312.23/171.89 MAYBE 312.23/171.90 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 312.23/171.90 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 312.23/171.90 312.23/171.90 312.23/171.90 Outermost Termination of the given OTRS could not be shown: 312.23/171.90 312.23/171.90 (0) OTRS 312.23/171.90 (1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms] 312.23/171.90 (2) QTRS 312.23/171.90 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 312.23/171.90 (4) QDP 312.23/171.90 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 312.23/171.90 (6) AND 312.23/171.90 (7) QDP 312.23/171.90 (8) UsableRulesProof [EQUIVALENT, 0 ms] 312.23/171.90 (9) QDP 312.23/171.90 (10) QReductionProof [EQUIVALENT, 0 ms] 312.23/171.90 (11) QDP 312.23/171.90 (12) UsableRulesReductionPairsProof [EQUIVALENT, 5 ms] 312.23/171.90 (13) QDP 312.23/171.90 (14) DependencyGraphProof [EQUIVALENT, 0 ms] 312.23/171.90 (15) TRUE 312.23/171.90 (16) QDP 312.23/171.90 (17) UsableRulesProof [EQUIVALENT, 0 ms] 312.23/171.90 (18) QDP 312.23/171.90 (19) QReductionProof [EQUIVALENT, 0 ms] 312.23/171.90 (20) QDP 312.23/171.90 (21) UsableRulesReductionPairsProof [EQUIVALENT, 5 ms] 312.23/171.90 (22) QDP 312.23/171.90 (23) DependencyGraphProof [EQUIVALENT, 0 ms] 312.23/171.90 (24) TRUE 312.23/171.90 (25) QDP 312.23/171.90 (26) UsableRulesProof [EQUIVALENT, 0 ms] 312.23/171.90 (27) QDP 312.23/171.90 (28) QReductionProof [EQUIVALENT, 0 ms] 312.23/171.90 (29) QDP 312.23/171.90 (30) TransformationProof [EQUIVALENT, 0 ms] 312.23/171.90 (31) QDP 312.23/171.90 (32) QDPOrderProof [EQUIVALENT, 12 ms] 312.23/171.90 (33) QDP 312.23/171.90 (34) UsableRulesProof [EQUIVALENT, 0 ms] 312.23/171.90 (35) QDP 312.23/171.90 (36) QReductionProof [EQUIVALENT, 0 ms] 312.23/171.90 (37) QDP 312.23/171.90 (38) Trivial-Transformation [SOUND, 0 ms] 312.23/171.90 (39) QTRS 312.23/171.90 (40) DependencyPairsProof [EQUIVALENT, 0 ms] 312.23/171.90 (41) QDP 312.23/171.90 (42) DependencyGraphProof [EQUIVALENT, 3 ms] 312.23/171.90 (43) QDP 312.23/171.90 (44) UsableRulesProof [EQUIVALENT, 0 ms] 312.23/171.90 (45) QDP 312.23/171.90 (46) NonTerminationLoopProof [COMPLETE, 0 ms] 312.23/171.90 (47) NO 312.23/171.90 (48) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] 312.23/171.90 (49) QTRS 312.23/171.90 (50) DependencyPairsProof [EQUIVALENT, 76 ms] 312.23/171.90 (51) QDP 312.23/171.90 (52) DependencyGraphProof [EQUIVALENT, 0 ms] 312.23/171.90 (53) QDP 312.23/171.90 (54) UsableRulesProof [EQUIVALENT, 0 ms] 312.23/171.90 (55) QDP 312.23/171.90 (56) TransformationProof [EQUIVALENT, 0 ms] 312.23/171.90 (57) QDP 312.23/171.90 (58) DependencyGraphProof [EQUIVALENT, 0 ms] 312.23/171.90 (59) QDP 312.23/171.90 (60) UsableRulesProof [EQUIVALENT, 11 ms] 312.23/171.90 (61) QDP 312.23/171.90 (62) TransformationProof [EQUIVALENT, 32 ms] 312.23/171.90 (63) QDP 312.23/171.90 (64) DependencyGraphProof [EQUIVALENT, 0 ms] 312.23/171.90 (65) QDP 312.23/171.90 (66) TransformationProof [EQUIVALENT, 0 ms] 312.23/171.90 (67) QDP 312.23/171.90 (68) DependencyGraphProof [EQUIVALENT, 0 ms] 312.23/171.90 (69) QDP 312.23/171.90 (70) TransformationProof [EQUIVALENT, 0 ms] 312.23/171.90 (71) QDP 312.23/171.90 (72) DependencyGraphProof [EQUIVALENT, 0 ms] 312.23/171.90 (73) QDP 312.23/171.90 (74) TransformationProof [EQUIVALENT, 27 ms] 312.23/171.90 (75) QDP 312.23/171.90 (76) DependencyGraphProof [EQUIVALENT, 0 ms] 312.23/171.90 (77) QDP 312.23/171.90 (78) MRRProof [EQUIVALENT, 332 ms] 312.23/171.90 (79) QDP 312.23/171.90 (80) UsableRulesProof [EQUIVALENT, 0 ms] 312.23/171.90 (81) QDP 312.23/171.90 (82) QDPOrderProof [EQUIVALENT, 30 ms] 312.23/171.90 (83) QDP 312.23/171.90 (84) QDPOrderProof [EQUIVALENT, 24 ms] 312.23/171.90 (85) QDP 312.23/171.90 (86) UsableRulesProof [EQUIVALENT, 0 ms] 312.23/171.90 (87) QDP 312.23/171.90 (88) QDPOrderProof [EQUIVALENT, 1343 ms] 312.23/171.90 (89) QDP 312.23/171.90 (90) QDPOrderProof [EQUIVALENT, 3137 ms] 312.23/171.90 (91) QDP 312.23/171.90 (92) SplitQDPProof [EQUIVALENT, 0 ms] 312.23/171.90 (93) AND 312.23/171.90 (94) QDP 312.23/171.90 (95) SemLabProof [SOUND, 0 ms] 312.23/171.90 (96) QDP 312.23/171.90 (97) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 312.23/171.90 (98) QDP 312.23/171.90 (99) DependencyGraphProof [EQUIVALENT, 0 ms] 312.23/171.90 (100) QDP 312.23/171.90 (101) MRRProof [EQUIVALENT, 9 ms] 312.23/171.90 (102) QDP 312.23/171.90 (103) DependencyGraphProof [EQUIVALENT, 0 ms] 312.23/171.90 (104) QDP 312.23/171.90 (105) QDPOrderProof [EQUIVALENT, 23 ms] 312.23/171.90 (106) QDP 312.23/171.90 (107) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 312.23/171.90 (108) QDP 312.23/171.90 (109) QDPOrderProof [EQUIVALENT, 18 ms] 312.23/171.90 (110) QDP 312.23/171.90 (111) PisEmptyProof [SOUND, 0 ms] 312.23/171.90 (112) TRUE 312.23/171.90 (113) QDP 312.23/171.90 (114) QDPOrderProof [EQUIVALENT, 2559 ms] 312.23/171.90 (115) QDP 312.23/171.90 (116) QDPOrderProof [EQUIVALENT, 1722 ms] 312.23/171.90 (117) QDP 312.23/171.90 (118) SplitQDPProof [EQUIVALENT, 0 ms] 312.23/171.90 (119) AND 312.23/171.90 (120) QDP 312.23/171.90 (121) SemLabProof [SOUND, 0 ms] 312.23/171.90 (122) QDP 312.23/171.90 (123) MRRProof [EQUIVALENT, 0 ms] 312.23/171.90 (124) QDP 312.23/171.90 (125) DependencyGraphProof [EQUIVALENT, 0 ms] 312.23/171.90 (126) QDP 312.23/171.90 (127) QDPOrderProof [EQUIVALENT, 0 ms] 312.23/171.90 (128) QDP 312.23/171.90 (129) UsableRulesReductionPairsProof [EQUIVALENT, 20 ms] 312.23/171.90 (130) QDP 312.23/171.90 (131) PisEmptyProof [SOUND, 0 ms] 312.23/171.90 (132) TRUE 312.23/171.90 (133) QDP 312.23/171.90 (134) SplitQDPProof [EQUIVALENT, 0 ms] 312.23/171.90 (135) AND 312.23/171.90 (136) QDP 312.23/171.90 (137) SemLabProof [SOUND, 0 ms] 312.23/171.90 (138) QDP 312.23/171.90 (139) MRRProof [EQUIVALENT, 5 ms] 312.23/171.90 (140) QDP 312.23/171.90 (141) DependencyGraphProof [EQUIVALENT, 0 ms] 312.23/171.90 (142) QDP 312.23/171.90 (143) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 312.23/171.90 (144) QDP 312.23/171.90 (145) MRRProof [EQUIVALENT, 0 ms] 312.23/171.90 (146) QDP 312.23/171.90 (147) QDPOrderProof [EQUIVALENT, 0 ms] 312.23/171.90 (148) QDP 312.23/171.90 (149) UsableRulesReductionPairsProof [EQUIVALENT, 8 ms] 312.23/171.90 (150) QDP 312.23/171.90 (151) PisEmptyProof [SOUND, 0 ms] 312.23/171.90 (152) TRUE 312.23/171.90 (153) QDP 312.23/171.90 312.23/171.90 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (0) 312.23/171.90 Obligation: 312.23/171.90 Term rewrite system R: 312.23/171.90 The TRS R consists of the following rules: 312.23/171.90 312.23/171.90 a -> f(a) 312.23/171.90 a -> g(a) 312.23/171.90 f(f(f(f(f(f(x)))))) -> b 312.23/171.90 g(g(g(g(g(g(x)))))) -> b 312.23/171.90 g(f(x)) -> b 312.23/171.90 312.23/171.90 312.23/171.90 312.23/171.90 Outermost Strategy. 312.23/171.90 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (1) Thiemann-SpecialC-Transformation (EQUIVALENT) 312.23/171.90 We applied the Thiemann-SpecialC transformation to transform the outermost TRS to an innermost TRS. 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (2) 312.23/171.90 Obligation: 312.23/171.90 Q restricted rewrite system: 312.23/171.90 The TRS R consists of the following rules: 312.23/171.90 312.23/171.90 top(go_up(x)) -> top(reduce(x)) 312.23/171.90 reduce(f(x_1)) -> check_f(redex_f(x_1)) 312.23/171.90 reduce(g(x_1)) -> check_g(redex_g(x_1)) 312.23/171.90 reduce(a) -> go_up(f(a)) 312.23/171.90 reduce(a) -> go_up(g(a)) 312.23/171.90 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.90 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.90 redex_g(f(x)) -> result_g(b) 312.23/171.90 check_f(result_f(x)) -> go_up(x) 312.23/171.90 check_g(result_g(x)) -> go_up(x) 312.23/171.90 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 312.23/171.90 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 312.23/171.90 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 312.23/171.90 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 312.23/171.90 312.23/171.90 The set Q consists of the following terms: 312.23/171.90 312.23/171.90 top(go_up(x0)) 312.23/171.90 reduce(f(x0)) 312.23/171.90 reduce(g(x0)) 312.23/171.90 reduce(a) 312.23/171.90 redex_f(f(f(f(f(f(x0)))))) 312.23/171.90 redex_g(g(g(g(g(g(x0)))))) 312.23/171.90 redex_g(f(x0)) 312.23/171.90 check_f(result_f(x0)) 312.23/171.90 check_g(result_g(x0)) 312.23/171.90 check_f(redex_f(x0)) 312.23/171.90 check_g(redex_g(x0)) 312.23/171.90 in_f_1(go_up(x0)) 312.23/171.90 in_g_1(go_up(x0)) 312.23/171.90 312.23/171.90 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (3) DependencyPairsProof (EQUIVALENT) 312.23/171.90 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (4) 312.23/171.90 Obligation: 312.23/171.90 Q DP problem: 312.23/171.90 The TRS P consists of the following rules: 312.23/171.90 312.23/171.90 TOP(go_up(x)) -> TOP(reduce(x)) 312.23/171.90 TOP(go_up(x)) -> REDUCE(x) 312.23/171.90 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 312.23/171.90 REDUCE(f(x_1)) -> REDEX_F(x_1) 312.23/171.90 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 312.23/171.90 REDUCE(g(x_1)) -> REDEX_G(x_1) 312.23/171.90 CHECK_F(redex_f(x_1)) -> IN_F_1(reduce(x_1)) 312.23/171.90 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 312.23/171.90 CHECK_G(redex_g(x_1)) -> IN_G_1(reduce(x_1)) 312.23/171.90 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 312.23/171.90 312.23/171.90 The TRS R consists of the following rules: 312.23/171.90 312.23/171.90 top(go_up(x)) -> top(reduce(x)) 312.23/171.90 reduce(f(x_1)) -> check_f(redex_f(x_1)) 312.23/171.90 reduce(g(x_1)) -> check_g(redex_g(x_1)) 312.23/171.90 reduce(a) -> go_up(f(a)) 312.23/171.90 reduce(a) -> go_up(g(a)) 312.23/171.90 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.90 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.90 redex_g(f(x)) -> result_g(b) 312.23/171.90 check_f(result_f(x)) -> go_up(x) 312.23/171.90 check_g(result_g(x)) -> go_up(x) 312.23/171.90 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 312.23/171.90 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 312.23/171.90 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 312.23/171.90 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 312.23/171.90 312.23/171.90 The set Q consists of the following terms: 312.23/171.90 312.23/171.90 top(go_up(x0)) 312.23/171.90 reduce(f(x0)) 312.23/171.90 reduce(g(x0)) 312.23/171.90 reduce(a) 312.23/171.90 redex_f(f(f(f(f(f(x0)))))) 312.23/171.90 redex_g(g(g(g(g(g(x0)))))) 312.23/171.90 redex_g(f(x0)) 312.23/171.90 check_f(result_f(x0)) 312.23/171.90 check_g(result_g(x0)) 312.23/171.90 check_f(redex_f(x0)) 312.23/171.90 check_g(redex_g(x0)) 312.23/171.90 in_f_1(go_up(x0)) 312.23/171.90 in_g_1(go_up(x0)) 312.23/171.90 312.23/171.90 We have to consider all minimal (P,Q,R)-chains. 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (5) DependencyGraphProof (EQUIVALENT) 312.23/171.90 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 5 less nodes. 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (6) 312.23/171.90 Complex Obligation (AND) 312.23/171.90 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (7) 312.23/171.90 Obligation: 312.23/171.90 Q DP problem: 312.23/171.90 The TRS P consists of the following rules: 312.23/171.90 312.23/171.90 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 312.23/171.90 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 312.23/171.90 312.23/171.90 The TRS R consists of the following rules: 312.23/171.90 312.23/171.90 top(go_up(x)) -> top(reduce(x)) 312.23/171.90 reduce(f(x_1)) -> check_f(redex_f(x_1)) 312.23/171.90 reduce(g(x_1)) -> check_g(redex_g(x_1)) 312.23/171.90 reduce(a) -> go_up(f(a)) 312.23/171.90 reduce(a) -> go_up(g(a)) 312.23/171.90 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.90 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.90 redex_g(f(x)) -> result_g(b) 312.23/171.90 check_f(result_f(x)) -> go_up(x) 312.23/171.90 check_g(result_g(x)) -> go_up(x) 312.23/171.90 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 312.23/171.90 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 312.23/171.90 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 312.23/171.90 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 312.23/171.90 312.23/171.90 The set Q consists of the following terms: 312.23/171.90 312.23/171.90 top(go_up(x0)) 312.23/171.90 reduce(f(x0)) 312.23/171.90 reduce(g(x0)) 312.23/171.90 reduce(a) 312.23/171.90 redex_f(f(f(f(f(f(x0)))))) 312.23/171.90 redex_g(g(g(g(g(g(x0)))))) 312.23/171.90 redex_g(f(x0)) 312.23/171.90 check_f(result_f(x0)) 312.23/171.90 check_g(result_g(x0)) 312.23/171.90 check_f(redex_f(x0)) 312.23/171.90 check_g(redex_g(x0)) 312.23/171.90 in_f_1(go_up(x0)) 312.23/171.90 in_g_1(go_up(x0)) 312.23/171.90 312.23/171.90 We have to consider all minimal (P,Q,R)-chains. 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (8) UsableRulesProof (EQUIVALENT) 312.23/171.90 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. 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (9) 312.23/171.90 Obligation: 312.23/171.90 Q DP problem: 312.23/171.90 The TRS P consists of the following rules: 312.23/171.90 312.23/171.90 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 312.23/171.90 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 312.23/171.90 312.23/171.90 The TRS R consists of the following rules: 312.23/171.90 312.23/171.90 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.90 redex_g(f(x)) -> result_g(b) 312.23/171.90 312.23/171.90 The set Q consists of the following terms: 312.23/171.90 312.23/171.90 top(go_up(x0)) 312.23/171.90 reduce(f(x0)) 312.23/171.90 reduce(g(x0)) 312.23/171.90 reduce(a) 312.23/171.90 redex_f(f(f(f(f(f(x0)))))) 312.23/171.90 redex_g(g(g(g(g(g(x0)))))) 312.23/171.90 redex_g(f(x0)) 312.23/171.90 check_f(result_f(x0)) 312.23/171.90 check_g(result_g(x0)) 312.23/171.90 check_f(redex_f(x0)) 312.23/171.90 check_g(redex_g(x0)) 312.23/171.90 in_f_1(go_up(x0)) 312.23/171.90 in_g_1(go_up(x0)) 312.23/171.90 312.23/171.90 We have to consider all minimal (P,Q,R)-chains. 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (10) QReductionProof (EQUIVALENT) 312.23/171.90 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 312.23/171.90 312.23/171.90 top(go_up(x0)) 312.23/171.90 reduce(f(x0)) 312.23/171.90 reduce(g(x0)) 312.23/171.90 reduce(a) 312.23/171.90 redex_f(f(f(f(f(f(x0)))))) 312.23/171.90 check_f(result_f(x0)) 312.23/171.90 check_g(result_g(x0)) 312.23/171.90 check_f(redex_f(x0)) 312.23/171.90 check_g(redex_g(x0)) 312.23/171.90 in_f_1(go_up(x0)) 312.23/171.90 in_g_1(go_up(x0)) 312.23/171.90 312.23/171.90 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (11) 312.23/171.90 Obligation: 312.23/171.90 Q DP problem: 312.23/171.90 The TRS P consists of the following rules: 312.23/171.90 312.23/171.90 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 312.23/171.90 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 312.23/171.90 312.23/171.90 The TRS R consists of the following rules: 312.23/171.90 312.23/171.90 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.90 redex_g(f(x)) -> result_g(b) 312.23/171.90 312.23/171.90 The set Q consists of the following terms: 312.23/171.90 312.23/171.90 redex_g(g(g(g(g(g(x0)))))) 312.23/171.90 redex_g(f(x0)) 312.23/171.90 312.23/171.90 We have to consider all minimal (P,Q,R)-chains. 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (12) UsableRulesReductionPairsProof (EQUIVALENT) 312.23/171.90 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. 312.23/171.90 312.23/171.90 The following dependency pairs can be deleted: 312.23/171.90 312.23/171.90 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 312.23/171.90 The following rules are removed from R: 312.23/171.90 312.23/171.90 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.90 redex_g(f(x)) -> result_g(b) 312.23/171.90 Used ordering: POLO with Polynomial interpretation [POLO]: 312.23/171.90 312.23/171.90 POL(CHECK_G(x_1)) = 2*x_1 312.23/171.90 POL(REDUCE(x_1)) = 2*x_1 312.23/171.90 POL(b) = 0 312.23/171.90 POL(f(x_1)) = x_1 312.23/171.90 POL(g(x_1)) = 2*x_1 312.23/171.90 POL(redex_g(x_1)) = x_1 312.23/171.90 POL(result_g(x_1)) = x_1 312.23/171.90 312.23/171.90 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (13) 312.23/171.90 Obligation: 312.23/171.90 Q DP problem: 312.23/171.90 The TRS P consists of the following rules: 312.23/171.90 312.23/171.90 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 312.23/171.90 312.23/171.90 R is empty. 312.23/171.90 The set Q consists of the following terms: 312.23/171.90 312.23/171.90 redex_g(g(g(g(g(g(x0)))))) 312.23/171.90 redex_g(f(x0)) 312.23/171.90 312.23/171.90 We have to consider all minimal (P,Q,R)-chains. 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (14) DependencyGraphProof (EQUIVALENT) 312.23/171.90 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (15) 312.23/171.90 TRUE 312.23/171.90 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (16) 312.23/171.90 Obligation: 312.23/171.90 Q DP problem: 312.23/171.90 The TRS P consists of the following rules: 312.23/171.90 312.23/171.90 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 312.23/171.90 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 312.23/171.90 312.23/171.90 The TRS R consists of the following rules: 312.23/171.90 312.23/171.90 top(go_up(x)) -> top(reduce(x)) 312.23/171.90 reduce(f(x_1)) -> check_f(redex_f(x_1)) 312.23/171.90 reduce(g(x_1)) -> check_g(redex_g(x_1)) 312.23/171.90 reduce(a) -> go_up(f(a)) 312.23/171.90 reduce(a) -> go_up(g(a)) 312.23/171.90 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.90 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.90 redex_g(f(x)) -> result_g(b) 312.23/171.90 check_f(result_f(x)) -> go_up(x) 312.23/171.90 check_g(result_g(x)) -> go_up(x) 312.23/171.90 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 312.23/171.90 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 312.23/171.90 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 312.23/171.90 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 312.23/171.90 312.23/171.90 The set Q consists of the following terms: 312.23/171.90 312.23/171.90 top(go_up(x0)) 312.23/171.90 reduce(f(x0)) 312.23/171.90 reduce(g(x0)) 312.23/171.90 reduce(a) 312.23/171.90 redex_f(f(f(f(f(f(x0)))))) 312.23/171.90 redex_g(g(g(g(g(g(x0)))))) 312.23/171.90 redex_g(f(x0)) 312.23/171.90 check_f(result_f(x0)) 312.23/171.90 check_g(result_g(x0)) 312.23/171.90 check_f(redex_f(x0)) 312.23/171.90 check_g(redex_g(x0)) 312.23/171.90 in_f_1(go_up(x0)) 312.23/171.90 in_g_1(go_up(x0)) 312.23/171.90 312.23/171.90 We have to consider all minimal (P,Q,R)-chains. 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (17) UsableRulesProof (EQUIVALENT) 312.23/171.90 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. 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (18) 312.23/171.90 Obligation: 312.23/171.90 Q DP problem: 312.23/171.90 The TRS P consists of the following rules: 312.23/171.90 312.23/171.90 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 312.23/171.90 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 312.23/171.90 312.23/171.90 The TRS R consists of the following rules: 312.23/171.90 312.23/171.90 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.90 312.23/171.90 The set Q consists of the following terms: 312.23/171.90 312.23/171.90 top(go_up(x0)) 312.23/171.90 reduce(f(x0)) 312.23/171.90 reduce(g(x0)) 312.23/171.90 reduce(a) 312.23/171.90 redex_f(f(f(f(f(f(x0)))))) 312.23/171.90 redex_g(g(g(g(g(g(x0)))))) 312.23/171.90 redex_g(f(x0)) 312.23/171.90 check_f(result_f(x0)) 312.23/171.90 check_g(result_g(x0)) 312.23/171.90 check_f(redex_f(x0)) 312.23/171.90 check_g(redex_g(x0)) 312.23/171.90 in_f_1(go_up(x0)) 312.23/171.90 in_g_1(go_up(x0)) 312.23/171.90 312.23/171.90 We have to consider all minimal (P,Q,R)-chains. 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (19) QReductionProof (EQUIVALENT) 312.23/171.90 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 312.23/171.90 312.23/171.90 top(go_up(x0)) 312.23/171.90 reduce(f(x0)) 312.23/171.90 reduce(g(x0)) 312.23/171.90 reduce(a) 312.23/171.90 redex_g(g(g(g(g(g(x0)))))) 312.23/171.90 redex_g(f(x0)) 312.23/171.90 check_f(result_f(x0)) 312.23/171.90 check_g(result_g(x0)) 312.23/171.90 check_f(redex_f(x0)) 312.23/171.90 check_g(redex_g(x0)) 312.23/171.90 in_f_1(go_up(x0)) 312.23/171.90 in_g_1(go_up(x0)) 312.23/171.90 312.23/171.90 312.23/171.90 ---------------------------------------- 312.23/171.90 312.23/171.90 (20) 312.23/171.90 Obligation: 312.23/171.90 Q DP problem: 312.23/171.90 The TRS P consists of the following rules: 312.23/171.90 312.23/171.90 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 312.23/171.90 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 312.23/171.90 312.23/171.90 The TRS R consists of the following rules: 312.23/171.90 312.23/171.90 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.91 312.23/171.91 The set Q consists of the following terms: 312.23/171.91 312.23/171.91 redex_f(f(f(f(f(f(x0)))))) 312.23/171.91 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (21) UsableRulesReductionPairsProof (EQUIVALENT) 312.23/171.91 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. 