contains the following edges 1 > 1 274.75/150.95 274.75/150.95 274.75/150.95 *DOWN(f(f(g(y12)))) -> DOWN(f(g(y12))) 274.75/150.95 The graph contains the following edges 1 > 1 274.75/150.95 274.75/150.95 274.75/150.95 *DOWN(f(f(f(y13)))) -> DOWN(f(f(y13))) 274.75/150.95 The graph contains the following edges 1 > 1 274.75/150.95 274.75/150.95 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (17) 274.75/150.95 YES 274.75/150.95 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (18) 274.75/150.95 Obligation: 274.75/150.95 Q DP problem: 274.75/150.95 The TRS P consists of the following rules: 274.75/150.95 274.75/150.95 TOP(up(x)) -> TOP(down(x)) 274.75/150.95 274.75/150.95 The TRS R consists of the following rules: 274.75/150.95 274.75/150.95 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.95 down(f(f(b))) -> up(b) 274.75/150.95 down(f(b)) -> up(b) 274.75/150.95 top(up(x)) -> top(down(x)) 274.75/150.95 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.95 down(g(c)) -> g_flat(down(c)) 274.75/150.95 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.95 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.95 down(f(c)) -> f_flat(down(c)) 274.75/150.95 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.95 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.95 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.95 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.95 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.95 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.95 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.95 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.95 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.95 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.95 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.95 274.75/150.95 The set Q consists of the following terms: 274.75/150.95 274.75/150.95 down(g(b)) 274.75/150.95 down(f(f(b))) 274.75/150.95 down(f(b)) 274.75/150.95 top(up(x0)) 274.75/150.95 down(g(g(x0))) 274.75/150.95 down(g(c)) 274.75/150.95 down(g(fresh_constant)) 274.75/150.95 down(f(g(x0))) 274.75/150.95 down(f(c)) 274.75/150.95 down(f(fresh_constant)) 274.75/150.95 down(g(f(g(x0)))) 274.75/150.95 down(g(f(f(x0)))) 274.75/150.95 down(g(f(c))) 274.75/150.95 down(g(f(fresh_constant))) 274.75/150.95 down(f(f(g(x0)))) 274.75/150.95 down(f(f(f(x0)))) 274.75/150.95 down(f(f(c))) 274.75/150.95 down(f(f(fresh_constant))) 274.75/150.95 g_flat(up(x0)) 274.75/150.95 f_flat(up(x0)) 274.75/150.95 274.75/150.95 We have to consider all minimal (P,Q,R)-chains. 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (19) UsableRulesProof (EQUIVALENT) 274.75/150.95 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. 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (20) 274.75/150.95 Obligation: 274.75/150.95 Q DP problem: 274.75/150.95 The TRS P consists of the following rules: 274.75/150.95 274.75/150.95 TOP(up(x)) -> TOP(down(x)) 274.75/150.95 274.75/150.95 The TRS R consists of the following rules: 274.75/150.95 274.75/150.95 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.95 down(f(f(b))) -> up(b) 274.75/150.95 down(f(b)) -> up(b) 274.75/150.95 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.95 down(g(c)) -> g_flat(down(c)) 274.75/150.95 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.95 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.95 down(f(c)) -> f_flat(down(c)) 274.75/150.95 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.95 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.95 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.95 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.95 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.95 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.95 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.95 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.95 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.95 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.95 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.95 274.75/150.95 The set Q consists of the following terms: 274.75/150.95 274.75/150.95 down(g(b)) 274.75/150.95 down(f(f(b))) 274.75/150.95 down(f(b)) 274.75/150.95 top(up(x0)) 274.75/150.95 down(g(g(x0))) 274.75/150.95 down(g(c)) 274.75/150.95 down(g(fresh_constant)) 274.75/150.95 down(f(g(x0))) 274.75/150.95 down(f(c)) 274.75/150.95 down(f(fresh_constant)) 274.75/150.95 down(g(f(g(x0)))) 274.75/150.95 down(g(f(f(x0)))) 274.75/150.95 down(g(f(c))) 274.75/150.95 down(g(f(fresh_constant))) 274.75/150.95 down(f(f(g(x0)))) 274.75/150.95 down(f(f(f(x0)))) 274.75/150.95 down(f(f(c))) 274.75/150.95 down(f(f(fresh_constant))) 274.75/150.95 g_flat(up(x0)) 274.75/150.95 f_flat(up(x0)) 274.75/150.95 274.75/150.95 We have to consider all minimal (P,Q,R)-chains. 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (21) QReductionProof (EQUIVALENT) 274.75/150.95 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 274.75/150.95 274.75/150.95 top(up(x0)) 274.75/150.95 274.75/150.95 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (22) 274.75/150.95 Obligation: 274.75/150.95 Q DP problem: 274.75/150.95 The TRS P consists of the following rules: 274.75/150.95 274.75/150.95 TOP(up(x)) -> TOP(down(x)) 274.75/150.95 274.75/150.95 The TRS R consists of the following rules: 274.75/150.95 274.75/150.95 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.95 down(f(f(b))) -> up(b) 274.75/150.95 down(f(b)) -> up(b) 274.75/150.95 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.95 down(g(c)) -> g_flat(down(c)) 274.75/150.95 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.95 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.95 down(f(c)) -> f_flat(down(c)) 274.75/150.95 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.95 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.95 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.95 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.95 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.95 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.95 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.95 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.95 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.95 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.95 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.95 274.75/150.95 The set Q consists of the following terms: 274.75/150.95 274.75/150.95 down(g(b)) 274.75/150.95 down(f(f(b))) 274.75/150.95 down(f(b)) 274.75/150.95 down(g(g(x0))) 274.75/150.95 down(g(c)) 274.75/150.95 down(g(fresh_constant)) 274.75/150.95 down(f(g(x0))) 274.75/150.95 down(f(c)) 274.75/150.95 down(f(fresh_constant)) 274.75/150.95 down(g(f(g(x0)))) 274.75/150.95 down(g(f(f(x0)))) 274.75/150.95 down(g(f(c))) 274.75/150.95 down(g(f(fresh_constant))) 274.75/150.95 down(f(f(g(x0)))) 274.75/150.95 down(f(f(f(x0)))) 274.75/150.95 down(f(f(c))) 274.75/150.95 down(f(f(fresh_constant))) 274.75/150.95 g_flat(up(x0)) 274.75/150.95 f_flat(up(x0)) 274.75/150.95 274.75/150.95 We have to consider all minimal (P,Q,R)-chains. 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (23) TransformationProof (EQUIVALENT) 274.75/150.95 By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]: 274.75/150.95 274.75/150.95 (TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))),TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b))))))))) 274.75/150.95 (TOP(up(f(f(b)))) -> TOP(up(b)),TOP(up(f(f(b)))) -> TOP(up(b))) 274.75/150.95 (TOP(up(f(b))) -> TOP(up(b)),TOP(up(f(b))) -> TOP(up(b))) 274.75/150.95 (TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))),TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))) 274.75/150.95 (TOP(up(g(c))) -> TOP(g_flat(down(c))),TOP(up(g(c))) -> TOP(g_flat(down(c)))) 274.75/150.95 (TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))),TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant)))) 274.75/150.95 (TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))),TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))) 274.75/150.95 (TOP(up(f(c))) -> TOP(f_flat(down(c))),TOP(up(f(c))) -> TOP(f_flat(down(c)))) 274.75/150.95 (TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))),TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant)))) 274.75/150.95 (TOP(up(g(f(g(x0))))) -> TOP(g_flat(down(f(g(x0))))),TOP(up(g(f(g(x0))))) -> TOP(g_flat(down(f(g(x0)))))) 274.75/150.95 (TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))),TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0)))))) 274.75/150.95 (TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c)))),TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c))))) 274.75/150.95 (TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))),TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))) 274.75/150.95 (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)))))) 274.75/150.95 (TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))),TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0)))))) 274.75/150.95 (TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))),TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c))))) 274.75/150.95 (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))))) 274.75/150.95 274.75/150.95 274.75/150.95 ---------------------------------------- 274.75/150.95 274.75/150.95 (24) 274.75/150.95 Obligation: 274.75/150.95 Q DP problem: 274.75/150.95 The TRS P consists of the following rules: 274.75/150.95 274.75/150.95 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.95 TOP(up(f(f(b)))) -> TOP(up(b)) 274.75/150.95 TOP(up(f(b))) -> TOP(up(b)) 274.75/150.95 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.95 TOP(up(g(c))) -> TOP(g_flat(down(c))) 274.75/150.95 TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))) 274.75/150.95 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.95 TOP(up(f(c))) -> TOP(f_flat(down(c))) 274.75/150.95 TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))) 274.75/150.95 TOP(up(g(f(g(x0))))) -> TOP(g_flat(down(f(g(x0))))) 274.75/150.95 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.95 TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c)))) 274.75/150.95 TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))) 274.75/150.95 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 274.75/150.95 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.95 TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) 274.75/150.95 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 274.75/150.95 274.75/150.95 The TRS R consists of the following rules: 274.75/150.95 274.75/150.95 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.95 down(f(f(b))) -> up(b) 274.75/150.95 down(f(b)) -> up(b) 274.75/150.95 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.95 down(g(c)) -> g_flat(down(c)) 274.75/150.95 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.95 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.95 down(f(c)) -> f_flat(down(c)) 274.75/150.95 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.95 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.95 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.95 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.95 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.95 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.95 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.95 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.96 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.96 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.96 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.96 274.75/150.96 The set Q consists of the following terms: 274.75/150.96 274.75/150.96 down(g(b)) 274.75/150.96 down(f(f(b))) 274.75/150.96 down(f(b)) 274.75/150.96 down(g(g(x0))) 274.75/150.96 down(g(c)) 274.75/150.96 down(g(fresh_constant)) 274.75/150.96 down(f(g(x0))) 274.75/150.96 down(f(c)) 274.75/150.96 down(f(fresh_constant)) 274.75/150.96 down(g(f(g(x0)))) 274.75/150.96 down(g(f(f(x0)))) 274.75/150.96 down(g(f(c))) 274.75/150.96 down(g(f(fresh_constant))) 274.75/150.96 down(f(f(g(x0)))) 274.75/150.96 down(f(f(f(x0)))) 274.75/150.96 down(f(f(c))) 274.75/150.96 down(f(f(fresh_constant))) 274.75/150.96 g_flat(up(x0)) 274.75/150.96 f_flat(up(x0)) 274.75/150.96 274.75/150.96 We have to consider all minimal (P,Q,R)-chains. 274.75/150.96 ---------------------------------------- 274.75/150.96 274.75/150.96 (25) DependencyGraphProof (EQUIVALENT) 274.75/150.96 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 274.75/150.96 ---------------------------------------- 274.75/150.96 274.75/150.96 (26) 274.75/150.96 Obligation: 274.75/150.96 Q DP problem: 274.75/150.96 The TRS P consists of the following rules: 274.75/150.96 274.75/150.96 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.96 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.96 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.96 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.96 TOP(up(g(f(g(x0))))) -> TOP(g_flat(down(f(g(x0))))) 274.75/150.96 TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c)))) 274.75/150.96 TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))) 274.75/150.96 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 274.75/150.96 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.96 TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) 274.75/150.96 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 274.75/150.96 274.75/150.96 The TRS R consists of the following rules: 274.75/150.96 274.75/150.96 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.96 down(f(f(b))) -> up(b) 274.75/150.96 down(f(b)) -> up(b) 274.75/150.96 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.96 down(g(c)) -> g_flat(down(c)) 274.75/150.96 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.96 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.96 down(f(c)) -> f_flat(down(c)) 274.75/150.96 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.96 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.96 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.96 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.96 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.96 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.96 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.96 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.96 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.96 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.96 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.96 274.75/150.96 The set Q consists of the following terms: 274.75/150.96 274.75/150.96 down(g(b)) 274.75/150.96 down(f(f(b))) 274.75/150.96 down(f(b)) 274.75/150.96 down(g(g(x0))) 274.75/150.96 down(g(c)) 274.75/150.96 down(g(fresh_constant)) 274.75/150.96 down(f(g(x0))) 274.75/150.96 down(f(c)) 274.75/150.96 down(f(fresh_constant)) 274.75/150.96 down(g(f(g(x0)))) 274.75/150.96 down(g(f(f(x0)))) 274.75/150.96 down(g(f(c))) 274.75/150.96 down(g(f(fresh_constant))) 274.75/150.96 down(f(f(g(x0)))) 274.75/150.96 down(f(f(f(x0)))) 274.75/150.96 down(f(f(c))) 274.75/150.96 down(f(f(fresh_constant))) 274.75/150.96 g_flat(up(x0)) 274.75/150.96 f_flat(up(x0)) 274.75/150.96 274.75/150.96 We have to consider all minimal (P,Q,R)-chains. 274.75/150.96 ---------------------------------------- 274.75/150.96 274.75/150.96 (27) UsableRulesProof (EQUIVALENT) 274.75/150.96 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. 