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274.75/150.95 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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