312.23/171.91 312.23/171.91 The following dependency pairs can be deleted: 312.23/171.91 312.23/171.91 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 312.23/171.91 The following rules are removed from R: 312.23/171.91 312.23/171.91 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.91 Used ordering: POLO with Polynomial interpretation [POLO]: 312.23/171.91 312.23/171.91 POL(CHECK_F(x_1)) = x_1 312.23/171.91 POL(REDUCE(x_1)) = 2*x_1 312.23/171.91 POL(b) = 0 312.23/171.91 POL(f(x_1)) = 2*x_1 312.23/171.91 POL(redex_f(x_1)) = 2*x_1 312.23/171.91 POL(result_f(x_1)) = x_1 312.23/171.91 312.23/171.91 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (22) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 312.23/171.91 312.23/171.91 R is empty. 312.23/171.91 The set Q consists of the following terms: 312.23/171.91 312.23/171.91 redex_f(f(f(f(f(f(x0)))))) 312.23/171.91 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (23) DependencyGraphProof (EQUIVALENT) 312.23/171.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (24) 312.23/171.91 TRUE 312.23/171.91 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (25) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(go_up(x)) -> TOP(reduce(x)) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 top(go_up(x)) -> top(reduce(x)) 312.23/171.91 reduce(f(x_1)) -> check_f(redex_f(x_1)) 312.23/171.91 reduce(g(x_1)) -> check_g(redex_g(x_1)) 312.23/171.91 reduce(a) -> go_up(f(a)) 312.23/171.91 reduce(a) -> go_up(g(a)) 312.23/171.91 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.91 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.91 redex_g(f(x)) -> result_g(b) 312.23/171.91 check_f(result_f(x)) -> go_up(x) 312.23/171.91 check_g(result_g(x)) -> go_up(x) 312.23/171.91 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 312.23/171.91 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 312.23/171.91 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 312.23/171.91 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 312.23/171.91 312.23/171.91 The set Q consists of the following terms: 312.23/171.91 312.23/171.91 top(go_up(x0)) 312.23/171.91 reduce(f(x0)) 312.23/171.91 reduce(g(x0)) 312.23/171.91 reduce(a) 312.23/171.91 redex_f(f(f(f(f(f(x0)))))) 312.23/171.91 redex_g(g(g(g(g(g(x0)))))) 312.23/171.91 redex_g(f(x0)) 312.23/171.91 check_f(result_f(x0)) 312.23/171.91 check_g(result_g(x0)) 312.23/171.91 check_f(redex_f(x0)) 312.23/171.91 check_g(redex_g(x0)) 312.23/171.91 in_f_1(go_up(x0)) 312.23/171.91 in_g_1(go_up(x0)) 312.23/171.91 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (26) UsableRulesProof (EQUIVALENT) 312.23/171.91 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. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (27) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(go_up(x)) -> TOP(reduce(x)) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 reduce(f(x_1)) -> check_f(redex_f(x_1)) 312.23/171.91 reduce(g(x_1)) -> check_g(redex_g(x_1)) 312.23/171.91 reduce(a) -> go_up(f(a)) 312.23/171.91 reduce(a) -> go_up(g(a)) 312.23/171.91 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.91 redex_g(f(x)) -> result_g(b) 312.23/171.91 check_g(result_g(x)) -> go_up(x) 312.23/171.91 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 312.23/171.91 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 312.23/171.91 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.91 check_f(result_f(x)) -> go_up(x) 312.23/171.91 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 312.23/171.91 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 312.23/171.91 312.23/171.91 The set Q consists of the following terms: 312.23/171.91 312.23/171.91 top(go_up(x0)) 312.23/171.91 reduce(f(x0)) 312.23/171.91 reduce(g(x0)) 312.23/171.91 reduce(a) 312.23/171.91 redex_f(f(f(f(f(f(x0)))))) 312.23/171.91 redex_g(g(g(g(g(g(x0)))))) 312.23/171.91 redex_g(f(x0)) 312.23/171.91 check_f(result_f(x0)) 312.23/171.91 check_g(result_g(x0)) 312.23/171.91 check_f(redex_f(x0)) 312.23/171.91 check_g(redex_g(x0)) 312.23/171.91 in_f_1(go_up(x0)) 312.23/171.91 in_g_1(go_up(x0)) 312.23/171.91 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (28) QReductionProof (EQUIVALENT) 312.23/171.91 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 312.23/171.91 312.23/171.91 top(go_up(x0)) 312.23/171.91 312.23/171.91 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (29) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(go_up(x)) -> TOP(reduce(x)) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 reduce(f(x_1)) -> check_f(redex_f(x_1)) 312.23/171.91 reduce(g(x_1)) -> check_g(redex_g(x_1)) 312.23/171.91 reduce(a) -> go_up(f(a)) 312.23/171.91 reduce(a) -> go_up(g(a)) 312.23/171.91 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.91 redex_g(f(x)) -> result_g(b) 312.23/171.91 check_g(result_g(x)) -> go_up(x) 312.23/171.91 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 312.23/171.91 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 312.23/171.91 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.91 check_f(result_f(x)) -> go_up(x) 312.23/171.91 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 312.23/171.91 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 312.23/171.91 312.23/171.91 The set Q consists of the following terms: 312.23/171.91 312.23/171.91 reduce(f(x0)) 312.23/171.91 reduce(g(x0)) 312.23/171.91 reduce(a) 312.23/171.91 redex_f(f(f(f(f(f(x0)))))) 312.23/171.91 redex_g(g(g(g(g(g(x0)))))) 312.23/171.91 redex_g(f(x0)) 312.23/171.91 check_f(result_f(x0)) 312.23/171.91 check_g(result_g(x0)) 312.23/171.91 check_f(redex_f(x0)) 312.23/171.91 check_g(redex_g(x0)) 312.23/171.91 in_f_1(go_up(x0)) 312.23/171.91 in_g_1(go_up(x0)) 312.23/171.91 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (30) TransformationProof (EQUIVALENT) 312.23/171.91 By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]: 312.23/171.91 312.23/171.91 (TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))),TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))) 312.23/171.91 (TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))),TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))) 312.23/171.91 (TOP(go_up(a)) -> TOP(go_up(f(a))),TOP(go_up(a)) -> TOP(go_up(f(a)))) 312.23/171.91 (TOP(go_up(a)) -> TOP(go_up(g(a))),TOP(go_up(a)) -> TOP(go_up(g(a)))) 312.23/171.91 312.23/171.91 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (31) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 312.23/171.91 TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))) 312.23/171.91 TOP(go_up(a)) -> TOP(go_up(f(a))) 312.23/171.91 TOP(go_up(a)) -> TOP(go_up(g(a))) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 reduce(f(x_1)) -> check_f(redex_f(x_1)) 312.23/171.91 reduce(g(x_1)) -> check_g(redex_g(x_1)) 312.23/171.91 reduce(a) -> go_up(f(a)) 312.23/171.91 reduce(a) -> go_up(g(a)) 312.23/171.91 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.91 redex_g(f(x)) -> result_g(b) 312.23/171.91 check_g(result_g(x)) -> go_up(x) 312.23/171.91 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 312.23/171.91 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 312.23/171.91 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.91 check_f(result_f(x)) -> go_up(x) 312.23/171.91 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 312.23/171.91 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 312.23/171.91 312.23/171.91 The set Q consists of the following terms: 312.23/171.91 312.23/171.91 reduce(f(x0)) 312.23/171.91 reduce(g(x0)) 312.23/171.91 reduce(a) 312.23/171.91 redex_f(f(f(f(f(f(x0)))))) 312.23/171.91 redex_g(g(g(g(g(g(x0)))))) 312.23/171.91 redex_g(f(x0)) 312.23/171.91 check_f(result_f(x0)) 312.23/171.91 check_g(result_g(x0)) 312.23/171.91 check_f(redex_f(x0)) 312.23/171.91 check_g(redex_g(x0)) 312.23/171.91 in_f_1(go_up(x0)) 312.23/171.91 in_g_1(go_up(x0)) 312.23/171.91 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (32) QDPOrderProof (EQUIVALENT) 312.23/171.91 We use the reduction pair processor [LPAR04,JAR06]. 312.23/171.91 312.23/171.91 312.23/171.91 The following pairs can be oriented strictly and are deleted. 312.23/171.91 312.23/171.91 TOP(go_up(a)) -> TOP(go_up(f(a))) 312.23/171.91 TOP(go_up(a)) -> TOP(go_up(g(a))) 312.23/171.91 The remaining pairs can at least be oriented weakly. 312.23/171.91 Used ordering: Polynomial interpretation [POLO]: 312.23/171.91 312.23/171.91 POL(TOP(x_1)) = x_1 312.23/171.91 POL(a) = 1 312.23/171.91 POL(b) = 0 312.23/171.91 POL(check_f(x_1)) = x_1 312.23/171.91 POL(check_g(x_1)) = x_1 312.23/171.91 POL(f(x_1)) = 0 312.23/171.91 POL(g(x_1)) = 0 312.23/171.91 POL(go_up(x_1)) = x_1 312.23/171.91 POL(in_f_1(x_1)) = 0 312.23/171.91 POL(in_g_1(x_1)) = 0 312.23/171.91 POL(redex_f(x_1)) = 0 312.23/171.91 POL(redex_g(x_1)) = 0 312.23/171.91 POL(reduce(x_1)) = 0 312.23/171.91 POL(result_f(x_1)) = x_1 312.23/171.91 POL(result_g(x_1)) = x_1 312.23/171.91 312.23/171.91 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 312.23/171.91 312.23/171.91 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.91 check_f(result_f(x)) -> go_up(x) 312.23/171.91 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 312.23/171.91 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.91 redex_g(f(x)) -> result_g(b) 312.23/171.91 check_g(result_g(x)) -> go_up(x) 312.23/171.91 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 312.23/171.91 reduce(g(x_1)) -> check_g(redex_g(x_1)) 312.23/171.91 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 312.23/171.91 reduce(f(x_1)) -> check_f(redex_f(x_1)) 312.23/171.91 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 312.23/171.91 312.23/171.91 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (33) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 312.23/171.91 TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 reduce(f(x_1)) -> check_f(redex_f(x_1)) 312.23/171.91 reduce(g(x_1)) -> check_g(redex_g(x_1)) 312.23/171.91 reduce(a) -> go_up(f(a)) 312.23/171.91 reduce(a) -> go_up(g(a)) 312.23/171.91 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.91 redex_g(f(x)) -> result_g(b) 312.23/171.91 check_g(result_g(x)) -> go_up(x) 312.23/171.91 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 312.23/171.91 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 312.23/171.91 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.91 check_f(result_f(x)) -> go_up(x) 312.23/171.91 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 312.23/171.91 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 312.23/171.91 312.23/171.91 The set Q consists of the following terms: 312.23/171.91 312.23/171.91 reduce(f(x0)) 312.23/171.91 reduce(g(x0)) 312.23/171.91 reduce(a) 312.23/171.91 redex_f(f(f(f(f(f(x0)))))) 312.23/171.91 redex_g(g(g(g(g(g(x0)))))) 312.23/171.91 redex_g(f(x0)) 312.23/171.91 check_f(result_f(x0)) 312.23/171.91 check_g(result_g(x0)) 312.23/171.91 check_f(redex_f(x0)) 312.23/171.91 check_g(redex_g(x0)) 312.23/171.91 in_f_1(go_up(x0)) 312.23/171.91 in_g_1(go_up(x0)) 312.23/171.91 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (34) UsableRulesProof (EQUIVALENT) 312.23/171.91 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. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (35) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(go_up(x)) -> TOP(reduce(x)) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 reduce(f(x_1)) -> check_f(redex_f(x_1)) 312.23/171.91 reduce(g(x_1)) -> check_g(redex_g(x_1)) 312.23/171.91 reduce(a) -> go_up(f(a)) 312.23/171.91 reduce(a) -> go_up(g(a)) 312.23/171.91 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.91 redex_g(f(x)) -> result_g(b) 312.23/171.91 check_g(result_g(x)) -> go_up(x) 312.23/171.91 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 312.23/171.91 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 312.23/171.91 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.91 check_f(result_f(x)) -> go_up(x) 312.23/171.91 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 312.23/171.91 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 312.23/171.91 312.23/171.91 The set Q consists of the following terms: 312.23/171.91 312.23/171.91 top(go_up(x0)) 312.23/171.91 reduce(f(x0)) 312.23/171.91 reduce(g(x0)) 312.23/171.91 reduce(a) 312.23/171.91 redex_f(f(f(f(f(f(x0)))))) 312.23/171.91 redex_g(g(g(g(g(g(x0)))))) 312.23/171.91 redex_g(f(x0)) 312.23/171.91 check_f(result_f(x0)) 312.23/171.91 check_g(result_g(x0)) 312.23/171.91 check_f(redex_f(x0)) 312.23/171.91 check_g(redex_g(x0)) 312.23/171.91 in_f_1(go_up(x0)) 312.23/171.91 in_g_1(go_up(x0)) 312.23/171.91 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (36) QReductionProof (EQUIVALENT) 312.23/171.91 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 312.23/171.91 312.23/171.91 top(go_up(x0)) 312.23/171.91 312.23/171.91 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (37) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(go_up(x)) -> TOP(reduce(x)) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 reduce(f(x_1)) -> check_f(redex_f(x_1)) 312.23/171.91 reduce(g(x_1)) -> check_g(redex_g(x_1)) 312.23/171.91 reduce(a) -> go_up(f(a)) 312.23/171.91 reduce(a) -> go_up(g(a)) 312.23/171.91 redex_g(g(g(g(g(g(x)))))) -> result_g(b) 312.23/171.91 redex_g(f(x)) -> result_g(b) 312.23/171.91 check_g(result_g(x)) -> go_up(x) 312.23/171.91 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 312.23/171.91 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 312.23/171.91 redex_f(f(f(f(f(f(x)))))) -> result_f(b) 312.23/171.91 check_f(result_f(x)) -> go_up(x) 312.23/171.91 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 312.23/171.91 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 312.23/171.91 312.23/171.91 The set Q consists of the following terms: 312.23/171.91 312.23/171.91 reduce(f(x0)) 312.23/171.91 reduce(g(x0)) 312.23/171.91 reduce(a) 312.23/171.91 redex_f(f(f(f(f(f(x0)))))) 312.23/171.91 redex_g(g(g(g(g(g(x0)))))) 312.23/171.91 redex_g(f(x0)) 312.23/171.91 check_f(result_f(x0)) 312.23/171.91 check_g(result_g(x0)) 312.23/171.91 check_f(redex_f(x0)) 312.23/171.91 check_g(redex_g(x0)) 312.23/171.91 in_f_1(go_up(x0)) 312.23/171.91 in_g_1(go_up(x0)) 312.23/171.91 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (38) Trivial-Transformation (SOUND) 312.23/171.91 We applied the Trivial transformation to transform the outermost TRS to a standard TRS. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (39) 312.23/171.91 Obligation: 312.23/171.91 Q restricted rewrite system: 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 a -> f(a) 312.23/171.91 a -> g(a) 312.23/171.91 f(f(f(f(f(f(x)))))) -> b 312.23/171.91 g(g(g(g(g(g(x)))))) -> b 312.23/171.91 g(f(x)) -> b 312.23/171.91 312.23/171.91 Q is empty. 312.23/171.91 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (40) DependencyPairsProof (EQUIVALENT) 312.23/171.91 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (41) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 A -> F(a) 312.23/171.91 A -> A 312.23/171.91 A -> G(a) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 a -> f(a) 312.23/171.91 a -> g(a) 312.23/171.91 f(f(f(f(f(f(x)))))) -> b 312.23/171.91 g(g(g(g(g(g(x)))))) -> b 312.23/171.91 g(f(x)) -> b 312.23/171.91 312.23/171.91 Q is empty. 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (42) DependencyGraphProof (EQUIVALENT) 312.23/171.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (43) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 A -> A 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 a -> f(a) 312.23/171.91 a -> g(a) 312.23/171.91 f(f(f(f(f(f(x)))))) -> b 312.23/171.91 g(g(g(g(g(g(x)))))) -> b 312.23/171.91 g(f(x)) -> b 312.23/171.91 312.23/171.91 Q is empty. 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (44) UsableRulesProof (EQUIVALENT) 312.23/171.91 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (45) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 A -> A 312.23/171.91 312.23/171.91 R is empty. 312.23/171.91 Q is empty. 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (46) NonTerminationLoopProof (COMPLETE) 312.23/171.91 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 312.23/171.91 Found a loop by semiunifying a rule from P directly. 312.23/171.91 312.23/171.91 s = A evaluates to t =A 312.23/171.91 312.23/171.91 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 312.23/171.91 * Matcher: [ ] 312.23/171.91 * Semiunifier: [ ] 312.23/171.91 312.23/171.91 -------------------------------------------------------------------------------- 312.23/171.91 Rewriting sequence 312.23/171.91 312.23/171.91 The DP semiunifies directly so there is only one rewrite step from A to A. 312.23/171.91 312.23/171.91 312.23/171.91 312.23/171.91 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (47) 312.23/171.91 NO 312.23/171.91 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (48) Raffelsieper-Zantema-Transformation (SOUND) 312.23/171.91 We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (49) 312.23/171.91 Obligation: 312.23/171.91 Q restricted rewrite system: 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 down(a) -> up(f(a)) 312.23/171.91 down(a) -> up(g(a)) 312.23/171.91 down(f(f(f(f(f(f(x))))))) -> up(b) 312.23/171.91 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.91 down(g(f(x))) -> up(b) 312.23/171.91 top(up(x)) -> top(down(x)) 312.23/171.91 down(f(a)) -> f_flat(down(a)) 312.23/171.91 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.91 down(f(b)) -> f_flat(down(b)) 312.23/171.91 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.91 down(g(a)) -> g_flat(down(a)) 312.23/171.91 down(g(b)) -> g_flat(down(b)) 312.23/171.91 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.91 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.91 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.91 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.91 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.91 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.91 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.91 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.91 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.91 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.91 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.91 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.91 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.91 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.91 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.91 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.91 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.91 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.91 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.91 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.91 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.91 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.91 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.91 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.91 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.91 down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) 312.23/171.91 down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) 312.23/171.91 down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) 312.23/171.91 down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) 312.23/171.91 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.91 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.91 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.91 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.91 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.91 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.91 312.23/171.91 Q is empty. 