274.75/150.96 ---------------------------------------- 274.75/150.96 274.75/150.96 (28) 274.75/150.96 Obligation: 274.75/150.96 Q DP problem: 274.75/150.96 The TRS P consists of the following rules: 274.75/150.96 274.75/150.96 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.96 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.96 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.96 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.96 TOP(up(g(f(g(x0))))) -> TOP(g_flat(down(f(g(x0))))) 274.75/150.96 TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c)))) 274.75/150.96 TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))) 274.75/150.96 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 274.75/150.96 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.96 TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) 274.75/150.96 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 274.75/150.96 274.75/150.96 The TRS R consists of the following rules: 274.75/150.96 274.75/150.96 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.96 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.96 down(f(c)) -> f_flat(down(c)) 274.75/150.96 down(f(f(b))) -> up(b) 274.75/150.96 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.96 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.96 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.96 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.96 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.96 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.96 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.96 down(g(c)) -> g_flat(down(c)) 274.75/150.96 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.96 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.96 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.96 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.96 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.96 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.96 274.75/150.96 The set Q consists of the following terms: 274.75/150.96 274.75/150.96 down(g(b)) 274.75/150.96 down(f(f(b))) 274.75/150.96 down(f(b)) 274.75/150.96 down(g(g(x0))) 274.75/150.96 down(g(c)) 274.75/150.96 down(g(fresh_constant)) 274.75/150.96 down(f(g(x0))) 274.75/150.96 down(f(c)) 274.75/150.96 down(f(fresh_constant)) 274.75/150.96 down(g(f(g(x0)))) 274.75/150.96 down(g(f(f(x0)))) 274.75/150.96 down(g(f(c))) 274.75/150.96 down(g(f(fresh_constant))) 274.75/150.96 down(f(f(g(x0)))) 274.75/150.96 down(f(f(f(x0)))) 274.75/150.96 down(f(f(c))) 274.75/150.96 down(f(f(fresh_constant))) 274.75/150.96 g_flat(up(x0)) 274.75/150.96 f_flat(up(x0)) 274.75/150.96 274.75/150.96 We have to consider all minimal (P,Q,R)-chains. 274.75/150.96 ---------------------------------------- 274.75/150.96 274.75/150.96 (29) TransformationProof (EQUIVALENT) 274.75/150.96 By rewriting [LPAR04] the rule TOP(up(g(f(g(x0))))) -> TOP(g_flat(down(f(g(x0))))) at position [0,0] we obtained the following new rules [LPAR04]: 274.75/150.96 274.75/150.96 (TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))),TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))) 274.75/150.96 274.75/150.96 274.75/150.96 ---------------------------------------- 274.75/150.96 274.75/150.96 (30) 274.75/150.96 Obligation: 274.75/150.96 Q DP problem: 274.75/150.96 The TRS P consists of the following rules: 274.75/150.96 274.75/150.96 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.96 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.96 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.96 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.96 TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c)))) 274.75/150.96 TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))) 274.75/150.96 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 274.75/150.96 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.96 TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) 274.75/150.96 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 274.75/150.96 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.96 274.75/150.96 The TRS R consists of the following rules: 274.75/150.96 274.75/150.96 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.96 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.96 down(f(c)) -> f_flat(down(c)) 274.75/150.96 down(f(f(b))) -> up(b) 274.75/150.96 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.96 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.96 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.96 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.96 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.96 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.96 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.96 down(g(c)) -> g_flat(down(c)) 274.75/150.96 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.96 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.96 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.96 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.96 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.96 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.96 274.75/150.96 The set Q consists of the following terms: 274.75/150.96 274.75/150.96 down(g(b)) 274.75/150.96 down(f(f(b))) 274.75/150.96 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (31) TransformationProof (EQUIVALENT) 274.75/150.97 By rewriting [LPAR04] the rule TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c)))) at position [0,0] we obtained the following new rules [LPAR04]: 274.75/150.97 274.75/150.97 (TOP(up(g(f(c)))) -> TOP(g_flat(f_flat(down(c)))),TOP(up(g(f(c)))) -> TOP(g_flat(f_flat(down(c))))) 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (32) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.97 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.97 TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))) 274.75/150.97 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 274.75/150.97 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) 274.75/150.97 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 274.75/150.97 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(g(f(c)))) -> TOP(g_flat(f_flat(down(c)))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(f(c)) -> f_flat(down(c)) 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(c)) -> g_flat(down(c)) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down(g(b)) 274.75/150.97 down(f(f(b))) 274.75/150.97 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (33) DependencyGraphProof (EQUIVALENT) 274.75/150.97 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (34) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.97 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.97 TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))) 274.75/150.97 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 274.75/150.97 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) 274.75/150.97 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(f(c)) -> f_flat(down(c)) 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(c)) -> g_flat(down(c)) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down(g(b)) 274.75/150.97 down(f(f(b))) 274.75/150.97 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (35) TransformationProof (EQUIVALENT) 274.75/150.97 By rewriting [LPAR04] the rule TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))) at position [0,0] we obtained the following new rules [LPAR04]: 274.75/150.97 274.75/150.97 (TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(f_flat(down(fresh_constant)))),TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(f_flat(down(fresh_constant))))) 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (36) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.97 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.97 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 274.75/150.97 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) 274.75/150.97 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 274.75/150.97 TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(f_flat(down(fresh_constant)))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(f(c)) -> f_flat(down(c)) 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(c)) -> g_flat(down(c)) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down(g(b)) 274.75/150.97 down(f(f(b))) 274.75/150.97 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (37) DependencyGraphProof (EQUIVALENT) 274.75/150.97 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (38) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.97 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 274.75/150.97 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) 274.75/150.97 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 274.75/150.97 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(f(c)) -> f_flat(down(c)) 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(c)) -> g_flat(down(c)) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down(g(b)) 274.75/150.97 down(f(f(b))) 274.75/150.97 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (39) TransformationProof (EQUIVALENT) 274.75/150.97 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]: 274.75/150.97 274.75/150.97 (TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))),TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))) 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (40) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.97 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) 274.75/150.97 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 274.75/150.97 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(f(c)) -> f_flat(down(c)) 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(c)) -> g_flat(down(c)) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down(g(b)) 274.75/150.97 down(f(f(b))) 274.75/150.97 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (41) TransformationProof (EQUIVALENT) 274.75/150.97 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]: 274.75/150.97 274.75/150.97 (TOP(up(f(f(c)))) -> TOP(f_flat(f_flat(down(c)))),TOP(up(f(f(c)))) -> TOP(f_flat(f_flat(down(c))))) 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (42) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.97 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 274.75/150.97 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(f(f(c)))) -> TOP(f_flat(f_flat(down(c)))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(f(c)) -> f_flat(down(c)) 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(c)) -> g_flat(down(c)) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down(g(b)) 274.75/150.97 down(f(f(b))) 274.75/150.97 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (43) DependencyGraphProof (EQUIVALENT) 274.75/150.97 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (44) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.97 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 274.75/150.97 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(f(c)) -> f_flat(down(c)) 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(c)) -> g_flat(down(c)) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down(g(b)) 274.75/150.97 down(f(f(b))) 274.75/150.97 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (45) TransformationProof (EQUIVALENT) 274.75/150.97 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]: 274.75/150.97 274.75/150.97 (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))))) 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (46) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.97 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant)))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(f(c)) -> f_flat(down(c)) 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(c)) -> g_flat(down(c)) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down(g(b)) 274.75/150.97 down(f(f(b))) 274.75/150.97 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (47) DependencyGraphProof (EQUIVALENT) 274.75/150.97 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (48) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.97 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(f(c)) -> f_flat(down(c)) 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(c)) -> g_flat(down(c)) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down(g(b)) 274.75/150.97 down(f(f(b))) 274.75/150.97 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (49) QDPOrderProof (EQUIVALENT) 274.75/150.97 We use the reduction pair processor [LPAR04,JAR06]. 274.75/150.97 274.75/150.97 274.75/150.97 The following pairs can be oriented strictly and are deleted. 274.75/150.97 274.75/150.97 TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))) 274.75/150.97 The remaining pairs can at least be oriented weakly. 274.75/150.97 Used ordering: Matrix interpretation [MATRO]: 274.75/150.97 274.75/150.97 Non-tuple symbols: 274.75/150.97 <<< 274.75/150.97 M( b ) = [[1], [0]] 274.75/150.97 >>> 274.75/150.97 274.75/150.97 <<< 274.75/150.97 M( c ) = [[0], [0]] 274.75/150.97 >>> 274.75/150.97 274.75/150.97 <<< 274.75/150.97 M( down_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 274.75/150.97 >>> 274.75/150.97 274.75/150.97 <<< 274.75/150.97 M( f_1(x_1) ) = [[0], [0]] + [[0, 1], [1, 0]] * x_1 274.75/150.97 >>> 274.75/150.97 274.75/150.97 <<< 274.75/150.97 M( fresh_constant ) = [[0], [0]] 274.75/150.97 >>> 274.75/150.97 274.75/150.97 <<< 274.75/150.97 M( up_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 274.75/150.97 >>> 274.75/150.97 274.75/150.97 <<< 274.75/150.97 M( f_flat_1(x_1) ) = [[0], [0]] + [[0, 1], [1, 0]] * x_1 274.75/150.97 >>> 274.75/150.97 274.75/150.97 <<< 274.75/150.97 M( g_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 274.75/150.97 >>> 274.75/150.97 274.75/150.97 <<< 274.75/150.97 M( g_flat_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 274.75/150.97 >>> 274.75/150.97 274.75/150.97 Tuple symbols: 274.75/150.97 <<< 274.75/150.97 M( TOP_1(x_1) ) = [[0]] + [[1, 0]] * x_1 274.75/150.97 >>> 274.75/150.97 274.75/150.97 274.75/150.97 274.75/150.97 Matrix type: 274.75/150.97 274.75/150.97 We used a basic matrix type which is not further parametrizeable. 274.75/150.97 274.75/150.97 274.75/150.97 274.75/150.97 274.75/150.97 274.75/150.97 As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. 274.75/150.97 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 274.75/150.97 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(c)) -> g_flat(down(c)) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 down(f(c)) -> f_flat(down(c)) 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (50) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(f(c)) -> f_flat(down(c)) 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(c)) -> g_flat(down(c)) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down(g(b)) 274.75/150.97 down(f(f(b))) 274.75/150.97 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (51) SplitQDPProof (EQUIVALENT) 274.75/150.97 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (52) 274.75/150.97 Complex Obligation (AND) 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (53) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(f(c)) -> f_flat(down(c)) 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(c)) -> g_flat(down(c)) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down(g(b)) 274.75/150.97 down(f(f(b))) 274.75/150.97 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (54) SemLabProof (SOUND) 274.75/150.97 We found the following model for the rules of the TRSs R and P. 274.75/150.97 Interpretation over the domain with elements from 0 to 1. 