312.23/171.91 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (50) DependencyPairsProof (EQUIVALENT) 312.23/171.91 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (51) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(up(x)) -> TOP(down(x)) 312.23/171.91 TOP(up(x)) -> DOWN(x) 312.23/171.91 DOWN(f(a)) -> F_FLAT(down(a)) 312.23/171.91 DOWN(f(a)) -> DOWN(a) 312.23/171.91 DOWN(f(g(y4))) -> F_FLAT(down(g(y4))) 312.23/171.91 DOWN(f(g(y4))) -> DOWN(g(y4)) 312.23/171.91 DOWN(f(b)) -> F_FLAT(down(b)) 312.23/171.91 DOWN(f(b)) -> DOWN(b) 312.23/171.91 DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant)) 312.23/171.91 DOWN(f(fresh_constant)) -> DOWN(fresh_constant) 312.23/171.91 DOWN(g(a)) -> G_FLAT(down(a)) 312.23/171.91 DOWN(g(a)) -> DOWN(a) 312.23/171.91 DOWN(g(b)) -> G_FLAT(down(b)) 312.23/171.91 DOWN(g(b)) -> DOWN(b) 312.23/171.91 DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant)) 312.23/171.91 DOWN(g(fresh_constant)) -> DOWN(fresh_constant) 312.23/171.91 DOWN(f(f(a))) -> F_FLAT(down(f(a))) 312.23/171.91 DOWN(f(f(a))) -> DOWN(f(a)) 312.23/171.91 DOWN(f(f(g(y10)))) -> F_FLAT(down(f(g(y10)))) 312.23/171.91 DOWN(f(f(g(y10)))) -> DOWN(f(g(y10))) 312.23/171.91 DOWN(f(f(b))) -> F_FLAT(down(f(b))) 312.23/171.91 DOWN(f(f(b))) -> DOWN(f(b)) 312.23/171.91 DOWN(f(f(fresh_constant))) -> F_FLAT(down(f(fresh_constant))) 312.23/171.91 DOWN(f(f(fresh_constant))) -> DOWN(f(fresh_constant)) 312.23/171.91 DOWN(g(g(a))) -> G_FLAT(down(g(a))) 312.23/171.91 DOWN(g(g(a))) -> DOWN(g(a)) 312.23/171.91 DOWN(g(g(f(y12)))) -> G_FLAT(down(g(f(y12)))) 312.23/171.91 DOWN(g(g(f(y12)))) -> DOWN(g(f(y12))) 312.23/171.91 DOWN(g(g(b))) -> G_FLAT(down(g(b))) 312.23/171.91 DOWN(g(g(b))) -> DOWN(g(b)) 312.23/171.91 DOWN(g(g(fresh_constant))) -> G_FLAT(down(g(fresh_constant))) 312.23/171.91 DOWN(g(g(fresh_constant))) -> DOWN(g(fresh_constant)) 312.23/171.91 DOWN(f(f(f(a)))) -> F_FLAT(down(f(f(a)))) 312.23/171.91 DOWN(f(f(f(a)))) -> DOWN(f(f(a))) 312.23/171.91 DOWN(f(f(f(g(y16))))) -> F_FLAT(down(f(f(g(y16))))) 312.23/171.91 DOWN(f(f(f(g(y16))))) -> DOWN(f(f(g(y16)))) 312.23/171.91 DOWN(f(f(f(b)))) -> F_FLAT(down(f(f(b)))) 312.23/171.91 DOWN(f(f(f(b)))) -> DOWN(f(f(b))) 312.23/171.91 DOWN(f(f(f(fresh_constant)))) -> F_FLAT(down(f(f(fresh_constant)))) 312.23/171.91 DOWN(f(f(f(fresh_constant)))) -> DOWN(f(f(fresh_constant))) 312.23/171.91 DOWN(g(g(g(a)))) -> G_FLAT(down(g(g(a)))) 312.23/171.91 DOWN(g(g(g(a)))) -> DOWN(g(g(a))) 312.23/171.91 DOWN(g(g(g(f(y18))))) -> G_FLAT(down(g(g(f(y18))))) 312.23/171.91 DOWN(g(g(g(f(y18))))) -> DOWN(g(g(f(y18)))) 312.23/171.91 DOWN(g(g(g(b)))) -> G_FLAT(down(g(g(b)))) 312.23/171.91 DOWN(g(g(g(b)))) -> DOWN(g(g(b))) 312.23/171.91 DOWN(g(g(g(fresh_constant)))) -> G_FLAT(down(g(g(fresh_constant)))) 312.23/171.91 DOWN(g(g(g(fresh_constant)))) -> DOWN(g(g(fresh_constant))) 312.23/171.91 DOWN(f(f(f(f(a))))) -> F_FLAT(down(f(f(f(a))))) 312.23/171.91 DOWN(f(f(f(f(a))))) -> DOWN(f(f(f(a)))) 312.23/171.91 DOWN(f(f(f(f(g(y22)))))) -> F_FLAT(down(f(f(f(g(y22)))))) 312.23/171.91 DOWN(f(f(f(f(g(y22)))))) -> DOWN(f(f(f(g(y22))))) 312.23/171.91 DOWN(f(f(f(f(b))))) -> F_FLAT(down(f(f(f(b))))) 312.23/171.91 DOWN(f(f(f(f(b))))) -> DOWN(f(f(f(b)))) 312.23/171.91 DOWN(f(f(f(f(fresh_constant))))) -> F_FLAT(down(f(f(f(fresh_constant))))) 312.23/171.91 DOWN(f(f(f(f(fresh_constant))))) -> DOWN(f(f(f(fresh_constant)))) 312.23/171.91 DOWN(g(g(g(g(a))))) -> G_FLAT(down(g(g(g(a))))) 312.23/171.91 DOWN(g(g(g(g(a))))) -> DOWN(g(g(g(a)))) 312.23/171.91 DOWN(g(g(g(g(f(y24)))))) -> G_FLAT(down(g(g(g(f(y24)))))) 312.23/171.91 DOWN(g(g(g(g(f(y24)))))) -> DOWN(g(g(g(f(y24))))) 312.23/171.91 DOWN(g(g(g(g(b))))) -> G_FLAT(down(g(g(g(b))))) 312.23/171.91 DOWN(g(g(g(g(b))))) -> DOWN(g(g(g(b)))) 312.23/171.91 DOWN(g(g(g(g(fresh_constant))))) -> G_FLAT(down(g(g(g(fresh_constant))))) 312.23/171.91 DOWN(g(g(g(g(fresh_constant))))) -> DOWN(g(g(g(fresh_constant)))) 312.23/171.91 DOWN(f(f(f(f(f(a)))))) -> F_FLAT(down(f(f(f(f(a)))))) 312.23/171.91 DOWN(f(f(f(f(f(a)))))) -> DOWN(f(f(f(f(a))))) 312.23/171.91 DOWN(f(f(f(f(f(g(y28))))))) -> F_FLAT(down(f(f(f(f(g(y28))))))) 312.23/171.91 DOWN(f(f(f(f(f(g(y28))))))) -> DOWN(f(f(f(f(g(y28)))))) 312.23/171.91 DOWN(f(f(f(f(f(b)))))) -> F_FLAT(down(f(f(f(f(b)))))) 312.23/171.91 DOWN(f(f(f(f(f(b)))))) -> DOWN(f(f(f(f(b))))) 312.23/171.91 DOWN(f(f(f(f(f(fresh_constant)))))) -> F_FLAT(down(f(f(f(f(fresh_constant)))))) 312.23/171.91 DOWN(f(f(f(f(f(fresh_constant)))))) -> DOWN(f(f(f(f(fresh_constant))))) 312.23/171.91 DOWN(g(g(g(g(g(a)))))) -> G_FLAT(down(g(g(g(g(a)))))) 312.23/171.91 DOWN(g(g(g(g(g(a)))))) -> DOWN(g(g(g(g(a))))) 312.23/171.91 DOWN(g(g(g(g(g(f(y30))))))) -> G_FLAT(down(g(g(g(g(f(y30))))))) 312.23/171.91 DOWN(g(g(g(g(g(f(y30))))))) -> DOWN(g(g(g(g(f(y30)))))) 312.23/171.91 DOWN(g(g(g(g(g(b)))))) -> G_FLAT(down(g(g(g(g(b)))))) 312.23/171.91 DOWN(g(g(g(g(g(b)))))) -> DOWN(g(g(g(g(b))))) 312.23/171.91 DOWN(g(g(g(g(g(fresh_constant)))))) -> G_FLAT(down(g(g(g(g(fresh_constant)))))) 312.23/171.91 DOWN(g(g(g(g(g(fresh_constant)))))) -> DOWN(g(g(g(g(fresh_constant))))) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 down(a) -> up(f(a)) 312.23/171.91 down(a) -> up(g(a)) 312.23/171.91 down(f(f(f(f(f(f(x))))))) -> up(b) 312.23/171.91 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.91 down(g(f(x))) -> up(b) 312.23/171.91 top(up(x)) -> top(down(x)) 312.23/171.91 down(f(a)) -> f_flat(down(a)) 312.23/171.91 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.91 down(f(b)) -> f_flat(down(b)) 312.23/171.91 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.91 down(g(a)) -> g_flat(down(a)) 312.23/171.91 down(g(b)) -> g_flat(down(b)) 312.23/171.91 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.91 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.91 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.91 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.91 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.91 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.91 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.91 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.91 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.91 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.91 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.91 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.91 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.91 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.91 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.91 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.91 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.91 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.91 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.91 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.91 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.91 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.91 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.91 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.91 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.91 down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) 312.23/171.91 down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) 312.23/171.91 down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) 312.23/171.91 down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) 312.23/171.91 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.91 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.91 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.91 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.91 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.91 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.91 312.23/171.91 Q is empty. 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (52) DependencyGraphProof (EQUIVALENT) 312.23/171.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 79 less nodes. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (53) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(up(x)) -> TOP(down(x)) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 down(a) -> up(f(a)) 312.23/171.91 down(a) -> up(g(a)) 312.23/171.91 down(f(f(f(f(f(f(x))))))) -> up(b) 312.23/171.91 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.91 down(g(f(x))) -> up(b) 312.23/171.91 top(up(x)) -> top(down(x)) 312.23/171.91 down(f(a)) -> f_flat(down(a)) 312.23/171.91 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.91 down(f(b)) -> f_flat(down(b)) 312.23/171.91 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.91 down(g(a)) -> g_flat(down(a)) 312.23/171.91 down(g(b)) -> g_flat(down(b)) 312.23/171.91 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.91 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.91 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.91 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.91 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.91 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.91 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.91 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.91 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.91 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.91 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.91 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.91 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.91 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.91 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.91 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.91 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.91 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.91 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.91 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.91 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.91 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.91 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.91 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.91 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.91 down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) 312.23/171.91 down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) 312.23/171.91 down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) 312.23/171.91 down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) 312.23/171.91 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.91 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.91 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.91 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.91 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.91 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.91 312.23/171.91 Q is empty. 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (54) UsableRulesProof (EQUIVALENT) 312.23/171.91 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (55) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(up(x)) -> TOP(down(x)) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 down(a) -> up(f(a)) 312.23/171.91 down(a) -> up(g(a)) 312.23/171.91 down(f(f(f(f(f(f(x))))))) -> up(b) 312.23/171.91 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.91 down(g(f(x))) -> up(b) 312.23/171.91 down(f(a)) -> f_flat(down(a)) 312.23/171.91 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.91 down(f(b)) -> f_flat(down(b)) 312.23/171.91 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.91 down(g(a)) -> g_flat(down(a)) 312.23/171.91 down(g(b)) -> g_flat(down(b)) 312.23/171.91 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.91 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.91 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.91 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.91 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.91 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.91 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.91 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.91 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.91 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.91 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.91 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.91 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.91 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.91 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.91 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.91 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.91 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.91 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.91 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.91 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.91 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.91 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.91 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.91 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.91 down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) 312.23/171.91 down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) 312.23/171.91 down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) 312.23/171.91 down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) 312.23/171.91 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.91 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.91 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.91 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.91 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.91 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.91 312.23/171.91 Q is empty. 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (56) TransformationProof (EQUIVALENT) 312.23/171.91 By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]: 312.23/171.91 312.23/171.91 (TOP(up(a)) -> TOP(up(f(a))),TOP(up(a)) -> TOP(up(f(a)))) 312.23/171.91 (TOP(up(a)) -> TOP(up(g(a))),TOP(up(a)) -> TOP(up(g(a)))) 312.23/171.91 (TOP(up(f(f(f(f(f(f(x0)))))))) -> TOP(up(b)),TOP(up(f(f(f(f(f(f(x0)))))))) -> TOP(up(b))) 312.23/171.91 (TOP(up(g(g(g(g(g(g(x0)))))))) -> TOP(up(b)),TOP(up(g(g(g(g(g(g(x0)))))))) -> TOP(up(b))) 312.23/171.91 (TOP(up(g(f(x0)))) -> TOP(up(b)),TOP(up(g(f(x0)))) -> TOP(up(b))) 312.23/171.91 (TOP(up(f(a))) -> TOP(f_flat(down(a))),TOP(up(f(a))) -> TOP(f_flat(down(a)))) 312.23/171.91 (TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))),TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))) 312.23/171.91 (TOP(up(f(b))) -> TOP(f_flat(down(b))),TOP(up(f(b))) -> TOP(f_flat(down(b)))) 312.23/171.91 (TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))),TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant)))) 312.23/171.91 (TOP(up(g(a))) -> TOP(g_flat(down(a))),TOP(up(g(a))) -> TOP(g_flat(down(a)))) 312.23/171.91 (TOP(up(g(b))) -> TOP(g_flat(down(b))),TOP(up(g(b))) -> TOP(g_flat(down(b)))) 312.23/171.91 (TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))),TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant)))) 312.23/171.91 (TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))),TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))) 312.23/171.91 (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)))))) 312.23/171.91 (TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))),TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))) 312.23/171.91 (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))))) 312.23/171.91 (TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))),TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))) 312.23/171.91 (TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))),TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))) 312.23/171.91 (TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))),TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))) 312.23/171.91 (TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))),TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))) 312.23/171.91 (TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))) 312.23/171.91 (TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))),TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))) 312.23/171.91 (TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))) 312.23/171.91 (TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))),TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))) 312.23/171.91 (TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))),TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a)))))) 312.23/171.91 (TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))),TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0))))))) 312.23/171.91 (TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))),TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b)))))) 312.23/171.91 (TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))),TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant)))))) 312.23/171.91 (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))) 312.23/171.91 (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))) 312.23/171.91 (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))) 312.23/171.91 (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))) 312.23/171.91 (TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))),TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a))))))) 312.23/171.91 (TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))),TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0)))))))) 312.23/171.91 (TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))),TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b))))))) 312.23/171.91 (TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))),TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant))))))) 312.23/171.91 (TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))),TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a)))))))) 312.23/171.91 (TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))),TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0))))))))) 312.23/171.91 (TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))),TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b)))))))) 312.23/171.91 (TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))),TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant)))))))) 312.23/171.91 (TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))),TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a)))))))) 312.23/171.91 (TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))),TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0))))))))) 312.23/171.91 (TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))),TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b)))))))) 312.23/171.91 (TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))),TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant)))))))) 312.23/171.91 312.23/171.91 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (57) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(up(a)) -> TOP(up(f(a))) 312.23/171.91 TOP(up(a)) -> TOP(up(g(a))) 312.23/171.91 TOP(up(f(f(f(f(f(f(x0)))))))) -> TOP(up(b)) 312.23/171.91 TOP(up(g(g(g(g(g(g(x0)))))))) -> TOP(up(b)) 312.23/171.91 TOP(up(g(f(x0)))) -> TOP(up(b)) 312.23/171.91 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.91 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.91 TOP(up(f(b))) -> TOP(f_flat(down(b))) 312.23/171.91 TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))) 312.23/171.91 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.91 TOP(up(g(b))) -> TOP(g_flat(down(b))) 312.23/171.91 TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))) 312.23/171.91 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.91 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.91 TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) 312.23/171.91 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 312.23/171.91 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.91 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.91 TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) 312.23/171.91 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 312.23/171.91 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.91 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.91 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.91 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 312.23/171.91 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.91 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.91 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.91 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 312.23/171.91 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.91 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.91 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.91 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 312.23/171.91 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.91 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.91 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.91 TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) 312.23/171.91 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.91 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.91 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.91 TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) 312.23/171.91 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.91 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.91 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.91 TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 down(a) -> up(f(a)) 312.23/171.91 down(a) -> up(g(a)) 312.23/171.91 down(f(f(f(f(f(f(x))))))) -> up(b) 312.23/171.91 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.91 down(g(f(x))) -> up(b) 312.23/171.91 down(f(a)) -> f_flat(down(a)) 312.23/171.91 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.91 down(f(b)) -> f_flat(down(b)) 312.23/171.91 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.91 down(g(a)) -> g_flat(down(a)) 312.23/171.91 down(g(b)) -> g_flat(down(b)) 312.23/171.91 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.91 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.91 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.91 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.91 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.91 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.91 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.91 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.91 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.91 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.91 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.91 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.91 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.91 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.91 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.91 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.91 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.91 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.91 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.91 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.91 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.91 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.91 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.91 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.91 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.91 down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) 312.23/171.91 down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) 312.23/171.91 down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) 312.23/171.91 down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) 312.23/171.91 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.91 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.91 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.91 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.91 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.91 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.91 312.23/171.91 Q is empty. 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (58) DependencyGraphProof (EQUIVALENT) 312.23/171.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (59) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.91 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.91 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.91 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.91 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.91 TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) 312.23/171.91 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 312.23/171.91 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.91 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.91 TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) 312.23/171.91 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 312.23/171.91 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.91 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.91 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.91 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 312.23/171.91 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.91 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.91 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.91 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 312.23/171.91 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.91 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.91 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.91 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 312.23/171.91 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.91 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.91 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.91 TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) 312.23/171.91 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.91 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.91 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.91 TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) 312.23/171.91 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.91 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.91 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.91 TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 down(a) -> up(f(a)) 312.23/171.91 down(a) -> up(g(a)) 312.23/171.91 down(f(f(f(f(f(f(x))))))) -> up(b) 312.23/171.91 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.91 down(g(f(x))) -> up(b) 312.23/171.91 down(f(a)) -> f_flat(down(a)) 312.23/171.91 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.91 down(f(b)) -> f_flat(down(b)) 312.23/171.91 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.91 down(g(a)) -> g_flat(down(a)) 312.23/171.91 down(g(b)) -> g_flat(down(b)) 312.23/171.91 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.91 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.91 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.91 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.91 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.91 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.91 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.91 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.91 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.91 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.91 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.91 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.91 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.91 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.91 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.91 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.91 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.91 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.91 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.91 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.91 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.91 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.91 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.91 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.91 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.91 down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) 312.23/171.91 down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) 312.23/171.91 down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) 312.23/171.91 down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) 312.23/171.91 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.91 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.91 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.91 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.91 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.91 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.91 312.23/171.91 Q is empty. 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (60) UsableRulesProof (EQUIVALENT) 312.23/171.91 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (61) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.91 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.91 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.91 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.91 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.91 TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) 312.23/171.91 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 312.23/171.91 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.91 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.91 TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) 312.23/171.91 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 312.23/171.91 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.91 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.91 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.91 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 312.23/171.91 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.91 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.91 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.91 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 312.23/171.91 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.91 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.91 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.91 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 312.23/171.91 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.91 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.91 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.91 TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) 312.23/171.91 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.91 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.91 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.91 TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) 312.23/171.91 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.91 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.91 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.91 TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.91 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.91 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.91 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.91 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.91 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.91 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.91 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.91 down(g(b)) -> g_flat(down(b)) 312.23/171.91 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.91 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.91 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.91 down(g(f(x))) -> up(b) 312.23/171.91 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.91 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.91 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.91 down(g(a)) -> g_flat(down(a)) 312.23/171.91 down(a) -> up(f(a)) 312.23/171.91 down(a) -> up(g(a)) 312.23/171.91 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.91 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.91 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.91 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.91 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.91 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.91 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.91 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.91 down(f(b)) -> f_flat(down(b)) 312.23/171.91 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.91 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.91 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.91 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.91 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.91 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.91 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.91 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.91 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.91 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.91 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.91 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.91 down(f(a)) -> f_flat(down(a)) 312.23/171.91 312.23/171.91 Q is empty. 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (62) TransformationProof (EQUIVALENT) 312.23/171.91 By narrowing [LPAR04] the rule TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) at position [0] we obtained the following new rules [LPAR04]: 312.23/171.91 312.23/171.91 (TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))),TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b))))) 312.23/171.91 312.23/171.91 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (63) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.91 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.91 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.91 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.91 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.91 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 312.23/171.91 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.91 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.91 TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) 312.23/171.91 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 312.23/171.91 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.91 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.91 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.91 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 312.23/171.91 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.91 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.91 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.91 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 312.23/171.91 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.91 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.91 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.91 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 312.23/171.91 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.91 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.91 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.91 TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) 312.23/171.91 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.91 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.91 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.91 TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) 312.23/171.91 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.91 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.91 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.91 TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) 312.23/171.91 TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.91 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.91 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.91 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.91 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.91 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.91 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.91 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.91 down(g(b)) -> g_flat(down(b)) 312.23/171.91 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.91 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.91 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.91 down(g(f(x))) -> up(b) 312.23/171.91 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.91 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.91 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.91 down(g(a)) -> g_flat(down(a)) 312.23/171.91 down(a) -> up(f(a)) 312.23/171.91 down(a) -> up(g(a)) 312.23/171.91 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.91 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.91 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.91 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.91 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.91 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.91 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.91 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.91 down(f(b)) -> f_flat(down(b)) 312.23/171.91 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.91 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.91 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.91 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.91 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.91 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.91 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.91 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.91 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.91 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.91 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.91 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.91 down(f(a)) -> f_flat(down(a)) 312.23/171.91 312.23/171.91 Q is empty. 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (64) DependencyGraphProof (EQUIVALENT) 312.23/171.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (65) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.91 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.91 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.91 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.91 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.91 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) 312.23/171.91 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.91 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.91 TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) 312.23/171.91 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 312.23/171.91 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.91 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.91 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.91 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 312.23/171.91 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.91 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.91 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.91 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 312.23/171.91 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.91 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.91 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.91 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 312.23/171.91 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.91 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.91 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.91 TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) 312.23/171.91 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.91 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.91 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.91 TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) 312.23/171.91 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.91 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.91 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.91 TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.91 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.91 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.91 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.91 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.91 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.91 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.91 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.91 down(g(b)) -> g_flat(down(b)) 312.23/171.91 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.91 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.91 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.91 down(g(f(x))) -> up(b) 312.23/171.91 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.91 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.91 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.91 down(g(a)) -> g_flat(down(a)) 312.23/171.91 down(a) -> up(f(a)) 312.23/171.91 down(a) -> up(g(a)) 312.23/171.91 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.91 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.91 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.91 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.91 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.91 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.91 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.91 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.91 down(f(b)) -> f_flat(down(b)) 312.23/171.91 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.91 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.91 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.91 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.91 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.91 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.91 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.91 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.91 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.91 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.91 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.91 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.91 down(f(a)) -> f_flat(down(a)) 312.23/171.91 312.23/171.91 Q is empty. 