274.75/150.97 b: 0 274.75/150.97 c: 1 274.75/150.97 down: 0 274.75/150.97 f: 0 274.75/150.97 fresh_constant: 0 274.75/150.97 up: 0 274.75/150.97 f_flat: 0 274.75/150.97 TOP: 0 274.75/150.97 g_flat: 0 274.75/150.97 g: 0 274.75/150.97 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (55) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.)) 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 274.75/150.97 down.0(f.1(c.)) -> f_flat.0(down.1(c.)) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.1(c.)) -> g_flat.0(down.1(c.)) 274.75/150.97 down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.1(c.))) -> g_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) 274.75/150.97 down.0(f.0(f.0(b.))) 274.75/150.97 down.0(f.0(b.)) 274.75/150.97 down.0(g.0(g.0(x0))) 274.75/150.97 down.0(g.0(g.1(x0))) 274.75/150.97 down.0(g.1(c.)) 274.75/150.97 down.0(g.0(fresh_constant.)) 274.75/150.97 down.0(f.0(g.0(x0))) 274.75/150.97 down.0(f.0(g.1(x0))) 274.75/150.97 down.0(f.1(c.)) 274.75/150.97 down.0(f.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(x0)))) 274.75/150.97 down.0(g.0(f.0(g.1(x0)))) 274.75/150.97 down.0(g.0(f.0(f.0(x0)))) 274.75/150.97 down.0(g.0(f.0(f.1(x0)))) 274.75/150.97 down.0(g.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(g.0(x0)))) 274.75/150.97 down.0(f.0(f.0(g.1(x0)))) 274.75/150.97 down.0(f.0(f.0(f.0(x0)))) 274.75/150.97 down.0(f.0(f.0(f.1(x0)))) 274.75/150.97 down.0(f.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x0)) 274.75/150.97 g_flat.0(up.1(x0)) 274.75/150.97 f_flat.0(up.0(x0)) 274.75/150.97 f_flat.0(up.1(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (56) UsableRulesReductionPairsProof (EQUIVALENT) 274.75/150.97 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. 274.75/150.97 274.75/150.97 No dependency pairs are removed. 274.75/150.97 274.75/150.97 The following rules are removed from R: 274.75/150.97 274.75/150.97 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 274.75/150.97 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.97 Used ordering: POLO with Polynomial interpretation [POLO]: 274.75/150.97 274.75/150.97 POL(TOP.0(x_1)) = x_1 274.75/150.97 POL(b.) = 0 274.75/150.97 POL(c.) = 0 274.75/150.97 POL(down.0(x_1)) = 1 + x_1 274.75/150.97 POL(down.1(x_1)) = 1 + x_1 274.75/150.97 POL(f.0(x_1)) = x_1 274.75/150.97 POL(f.1(x_1)) = x_1 274.75/150.97 POL(f_flat.0(x_1)) = x_1 274.75/150.97 POL(fresh_constant.) = 0 274.75/150.97 POL(g.0(x_1)) = 1 + x_1 274.75/150.97 POL(g.1(x_1)) = 1 + x_1 274.75/150.97 POL(g_flat.0(x_1)) = 1 + x_1 274.75/150.97 POL(up.0(x_1)) = 1 + x_1 274.75/150.97 POL(up.1(x_1)) = 1 + x_1 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (57) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down.0(g.1(c.)) -> g_flat.0(down.1(c.)) 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.1(c.))) -> g_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.)) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 down.0(f.1(c.)) -> f_flat.0(down.1(c.)) 274.75/150.97 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) 274.75/150.97 down.0(f.0(f.0(b.))) 274.75/150.97 down.0(f.0(b.)) 274.75/150.97 down.0(g.0(g.0(x0))) 274.75/150.97 down.0(g.0(g.1(x0))) 274.75/150.97 down.0(g.1(c.)) 274.75/150.97 down.0(g.0(fresh_constant.)) 274.75/150.97 down.0(f.0(g.0(x0))) 274.75/150.97 down.0(f.0(g.1(x0))) 274.75/150.97 down.0(f.1(c.)) 274.75/150.97 down.0(f.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(x0)))) 274.75/150.97 down.0(g.0(f.0(g.1(x0)))) 274.75/150.97 down.0(g.0(f.0(f.0(x0)))) 274.75/150.97 down.0(g.0(f.0(f.1(x0)))) 274.75/150.97 down.0(g.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(g.0(x0)))) 274.75/150.97 down.0(f.0(f.0(g.1(x0)))) 274.75/150.97 down.0(f.0(f.0(f.0(x0)))) 274.75/150.97 down.0(f.0(f.0(f.1(x0)))) 274.75/150.97 down.0(f.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x0)) 274.75/150.97 g_flat.0(up.1(x0)) 274.75/150.97 f_flat.0(up.0(x0)) 274.75/150.97 f_flat.0(up.1(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (58) MRRProof (EQUIVALENT) 274.75/150.97 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. 274.75/150.97 274.75/150.97 274.75/150.97 Strictly oriented rules of the TRS R: 274.75/150.97 274.75/150.97 down.0(g.1(c.)) -> g_flat.0(down.1(c.)) 274.75/150.97 274.75/150.97 Used ordering: Polynomial interpretation [POLO]: 274.75/150.97 274.75/150.97 POL(TOP.0(x_1)) = x_1 274.75/150.97 POL(b.) = 0 274.75/150.97 POL(c.) = 0 274.75/150.97 POL(down.0(x_1)) = x_1 274.75/150.97 POL(down.1(x_1)) = x_1 274.75/150.97 POL(f.0(x_1)) = x_1 274.75/150.97 POL(f.1(x_1)) = x_1 274.75/150.97 POL(f_flat.0(x_1)) = x_1 274.75/150.97 POL(fresh_constant.) = 0 274.75/150.97 POL(g.0(x_1)) = x_1 274.75/150.97 POL(g.1(x_1)) = 1 + x_1 274.75/150.97 POL(g_flat.0(x_1)) = x_1 274.75/150.97 POL(up.0(x_1)) = x_1 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (59) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.1(c.))) -> g_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.)) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 down.0(f.1(c.)) -> f_flat.0(down.1(c.)) 274.75/150.97 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) 274.75/150.97 down.0(f.0(f.0(b.))) 274.75/150.97 down.0(f.0(b.)) 274.75/150.97 down.0(g.0(g.0(x0))) 274.75/150.97 down.0(g.0(g.1(x0))) 274.75/150.97 down.0(g.1(c.)) 274.75/150.97 down.0(g.0(fresh_constant.)) 274.75/150.97 down.0(f.0(g.0(x0))) 274.75/150.97 down.0(f.0(g.1(x0))) 274.75/150.97 down.0(f.1(c.)) 274.75/150.97 down.0(f.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(x0)))) 274.75/150.97 down.0(g.0(f.0(g.1(x0)))) 274.75/150.97 down.0(g.0(f.0(f.0(x0)))) 274.75/150.97 down.0(g.0(f.0(f.1(x0)))) 274.75/150.97 down.0(g.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(g.0(x0)))) 274.75/150.97 down.0(f.0(f.0(g.1(x0)))) 274.75/150.97 down.0(f.0(f.0(f.0(x0)))) 274.75/150.97 down.0(f.0(f.0(f.1(x0)))) 274.75/150.97 down.0(f.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x0)) 274.75/150.97 g_flat.0(up.1(x0)) 274.75/150.97 f_flat.0(up.0(x0)) 274.75/150.97 f_flat.0(up.1(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (60) DependencyGraphProof (EQUIVALENT) 274.75/150.97 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (61) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.1(c.))) -> g_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.)) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 down.0(f.1(c.)) -> f_flat.0(down.1(c.)) 274.75/150.97 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) 274.75/150.97 down.0(f.0(f.0(b.))) 274.75/150.97 down.0(f.0(b.)) 274.75/150.97 down.0(g.0(g.0(x0))) 274.75/150.97 down.0(g.0(g.1(x0))) 274.75/150.97 down.0(g.1(c.)) 274.75/150.97 down.0(g.0(fresh_constant.)) 274.75/150.97 down.0(f.0(g.0(x0))) 274.75/150.97 down.0(f.0(g.1(x0))) 274.75/150.97 down.0(f.1(c.)) 274.75/150.97 down.0(f.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(x0)))) 274.75/150.97 down.0(g.0(f.0(g.1(x0)))) 274.75/150.97 down.0(g.0(f.0(f.0(x0)))) 274.75/150.97 down.0(g.0(f.0(f.1(x0)))) 274.75/150.97 down.0(g.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(g.0(x0)))) 274.75/150.97 down.0(f.0(f.0(g.1(x0)))) 274.75/150.97 down.0(f.0(f.0(f.0(x0)))) 274.75/150.97 down.0(f.0(f.0(f.1(x0)))) 274.75/150.97 down.0(f.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x0)) 274.75/150.97 g_flat.0(up.1(x0)) 274.75/150.97 f_flat.0(up.0(x0)) 274.75/150.97 f_flat.0(up.1(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (62) MRRProof (EQUIVALENT) 274.75/150.97 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. 274.75/150.97 274.75/150.97 274.75/150.97 Strictly oriented rules of the TRS R: 274.75/150.97 274.75/150.97 down.0(f.1(c.)) -> f_flat.0(down.1(c.)) 274.75/150.97 274.75/150.97 Used ordering: Polynomial interpretation [POLO]: 274.75/150.97 274.75/150.97 POL(TOP.0(x_1)) = x_1 274.75/150.97 POL(b.) = 0 274.75/150.97 POL(c.) = 0 274.75/150.97 POL(down.0(x_1)) = x_1 274.75/150.97 POL(down.1(x_1)) = x_1 274.75/150.97 POL(f.0(x_1)) = x_1 274.75/150.97 POL(f.1(x_1)) = 1 + x_1 274.75/150.97 POL(f_flat.0(x_1)) = x_1 274.75/150.97 POL(fresh_constant.) = 0 274.75/150.97 POL(g.0(x_1)) = x_1 274.75/150.97 POL(g.1(x_1)) = x_1 274.75/150.97 POL(g_flat.0(x_1)) = x_1 274.75/150.97 POL(up.0(x_1)) = x_1 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (63) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.1(c.))) -> g_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.)) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) 274.75/150.97 down.0(f.0(f.0(b.))) 274.75/150.97 down.0(f.0(b.)) 274.75/150.97 down.0(g.0(g.0(x0))) 274.75/150.97 down.0(g.0(g.1(x0))) 274.75/150.97 down.0(g.1(c.)) 274.75/150.97 down.0(g.0(fresh_constant.)) 274.75/150.97 down.0(f.0(g.0(x0))) 274.75/150.97 down.0(f.0(g.1(x0))) 274.75/150.97 down.0(f.1(c.)) 274.75/150.97 down.0(f.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(x0)))) 274.75/150.97 down.0(g.0(f.0(g.1(x0)))) 274.75/150.97 down.0(g.0(f.0(f.0(x0)))) 274.75/150.97 down.0(g.0(f.0(f.1(x0)))) 274.75/150.97 down.0(g.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(g.0(x0)))) 274.75/150.97 down.0(f.0(f.0(g.1(x0)))) 274.75/150.97 down.0(f.0(f.0(f.0(x0)))) 274.75/150.97 down.0(f.0(f.0(f.1(x0)))) 274.75/150.97 down.0(f.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x0)) 274.75/150.97 g_flat.0(up.1(x0)) 274.75/150.97 f_flat.0(up.0(x0)) 274.75/150.97 f_flat.0(up.1(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (64) QDPOrderProof (EQUIVALENT) 274.75/150.97 We use the reduction pair processor [LPAR04,JAR06]. 274.75/150.97 274.75/150.97 274.75/150.97 The following pairs can be oriented strictly and are deleted. 274.75/150.97 274.75/150.97 TOP.0(up.0(f.0(f.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 The remaining pairs can at least be oriented weakly. 274.75/150.97 Used ordering: Polynomial interpretation [POLO]: 274.75/150.97 274.75/150.97 POL(TOP.0(x_1)) = x_1 274.75/150.97 POL(b.) = 0 274.75/150.97 POL(c.) = 0 274.75/150.97 POL(down.0(x_1)) = 0 274.75/150.97 POL(f.0(x_1)) = x_1 274.75/150.97 POL(f.1(x_1)) = 1 274.75/150.97 POL(f_flat.0(x_1)) = x_1 274.75/150.97 POL(fresh_constant.) = 0 274.75/150.97 POL(g.0(x_1)) = 0 274.75/150.97 POL(g.1(x_1)) = x_1 274.75/150.97 POL(g_flat.0(x_1)) = 0 274.75/150.97 POL(up.0(x_1)) = x_1 274.75/150.97 274.75/150.97 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.1(c.))) -> g_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.)) 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (65) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.1(c.))) -> g_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.)) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) 274.75/150.97 down.0(f.0(f.0(b.))) 274.75/150.97 down.0(f.0(b.)) 274.75/150.97 down.0(g.0(g.0(x0))) 274.75/150.97 down.0(g.0(g.1(x0))) 274.75/150.97 down.0(g.1(c.)) 274.75/150.97 down.0(g.0(fresh_constant.)) 274.75/150.97 down.0(f.0(g.0(x0))) 274.75/150.97 down.0(f.0(g.1(x0))) 274.75/150.97 down.0(f.1(c.)) 274.75/150.97 down.0(f.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(x0)))) 274.75/150.97 down.0(g.0(f.0(g.1(x0)))) 274.75/150.97 down.0(g.0(f.0(f.0(x0)))) 274.75/150.97 down.0(g.0(f.0(f.1(x0)))) 274.75/150.97 down.0(g.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(g.0(x0)))) 274.75/150.97 down.0(f.0(f.0(g.1(x0)))) 274.75/150.97 down.0(f.0(f.0(f.0(x0)))) 274.75/150.97 down.0(f.0(f.0(f.1(x0)))) 274.75/150.97 down.0(f.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x0)) 274.75/150.97 g_flat.0(up.1(x0)) 274.75/150.97 f_flat.0(up.0(x0)) 274.75/150.97 f_flat.0(up.1(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (66) QDPOrderProof (EQUIVALENT) 274.75/150.97 We use the reduction pair processor [LPAR04,JAR06]. 274.75/150.97 274.75/150.97 274.75/150.97 The following pairs can be oriented strictly and are deleted. 274.75/150.97 274.75/150.97 TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 The remaining pairs can at least be oriented weakly. 274.75/150.97 Used ordering: Polynomial interpretation [POLO]: 274.75/150.97 274.75/150.97 POL(TOP.0(x_1)) = x_1 274.75/150.97 POL(b.) = 0 274.75/150.97 POL(c.) = 0 274.75/150.97 POL(down.0(x_1)) = 0 274.75/150.97 POL(f.0(x_1)) = x_1 274.75/150.97 POL(f.1(x_1)) = 1 274.75/150.97 POL(f_flat.0(x_1)) = x_1 274.75/150.97 POL(fresh_constant.) = 0 274.75/150.97 POL(g.0(x_1)) = x_1 274.75/150.97 POL(g.1(x_1)) = x_1 274.75/150.97 POL(g_flat.0(x_1)) = x_1 274.75/150.97 POL(up.0(x_1)) = x_1 274.75/150.97 274.75/150.97 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.1(c.))) -> g_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.)) 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (67) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.1(c.))) -> g_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.)) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) 274.75/150.97 down.0(f.0(f.0(b.))) 274.75/150.97 down.0(f.0(b.)) 274.75/150.97 down.0(g.0(g.0(x0))) 274.75/150.97 down.0(g.0(g.1(x0))) 274.75/150.97 down.0(g.1(c.)) 274.75/150.97 down.0(g.0(fresh_constant.)) 274.75/150.97 down.0(f.0(g.0(x0))) 274.75/150.97 down.0(f.0(g.1(x0))) 274.75/150.97 down.0(f.1(c.)) 274.75/150.97 down.0(f.0(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(x0)))) 274.75/150.97 down.0(g.0(f.0(g.1(x0)))) 274.75/150.97 down.0(g.0(f.0(f.0(x0)))) 274.75/150.97 down.0(g.0(f.0(f.1(x0)))) 274.75/150.97 down.0(g.0(f.1(c.))) 274.75/150.97 down.0(g.0(f.0(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(g.0(x0)))) 274.75/150.97 down.0(f.0(f.0(g.1(x0)))) 274.75/150.97 down.0(f.0(f.0(f.0(x0)))) 274.75/150.97 down.0(f.0(f.0(f.1(x0)))) 274.75/150.97 down.0(f.0(f.1(c.))) 274.75/150.97 down.0(f.0(f.0(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x0)) 274.75/150.97 g_flat.0(up.1(x0)) 274.75/150.97 f_flat.0(up.0(x0)) 274.75/150.97 f_flat.0(up.1(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (68) PisEmptyProof (SOUND) 274.75/150.97 The TRS P is empty. Hence, there is no (P,Q,R) chain. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (69) 274.75/150.97 TRUE 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (70) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.97 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down(g(b)) 274.75/150.97 down(f(f(b))) 274.75/150.97 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (71) SplitQDPProof (EQUIVALENT) 274.75/150.97 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (72) 274.75/150.97 Complex Obligation (AND) 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (73) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.97 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.97 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.97 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.97 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.97 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.97 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.97 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 274.75/150.97 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.97 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.97 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.97 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.97 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 274.75/150.97 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.97 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.97 down(f(f(b))) -> up(b) 274.75/150.97 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.97 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.97 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.97 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down(g(b)) 274.75/150.97 down(f(f(b))) 274.75/150.97 down(f(b)) 274.75/150.97 down(g(g(x0))) 274.75/150.97 down(g(c)) 274.75/150.97 down(g(fresh_constant)) 274.75/150.97 down(f(g(x0))) 274.75/150.97 down(f(c)) 274.75/150.97 down(f(fresh_constant)) 274.75/150.97 down(g(f(g(x0)))) 274.75/150.97 down(g(f(f(x0)))) 274.75/150.97 down(g(f(c))) 274.75/150.97 down(g(f(fresh_constant))) 274.75/150.97 down(f(f(g(x0)))) 274.75/150.97 down(f(f(f(x0)))) 274.75/150.97 down(f(f(c))) 274.75/150.97 down(f(f(fresh_constant))) 274.75/150.97 g_flat(up(x0)) 274.75/150.97 f_flat(up(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (74) SemLabProof (SOUND) 274.75/150.97 We found the following model for the rules of the TRSs R and P. 274.75/150.97 Interpretation over the domain with elements from 0 to 1. 