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (66) TransformationProof (EQUIVALENT) 312.23/171.91 By narrowing [LPAR04] the rule TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) at position [0] we obtained the following new rules [LPAR04]: 312.23/171.91 312.23/171.91 (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))))) 312.23/171.91 312.23/171.91 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (67) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.91 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.91 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.91 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.91 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.91 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.91 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.91 TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) 312.23/171.91 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 312.23/171.91 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.91 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.91 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.91 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 312.23/171.91 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.91 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.91 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.91 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 312.23/171.91 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.91 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.91 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.91 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 312.23/171.91 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.91 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.91 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.91 TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) 312.23/171.91 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.91 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.91 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.91 TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) 312.23/171.91 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.91 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.91 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.91 TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) 312.23/171.91 TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant)))) 312.23/171.91 312.23/171.91 The TRS R consists of the following rules: 312.23/171.91 312.23/171.91 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.91 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.91 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.91 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.91 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.91 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.91 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.91 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.91 down(g(b)) -> g_flat(down(b)) 312.23/171.91 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.91 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.91 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.91 down(g(f(x))) -> up(b) 312.23/171.91 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.91 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.91 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.91 down(g(a)) -> g_flat(down(a)) 312.23/171.91 down(a) -> up(f(a)) 312.23/171.91 down(a) -> up(g(a)) 312.23/171.91 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.91 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.91 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.91 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.91 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.91 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.91 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.91 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.91 down(f(b)) -> f_flat(down(b)) 312.23/171.91 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.91 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.91 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.91 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.91 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.91 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.91 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.91 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.91 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.91 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.91 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.91 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.91 down(f(a)) -> f_flat(down(a)) 312.23/171.91 312.23/171.91 Q is empty. 312.23/171.91 We have to consider all minimal (P,Q,R)-chains. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (68) DependencyGraphProof (EQUIVALENT) 312.23/171.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 312.23/171.91 ---------------------------------------- 312.23/171.91 312.23/171.91 (69) 312.23/171.91 Obligation: 312.23/171.91 Q DP problem: 312.23/171.91 The TRS P consists of the following rules: 312.23/171.91 312.23/171.91 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.91 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.91 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.91 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.91 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.91 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.91 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.91 TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) 312.23/171.91 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 312.23/171.91 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.91 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.91 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.91 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 312.23/171.91 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.91 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.91 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.91 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 312.23/171.91 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.91 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.91 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.92 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 312.23/171.92 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.92 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.92 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.92 TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) 312.23/171.92 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.92 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.92 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.92 TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) 312.23/171.92 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.92 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.92 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.92 TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) 312.23/171.92 312.23/171.92 The TRS R consists of the following rules: 312.23/171.92 312.23/171.92 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.92 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.92 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.92 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.92 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.92 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.92 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.92 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.92 down(g(b)) -> g_flat(down(b)) 312.23/171.92 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.92 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.92 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.92 down(g(f(x))) -> up(b) 312.23/171.92 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.92 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.92 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.92 down(g(a)) -> g_flat(down(a)) 312.23/171.92 down(a) -> up(f(a)) 312.23/171.92 down(a) -> up(g(a)) 312.23/171.92 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.92 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.92 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.92 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.92 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.92 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.92 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.92 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.92 down(f(b)) -> f_flat(down(b)) 312.23/171.92 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.92 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.92 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.92 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.92 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.92 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.92 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.92 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.92 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.92 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.92 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.92 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.92 down(f(a)) -> f_flat(down(a)) 312.23/171.92 312.23/171.92 Q is empty. 312.23/171.92 We have to consider all minimal (P,Q,R)-chains. 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (70) TransformationProof (EQUIVALENT) 312.23/171.92 By narrowing [LPAR04] the rule TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) at position [0] we obtained the following new rules [LPAR04]: 312.23/171.92 312.23/171.92 (TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))),TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b))))) 312.23/171.92 312.23/171.92 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (71) 312.23/171.92 Obligation: 312.23/171.92 Q DP problem: 312.23/171.92 The TRS P consists of the following rules: 312.23/171.92 312.23/171.92 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.92 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.92 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.92 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.92 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.92 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.92 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.92 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 312.23/171.92 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.92 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.92 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.92 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 312.23/171.92 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.92 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.92 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.92 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 312.23/171.92 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.92 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.92 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.92 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 312.23/171.92 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.92 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.92 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.92 TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) 312.23/171.92 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.92 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.92 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.92 TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) 312.23/171.92 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.92 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.92 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.92 TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) 312.23/171.92 TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) 312.23/171.92 312.23/171.92 The TRS R consists of the following rules: 312.23/171.92 312.23/171.92 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.92 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.92 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.92 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.92 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.92 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.92 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.92 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.92 down(g(b)) -> g_flat(down(b)) 312.23/171.92 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.92 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.92 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.92 down(g(f(x))) -> up(b) 312.23/171.92 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.92 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.92 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.92 down(g(a)) -> g_flat(down(a)) 312.23/171.92 down(a) -> up(f(a)) 312.23/171.92 down(a) -> up(g(a)) 312.23/171.92 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.92 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.92 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.92 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.92 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.92 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.92 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.92 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.92 down(f(b)) -> f_flat(down(b)) 312.23/171.92 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.92 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.92 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.92 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.92 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.92 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.92 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.92 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.92 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.92 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.92 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.92 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.92 down(f(a)) -> f_flat(down(a)) 312.23/171.92 312.23/171.92 Q is empty. 312.23/171.92 We have to consider all minimal (P,Q,R)-chains. 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (72) DependencyGraphProof (EQUIVALENT) 312.23/171.92 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (73) 312.23/171.92 Obligation: 312.23/171.92 Q DP problem: 312.23/171.92 The TRS P consists of the following rules: 312.23/171.92 312.23/171.92 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.92 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.92 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.92 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.92 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.92 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.92 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.92 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) 312.23/171.92 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.92 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.92 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.92 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 312.23/171.92 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.92 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.92 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.92 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 312.23/171.92 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.92 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.92 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.92 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 312.23/171.92 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.92 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.92 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.92 TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) 312.23/171.92 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.92 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.92 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.92 TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) 312.23/171.92 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.92 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.92 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.92 TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) 312.23/171.92 312.23/171.92 The TRS R consists of the following rules: 312.23/171.92 312.23/171.92 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.92 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.92 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.92 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.92 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.92 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.92 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.92 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.92 down(g(b)) -> g_flat(down(b)) 312.23/171.92 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.92 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.92 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.92 down(g(f(x))) -> up(b) 312.23/171.92 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.92 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.92 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.92 down(g(a)) -> g_flat(down(a)) 312.23/171.92 down(a) -> up(f(a)) 312.23/171.92 down(a) -> up(g(a)) 312.23/171.92 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.92 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.92 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.92 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.92 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.92 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.92 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.92 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.92 down(f(b)) -> f_flat(down(b)) 312.23/171.92 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.92 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.92 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.92 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.92 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.92 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.92 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.92 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.92 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.92 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.92 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.92 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.92 down(f(a)) -> f_flat(down(a)) 312.23/171.92 312.23/171.92 Q is empty. 312.23/171.92 We have to consider all minimal (P,Q,R)-chains. 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (74) TransformationProof (EQUIVALENT) 312.23/171.92 By narrowing [LPAR04] the rule TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) at position [0] we obtained the following new rules [LPAR04]: 312.23/171.92 312.23/171.92 (TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(g_flat(down(fresh_constant)))),TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(g_flat(down(fresh_constant))))) 312.23/171.92 312.23/171.92 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (75) 312.23/171.92 Obligation: 312.23/171.92 Q DP problem: 312.23/171.92 The TRS P consists of the following rules: 312.23/171.92 312.23/171.92 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.92 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.92 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.92 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.92 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.92 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.92 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.92 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.92 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.92 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.92 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 312.23/171.92 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.92 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.92 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.92 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 312.23/171.92 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.92 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.92 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.92 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 312.23/171.92 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.92 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.92 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.92 TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) 312.23/171.92 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.92 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.92 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.92 TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) 312.23/171.92 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.92 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.92 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.92 TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) 312.23/171.92 TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(g_flat(down(fresh_constant)))) 312.23/171.92 312.23/171.92 The TRS R consists of the following rules: 312.23/171.92 312.23/171.92 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.92 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.92 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.92 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.92 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.92 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.92 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.92 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.92 down(g(b)) -> g_flat(down(b)) 312.23/171.92 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.92 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.92 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.92 down(g(f(x))) -> up(b) 312.23/171.92 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.92 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.92 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.92 down(g(a)) -> g_flat(down(a)) 312.23/171.92 down(a) -> up(f(a)) 312.23/171.92 down(a) -> up(g(a)) 312.23/171.92 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.92 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.92 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.92 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.92 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.92 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.92 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.92 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.92 down(f(b)) -> f_flat(down(b)) 312.23/171.92 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.92 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.92 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.92 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.92 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.92 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.92 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.92 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.92 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.92 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.92 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.92 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.92 down(f(a)) -> f_flat(down(a)) 312.23/171.92 312.23/171.92 Q is empty. 312.23/171.92 We have to consider all minimal (P,Q,R)-chains. 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (76) DependencyGraphProof (EQUIVALENT) 312.23/171.92 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (77) 312.23/171.92 Obligation: 312.23/171.92 Q DP problem: 312.23/171.92 The TRS P consists of the following rules: 312.23/171.92 312.23/171.92 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.92 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.92 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.92 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.92 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.92 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.92 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.92 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.92 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.92 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.92 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 312.23/171.92 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.92 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.92 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.92 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 312.23/171.92 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.92 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.92 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.92 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 312.23/171.92 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.92 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.92 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.92 TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) 312.23/171.92 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.92 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.92 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.92 TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) 312.23/171.92 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.92 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.92 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.92 TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) 312.23/171.92 312.23/171.92 The TRS R consists of the following rules: 312.23/171.92 312.23/171.92 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.92 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.92 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.92 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.92 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.92 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.92 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.92 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.92 down(g(b)) -> g_flat(down(b)) 312.23/171.92 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.92 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.92 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.92 down(g(f(x))) -> up(b) 312.23/171.92 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.92 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.92 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.92 down(g(a)) -> g_flat(down(a)) 312.23/171.92 down(a) -> up(f(a)) 312.23/171.92 down(a) -> up(g(a)) 312.23/171.92 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.92 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.92 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.92 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.92 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.92 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.92 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.92 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.92 down(f(b)) -> f_flat(down(b)) 312.23/171.92 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.92 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.92 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.92 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.92 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.92 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.92 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.92 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.92 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.92 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.92 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.92 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.92 down(f(a)) -> f_flat(down(a)) 312.23/171.92 312.23/171.92 Q is empty. 312.23/171.92 We have to consider all minimal (P,Q,R)-chains. 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (78) MRRProof (EQUIVALENT) 312.23/171.92 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. 312.23/171.92 312.23/171.92 Strictly oriented dependency pairs: 312.23/171.92 312.23/171.92 TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) 312.23/171.92 TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) 312.23/171.92 TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) 312.23/171.92 TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) 312.23/171.92 TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) 312.23/171.92 TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) 312.23/171.92 312.23/171.92 312.23/171.92 Used ordering: Polynomial interpretation [POLO]: 312.23/171.92 312.23/171.92 POL(TOP(x_1)) = 2*x_1 312.23/171.92 POL(a) = 0 312.23/171.92 POL(b) = 0 312.23/171.92 POL(down(x_1)) = x_1 312.23/171.92 POL(f(x_1)) = 2*x_1 312.23/171.92 POL(f_flat(x_1)) = 2*x_1 312.23/171.92 POL(fresh_constant) = 2 312.23/171.92 POL(g(x_1)) = 2*x_1 312.23/171.92 POL(g_flat(x_1)) = 2*x_1 312.23/171.92 POL(up(x_1)) = 2*x_1 312.23/171.92 312.23/171.92 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (79) 312.23/171.92 Obligation: 312.23/171.92 Q DP problem: 312.23/171.92 The TRS P consists of the following rules: 312.23/171.92 312.23/171.92 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.92 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.92 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.92 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.92 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.92 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.92 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.92 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.92 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.92 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.92 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.92 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.92 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.92 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.92 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.92 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.92 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.92 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.92 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.92 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.92 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.92 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.92 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.92 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.92 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.92 312.23/171.92 The TRS R consists of the following rules: 312.23/171.92 312.23/171.92 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.92 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.92 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.92 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.92 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.92 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.92 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.92 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.92 down(g(b)) -> g_flat(down(b)) 312.23/171.92 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.92 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.92 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.92 down(g(f(x))) -> up(b) 312.23/171.92 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.92 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.92 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.92 down(g(a)) -> g_flat(down(a)) 312.23/171.92 down(a) -> up(f(a)) 312.23/171.92 down(a) -> up(g(a)) 312.23/171.92 down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) 312.23/171.92 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.92 down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) 312.23/171.92 down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) 312.23/171.92 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 312.23/171.92 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.92 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.92 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.92 down(f(b)) -> f_flat(down(b)) 312.23/171.92 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.92 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.92 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.92 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.92 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.92 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.92 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.92 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.92 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.92 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.92 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.92 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.92 down(f(a)) -> f_flat(down(a)) 312.23/171.92 312.23/171.92 Q is empty. 312.23/171.92 We have to consider all minimal (P,Q,R)-chains. 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (80) UsableRulesProof (EQUIVALENT) 312.23/171.92 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (81) 312.23/171.92 Obligation: 312.23/171.92 Q DP problem: 312.23/171.92 The TRS P consists of the following rules: 312.23/171.92 312.23/171.92 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.92 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.92 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.92 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.92 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.92 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.92 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.92 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.92 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.92 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.92 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.92 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.92 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.92 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.92 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.92 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.92 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.92 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.92 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.92 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.92 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.92 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.92 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.92 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.92 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.92 312.23/171.92 The TRS R consists of the following rules: 312.23/171.92 312.23/171.92 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.92 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.92 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.92 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.92 down(g(b)) -> g_flat(down(b)) 312.23/171.92 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.92 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.92 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.92 down(g(f(x))) -> up(b) 312.23/171.92 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.92 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.92 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.92 down(g(a)) -> g_flat(down(a)) 312.23/171.92 down(a) -> up(f(a)) 312.23/171.92 down(a) -> up(g(a)) 312.23/171.92 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.23/171.92 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.92 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.23/171.92 down(f(f(b))) -> f_flat(down(f(b))) 312.23/171.92 down(f(b)) -> f_flat(down(b)) 312.23/171.92 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.23/171.92 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.23/171.92 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.23/171.92 down(f(g(y4))) -> f_flat(down(g(y4))) 312.23/171.92 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.23/171.92 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.23/171.92 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.23/171.92 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.23/171.92 down(g(g(g(g(g(g(x))))))) -> up(b) 312.23/171.92 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.23/171.92 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.23/171.92 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.23/171.92 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.23/171.92 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.23/171.92 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.23/171.92 down(f(f(a))) -> f_flat(down(f(a))) 312.23/171.92 down(f(a)) -> f_flat(down(a)) 312.23/171.92 312.23/171.92 Q is empty. 312.23/171.92 We have to consider all minimal (P,Q,R)-chains. 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (82) QDPOrderProof (EQUIVALENT) 312.23/171.92 We use the reduction pair processor [LPAR04,JAR06]. 312.23/171.92 312.23/171.92 312.23/171.92 The following pairs can be oriented strictly and are deleted. 312.23/171.92 312.23/171.92 TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) 312.23/171.92 TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) 312.23/171.92 TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) 312.23/171.92 The remaining pairs can at least be oriented weakly. 312.23/171.92 Used ordering: Polynomial interpretation [POLO]: 312.23/171.92 312.23/171.92 POL(TOP(x_1)) = x_1 312.23/171.92 POL(a) = 1 312.23/171.92 POL(b) = 0 312.23/171.92 POL(down(x_1)) = x_1 312.23/171.92 POL(f(x_1)) = 1 312.23/171.92 POL(f_flat(x_1)) = 1 312.23/171.92 POL(fresh_constant) = 0 312.23/171.92 POL(g(x_1)) = x_1 312.23/171.92 POL(g_flat(x_1)) = x_1 312.23/171.92 POL(up(x_1)) = 1 312.23/171.92 312.23/171.92 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 312.23/171.92 312.23/171.92 down(a) -> up(f(a)) 312.23/171.92 down(a) -> up(g(a)) 312.23/171.92 f_flat(up(x_1)) -> up(f(x_1)) 312.23/171.92 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.92 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.23/171.92 down(g(g(b))) -> g_flat(down(g(b))) 312.23/171.92 down(g(b)) -> g_flat(down(b)) 312.23/171.92 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.23/171.92 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.23/171.92 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.23/171.92 down(g(f(x))) -> up(b) 312.23/171.92 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.23/171.92 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.23/171.92 down(g(g(a))) -> g_flat(down(g(a))) 312.23/171.92 down(g(a)) -> g_flat(down(a)) 312.23/171.92 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.92 312.23/171.92 312.23/171.92 ---------------------------------------- 312.23/171.92 312.23/171.92 (83) 312.23/171.92 Obligation: 312.23/171.92 Q DP problem: 312.23/171.92 The TRS P consists of the following rules: 312.23/171.92 312.23/171.92 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.23/171.92 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.23/171.92 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.23/171.92 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.23/171.92 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.23/171.92 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.23/171.92 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.23/171.92 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.23/171.92 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.23/171.92 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.23/171.92 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.23/171.92 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.23/171.92 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.23/171.92 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.23/171.92 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.23/171.92 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.23/171.92 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.23/171.92 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.23/171.92 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.23/171.92 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.23/171.92 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.23/171.92 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.23/171.92 312.23/171.92 The TRS R consists of the following rules: 312.23/171.92 312.23/171.92 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.23/171.92 g_flat(up(x_1)) -> up(g(x_1)) 312.23/171.92 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.93 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.93 down(g(b)) -> g_flat(down(b)) 312.36/171.93 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.93 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.93 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.93 down(g(f(x))) -> up(b) 312.36/171.93 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.93 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.93 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.93 down(g(a)) -> g_flat(down(a)) 312.36/171.93 down(a) -> up(f(a)) 312.36/171.93 down(a) -> up(g(a)) 312.36/171.93 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.36/171.93 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.93 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.36/171.93 down(f(f(b))) -> f_flat(down(f(b))) 312.36/171.93 down(f(b)) -> f_flat(down(b)) 312.36/171.93 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.93 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.93 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.93 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.93 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.93 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.93 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.93 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.36/171.93 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.93 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.93 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.93 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.93 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.93 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.93 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.93 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.93 down(f(a)) -> f_flat(down(a)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (84) QDPOrderProof (EQUIVALENT) 312.36/171.93 We use the reduction pair processor [LPAR04,JAR06]. 