274.75/150.97 b: 0 274.75/150.97 c: 0 274.75/150.97 down: 0 274.75/150.97 f: 0 274.75/150.97 fresh_constant: 1 274.75/150.97 up: 0 274.75/150.97 f_flat: 0 274.75/150.97 TOP: 0 274.75/150.97 g_flat: 0 274.75/150.97 g: 0 274.75/150.97 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (75) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.))) 274.75/150.97 down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.97 down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.)) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.97 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) 274.75/150.97 down.0(f.0(f.0(b.))) 274.75/150.97 down.0(f.0(b.)) 274.75/150.97 down.0(g.0(g.0(x0))) 274.75/150.97 down.0(g.0(g.1(x0))) 274.75/150.97 down.0(g.0(c.)) 274.75/150.97 down.0(g.1(fresh_constant.)) 274.75/150.97 down.0(f.0(g.0(x0))) 274.75/150.97 down.0(f.0(g.1(x0))) 274.75/150.97 down.0(f.0(c.)) 274.75/150.97 down.0(f.1(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(x0)))) 274.75/150.97 down.0(g.0(f.0(g.1(x0)))) 274.75/150.97 down.0(g.0(f.0(f.0(x0)))) 274.75/150.97 down.0(g.0(f.0(f.1(x0)))) 274.75/150.97 down.0(g.0(f.0(c.))) 274.75/150.97 down.0(g.0(f.1(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(g.0(x0)))) 274.75/150.97 down.0(f.0(f.0(g.1(x0)))) 274.75/150.97 down.0(f.0(f.0(f.0(x0)))) 274.75/150.97 down.0(f.0(f.0(f.1(x0)))) 274.75/150.97 down.0(f.0(f.0(c.))) 274.75/150.97 down.0(f.0(f.1(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x0)) 274.75/150.97 g_flat.0(up.1(x0)) 274.75/150.97 f_flat.0(up.0(x0)) 274.75/150.97 f_flat.0(up.1(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (76) UsableRulesReductionPairsProof (EQUIVALENT) 274.75/150.97 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. 274.75/150.97 274.75/150.97 No dependency pairs are removed. 274.75/150.97 274.75/150.97 The following rules are removed from R: 274.75/150.97 274.75/150.97 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 274.75/150.97 down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.)) 274.75/150.97 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.97 Used ordering: POLO with Polynomial interpretation [POLO]: 274.75/150.97 274.75/150.97 POL(TOP.0(x_1)) = x_1 274.75/150.97 POL(b.) = 0 274.75/150.97 POL(c.) = 0 274.75/150.97 POL(down.0(x_1)) = 1 + x_1 274.75/150.97 POL(down.1(x_1)) = x_1 274.75/150.97 POL(f.0(x_1)) = x_1 274.75/150.97 POL(f.1(x_1)) = x_1 274.75/150.97 POL(f_flat.0(x_1)) = x_1 274.75/150.97 POL(fresh_constant.) = 0 274.75/150.97 POL(g.0(x_1)) = 1 + x_1 274.75/150.97 POL(g.1(x_1)) = x_1 274.75/150.97 POL(g_flat.0(x_1)) = 1 + x_1 274.75/150.97 POL(up.0(x_1)) = 1 + x_1 274.75/150.97 POL(up.1(x_1)) = 1 + x_1 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (77) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.)) 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.))) 274.75/150.97 down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) 274.75/150.97 down.0(f.0(f.0(b.))) 274.75/150.97 down.0(f.0(b.)) 274.75/150.97 down.0(g.0(g.0(x0))) 274.75/150.97 down.0(g.0(g.1(x0))) 274.75/150.97 down.0(g.0(c.)) 274.75/150.97 down.0(g.1(fresh_constant.)) 274.75/150.97 down.0(f.0(g.0(x0))) 274.75/150.97 down.0(f.0(g.1(x0))) 274.75/150.97 down.0(f.0(c.)) 274.75/150.97 down.0(f.1(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(x0)))) 274.75/150.97 down.0(g.0(f.0(g.1(x0)))) 274.75/150.97 down.0(g.0(f.0(f.0(x0)))) 274.75/150.97 down.0(g.0(f.0(f.1(x0)))) 274.75/150.97 down.0(g.0(f.0(c.))) 274.75/150.97 down.0(g.0(f.1(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(g.0(x0)))) 274.75/150.97 down.0(f.0(f.0(g.1(x0)))) 274.75/150.97 down.0(f.0(f.0(f.0(x0)))) 274.75/150.97 down.0(f.0(f.0(f.1(x0)))) 274.75/150.97 down.0(f.0(f.0(c.))) 274.75/150.97 down.0(f.0(f.1(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x0)) 274.75/150.97 g_flat.0(up.1(x0)) 274.75/150.97 f_flat.0(up.0(x0)) 274.75/150.97 f_flat.0(up.1(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (78) MRRProof (EQUIVALENT) 274.75/150.97 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. 274.75/150.97 274.75/150.97 274.75/150.97 Strictly oriented rules of the TRS R: 274.75/150.97 274.75/150.97 down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.)) 274.75/150.97 274.75/150.97 Used ordering: Polynomial interpretation [POLO]: 274.75/150.97 274.75/150.97 POL(TOP.0(x_1)) = x_1 274.75/150.97 POL(b.) = 0 274.75/150.97 POL(c.) = 0 274.75/150.97 POL(down.0(x_1)) = 1 + x_1 274.75/150.97 POL(down.1(x_1)) = x_1 274.75/150.97 POL(f.0(x_1)) = x_1 274.75/150.97 POL(f.1(x_1)) = x_1 274.75/150.97 POL(f_flat.0(x_1)) = x_1 274.75/150.97 POL(fresh_constant.) = 0 274.75/150.97 POL(g.0(x_1)) = x_1 274.75/150.97 POL(g.1(x_1)) = 1 + x_1 274.75/150.97 POL(g_flat.0(x_1)) = x_1 274.75/150.97 POL(up.0(x_1)) = 1 + x_1 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (79) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.))) 274.75/150.97 down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) 274.75/150.97 down.0(f.0(f.0(b.))) 274.75/150.97 down.0(f.0(b.)) 274.75/150.97 down.0(g.0(g.0(x0))) 274.75/150.97 down.0(g.0(g.1(x0))) 274.75/150.97 down.0(g.0(c.)) 274.75/150.97 down.0(g.1(fresh_constant.)) 274.75/150.97 down.0(f.0(g.0(x0))) 274.75/150.97 down.0(f.0(g.1(x0))) 274.75/150.97 down.0(f.0(c.)) 274.75/150.97 down.0(f.1(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(x0)))) 274.75/150.97 down.0(g.0(f.0(g.1(x0)))) 274.75/150.97 down.0(g.0(f.0(f.0(x0)))) 274.75/150.97 down.0(g.0(f.0(f.1(x0)))) 274.75/150.97 down.0(g.0(f.0(c.))) 274.75/150.97 down.0(g.0(f.1(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(g.0(x0)))) 274.75/150.97 down.0(f.0(f.0(g.1(x0)))) 274.75/150.97 down.0(f.0(f.0(f.0(x0)))) 274.75/150.97 down.0(f.0(f.0(f.1(x0)))) 274.75/150.97 down.0(f.0(f.0(c.))) 274.75/150.97 down.0(f.0(f.1(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x0)) 274.75/150.97 g_flat.0(up.1(x0)) 274.75/150.97 f_flat.0(up.0(x0)) 274.75/150.97 f_flat.0(up.1(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (80) DependencyGraphProof (EQUIVALENT) 274.75/150.97 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (81) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.97 274.75/150.97 The TRS R consists of the following rules: 274.75/150.97 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.))) 274.75/150.97 down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 274.75/150.97 The set Q consists of the following terms: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) 274.75/150.97 down.0(f.0(f.0(b.))) 274.75/150.97 down.0(f.0(b.)) 274.75/150.97 down.0(g.0(g.0(x0))) 274.75/150.97 down.0(g.0(g.1(x0))) 274.75/150.97 down.0(g.0(c.)) 274.75/150.97 down.0(g.1(fresh_constant.)) 274.75/150.97 down.0(f.0(g.0(x0))) 274.75/150.97 down.0(f.0(g.1(x0))) 274.75/150.97 down.0(f.0(c.)) 274.75/150.97 down.0(f.1(fresh_constant.)) 274.75/150.97 down.0(g.0(f.0(g.0(x0)))) 274.75/150.97 down.0(g.0(f.0(g.1(x0)))) 274.75/150.97 down.0(g.0(f.0(f.0(x0)))) 274.75/150.97 down.0(g.0(f.0(f.1(x0)))) 274.75/150.97 down.0(g.0(f.0(c.))) 274.75/150.97 down.0(g.0(f.1(fresh_constant.))) 274.75/150.97 down.0(f.0(f.0(g.0(x0)))) 274.75/150.97 down.0(f.0(f.0(g.1(x0)))) 274.75/150.97 down.0(f.0(f.0(f.0(x0)))) 274.75/150.97 down.0(f.0(f.0(f.1(x0)))) 274.75/150.97 down.0(f.0(f.0(c.))) 274.75/150.97 down.0(f.0(f.1(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x0)) 274.75/150.97 g_flat.0(up.1(x0)) 274.75/150.97 f_flat.0(up.0(x0)) 274.75/150.97 f_flat.0(up.1(x0)) 274.75/150.97 274.75/150.97 We have to consider all minimal (P,Q,R)-chains. 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (82) QDPOrderProof (EQUIVALENT) 274.75/150.97 We use the reduction pair processor [LPAR04,JAR06]. 274.75/150.97 274.75/150.97 274.75/150.97 The following pairs can be oriented strictly and are deleted. 274.75/150.97 274.75/150.97 TOP.0(up.0(f.0(f.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.97 The remaining pairs can at least be oriented weakly. 274.75/150.97 Used ordering: Polynomial interpretation [POLO]: 274.75/150.97 274.75/150.97 POL(TOP.0(x_1)) = x_1 274.75/150.97 POL(b.) = 0 274.75/150.97 POL(c.) = 0 274.75/150.97 POL(down.0(x_1)) = 0 274.75/150.97 POL(f.0(x_1)) = x_1 274.75/150.97 POL(f.1(x_1)) = 1 274.75/150.97 POL(f_flat.0(x_1)) = x_1 274.75/150.97 POL(fresh_constant.) = 0 274.75/150.97 POL(g.0(x_1)) = 0 274.75/150.97 POL(g.1(x_1)) = x_1 274.75/150.97 POL(g_flat.0(x_1)) = 0 274.75/150.97 POL(up.0(x_1)) = x_1 274.75/150.97 274.75/150.97 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 274.75/150.97 274.75/150.97 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.97 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.97 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.97 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.97 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.97 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.97 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.97 down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.))) 274.75/150.97 down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.97 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.97 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.97 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 274.75/150.97 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.97 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.97 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.97 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.97 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.97 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.97 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.97 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.97 274.75/150.97 274.75/150.97 ---------------------------------------- 274.75/150.97 274.75/150.97 (83) 274.75/150.97 Obligation: 274.75/150.97 Q DP problem: 274.75/150.97 The TRS P consists of the following rules: 274.75/150.97 274.75/150.97 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.97 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.98 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.98 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.98 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.98 down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.))) 274.75/150.98 down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 274.75/150.98 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.98 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.98 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.98 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.98 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.0(f.0(b.))) 274.75/150.98 down.0(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.1(fresh_constant.)) 274.75/150.98 down.0(f.0(g.0(x0))) 274.75/150.98 down.0(f.0(g.1(x0))) 274.75/150.98 down.0(f.0(c.)) 274.75/150.98 down.0(f.1(fresh_constant.)) 274.75/150.98 down.0(g.0(f.0(g.0(x0)))) 274.75/150.98 down.0(g.0(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.0(f.0(x0)))) 274.75/150.98 down.0(g.0(f.0(f.1(x0)))) 274.75/150.98 down.0(g.0(f.0(c.))) 274.75/150.98 down.0(g.0(f.1(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(g.0(x0)))) 274.75/150.98 down.0(f.0(f.0(g.1(x0)))) 274.75/150.98 down.0(f.0(f.0(f.0(x0)))) 274.75/150.98 down.0(f.0(f.0(f.1(x0)))) 274.75/150.98 down.0(f.0(f.0(c.))) 274.75/150.98 down.0(f.0(f.1(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.0(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.0(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (84) QDPOrderProof (EQUIVALENT) 274.75/150.98 We use the reduction pair processor [LPAR04,JAR06]. 274.75/150.98 274.75/150.98 274.75/150.98 The following pairs can be oriented strictly and are deleted. 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(x0))))) 274.75/150.98 The remaining pairs can at least be oriented weakly. 274.75/150.98 Used ordering: Polynomial interpretation [POLO]: 274.75/150.98 274.75/150.98 POL(TOP.0(x_1)) = x_1 274.75/150.98 POL(b.) = 0 274.75/150.98 POL(c.) = 0 274.75/150.98 POL(down.0(x_1)) = 0 274.75/150.98 POL(f.0(x_1)) = x_1 274.75/150.98 POL(f.1(x_1)) = 1 274.75/150.98 POL(f_flat.0(x_1)) = x_1 274.75/150.98 POL(fresh_constant.) = 0 274.75/150.98 POL(g.0(x_1)) = x_1 274.75/150.98 POL(g.1(x_1)) = x_1 274.75/150.98 POL(g_flat.0(x_1)) = x_1 274.75/150.98 POL(up.0(x_1)) = x_1 274.75/150.98 274.75/150.98 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.98 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.98 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.98 down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.))) 274.75/150.98 down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.98 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 274.75/150.98 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.98 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.98 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.98 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.98 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.98 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (85) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.98 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.98 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.98 down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10)))) 274.75/150.98 down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.))) 274.75/150.98 down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 274.75/150.98 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.98 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.98 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.98 down.0(f.0(f.0(f.1(y13)))) -> f_flat.0(down.0(f.0(f.1(y13)))) 274.75/150.98 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.0(f.0(b.))) 274.75/150.98 down.0(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.1(fresh_constant.)) 274.75/150.98 down.0(f.0(g.0(x0))) 274.75/150.98 down.0(f.0(g.1(x0))) 274.75/150.98 down.0(f.0(c.)) 274.75/150.98 down.0(f.1(fresh_constant.)) 274.75/150.98 down.0(g.0(f.0(g.0(x0)))) 274.75/150.98 down.0(g.0(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.0(f.0(x0)))) 274.75/150.98 down.0(g.0(f.0(f.1(x0)))) 274.75/150.98 down.0(g.0(f.0(c.))) 274.75/150.98 down.0(g.0(f.1(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(g.0(x0)))) 274.75/150.98 down.0(f.0(f.0(g.1(x0)))) 274.75/150.98 down.0(f.0(f.0(f.0(x0)))) 274.75/150.98 down.0(f.0(f.0(f.1(x0)))) 274.75/150.98 down.0(f.0(f.0(c.))) 274.75/150.98 down.0(f.0(f.1(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.0(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.0(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (86) PisEmptyProof (SOUND) 274.75/150.98 The TRS P is empty. Hence, there is no (P,Q,R) chain. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (87) 274.75/150.98 TRUE 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (88) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.98 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.98 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.98 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.98 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.98 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.98 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.98 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.98 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.98 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.98 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.98 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.98 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.98 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.98 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.98 down(f(f(b))) -> up(b) 274.75/150.98 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.98 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.98 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down(g(b)) 274.75/150.98 down(f(f(b))) 274.75/150.98 down(f(b)) 274.75/150.98 down(g(g(x0))) 274.75/150.98 down(g(c)) 274.75/150.98 down(g(fresh_constant)) 274.75/150.98 down(f(g(x0))) 274.75/150.98 down(f(c)) 274.75/150.98 down(f(fresh_constant)) 274.75/150.98 down(g(f(g(x0)))) 274.75/150.98 down(g(f(f(x0)))) 274.75/150.98 down(g(f(c))) 274.75/150.98 down(g(f(fresh_constant))) 274.75/150.98 down(f(f(g(x0)))) 274.75/150.98 down(f(f(f(x0)))) 274.75/150.98 down(f(f(c))) 274.75/150.98 down(f(f(fresh_constant))) 274.75/150.98 g_flat(up(x0)) 274.75/150.98 f_flat(up(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (89) SplitQDPProof (EQUIVALENT) 274.75/150.98 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (90) 274.75/150.98 Complex Obligation (AND) 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (91) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.98 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.98 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.98 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.98 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.98 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.98 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.98 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.98 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.98 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.98 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.98 down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant))) 274.75/150.98 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.98 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 274.75/150.98 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.98 down(f(f(b))) -> up(b) 274.75/150.98 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.98 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.98 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down(g(b)) 274.75/150.98 down(f(f(b))) 274.75/150.98 down(f(b)) 274.75/150.98 down(g(g(x0))) 274.75/150.98 down(g(c)) 274.75/150.98 down(g(fresh_constant)) 274.75/150.98 down(f(g(x0))) 274.75/150.98 down(f(c)) 274.75/150.98 down(f(fresh_constant)) 274.75/150.98 down(g(f(g(x0)))) 274.75/150.98 down(g(f(f(x0)))) 274.75/150.98 down(g(f(c))) 274.75/150.98 down(g(f(fresh_constant))) 274.75/150.98 down(f(f(g(x0)))) 274.75/150.98 down(f(f(f(x0)))) 274.75/150.98 down(f(f(c))) 274.75/150.98 down(f(f(fresh_constant))) 274.75/150.98 g_flat(up(x0)) 274.75/150.98 f_flat(up(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (92) SemLabProof (SOUND) 274.75/150.98 We found the following model for the rules of the TRSs R and P. 274.75/150.98 Interpretation over the domain with elements from 0 to 1. 274.75/150.98 b: 0 274.75/150.98 c: 0 274.75/150.98 down: 0 274.75/150.98 f: x0 274.75/150.98 fresh_constant: 1 274.75/150.98 up: 0 274.75/150.98 f_flat: 0 274.75/150.98 TOP: 0 274.75/150.98 g_flat: 0 274.75/150.98 g: 0 274.75/150.98 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (93) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.1(f.1(x0))))) -> TOP.0(g_flat.0(down.1(f.1(f.1(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.1(f.1(f.1(f.1(x0))))) -> TOP.0(f_flat.0(down.1(f.1(f.1(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.98 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 274.75/150.98 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.98 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.98 down.0(g.1(f.1(f.1(y10)))) -> g_flat.0(down.1(f.1(f.1(y10)))) 274.75/150.98 down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.))) 274.75/150.98 down.0(g.1(f.1(fresh_constant.))) -> g_flat.0(down.1(f.1(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 274.75/150.98 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.98 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.98 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.98 down.1(f.1(f.1(f.1(y13)))) -> f_flat.0(down.1(f.1(f.1(y13)))) 274.75/150.98 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.0(f.0(b.))) 274.75/150.98 down.0(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.1(fresh_constant.)) 274.75/150.98 down.0(f.0(g.0(x0))) 274.75/150.98 down.0(f.0(g.1(x0))) 274.75/150.98 down.0(f.0(c.)) 274.75/150.98 down.1(f.1(fresh_constant.)) 274.75/150.98 down.0(g.0(f.0(g.0(x0)))) 274.75/150.98 down.0(g.0(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.0(f.0(x0)))) 274.75/150.98 down.0(g.1(f.1(f.1(x0)))) 274.75/150.98 down.0(g.0(f.0(c.))) 274.75/150.98 down.0(g.1(f.1(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(g.0(x0)))) 274.75/150.98 down.0(f.0(f.0(g.1(x0)))) 274.75/150.98 down.0(f.0(f.0(f.0(x0)))) 274.75/150.98 down.1(f.1(f.1(f.1(x0)))) 274.75/150.98 down.0(f.0(f.0(c.))) 274.75/150.98 down.1(f.1(f.1(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.0(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.0(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (94) MRRProof (EQUIVALENT) 274.75/150.98 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. 274.75/150.98 274.75/150.98 Strictly oriented dependency pairs: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.1(f.1(f.1(x0))))) -> TOP.0(g_flat.0(down.1(f.1(f.1(x0))))) 274.75/150.98 TOP.0(up.1(f.1(f.1(f.1(x0))))) -> TOP.0(f_flat.0(down.1(f.1(f.1(x0))))) 274.75/150.98 274.75/150.98 Strictly oriented rules of the TRS R: 274.75/150.98 274.75/150.98 down.0(g.1(f.1(f.1(y10)))) -> g_flat.0(down.1(f.1(f.1(y10)))) 274.75/150.98 down.0(g.1(f.1(fresh_constant.))) -> g_flat.0(down.1(f.1(fresh_constant.))) 274.75/150.98 274.75/150.98 Used ordering: Polynomial interpretation [POLO]: 274.75/150.98 274.75/150.98 POL(TOP.0(x_1)) = x_1 274.75/150.98 POL(b.) = 0 274.75/150.98 POL(c.) = 0 274.75/150.98 POL(down.0(x_1)) = x_1 274.75/150.98 POL(down.1(x_1)) = x_1 274.75/150.98 POL(f.0(x_1)) = x_1 274.75/150.98 POL(f.1(x_1)) = x_1 274.75/150.98 POL(f_flat.0(x_1)) = x_1 274.75/150.98 POL(fresh_constant.) = 0 274.75/150.98 POL(g.0(x_1)) = x_1 274.75/150.98 POL(g.1(x_1)) = 1 + x_1 274.75/150.98 POL(g_flat.0(x_1)) = x_1 274.75/150.98 POL(up.0(x_1)) = x_1 274.75/150.98 POL(up.1(x_1)) = 1 + x_1 274.75/150.98 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (95) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.98 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 274.75/150.98 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.98 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.98 down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 274.75/150.98 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.98 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.98 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.98 down.1(f.1(f.1(f.1(y13)))) -> f_flat.0(down.1(f.1(f.1(y13)))) 274.75/150.98 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.0(f.0(b.))) 274.75/150.98 down.0(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.1(fresh_constant.)) 274.75/150.98 down.0(f.0(g.0(x0))) 274.75/150.98 down.0(f.0(g.1(x0))) 274.75/150.98 down.0(f.0(c.)) 274.75/150.98 down.1(f.1(fresh_constant.)) 274.75/150.98 down.0(g.0(f.0(g.0(x0)))) 274.75/150.98 down.0(g.0(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.0(f.0(x0)))) 274.75/150.98 down.0(g.1(f.1(f.1(x0)))) 274.75/150.98 down.0(g.0(f.0(c.))) 274.75/150.98 down.0(g.1(f.1(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(g.0(x0)))) 274.75/150.98 down.0(f.0(f.0(g.1(x0)))) 274.75/150.98 down.0(f.0(f.0(f.0(x0)))) 274.75/150.98 down.1(f.1(f.1(f.1(x0)))) 274.75/150.98 down.0(f.0(f.0(c.))) 274.75/150.98 down.1(f.1(f.1(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.0(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.0(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (96) DependencyGraphProof (EQUIVALENT) 274.75/150.98 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (97) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.98 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 274.75/150.98 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.98 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.98 down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 274.75/150.98 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.98 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.98 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.98 down.1(f.1(f.1(f.1(y13)))) -> f_flat.0(down.1(f.1(f.1(y13)))) 274.75/150.98 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.0(f.0(b.))) 274.75/150.98 down.0(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.1(fresh_constant.)) 274.75/150.98 down.0(f.0(g.0(x0))) 274.75/150.98 down.0(f.0(g.1(x0))) 274.75/150.98 down.0(f.0(c.)) 274.75/150.98 down.1(f.1(fresh_constant.)) 274.75/150.98 down.0(g.0(f.0(g.0(x0)))) 274.75/150.98 down.0(g.0(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.0(f.0(x0)))) 274.75/150.98 down.0(g.1(f.1(f.1(x0)))) 274.75/150.98 down.0(g.0(f.0(c.))) 274.75/150.98 down.0(g.1(f.1(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(g.0(x0)))) 274.75/150.98 down.0(f.0(f.0(g.1(x0)))) 274.75/150.98 down.0(f.0(f.0(f.0(x0)))) 274.75/150.98 down.1(f.1(f.1(f.1(x0)))) 274.75/150.98 down.0(f.0(f.0(c.))) 274.75/150.98 down.1(f.1(f.1(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.0(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.0(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (98) UsableRulesReductionPairsProof (EQUIVALENT) 274.75/150.98 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. 274.75/150.98 274.75/150.98 No dependency pairs are removed. 274.75/150.98 274.75/150.98 The following rules are removed from R: 274.75/150.98 274.75/150.98 down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.))) 274.75/150.98 down.1(f.1(f.1(f.1(y13)))) -> f_flat.0(down.1(f.1(f.1(y13)))) 274.75/150.98 Used ordering: POLO with Polynomial interpretation [POLO]: 274.75/150.98 274.75/150.98 POL(TOP.0(x_1)) = x_1 274.75/150.98 POL(b.) = 0 274.75/150.98 POL(c.) = 0 274.75/150.98 POL(down.0(x_1)) = 1 + x_1 274.75/150.98 POL(f.0(x_1)) = x_1 274.75/150.98 POL(f.1(x_1)) = x_1 274.75/150.98 POL(f_flat.0(x_1)) = x_1 274.75/150.98 POL(g.0(x_1)) = 1 + x_1 274.75/150.98 POL(g.1(x_1)) = 1 + x_1 274.75/150.98 POL(g_flat.0(x_1)) = 1 + x_1 274.75/150.98 POL(up.0(x_1)) = 1 + x_1 274.75/150.98 POL(up.1(x_1)) = 1 + x_1 274.75/150.98 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (99) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.98 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.98 down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.98 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 274.75/150.98 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 274.75/150.98 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.98 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.98 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.0(f.0(b.))) 274.75/150.98 down.0(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.1(fresh_constant.)) 274.75/150.98 down.0(f.0(g.0(x0))) 274.75/150.98 down.0(f.0(g.1(x0))) 274.75/150.98 down.0(f.0(c.)) 274.75/150.98 down.1(f.1(fresh_constant.)) 274.75/150.98 down.0(g.0(f.0(g.0(x0)))) 274.75/150.98 down.0(g.0(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.0(f.0(x0)))) 274.75/150.98 down.0(g.1(f.1(f.1(x0)))) 274.75/150.98 down.0(g.0(f.0(c.))) 274.75/150.98 down.0(g.1(f.1(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(g.0(x0)))) 274.75/150.98 down.0(f.0(f.0(g.1(x0)))) 274.75/150.98 down.0(f.0(f.0(f.0(x0)))) 274.75/150.98 down.1(f.1(f.1(f.1(x0)))) 274.75/150.98 down.0(f.0(f.0(c.))) 274.75/150.98 down.1(f.1(f.1(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.0(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.0(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (100) MRRProof (EQUIVALENT) 274.75/150.98 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. 274.75/150.98 274.75/150.98 274.75/150.98 Strictly oriented rules of the TRS R: 274.75/150.98 274.75/150.98 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 274.75/150.98 Used ordering: Polynomial interpretation [POLO]: 274.75/150.98 274.75/150.98 POL(TOP.0(x_1)) = x_1 274.75/150.98 POL(b.) = 0 274.75/150.98 POL(c.) = 0 274.75/150.98 POL(down.0(x_1)) = x_1 274.75/150.98 POL(f.0(x_1)) = x_1 274.75/150.98 POL(f.1(x_1)) = x_1 274.75/150.98 POL(f_flat.0(x_1)) = x_1 274.75/150.98 POL(g.0(x_1)) = 1 + x_1 274.75/150.98 POL(g.1(x_1)) = 1 + x_1 274.75/150.98 POL(g_flat.0(x_1)) = 1 + x_1 274.75/150.98 POL(up.0(x_1)) = x_1 274.75/150.98 POL(up.1(x_1)) = 1 + x_1 274.75/150.98 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (101) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.98 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.98 down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.98 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 274.75/150.98 down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) 274.75/150.98 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.98 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.98 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.0(f.0(b.))) 274.75/150.98 down.0(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.1(fresh_constant.)) 274.75/150.98 down.0(f.0(g.0(x0))) 274.75/150.98 down.0(f.0(g.1(x0))) 274.75/150.98 down.0(f.0(c.)) 274.75/150.98 down.1(f.1(fresh_constant.)) 274.75/150.98 down.0(g.0(f.0(g.0(x0)))) 274.75/150.98 down.0(g.0(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.0(f.0(x0)))) 274.75/150.98 down.0(g.1(f.1(f.1(x0)))) 274.75/150.98 down.0(g.0(f.0(c.))) 274.75/150.98 down.0(g.1(f.1(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(g.0(x0)))) 274.75/150.98 down.0(f.0(f.0(g.1(x0)))) 274.75/150.98 down.0(f.0(f.0(f.0(x0)))) 274.75/150.98 down.1(f.1(f.1(f.1(x0)))) 274.75/150.98 down.0(f.0(f.0(c.))) 274.75/150.98 down.1(f.1(f.1(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.0(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.0(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (102) PisEmptyProof (SOUND) 274.75/150.98 The TRS P is empty. Hence, there is no (P,Q,R) chain. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (103) 274.75/150.98 TRUE 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (104) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.98 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.98 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.98 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.98 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.98 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.98 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.98 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.98 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.98 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.98 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.98 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.98 down(f(f(b))) -> up(b) 274.75/150.98 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.98 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.98 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.98 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down(g(b)) 274.75/150.98 down(f(f(b))) 274.75/150.98 down(f(b)) 274.75/150.98 down(g(g(x0))) 274.75/150.98 down(g(c)) 274.75/150.98 down(g(fresh_constant)) 274.75/150.98 down(f(g(x0))) 274.75/150.98 down(f(c)) 274.75/150.98 down(f(fresh_constant)) 274.75/150.98 down(g(f(g(x0)))) 274.75/150.98 down(g(f(f(x0)))) 274.75/150.98 down(g(f(c))) 274.75/150.98 down(g(f(fresh_constant))) 274.75/150.98 down(f(f(g(x0)))) 274.75/150.98 down(f(f(f(x0)))) 274.75/150.98 down(f(f(c))) 274.75/150.98 down(f(f(fresh_constant))) 274.75/150.98 g_flat(up(x0)) 274.75/150.98 f_flat(up(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (105) SplitQDPProof (EQUIVALENT) 274.75/150.98 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (106) 274.75/150.98 Complex Obligation (AND) 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (107) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.98 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.98 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.98 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.98 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.98 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.98 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.98 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.98 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.98 down(g(f(c))) -> g_flat(down(f(c))) 274.75/150.98 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.98 down(f(f(c))) -> f_flat(down(f(c))) 274.75/150.98 down(f(f(b))) -> up(b) 274.75/150.98 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.98 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.98 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.98 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down(g(b)) 274.75/150.98 down(f(f(b))) 274.75/150.98 down(f(b)) 274.75/150.98 down(g(g(x0))) 274.75/150.98 down(g(c)) 274.75/150.98 down(g(fresh_constant)) 274.75/150.98 down(f(g(x0))) 274.75/150.98 down(f(c)) 274.75/150.98 down(f(fresh_constant)) 274.75/150.98 down(g(f(g(x0)))) 274.75/150.98 down(g(f(f(x0)))) 274.75/150.98 down(g(f(c))) 274.75/150.98 down(g(f(fresh_constant))) 274.75/150.98 down(f(f(g(x0)))) 274.75/150.98 down(f(f(f(x0)))) 274.75/150.98 down(f(f(c))) 274.75/150.98 down(f(f(fresh_constant))) 274.75/150.98 g_flat(up(x0)) 274.75/150.98 f_flat(up(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (108) SemLabProof (SOUND) 274.75/150.98 We found the following model for the rules of the TRSs R and P. 274.75/150.98 Interpretation over the domain with elements from 0 to 1. 274.75/150.98 b: 0 274.75/150.98 c: 1 274.75/150.98 down: 0 274.75/150.98 f: x0 274.75/150.98 fresh_constant: 0 274.75/150.98 up: 0 274.75/150.98 f_flat: 0 274.75/150.98 TOP: 0 274.75/150.98 g_flat: 0 274.75/150.98 g: 0 274.75/150.98 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (109) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.1(f.1(x0))))) -> TOP.0(g_flat.0(down.1(f.1(f.1(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.1(f.1(f.1(f.1(x0))))) -> TOP.0(f_flat.0(down.1(f.1(f.1(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.98 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.98 down.0(g.1(f.1(f.1(y10)))) -> g_flat.0(down.1(f.1(f.1(y10)))) 274.75/150.98 down.0(g.1(f.1(c.))) -> g_flat.0(down.1(f.1(c.))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.98 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 274.75/150.98 down.1(f.1(f.1(c.))) -> f_flat.0(down.1(f.1(c.))) 274.75/150.98 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.98 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.98 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.98 down.1(f.1(f.1(f.1(y13)))) -> f_flat.0(down.1(f.1(f.1(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.0(f.0(b.))) 274.75/150.98 down.0(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.1(c.)) 274.75/150.98 down.0(g.0(fresh_constant.)) 274.75/150.98 down.0(f.0(g.0(x0))) 274.75/150.98 down.0(f.0(g.1(x0))) 274.75/150.98 down.1(f.1(c.)) 274.75/150.98 down.0(f.0(fresh_constant.)) 274.75/150.98 down.0(g.0(f.0(g.0(x0)))) 274.75/150.98 down.0(g.0(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.0(f.0(x0)))) 274.75/150.98 down.0(g.1(f.1(f.1(x0)))) 274.75/150.98 down.0(g.1(f.1(c.))) 274.75/150.98 down.0(g.0(f.0(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(g.0(x0)))) 274.75/150.98 down.0(f.0(f.0(g.1(x0)))) 274.75/150.98 down.0(f.0(f.0(f.0(x0)))) 274.75/150.98 down.1(f.1(f.1(f.1(x0)))) 274.75/150.98 down.1(f.1(f.1(c.))) 274.75/150.98 down.0(f.0(f.0(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.0(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.0(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (110) MRRProof (EQUIVALENT) 274.75/150.98 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. 274.75/150.98 274.75/150.98 Strictly oriented dependency pairs: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.1(f.1(f.1(x0))))) -> TOP.0(g_flat.0(down.1(f.1(f.1(x0))))) 274.75/150.98 TOP.0(up.1(f.1(f.1(f.1(x0))))) -> TOP.0(f_flat.0(down.1(f.1(f.1(x0))))) 274.75/150.98 274.75/150.98 Strictly oriented rules of the TRS R: 274.75/150.98 274.75/150.98 down.0(g.1(f.1(f.1(y10)))) -> g_flat.0(down.1(f.1(f.1(y10)))) 274.75/150.98 down.0(g.1(f.1(c.))) -> g_flat.0(down.1(f.1(c.))) 274.75/150.98 274.75/150.98 Used ordering: Polynomial interpretation [POLO]: 274.75/150.98 274.75/150.98 POL(TOP.0(x_1)) = x_1 274.75/150.98 POL(b.) = 0 274.75/150.98 POL(c.) = 0 274.75/150.98 POL(down.0(x_1)) = 1 + x_1 274.75/150.98 POL(down.1(x_1)) = x_1 274.75/150.98 POL(f.0(x_1)) = x_1 274.75/150.98 POL(f.1(x_1)) = x_1 274.75/150.98 POL(f_flat.0(x_1)) = x_1 274.75/150.98 POL(g.0(x_1)) = x_1 274.75/150.98 POL(g.1(x_1)) = x_1 274.75/150.98 POL(g_flat.0(x_1)) = x_1 274.75/150.98 POL(up.0(x_1)) = 1 + x_1 274.75/150.98 POL(up.1(x_1)) = 1 + x_1 274.75/150.98 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (111) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.98 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.98 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 274.75/150.98 down.1(f.1(f.1(c.))) -> f_flat.0(down.1(f.1(c.))) 274.75/150.98 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.98 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.98 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.98 down.1(f.1(f.1(f.1(y13)))) -> f_flat.0(down.1(f.1(f.1(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.0(f.0(b.))) 274.75/150.98 down.0(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.1(c.)) 274.75/150.98 down.0(g.0(fresh_constant.)) 274.75/150.98 down.0(f.0(g.0(x0))) 274.75/150.98 down.0(f.0(g.1(x0))) 274.75/150.98 down.1(f.1(c.)) 274.75/150.98 down.0(f.0(fresh_constant.)) 274.75/150.98 down.0(g.0(f.0(g.0(x0)))) 274.75/150.98 down.0(g.0(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.0(f.0(x0)))) 274.75/150.98 down.0(g.1(f.1(f.1(x0)))) 274.75/150.98 down.0(g.1(f.1(c.))) 274.75/150.98 down.0(g.0(f.0(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(g.0(x0)))) 274.75/150.98 down.0(f.0(f.0(g.1(x0)))) 274.75/150.98 down.0(f.0(f.0(f.0(x0)))) 274.75/150.98 down.1(f.1(f.1(f.1(x0)))) 274.75/150.98 down.1(f.1(f.1(c.))) 274.75/150.98 down.0(f.0(f.0(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.0(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.0(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (112) DependencyGraphProof (EQUIVALENT) 274.75/150.98 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (113) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.98 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.98 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 274.75/150.98 down.1(f.1(f.1(c.))) -> f_flat.0(down.1(f.1(c.))) 274.75/150.98 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.98 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.98 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.98 down.1(f.1(f.1(f.1(y13)))) -> f_flat.0(down.1(f.1(f.1(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.0(f.0(b.))) 274.75/150.98 down.0(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.1(c.)) 274.75/150.98 down.0(g.0(fresh_constant.)) 274.75/150.98 down.0(f.0(g.0(x0))) 274.75/150.98 down.0(f.0(g.1(x0))) 274.75/150.98 down.1(f.1(c.)) 274.75/150.98 down.0(f.0(fresh_constant.)) 274.75/150.98 down.0(g.0(f.0(g.0(x0)))) 274.75/150.98 down.0(g.0(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.0(f.0(x0)))) 274.75/150.98 down.0(g.1(f.1(f.1(x0)))) 274.75/150.98 down.0(g.1(f.1(c.))) 274.75/150.98 down.0(g.0(f.0(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(g.0(x0)))) 274.75/150.98 down.0(f.0(f.0(g.1(x0)))) 274.75/150.98 down.0(f.0(f.0(f.0(x0)))) 274.75/150.98 down.1(f.1(f.1(f.1(x0)))) 274.75/150.98 down.1(f.1(f.1(c.))) 274.75/150.98 down.0(f.0(f.0(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.0(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.0(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (114) UsableRulesReductionPairsProof (EQUIVALENT) 274.75/150.98 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. 274.75/150.98 274.75/150.98 No dependency pairs are removed. 274.75/150.98 274.75/150.98 The following rules are removed from R: 274.75/150.98 274.75/150.98 down.1(f.1(f.1(c.))) -> f_flat.0(down.1(f.1(c.))) 274.75/150.98 down.1(f.1(f.1(f.1(y13)))) -> f_flat.0(down.1(f.1(f.1(y13)))) 274.75/150.98 Used ordering: POLO with Polynomial interpretation [POLO]: 274.75/150.98 274.75/150.98 POL(TOP.0(x_1)) = x_1 274.75/150.98 POL(b.) = 0 274.75/150.98 POL(down.0(x_1)) = 1 + x_1 274.75/150.98 POL(f.0(x_1)) = x_1 274.75/150.98 POL(f.1(x_1)) = x_1 274.75/150.98 POL(f_flat.0(x_1)) = x_1 274.75/150.98 POL(g.0(x_1)) = 1 + x_1 274.75/150.98 POL(g.1(x_1)) = 1 + x_1 274.75/150.98 POL(g_flat.0(x_1)) = 1 + x_1 274.75/150.98 POL(up.0(x_1)) = 1 + x_1 274.75/150.98 POL(up.1(x_1)) = 1 + x_1 274.75/150.98 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (115) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.98 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.98 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 274.75/150.98 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.98 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.98 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.0(f.0(b.))) 274.75/150.98 down.0(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.1(c.)) 274.75/150.98 down.0(g.0(fresh_constant.)) 274.75/150.98 down.0(f.0(g.0(x0))) 274.75/150.98 down.0(f.0(g.1(x0))) 274.75/150.98 down.1(f.1(c.)) 274.75/150.98 down.0(f.0(fresh_constant.)) 274.75/150.98 down.0(g.0(f.0(g.0(x0)))) 274.75/150.98 down.0(g.0(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.0(f.0(x0)))) 274.75/150.98 down.0(g.1(f.1(f.1(x0)))) 274.75/150.98 down.0(g.1(f.1(c.))) 274.75/150.98 down.0(g.0(f.0(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(g.0(x0)))) 274.75/150.98 down.0(f.0(f.0(g.1(x0)))) 274.75/150.98 down.0(f.0(f.0(f.0(x0)))) 274.75/150.98 down.1(f.1(f.1(f.1(x0)))) 274.75/150.98 down.1(f.1(f.1(c.))) 274.75/150.98 down.0(f.0(f.0(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.0(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.0(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (116) MRRProof (EQUIVALENT) 274.75/150.98 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. 274.75/150.98 274.75/150.98 274.75/150.98 Strictly oriented rules of the TRS R: 274.75/150.98 274.75/150.98 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 274.75/150.98 Used ordering: Polynomial interpretation [POLO]: 274.75/150.98 274.75/150.98 POL(TOP.0(x_1)) = x_1 274.75/150.98 POL(b.) = 0 274.75/150.98 POL(down.0(x_1)) = x_1 274.75/150.98 POL(f.0(x_1)) = x_1 274.75/150.98 POL(f.1(x_1)) = x_1 274.75/150.98 POL(f_flat.0(x_1)) = x_1 274.75/150.98 POL(g.0(x_1)) = 1 + x_1 274.75/150.98 POL(g.1(x_1)) = 1 + x_1 274.75/150.98 POL(g_flat.0(x_1)) = 1 + x_1 274.75/150.98 POL(up.0(x_1)) = x_1 274.75/150.98 POL(up.1(x_1)) = 1 + x_1 274.75/150.98 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (117) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9)))) 274.75/150.98 down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10)))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 274.75/150.98 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 274.75/150.98 down.0(f.0(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.0(f.0(g.0(y12)))) -> f_flat.0(down.0(f.0(g.0(y12)))) 274.75/150.98 down.0(f.0(f.0(g.1(y12)))) -> f_flat.0(down.0(f.0(g.1(y12)))) 274.75/150.98 down.0(f.0(f.0(f.0(y13)))) -> f_flat.0(down.0(f.0(f.0(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.0(f.0(b.))) 274.75/150.98 down.0(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.1(c.)) 274.75/150.98 down.0(g.0(fresh_constant.)) 274.75/150.98 down.0(f.0(g.0(x0))) 274.75/150.98 down.0(f.0(g.1(x0))) 274.75/150.98 down.1(f.1(c.)) 274.75/150.98 down.0(f.0(fresh_constant.)) 274.75/150.98 down.0(g.0(f.0(g.0(x0)))) 274.75/150.98 down.0(g.0(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.0(f.0(x0)))) 274.75/150.98 down.0(g.1(f.1(f.1(x0)))) 274.75/150.98 down.0(g.1(f.1(c.))) 274.75/150.98 down.0(g.0(f.0(fresh_constant.))) 274.75/150.98 down.0(f.0(f.0(g.0(x0)))) 274.75/150.98 down.0(f.0(f.0(g.1(x0)))) 274.75/150.98 down.0(f.0(f.0(f.0(x0)))) 274.75/150.98 down.1(f.1(f.1(f.1(x0)))) 274.75/150.98 down.1(f.1(f.1(c.))) 274.75/150.98 down.0(f.0(f.0(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.0(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.0(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (118) PisEmptyProof (SOUND) 274.75/150.98 The TRS P is empty. Hence, there is no (P,Q,R) chain. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (119) 274.75/150.98 TRUE 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (120) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) 274.75/150.98 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 274.75/150.98 TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) 274.75/150.98 TOP(up(f(f(f(x0))))) -> TOP(f_flat(down(f(f(x0))))) 274.75/150.98 TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))) 274.75/150.98 TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down(g(b)) -> up(g(f(f(f(f(f(b))))))) 274.75/150.98 down(g(g(y3))) -> g_flat(down(g(y3))) 274.75/150.98 down(g(f(g(y9)))) -> g_flat(down(f(g(y9)))) 274.75/150.98 down(g(f(f(y10)))) -> g_flat(down(f(f(y10)))) 274.75/150.98 f_flat(up(x_1)) -> up(f(x_1)) 274.75/150.98 down(f(f(b))) -> up(b) 274.75/150.98 down(f(f(g(y12)))) -> f_flat(down(f(g(y12)))) 274.75/150.98 down(f(f(f(y13)))) -> f_flat(down(f(f(y13)))) 274.75/150.98 g_flat(up(x_1)) -> up(g(x_1)) 274.75/150.98 down(f(g(y6))) -> f_flat(down(g(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down(g(b)) 274.75/150.98 down(f(f(b))) 274.75/150.98 down(f(b)) 274.75/150.98 down(g(g(x0))) 274.75/150.98 down(g(c)) 274.75/150.98 down(g(fresh_constant)) 274.75/150.98 down(f(g(x0))) 274.75/150.98 down(f(c)) 274.75/150.98 down(f(fresh_constant)) 274.75/150.98 down(g(f(g(x0)))) 274.75/150.98 down(g(f(f(x0)))) 274.75/150.98 down(g(f(c))) 274.75/150.98 down(g(f(fresh_constant))) 274.75/150.98 down(f(f(g(x0)))) 274.75/150.98 down(f(f(f(x0)))) 274.75/150.98 down(f(f(c))) 274.75/150.98 down(f(f(fresh_constant))) 274.75/150.98 g_flat(up(x0)) 274.75/150.98 f_flat(up(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (121) SemLabProof (SOUND) 274.75/150.98 We found the following model for the rules of the TRSs R and P. 274.75/150.98 Interpretation over the domain with elements from 0 to 1. 274.75/150.98 b: 0 274.75/150.98 c: 0 274.75/150.98 down: x0 274.75/150.98 f: 1 + x0 274.75/150.98 fresh_constant: 0 274.75/150.98 up: x0 274.75/150.98 f_flat: 1 + x0 274.75/150.98 TOP: 0 274.75/150.98 g: 0 274.75/150.98 g_flat: 0 274.75/150.98 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (122) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.1(up.1(f.0(g.0(x0)))) -> TOP.1(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.1(up.1(f.0(g.1(x0)))) -> TOP.1(f_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.1(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.1(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(f.1(x0))))) -> TOP.0(g_flat.1(down.1(f.0(f.1(x0))))) 274.75/150.98 TOP.1(up.1(f.0(f.1(f.0(x0))))) -> TOP.1(f_flat.0(down.0(f.1(f.0(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(f.1(x0))))) -> TOP.0(f_flat.1(down.1(f.0(f.1(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(g.0(x0))))) -> TOP.0(g_flat.1(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(g.1(x0))))) -> TOP.0(g_flat.1(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(g.0(x0))))) -> TOP.0(f_flat.1(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(g.1(x0))))) -> TOP.0(f_flat.1(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) -> up.0(g.1(f.0(f.1(f.0(f.1(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.1(f.0(g.0(y9)))) -> g_flat.1(down.1(f.0(g.0(y9)))) 274.75/150.98 down.0(g.1(f.0(g.1(y9)))) -> g_flat.1(down.1(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.1(f.0(y10)))) -> g_flat.0(down.0(f.1(f.0(y10)))) 274.75/150.98 down.0(g.1(f.0(f.1(y10)))) -> g_flat.1(down.1(f.0(f.1(y10)))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.1(f.0(x_1)) 274.75/150.98 f_flat.1(up.1(x_1)) -> up.0(f.1(x_1)) 274.75/150.98 down.0(f.1(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.1(f.0(g.0(y12)))) -> f_flat.1(down.1(f.0(g.0(y12)))) 274.75/150.98 down.0(f.1(f.0(g.1(y12)))) -> f_flat.1(down.1(f.0(g.1(y12)))) 274.75/150.98 down.1(f.0(f.1(f.0(y13)))) -> f_flat.0(down.0(f.1(f.0(y13)))) 274.75/150.98 down.0(f.1(f.0(f.1(y13)))) -> f_flat.1(down.1(f.0(f.1(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.1(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.1(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.1(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.1(f.0(b.))) 274.75/150.98 down.1(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.0(fresh_constant.)) 274.75/150.98 down.1(f.0(g.0(x0))) 274.75/150.98 down.1(f.0(g.1(x0))) 274.75/150.98 down.1(f.0(c.)) 274.75/150.98 down.1(f.0(fresh_constant.)) 274.75/150.98 down.0(g.1(f.0(g.0(x0)))) 274.75/150.98 down.0(g.1(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.1(f.0(x0)))) 274.75/150.98 down.0(g.1(f.0(f.1(x0)))) 274.75/150.98 down.0(g.1(f.0(c.))) 274.75/150.98 down.0(g.1(f.0(fresh_constant.))) 274.75/150.98 down.0(f.1(f.0(g.0(x0)))) 274.75/150.98 down.0(f.1(f.0(g.1(x0)))) 274.75/150.98 down.1(f.0(f.1(f.0(x0)))) 274.75/150.98 down.0(f.1(f.0(f.1(x0)))) 274.75/150.98 down.0(f.1(f.0(c.))) 274.75/150.98 down.0(f.1(f.0(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.1(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.1(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (123) DependencyGraphProof (EQUIVALENT) 274.75/150.98 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (124) 274.75/150.98 Complex Obligation (AND) 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (125) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.1(up.1(f.0(g.0(x0)))) -> TOP.1(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.1(up.1(f.0(g.1(x0)))) -> TOP.1(f_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.1(up.1(f.0(f.1(f.0(x0))))) -> TOP.1(f_flat.0(down.0(f.1(f.0(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) -> up.0(g.1(f.0(f.1(f.0(f.1(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.1(f.0(g.0(y9)))) -> g_flat.1(down.1(f.0(g.0(y9)))) 274.75/150.98 down.0(g.1(f.0(g.1(y9)))) -> g_flat.1(down.1(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.1(f.0(y10)))) -> g_flat.0(down.0(f.1(f.0(y10)))) 274.75/150.98 down.0(g.1(f.0(f.1(y10)))) -> g_flat.1(down.1(f.0(f.1(y10)))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.1(f.0(x_1)) 274.75/150.98 f_flat.1(up.1(x_1)) -> up.0(f.1(x_1)) 274.75/150.98 down.0(f.1(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.1(f.0(g.0(y12)))) -> f_flat.1(down.1(f.0(g.0(y12)))) 274.75/150.98 down.0(f.1(f.0(g.1(y12)))) -> f_flat.1(down.1(f.0(g.1(y12)))) 274.75/150.98 down.1(f.0(f.1(f.0(y13)))) -> f_flat.0(down.0(f.1(f.0(y13)))) 274.75/150.98 down.0(f.1(f.0(f.1(y13)))) -> f_flat.1(down.1(f.0(f.1(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.1(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.1(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.1(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.1(f.0(b.))) 274.75/150.98 down.1(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.0(fresh_constant.)) 274.75/150.98 down.1(f.0(g.0(x0))) 274.75/150.98 down.1(f.0(g.1(x0))) 274.75/150.98 down.1(f.0(c.)) 274.75/150.98 down.1(f.0(fresh_constant.)) 274.75/150.98 down.0(g.1(f.0(g.0(x0)))) 274.75/150.98 down.0(g.1(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.1(f.0(x0)))) 274.75/150.98 down.0(g.1(f.0(f.1(x0)))) 274.75/150.98 down.0(g.1(f.0(c.))) 274.75/150.98 down.0(g.1(f.0(fresh_constant.))) 274.75/150.98 down.0(f.1(f.0(g.0(x0)))) 274.75/150.98 down.0(f.1(f.0(g.1(x0)))) 274.75/150.98 down.1(f.0(f.1(f.0(x0)))) 274.75/150.98 down.0(f.1(f.0(f.1(x0)))) 274.75/150.98 down.0(f.1(f.0(c.))) 274.75/150.98 down.0(f.1(f.0(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.1(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.1(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (126) MRRProof (EQUIVALENT) 274.75/150.98 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. 274.75/150.98 274.75/150.98 274.75/150.98 Strictly oriented rules of the TRS R: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) -> up.0(g.1(f.0(f.1(f.0(f.1(f.0(b.))))))) 274.75/150.98 274.75/150.98 Used ordering: Polynomial interpretation [POLO]: 274.75/150.98 274.75/150.98 POL(TOP.1(x_1)) = x_1 274.75/150.98 POL(b.) = 0 274.75/150.98 POL(down.0(x_1)) = x_1 274.75/150.98 POL(down.1(x_1)) = x_1 274.75/150.98 POL(f.0(x_1)) = x_1 274.75/150.98 POL(f.1(x_1)) = x_1 274.75/150.98 POL(f_flat.0(x_1)) = x_1 274.75/150.98 POL(f_flat.1(x_1)) = x_1 274.75/150.98 POL(g.0(x_1)) = 1 + x_1 274.75/150.98 POL(g.1(x_1)) = x_1 274.75/150.98 POL(g_flat.0(x_1)) = 1 + x_1 274.75/150.98 POL(g_flat.1(x_1)) = x_1 274.75/150.98 POL(up.0(x_1)) = x_1 274.75/150.98 POL(up.1(x_1)) = x_1 274.75/150.98 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (127) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.1(up.1(f.0(g.0(x0)))) -> TOP.1(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.1(up.1(f.0(g.1(x0)))) -> TOP.1(f_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.1(up.1(f.0(f.1(f.0(x0))))) -> TOP.1(f_flat.0(down.0(f.1(f.0(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.1(f.0(g.0(y9)))) -> g_flat.1(down.1(f.0(g.0(y9)))) 274.75/150.98 down.0(g.1(f.0(g.1(y9)))) -> g_flat.1(down.1(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.1(f.0(y10)))) -> g_flat.0(down.0(f.1(f.0(y10)))) 274.75/150.98 down.0(g.1(f.0(f.1(y10)))) -> g_flat.1(down.1(f.0(f.1(y10)))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.1(f.0(x_1)) 274.75/150.98 f_flat.1(up.1(x_1)) -> up.0(f.1(x_1)) 274.75/150.98 down.0(f.1(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.1(f.0(g.0(y12)))) -> f_flat.1(down.1(f.0(g.0(y12)))) 274.75/150.98 down.0(f.1(f.0(g.1(y12)))) -> f_flat.1(down.1(f.0(g.1(y12)))) 274.75/150.98 down.1(f.0(f.1(f.0(y13)))) -> f_flat.0(down.0(f.1(f.0(y13)))) 274.75/150.98 down.0(f.1(f.0(f.1(y13)))) -> f_flat.1(down.1(f.0(f.1(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.1(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.1(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.1(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.1(f.0(b.))) 274.75/150.98 down.1(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.0(fresh_constant.)) 274.75/150.98 down.1(f.0(g.0(x0))) 274.75/150.98 down.1(f.0(g.1(x0))) 274.75/150.98 down.1(f.0(c.)) 274.75/150.98 down.1(f.0(fresh_constant.)) 274.75/150.98 down.0(g.1(f.0(g.0(x0)))) 274.75/150.98 down.0(g.1(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.1(f.0(x0)))) 274.75/150.98 down.0(g.1(f.0(f.1(x0)))) 274.75/150.98 down.0(g.1(f.0(c.))) 274.75/150.98 down.0(g.1(f.0(fresh_constant.))) 274.75/150.98 down.0(f.1(f.0(g.0(x0)))) 274.75/150.98 down.0(f.1(f.0(g.1(x0)))) 274.75/150.98 down.1(f.0(f.1(f.0(x0)))) 274.75/150.98 down.0(f.1(f.0(f.1(x0)))) 274.75/150.98 down.0(f.1(f.0(c.))) 274.75/150.98 down.0(f.1(f.0(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.1(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.1(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (128) MRRProof (EQUIVALENT) 274.75/150.98 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. 274.75/150.98 274.75/150.98 274.75/150.98 Strictly oriented rules of the TRS R: 274.75/150.98 274.75/150.98 down.0(f.1(f.0(b.))) -> up.0(b.) 274.75/150.98 274.75/150.98 Used ordering: Polynomial interpretation [POLO]: 274.75/150.98 274.75/150.98 POL(TOP.1(x_1)) = x_1 274.75/150.98 POL(b.) = 0 274.75/150.98 POL(down.0(x_1)) = x_1 274.75/150.98 POL(down.1(x_1)) = x_1 274.75/150.98 POL(f.0(x_1)) = x_1 274.75/150.98 POL(f.1(x_1)) = 1 + x_1 274.75/150.98 POL(f_flat.0(x_1)) = x_1 274.75/150.98 POL(f_flat.1(x_1)) = 1 + x_1 274.75/150.98 POL(g.0(x_1)) = x_1 274.75/150.98 POL(g.1(x_1)) = x_1 274.75/150.98 POL(g_flat.0(x_1)) = x_1 274.75/150.98 POL(g_flat.1(x_1)) = x_1 274.75/150.98 POL(up.0(x_1)) = x_1 274.75/150.98 POL(up.1(x_1)) = x_1 274.75/150.98 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (129) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.1(up.1(f.0(g.0(x0)))) -> TOP.1(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.1(up.1(f.0(g.1(x0)))) -> TOP.1(f_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.1(up.1(f.0(f.1(f.0(x0))))) -> TOP.1(f_flat.0(down.0(f.1(f.0(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.1(f.0(g.0(y9)))) -> g_flat.1(down.1(f.0(g.0(y9)))) 274.75/150.98 down.0(g.1(f.0(g.1(y9)))) -> g_flat.1(down.1(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.1(f.0(y10)))) -> g_flat.0(down.0(f.1(f.0(y10)))) 274.75/150.98 down.0(g.1(f.0(f.1(y10)))) -> g_flat.1(down.1(f.0(f.1(y10)))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.1(f.0(x_1)) 274.75/150.98 f_flat.1(up.1(x_1)) -> up.0(f.1(x_1)) 274.75/150.98 down.0(f.1(f.0(g.0(y12)))) -> f_flat.1(down.1(f.0(g.0(y12)))) 274.75/150.98 down.0(f.1(f.0(g.1(y12)))) -> f_flat.1(down.1(f.0(g.1(y12)))) 274.75/150.98 down.1(f.0(f.1(f.0(y13)))) -> f_flat.0(down.0(f.1(f.0(y13)))) 274.75/150.98 down.0(f.1(f.0(f.1(y13)))) -> f_flat.1(down.1(f.0(f.1(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.1(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.1(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.1(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.1(f.0(b.))) 274.75/150.98 down.1(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.0(fresh_constant.)) 274.75/150.98 down.1(f.0(g.0(x0))) 274.75/150.98 down.1(f.0(g.1(x0))) 274.75/150.98 down.1(f.0(c.)) 274.75/150.98 down.1(f.0(fresh_constant.)) 274.75/150.98 down.0(g.1(f.0(g.0(x0)))) 274.75/150.98 down.0(g.1(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.1(f.0(x0)))) 274.75/150.98 down.0(g.1(f.0(f.1(x0)))) 274.75/150.98 down.0(g.1(f.0(c.))) 274.75/150.98 down.0(g.1(f.0(fresh_constant.))) 274.75/150.98 down.0(f.1(f.0(g.0(x0)))) 274.75/150.98 down.0(f.1(f.0(g.1(x0)))) 274.75/150.98 down.1(f.0(f.1(f.0(x0)))) 274.75/150.98 down.0(f.1(f.0(f.1(x0)))) 274.75/150.98 down.0(f.1(f.0(c.))) 274.75/150.98 down.0(f.1(f.0(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.1(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.1(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (130) MRRProof (EQUIVALENT) 274.75/150.98 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. 274.75/150.98 274.75/150.98 Strictly oriented dependency pairs: 274.75/150.98 274.75/150.98 TOP.1(up.1(f.0(g.0(x0)))) -> TOP.1(f_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.1(up.1(f.0(g.1(x0)))) -> TOP.1(f_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.1(up.1(f.0(f.1(f.0(x0))))) -> TOP.1(f_flat.0(down.0(f.1(f.0(x0))))) 274.75/150.98 274.75/150.98 274.75/150.98 Used ordering: Polynomial interpretation [POLO]: 274.75/150.98 274.75/150.98 POL(TOP.1(x_1)) = x_1 274.75/150.98 POL(down.0(x_1)) = x_1 274.75/150.98 POL(down.1(x_1)) = x_1 274.75/150.98 POL(f.0(x_1)) = x_1 274.75/150.98 POL(f.1(x_1)) = x_1 274.75/150.98 POL(f_flat.0(x_1)) = x_1 274.75/150.98 POL(f_flat.1(x_1)) = x_1 274.75/150.98 POL(g.0(x_1)) = x_1 274.75/150.98 POL(g.1(x_1)) = x_1 274.75/150.98 POL(g_flat.0(x_1)) = x_1 274.75/150.98 POL(g_flat.1(x_1)) = x_1 274.75/150.98 POL(up.0(x_1)) = 1 + x_1 274.75/150.98 POL(up.1(x_1)) = 1 + x_1 274.75/150.98 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (131) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 P is empty. 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.1(f.0(g.0(y9)))) -> g_flat.1(down.1(f.0(g.0(y9)))) 274.75/150.98 down.0(g.1(f.0(g.1(y9)))) -> g_flat.1(down.1(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.1(f.0(y10)))) -> g_flat.0(down.0(f.1(f.0(y10)))) 274.75/150.98 down.0(g.1(f.0(f.1(y10)))) -> g_flat.1(down.1(f.0(f.1(y10)))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.1(f.0(x_1)) 274.75/150.98 f_flat.1(up.1(x_1)) -> up.0(f.1(x_1)) 274.75/150.98 down.0(f.1(f.0(g.0(y12)))) -> f_flat.1(down.1(f.0(g.0(y12)))) 274.75/150.98 down.0(f.1(f.0(g.1(y12)))) -> f_flat.1(down.1(f.0(g.1(y12)))) 274.75/150.98 down.1(f.0(f.1(f.0(y13)))) -> f_flat.0(down.0(f.1(f.0(y13)))) 274.75/150.98 down.0(f.1(f.0(f.1(y13)))) -> f_flat.1(down.1(f.0(f.1(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.1(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.1(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.1(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.1(f.0(b.))) 274.75/150.98 down.1(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.0(fresh_constant.)) 274.75/150.98 down.1(f.0(g.0(x0))) 274.75/150.98 down.1(f.0(g.1(x0))) 274.75/150.98 down.1(f.0(c.)) 274.75/150.98 down.1(f.0(fresh_constant.)) 