312.36/171.93 312.36/171.93 312.36/171.93 The following pairs can be oriented strictly and are deleted. 312.36/171.93 312.36/171.93 TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) 312.36/171.93 TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) 312.36/171.93 TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) 312.36/171.93 The remaining pairs can at least be oriented weakly. 312.36/171.93 Used ordering: Polynomial interpretation [POLO]: 312.36/171.93 312.36/171.93 POL(TOP(x_1)) = x_1 312.36/171.93 POL(a) = 1 312.36/171.93 POL(b) = 0 312.36/171.93 POL(down(x_1)) = x_1 312.36/171.93 POL(f(x_1)) = x_1 312.36/171.93 POL(f_flat(x_1)) = x_1 312.36/171.93 POL(fresh_constant) = 0 312.36/171.93 POL(g(x_1)) = 1 312.36/171.93 POL(g_flat(x_1)) = 1 312.36/171.93 POL(up(x_1)) = 1 312.36/171.93 312.36/171.93 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 312.36/171.93 312.36/171.93 down(a) -> up(f(a)) 312.36/171.93 down(a) -> up(g(a)) 312.36/171.93 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.93 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.93 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.93 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.93 down(g(b)) -> g_flat(down(b)) 312.36/171.93 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.93 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.93 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.93 down(g(f(x))) -> up(b) 312.36/171.93 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.93 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.93 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.93 down(g(a)) -> g_flat(down(a)) 312.36/171.93 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.93 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.93 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.93 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.36/171.93 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.93 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.93 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.93 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.93 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.93 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.93 down(f(a)) -> f_flat(down(a)) 312.36/171.93 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.93 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.93 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.93 down(f(f(b))) -> f_flat(down(f(b))) 312.36/171.93 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.93 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.93 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.36/171.93 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.93 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.93 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.36/171.93 down(f(b)) -> f_flat(down(b)) 312.36/171.93 312.36/171.93 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (85) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.36/171.93 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.36/171.93 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.36/171.93 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.36/171.93 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.36/171.93 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.36/171.93 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.36/171.93 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.36/171.93 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.36/171.93 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.36/171.93 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.36/171.93 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.36/171.93 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.36/171.93 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.36/171.93 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.36/171.93 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.36/171.93 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.36/171.93 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.36/171.93 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.93 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.93 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.93 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.93 down(g(b)) -> g_flat(down(b)) 312.36/171.93 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.93 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.93 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.93 down(g(f(x))) -> up(b) 312.36/171.93 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.93 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.93 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.93 down(g(a)) -> g_flat(down(a)) 312.36/171.93 down(a) -> up(f(a)) 312.36/171.93 down(a) -> up(g(a)) 312.36/171.93 down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) 312.36/171.93 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.93 down(f(f(f(b)))) -> f_flat(down(f(f(b)))) 312.36/171.93 down(f(f(b))) -> f_flat(down(f(b))) 312.36/171.93 down(f(b)) -> f_flat(down(b)) 312.36/171.93 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.93 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.93 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.93 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.93 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.93 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.93 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.93 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.36/171.93 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.93 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.93 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.93 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.93 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.93 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.93 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.93 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.93 down(f(a)) -> f_flat(down(a)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (86) UsableRulesProof (EQUIVALENT) 312.36/171.93 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (87) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.36/171.93 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.36/171.93 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.36/171.93 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.36/171.93 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.36/171.93 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.36/171.93 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.36/171.93 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.36/171.93 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.36/171.93 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.36/171.93 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.36/171.93 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.36/171.93 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.36/171.93 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.36/171.93 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.36/171.93 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.36/171.93 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.36/171.93 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.36/171.93 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.93 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.93 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.93 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.93 down(g(f(x))) -> up(b) 312.36/171.93 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.93 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.93 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.93 down(g(a)) -> g_flat(down(a)) 312.36/171.93 down(a) -> up(f(a)) 312.36/171.93 down(a) -> up(g(a)) 312.36/171.93 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.93 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.93 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.93 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.93 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.93 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.93 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.93 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.93 down(g(b)) -> g_flat(down(b)) 312.36/171.93 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.93 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.93 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.93 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.36/171.93 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.93 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.93 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.93 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.93 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.93 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.93 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.93 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.93 down(f(a)) -> f_flat(down(a)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (88) QDPOrderProof (EQUIVALENT) 312.36/171.93 We use the reduction pair processor [LPAR04,JAR06]. 312.36/171.93 312.36/171.93 312.36/171.93 The following pairs can be oriented strictly and are deleted. 312.36/171.93 312.36/171.93 TOP(up(g(a))) -> TOP(g_flat(down(a))) 312.36/171.93 The remaining pairs can at least be oriented weakly. 312.36/171.93 Used ordering: Matrix interpretation [MATRO]: 312.36/171.93 312.36/171.93 Non-tuple symbols: 312.36/171.93 <<< 312.36/171.93 M( a ) = [[0], [1]] 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( b ) = [[0], [0]] 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( down_1(x_1) ) = [[0], [0]] + [[0, 1], [0, 0]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( f_1(x_1) ) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( fresh_constant ) = [[0], [0]] 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( up_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( f_flat_1(x_1) ) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( g_1(x_1) ) = [[0], [0]] + [[0, 1], [0, 0]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( g_flat_1(x_1) ) = [[0], [0]] + [[0, 1], [0, 0]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 Tuple symbols: 312.36/171.93 <<< 312.36/171.93 M( TOP_1(x_1) ) = [[0]] + [[1, 0]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 312.36/171.93 312.36/171.93 Matrix type: 312.36/171.93 312.36/171.93 We used a basic matrix type which is not further parametrizeable. 312.36/171.93 312.36/171.93 312.36/171.93 312.36/171.93 312.36/171.93 312.36/171.93 As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. 312.36/171.93 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 312.36/171.93 312.36/171.93 down(a) -> up(f(a)) 312.36/171.93 down(a) -> up(g(a)) 312.36/171.93 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.93 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.93 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.93 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.93 down(g(f(x))) -> up(b) 312.36/171.93 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.93 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.93 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.93 down(g(a)) -> g_flat(down(a)) 312.36/171.93 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.93 312.36/171.93 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (89) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.36/171.93 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.36/171.93 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.36/171.93 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.36/171.93 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.36/171.93 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.36/171.93 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.36/171.93 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.36/171.93 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.36/171.93 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.36/171.93 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.36/171.93 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.36/171.93 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.36/171.93 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.36/171.93 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.36/171.93 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.36/171.93 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.36/171.93 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.93 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.93 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.93 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.93 down(g(f(x))) -> up(b) 312.36/171.93 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.93 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.93 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.93 down(g(a)) -> g_flat(down(a)) 312.36/171.93 down(a) -> up(f(a)) 312.36/171.93 down(a) -> up(g(a)) 312.36/171.93 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.93 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.93 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.93 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.93 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.93 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.93 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.93 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.93 down(g(b)) -> g_flat(down(b)) 312.36/171.93 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.93 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.93 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.93 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.36/171.93 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.93 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.93 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.93 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.93 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.93 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.93 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.93 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.93 down(f(a)) -> f_flat(down(a)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (90) QDPOrderProof (EQUIVALENT) 312.36/171.93 We use the reduction pair processor [LPAR04,JAR06]. 312.36/171.93 312.36/171.93 312.36/171.93 The following pairs can be oriented strictly and are deleted. 312.36/171.93 312.36/171.93 TOP(up(f(a))) -> TOP(f_flat(down(a))) 312.36/171.93 The remaining pairs can at least be oriented weakly. 312.36/171.93 Used ordering: Matrix interpretation [MATRO]: 312.36/171.93 312.36/171.93 Non-tuple symbols: 312.36/171.93 <<< 312.36/171.93 M( a ) = [[1], [0]] 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( b ) = [[0], [0]] 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( down_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( f_1(x_1) ) = [[0], [0]] + [[0, 0], [1, 0]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( fresh_constant ) = [[0], [0]] 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( up_1(x_1) ) = [[0], [0]] + [[0, 1], [1, 0]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( f_flat_1(x_1) ) = [[0], [0]] + [[0, 1], [0, 0]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( g_1(x_1) ) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 <<< 312.36/171.93 M( g_flat_1(x_1) ) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 Tuple symbols: 312.36/171.93 <<< 312.36/171.93 M( TOP_1(x_1) ) = [[0]] + [[1, 0]] * x_1 312.36/171.93 >>> 312.36/171.93 312.36/171.93 312.36/171.93 312.36/171.93 Matrix type: 312.36/171.93 312.36/171.93 We used a basic matrix type which is not further parametrizeable. 312.36/171.93 312.36/171.93 312.36/171.93 312.36/171.93 312.36/171.93 312.36/171.93 As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. 