274.75/150.98 down.0(g.1(f.0(g.0(x0)))) 274.75/150.98 down.0(g.1(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.1(f.0(x0)))) 274.75/150.98 down.0(g.1(f.0(f.1(x0)))) 274.75/150.98 down.0(g.1(f.0(c.))) 274.75/150.98 down.0(g.1(f.0(fresh_constant.))) 274.75/150.98 down.0(f.1(f.0(g.0(x0)))) 274.75/150.98 down.0(f.1(f.0(g.1(x0)))) 274.75/150.98 down.1(f.0(f.1(f.0(x0)))) 274.75/150.98 down.0(f.1(f.0(f.1(x0)))) 274.75/150.98 down.0(f.1(f.0(c.))) 274.75/150.98 down.0(f.1(f.0(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.1(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.1(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (132) PisEmptyProof (EQUIVALENT) 274.75/150.98 The TRS P is empty. Hence, there is no (P,Q,R) chain. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (133) 274.75/150.98 YES 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (134) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.1(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.1(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(f.1(x0))))) -> TOP.0(g_flat.1(down.1(f.0(f.1(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(f.1(x0))))) -> TOP.0(f_flat.1(down.1(f.0(f.1(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(g.0(x0))))) -> TOP.0(g_flat.1(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(g.1(x0))))) -> TOP.0(g_flat.1(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(g.0(x0))))) -> TOP.0(f_flat.1(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(g.1(x0))))) -> TOP.0(f_flat.1(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) -> up.0(g.1(f.0(f.1(f.0(f.1(f.0(b.))))))) 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.1(f.0(g.0(y9)))) -> g_flat.1(down.1(f.0(g.0(y9)))) 274.75/150.98 down.0(g.1(f.0(g.1(y9)))) -> g_flat.1(down.1(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.1(f.0(y10)))) -> g_flat.0(down.0(f.1(f.0(y10)))) 274.75/150.98 down.0(g.1(f.0(f.1(y10)))) -> g_flat.1(down.1(f.0(f.1(y10)))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.1(f.0(x_1)) 274.75/150.98 f_flat.1(up.1(x_1)) -> up.0(f.1(x_1)) 274.75/150.98 down.0(f.1(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.1(f.0(g.0(y12)))) -> f_flat.1(down.1(f.0(g.0(y12)))) 274.75/150.98 down.0(f.1(f.0(g.1(y12)))) -> f_flat.1(down.1(f.0(g.1(y12)))) 274.75/150.98 down.1(f.0(f.1(f.0(y13)))) -> f_flat.0(down.0(f.1(f.0(y13)))) 274.75/150.98 down.0(f.1(f.0(f.1(y13)))) -> f_flat.1(down.1(f.0(f.1(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.1(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.1(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.1(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.1(f.0(b.))) 274.75/150.98 down.1(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.0(fresh_constant.)) 274.75/150.98 down.1(f.0(g.0(x0))) 274.75/150.98 down.1(f.0(g.1(x0))) 274.75/150.98 down.1(f.0(c.)) 274.75/150.98 down.1(f.0(fresh_constant.)) 274.75/150.98 down.0(g.1(f.0(g.0(x0)))) 274.75/150.98 down.0(g.1(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.1(f.0(x0)))) 274.75/150.98 down.0(g.1(f.0(f.1(x0)))) 274.75/150.98 down.0(g.1(f.0(c.))) 274.75/150.98 down.0(g.1(f.0(fresh_constant.))) 274.75/150.98 down.0(f.1(f.0(g.0(x0)))) 274.75/150.98 down.0(f.1(f.0(g.1(x0)))) 274.75/150.98 down.1(f.0(f.1(f.0(x0)))) 274.75/150.98 down.0(f.1(f.0(f.1(x0)))) 274.75/150.98 down.0(f.1(f.0(c.))) 274.75/150.98 down.0(f.1(f.0(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.1(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.1(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (135) MRRProof (EQUIVALENT) 274.75/150.98 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. 274.75/150.98 274.75/150.98 274.75/150.98 Strictly oriented rules of the TRS R: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) -> up.0(g.1(f.0(f.1(f.0(f.1(f.0(b.))))))) 274.75/150.98 274.75/150.98 Used ordering: Polynomial interpretation [POLO]: 274.75/150.98 274.75/150.98 POL(TOP.0(x_1)) = x_1 274.75/150.98 POL(b.) = 0 274.75/150.98 POL(down.0(x_1)) = x_1 274.75/150.98 POL(down.1(x_1)) = x_1 274.75/150.98 POL(f.0(x_1)) = x_1 274.75/150.98 POL(f.1(x_1)) = x_1 274.75/150.98 POL(f_flat.0(x_1)) = x_1 274.75/150.98 POL(f_flat.1(x_1)) = x_1 274.75/150.98 POL(g.0(x_1)) = 1 + x_1 274.75/150.98 POL(g.1(x_1)) = x_1 274.75/150.98 POL(g_flat.0(x_1)) = 1 + x_1 274.75/150.98 POL(g_flat.1(x_1)) = x_1 274.75/150.98 POL(up.0(x_1)) = x_1 274.75/150.98 POL(up.1(x_1)) = x_1 274.75/150.98 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (136) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.1(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.1(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(f.1(x0))))) -> TOP.0(g_flat.1(down.1(f.0(f.1(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(f.1(x0))))) -> TOP.0(f_flat.1(down.1(f.0(f.1(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(g.0(x0))))) -> TOP.0(g_flat.1(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(g.1(x0))))) -> TOP.0(g_flat.1(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(g.0(x0))))) -> TOP.0(f_flat.1(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(g.1(x0))))) -> TOP.0(f_flat.1(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.1(f.0(g.0(y9)))) -> g_flat.1(down.1(f.0(g.0(y9)))) 274.75/150.98 down.0(g.1(f.0(g.1(y9)))) -> g_flat.1(down.1(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.1(f.0(y10)))) -> g_flat.0(down.0(f.1(f.0(y10)))) 274.75/150.98 down.0(g.1(f.0(f.1(y10)))) -> g_flat.1(down.1(f.0(f.1(y10)))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.1(f.0(x_1)) 274.75/150.98 f_flat.1(up.1(x_1)) -> up.0(f.1(x_1)) 274.75/150.98 down.0(f.1(f.0(b.))) -> up.0(b.) 274.75/150.98 down.0(f.1(f.0(g.0(y12)))) -> f_flat.1(down.1(f.0(g.0(y12)))) 274.75/150.98 down.0(f.1(f.0(g.1(y12)))) -> f_flat.1(down.1(f.0(g.1(y12)))) 274.75/150.98 down.1(f.0(f.1(f.0(y13)))) -> f_flat.0(down.0(f.1(f.0(y13)))) 274.75/150.98 down.0(f.1(f.0(f.1(y13)))) -> f_flat.1(down.1(f.0(f.1(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.1(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.1(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.1(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.1(f.0(b.))) 274.75/150.98 down.1(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.0(fresh_constant.)) 274.75/150.98 down.1(f.0(g.0(x0))) 274.75/150.98 down.1(f.0(g.1(x0))) 274.75/150.98 down.1(f.0(c.)) 274.75/150.98 down.1(f.0(fresh_constant.)) 274.75/150.98 down.0(g.1(f.0(g.0(x0)))) 274.75/150.98 down.0(g.1(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.1(f.0(x0)))) 274.75/150.98 down.0(g.1(f.0(f.1(x0)))) 274.75/150.98 down.0(g.1(f.0(c.))) 274.75/150.98 down.0(g.1(f.0(fresh_constant.))) 274.75/150.98 down.0(f.1(f.0(g.0(x0)))) 274.75/150.98 down.0(f.1(f.0(g.1(x0)))) 274.75/150.98 down.1(f.0(f.1(f.0(x0)))) 274.75/150.98 down.0(f.1(f.0(f.1(x0)))) 274.75/150.98 down.0(f.1(f.0(c.))) 274.75/150.98 down.0(f.1(f.0(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.1(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.1(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (137) MRRProof (EQUIVALENT) 274.75/150.98 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. 274.75/150.98 274.75/150.98 274.75/150.98 Strictly oriented rules of the TRS R: 274.75/150.98 274.75/150.98 down.0(f.1(f.0(b.))) -> up.0(b.) 274.75/150.98 274.75/150.98 Used ordering: Polynomial interpretation [POLO]: 274.75/150.98 274.75/150.98 POL(TOP.0(x_1)) = x_1 274.75/150.98 POL(b.) = 0 274.75/150.98 POL(down.0(x_1)) = x_1 274.75/150.98 POL(down.1(x_1)) = x_1 274.75/150.98 POL(f.0(x_1)) = x_1 274.75/150.98 POL(f.1(x_1)) = 1 + x_1 274.75/150.98 POL(f_flat.0(x_1)) = x_1 274.75/150.98 POL(f_flat.1(x_1)) = 1 + x_1 274.75/150.98 POL(g.0(x_1)) = x_1 274.75/150.98 POL(g.1(x_1)) = x_1 274.75/150.98 POL(g_flat.0(x_1)) = x_1 274.75/150.98 POL(g_flat.1(x_1)) = x_1 274.75/150.98 POL(up.0(x_1)) = x_1 274.75/150.98 POL(up.1(x_1)) = x_1 274.75/150.98 274.75/150.98 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (138) 274.75/150.98 Obligation: 274.75/150.98 Q DP problem: 274.75/150.98 The TRS P consists of the following rules: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.1(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.1(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(f.1(x0))))) -> TOP.0(g_flat.1(down.1(f.0(f.1(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(f.1(x0))))) -> TOP.0(f_flat.1(down.1(f.0(f.1(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(g.0(x0))))) -> TOP.0(g_flat.1(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(g.1(x0))))) -> TOP.0(g_flat.1(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(g.0(x0))))) -> TOP.0(f_flat.1(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(g.1(x0))))) -> TOP.0(f_flat.1(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 274.75/150.98 The TRS R consists of the following rules: 274.75/150.98 274.75/150.98 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.98 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.98 down.0(g.1(f.0(g.0(y9)))) -> g_flat.1(down.1(f.0(g.0(y9)))) 274.75/150.98 down.0(g.1(f.0(g.1(y9)))) -> g_flat.1(down.1(f.0(g.1(y9)))) 274.75/150.98 down.0(g.0(f.1(f.0(y10)))) -> g_flat.0(down.0(f.1(f.0(y10)))) 274.75/150.98 down.0(g.1(f.0(f.1(y10)))) -> g_flat.1(down.1(f.0(f.1(y10)))) 274.75/150.98 f_flat.0(up.0(x_1)) -> up.1(f.0(x_1)) 274.75/150.98 f_flat.1(up.1(x_1)) -> up.0(f.1(x_1)) 274.75/150.98 down.0(f.1(f.0(g.0(y12)))) -> f_flat.1(down.1(f.0(g.0(y12)))) 274.75/150.98 down.0(f.1(f.0(g.1(y12)))) -> f_flat.1(down.1(f.0(g.1(y12)))) 274.75/150.98 down.1(f.0(f.1(f.0(y13)))) -> f_flat.0(down.0(f.1(f.0(y13)))) 274.75/150.98 down.0(f.1(f.0(f.1(y13)))) -> f_flat.1(down.1(f.0(f.1(y13)))) 274.75/150.98 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.98 g_flat.1(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.98 down.1(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.98 down.1(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.98 274.75/150.98 The set Q consists of the following terms: 274.75/150.98 274.75/150.98 down.0(g.0(b.)) 274.75/150.98 down.0(f.1(f.0(b.))) 274.75/150.98 down.1(f.0(b.)) 274.75/150.98 down.0(g.0(g.0(x0))) 274.75/150.98 down.0(g.0(g.1(x0))) 274.75/150.98 down.0(g.0(c.)) 274.75/150.98 down.0(g.0(fresh_constant.)) 274.75/150.98 down.1(f.0(g.0(x0))) 274.75/150.98 down.1(f.0(g.1(x0))) 274.75/150.98 down.1(f.0(c.)) 274.75/150.98 down.1(f.0(fresh_constant.)) 274.75/150.98 down.0(g.1(f.0(g.0(x0)))) 274.75/150.98 down.0(g.1(f.0(g.1(x0)))) 274.75/150.98 down.0(g.0(f.1(f.0(x0)))) 274.75/150.98 down.0(g.1(f.0(f.1(x0)))) 274.75/150.98 down.0(g.1(f.0(c.))) 274.75/150.98 down.0(g.1(f.0(fresh_constant.))) 274.75/150.98 down.0(f.1(f.0(g.0(x0)))) 274.75/150.98 down.0(f.1(f.0(g.1(x0)))) 274.75/150.98 down.1(f.0(f.1(f.0(x0)))) 274.75/150.98 down.0(f.1(f.0(f.1(x0)))) 274.75/150.98 down.0(f.1(f.0(c.))) 274.75/150.98 down.0(f.1(f.0(fresh_constant.))) 274.75/150.98 g_flat.0(up.0(x0)) 274.75/150.98 g_flat.1(up.1(x0)) 274.75/150.98 f_flat.0(up.0(x0)) 274.75/150.98 f_flat.1(up.1(x0)) 274.75/150.98 274.75/150.98 We have to consider all minimal (P,Q,R)-chains. 274.75/150.98 ---------------------------------------- 274.75/150.98 274.75/150.98 (139) MRRProof (EQUIVALENT) 274.75/150.98 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. 274.75/150.98 274.75/150.98 Strictly oriented dependency pairs: 274.75/150.98 274.75/150.98 TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0)))) 274.75/150.98 TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0)))) 274.75/150.98 TOP.0(up.0(g.0(f.1(f.0(x0))))) -> TOP.0(g_flat.0(down.0(f.1(f.0(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(f.1(x0))))) -> TOP.0(g_flat.1(down.1(f.0(f.1(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(f.1(x0))))) -> TOP.0(f_flat.1(down.1(f.0(f.1(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(g.0(x0))))) -> TOP.0(g_flat.1(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(g.1(f.0(g.1(x0))))) -> TOP.0(g_flat.1(f_flat.0(down.0(g.1(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(g.0(x0))))) -> TOP.0(f_flat.1(f_flat.0(down.0(g.0(x0))))) 274.75/150.98 TOP.0(up.0(f.1(f.0(g.1(x0))))) -> TOP.0(f_flat.1(f_flat.0(down.0(g.1(x0))))) 274.75/150.99 274.75/150.99 274.75/150.99 Used ordering: Polynomial interpretation [POLO]: 274.75/150.99 274.75/150.99 POL(TOP.0(x_1)) = x_1 274.75/150.99 POL(down.0(x_1)) = x_1 274.75/150.99 POL(down.1(x_1)) = x_1 274.75/150.99 POL(f.0(x_1)) = x_1 274.75/150.99 POL(f.1(x_1)) = 1 + x_1 274.75/150.99 POL(f_flat.0(x_1)) = x_1 274.75/150.99 POL(f_flat.1(x_1)) = 1 + x_1 274.75/150.99 POL(g.0(x_1)) = x_1 274.75/150.99 POL(g.1(x_1)) = x_1 274.75/150.99 POL(g_flat.0(x_1)) = x_1 274.75/150.99 POL(g_flat.1(x_1)) = x_1 274.75/150.99 POL(up.0(x_1)) = 1 + x_1 274.75/150.99 POL(up.1(x_1)) = 1 + x_1 274.75/150.99 274.75/150.99 274.75/150.99 ---------------------------------------- 274.75/150.99 274.75/150.99 (140) 274.75/150.99 Obligation: 274.75/150.99 Q DP problem: 274.75/150.99 P is empty. 274.75/150.99 The TRS R consists of the following rules: 274.75/150.99 274.75/150.99 down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3))) 274.75/150.99 down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3))) 274.75/150.99 down.0(g.1(f.0(g.0(y9)))) -> g_flat.1(down.1(f.0(g.0(y9)))) 274.75/150.99 down.0(g.1(f.0(g.1(y9)))) -> g_flat.1(down.1(f.0(g.1(y9)))) 274.75/150.99 down.0(g.0(f.1(f.0(y10)))) -> g_flat.0(down.0(f.1(f.0(y10)))) 274.75/150.99 down.0(g.1(f.0(f.1(y10)))) -> g_flat.1(down.1(f.0(f.1(y10)))) 274.75/150.99 f_flat.0(up.0(x_1)) -> up.1(f.0(x_1)) 274.75/150.99 f_flat.1(up.1(x_1)) -> up.0(f.1(x_1)) 274.75/150.99 down.0(f.1(f.0(g.0(y12)))) -> f_flat.1(down.1(f.0(g.0(y12)))) 274.75/150.99 down.0(f.1(f.0(g.1(y12)))) -> f_flat.1(down.1(f.0(g.1(y12)))) 274.75/150.99 down.1(f.0(f.1(f.0(y13)))) -> f_flat.0(down.0(f.1(f.0(y13)))) 274.75/150.99 down.0(f.1(f.0(f.1(y13)))) -> f_flat.1(down.1(f.0(f.1(y13)))) 274.75/150.99 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 274.75/150.99 g_flat.1(up.1(x_1)) -> up.0(g.1(x_1)) 274.75/150.99 down.1(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6))) 274.75/150.99 down.1(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6))) 274.75/150.99 274.75/150.99 The set Q consists of the following terms: 274.75/150.99 274.75/150.99 down.0(g.0(b.)) 274.75/150.99 down.0(f.1(f.0(b.))) 274.75/150.99 down.1(f.0(b.)) 274.75/150.99 down.0(g.0(g.0(x0))) 274.75/150.99 down.0(g.0(g.1(x0))) 274.75/150.99 down.0(g.0(c.)) 274.75/150.99 down.0(g.0(fresh_constant.)) 274.75/150.99 down.1(f.0(g.0(x0))) 274.75/150.99 down.1(f.0(g.1(x0))) 274.75/150.99 down.1(f.0(c.)) 274.75/150.99 down.1(f.0(fresh_constant.)) 274.75/150.99 down.0(g.1(f.0(g.0(x0)))) 274.75/150.99 down.0(g.1(f.0(g.1(x0)))) 274.75/150.99 down.0(g.0(f.1(f.0(x0)))) 274.75/150.99 down.0(g.1(f.0(f.1(x0)))) 274.75/150.99 down.0(g.1(f.0(c.))) 274.75/150.99 down.0(g.1(f.0(fresh_constant.))) 274.75/150.99 down.0(f.1(f.0(g.0(x0)))) 274.75/150.99 down.0(f.1(f.0(g.1(x0)))) 274.75/150.99 down.1(f.0(f.1(f.0(x0)))) 274.75/150.99 down.0(f.1(f.0(f.1(x0)))) 274.75/150.99 down.0(f.1(f.0(c.))) 274.75/150.99 down.0(f.1(f.0(fresh_constant.))) 274.75/150.99 g_flat.0(up.0(x0)) 274.75/150.99 g_flat.1(up.1(x0)) 274.75/150.99 f_flat.0(up.0(x0)) 274.75/150.99 f_flat.1(up.1(x0)) 274.75/150.99 274.75/150.99 We have to consider all minimal (P,Q,R)-chains. 274.75/150.99 ---------------------------------------- 274.75/150.99 274.75/150.99 (141) PisEmptyProof (EQUIVALENT) 274.75/150.99 The TRS P is empty. Hence, there is no (P,Q,R) chain. 274.75/150.99 ---------------------------------------- 274.75/150.99 274.75/150.99 (142) 274.75/150.99 YES 276.17/152.22 EOF