312.36/171.93 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 312.36/171.93 312.36/171.93 down(a) -> up(f(a)) 312.36/171.93 down(a) -> up(g(a)) 312.36/171.93 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.93 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.93 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.93 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.93 down(g(f(x))) -> up(b) 312.36/171.93 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.93 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.93 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.93 down(g(a)) -> g_flat(down(a)) 312.36/171.93 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.93 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.93 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.93 down(g(b)) -> g_flat(down(b)) 312.36/171.93 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.93 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.93 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.93 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.36/171.93 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.93 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.93 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.93 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.93 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.93 down(f(a)) -> f_flat(down(a)) 312.36/171.93 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.93 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.93 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.93 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.93 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.93 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.93 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.93 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.93 312.36/171.93 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (91) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.36/171.93 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.36/171.93 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.36/171.93 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.36/171.93 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.36/171.93 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.36/171.93 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.36/171.93 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.36/171.93 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.36/171.93 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.36/171.93 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.36/171.93 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.36/171.93 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.36/171.93 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.36/171.93 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.36/171.93 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.36/171.93 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.93 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.93 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.93 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.93 down(g(f(x))) -> up(b) 312.36/171.93 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.93 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.93 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.93 down(g(a)) -> g_flat(down(a)) 312.36/171.93 down(a) -> up(f(a)) 312.36/171.93 down(a) -> up(g(a)) 312.36/171.93 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.93 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.93 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.93 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.93 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.93 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.93 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.93 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.93 down(g(b)) -> g_flat(down(b)) 312.36/171.93 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.93 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.93 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.93 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.36/171.93 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.93 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.93 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.93 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.93 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.93 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.93 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.93 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.93 down(f(a)) -> f_flat(down(a)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (92) SplitQDPProof (EQUIVALENT) 312.36/171.93 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 312.36/171.93 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (93) 312.36/171.93 Complex Obligation (AND) 312.36/171.93 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (94) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.36/171.93 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.36/171.93 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.36/171.93 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.36/171.93 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.36/171.93 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.36/171.93 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.36/171.93 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.36/171.93 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.36/171.93 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.36/171.93 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.36/171.93 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.36/171.93 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.36/171.93 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.36/171.93 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.36/171.93 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.36/171.93 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.93 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.93 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.93 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.93 down(g(f(x))) -> up(b) 312.36/171.93 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.93 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.93 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.93 down(g(a)) -> g_flat(down(a)) 312.36/171.93 down(a) -> up(f(a)) 312.36/171.93 down(a) -> up(g(a)) 312.36/171.93 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.93 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.93 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.93 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.93 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.93 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.93 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.93 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.93 down(g(b)) -> g_flat(down(b)) 312.36/171.93 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.93 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.93 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.93 down(g(fresh_constant)) -> g_flat(down(fresh_constant)) 312.36/171.93 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.93 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.93 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.93 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.93 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.93 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.93 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.93 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.93 down(f(a)) -> f_flat(down(a)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (95) SemLabProof (SOUND) 312.36/171.93 We found the following model for the rules of the TRSs R and P. 312.36/171.93 Interpretation over the domain with elements from 0 to 1. 312.36/171.93 a: 0 312.36/171.93 b: 0 312.36/171.93 down: 0 312.36/171.93 f: 0 312.36/171.93 fresh_constant: 1 312.36/171.93 up: 0 312.36/171.93 f_flat: 0 312.36/171.93 TOP: 0 312.36/171.93 g_flat: 0 312.36/171.93 g: 0 312.36/171.93 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (96) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.93 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.93 TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.1(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.0(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.1(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.1(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.0(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.1(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.93 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.93 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 312.36/171.93 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.93 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.93 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.93 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.93 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(f.1(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.93 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.93 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.93 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.93 down.0(a.) -> up.0(f.0(a.)) 312.36/171.93 down.0(a.) -> up.0(g.0(a.)) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.93 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.93 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 312.36/171.93 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.93 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.93 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.93 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.93 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.93 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.93 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.93 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.93 down.0(g.0(b.)) -> g_flat.0(down.0(b.)) 312.36/171.93 down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.1(fresh_constant.))))) 312.36/171.93 down.0(g.0(g.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.1(fresh_constant.)))) 312.36/171.93 down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.))) 312.36/171.93 down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.)) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.93 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.93 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.93 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (97) UsableRulesReductionPairsProof (EQUIVALENT) 312.36/171.93 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. 312.36/171.93 312.36/171.93 No dependency pairs are removed. 312.36/171.93 312.36/171.93 The following rules are removed from R: 312.36/171.93 312.36/171.93 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 312.36/171.93 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 312.36/171.93 down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.)) 312.36/171.93 Used ordering: POLO with Polynomial interpretation [POLO]: 312.36/171.93 312.36/171.93 POL(TOP.0(x_1)) = x_1 312.36/171.93 POL(a.) = 0 312.36/171.93 POL(b.) = 0 312.36/171.93 POL(down.0(x_1)) = 1 + x_1 312.36/171.93 POL(down.1(x_1)) = x_1 312.36/171.93 POL(f.0(x_1)) = x_1 312.36/171.93 POL(f.1(x_1)) = x_1 312.36/171.93 POL(f_flat.0(x_1)) = x_1 312.36/171.93 POL(fresh_constant.) = 0 312.36/171.93 POL(g.0(x_1)) = x_1 312.36/171.93 POL(g.1(x_1)) = x_1 312.36/171.93 POL(g_flat.0(x_1)) = x_1 312.36/171.93 POL(up.0(x_1)) = 1 + x_1 312.36/171.93 POL(up.1(x_1)) = 1 + x_1 312.36/171.93 312.36/171.93 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (98) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.93 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.93 TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.1(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.0(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.1(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.1(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.0(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.1(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.93 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.93 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.93 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.93 down.0(g.0(f.1(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.93 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.93 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.93 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.93 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.93 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.93 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.93 down.0(a.) -> up.0(f.0(a.)) 312.36/171.93 down.0(a.) -> up.0(g.0(a.)) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.93 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.93 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.93 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.93 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.93 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.93 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.93 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.93 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.93 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.93 down.0(g.0(b.)) -> g_flat.0(down.0(b.)) 312.36/171.93 down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.1(fresh_constant.))))) 312.36/171.93 down.0(g.0(g.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.1(fresh_constant.)))) 312.36/171.93 down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.93 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.93 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.93 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (99) DependencyGraphProof (EQUIVALENT) 312.36/171.93 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (100) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.93 TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.1(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.0(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.1(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.1(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.0(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.1(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.93 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.93 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.93 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.93 down.0(g.0(f.1(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.93 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.93 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.93 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.93 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.93 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.93 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.93 down.0(a.) -> up.0(f.0(a.)) 312.36/171.93 down.0(a.) -> up.0(g.0(a.)) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.93 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.93 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.93 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.93 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.93 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.93 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.93 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.93 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.93 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.93 down.0(g.0(b.)) -> g_flat.0(down.0(b.)) 312.36/171.93 down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.1(fresh_constant.))))) 312.36/171.93 down.0(g.0(g.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.1(fresh_constant.)))) 312.36/171.93 down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.93 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.93 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.93 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (101) MRRProof (EQUIVALENT) 312.36/171.93 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. 312.36/171.93 312.36/171.93 312.36/171.93 Strictly oriented rules of the TRS R: 312.36/171.93 312.36/171.93 down.0(g.0(f.1(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))) -> up.0(b.) 312.36/171.93 312.36/171.93 Used ordering: Polynomial interpretation [POLO]: 312.36/171.93 312.36/171.93 POL(TOP.0(x_1)) = x_1 312.36/171.93 POL(a.) = 0 312.36/171.93 POL(b.) = 0 312.36/171.93 POL(down.0(x_1)) = 1 + x_1 312.36/171.93 POL(f.0(x_1)) = x_1 312.36/171.93 POL(f.1(x_1)) = 1 + x_1 312.36/171.93 POL(f_flat.0(x_1)) = x_1 312.36/171.93 POL(fresh_constant.) = 0 312.36/171.93 POL(g.0(x_1)) = x_1 312.36/171.93 POL(g.1(x_1)) = 1 + x_1 312.36/171.93 POL(g_flat.0(x_1)) = x_1 312.36/171.93 POL(up.0(x_1)) = 1 + x_1 312.36/171.93 312.36/171.93 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (102) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.93 TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.1(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.0(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.1(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.1(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.0(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.1(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.93 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.93 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.93 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.93 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.93 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.93 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.93 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.93 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.93 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.93 down.0(a.) -> up.0(f.0(a.)) 312.36/171.93 down.0(a.) -> up.0(g.0(a.)) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.93 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.93 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.93 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.93 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.93 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.93 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.93 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.93 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.93 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.93 down.0(g.0(b.)) -> g_flat.0(down.0(b.)) 312.36/171.93 down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.1(fresh_constant.))))) 312.36/171.93 down.0(g.0(g.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.1(fresh_constant.)))) 312.36/171.93 down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.93 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.93 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.93 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (103) DependencyGraphProof (EQUIVALENT) 312.36/171.93 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (104) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.93 TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.1(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.0(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.1(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.1(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.0(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.1(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.93 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.93 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.93 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.93 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.93 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.93 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.93 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.93 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.93 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.93 down.0(a.) -> up.0(f.0(a.)) 312.36/171.93 down.0(a.) -> up.0(g.0(a.)) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.93 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.93 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.93 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.93 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.93 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.93 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.93 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.93 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.93 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.93 down.0(g.0(b.)) -> g_flat.0(down.0(b.)) 312.36/171.93 down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.1(fresh_constant.))))) 312.36/171.93 down.0(g.0(g.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.1(fresh_constant.)))) 312.36/171.93 down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.93 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.93 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.93 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (105) QDPOrderProof (EQUIVALENT) 312.36/171.93 We use the reduction pair processor [LPAR04,JAR06]. 312.36/171.93 312.36/171.93 312.36/171.93 The following pairs can be oriented strictly and are deleted. 312.36/171.93 312.36/171.93 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.1(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.1(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) 312.36/171.93 The remaining pairs can at least be oriented weakly. 312.36/171.93 Used ordering: Polynomial interpretation [POLO]: 312.36/171.93 312.36/171.93 POL(TOP.0(x_1)) = x_1 312.36/171.93 POL(a.) = 1 312.36/171.93 POL(b.) = 0 312.36/171.93 POL(down.0(x_1)) = x_1 312.36/171.93 POL(f.0(x_1)) = x_1 312.36/171.93 POL(f.1(x_1)) = x_1 312.36/171.93 POL(f_flat.0(x_1)) = x_1 312.36/171.93 POL(fresh_constant.) = 0 312.36/171.93 POL(g.0(x_1)) = 1 312.36/171.93 POL(g.1(x_1)) = 0 312.36/171.93 POL(g_flat.0(x_1)) = 1 312.36/171.93 POL(up.0(x_1)) = 1 312.36/171.93 312.36/171.93 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 312.36/171.93 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.93 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.93 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.93 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.93 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.93 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.93 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.93 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.93 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.93 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.93 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.93 down.0(g.0(b.)) -> g_flat.0(down.0(b.)) 312.36/171.93 down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.1(fresh_constant.))))) 312.36/171.93 down.0(g.0(g.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.1(fresh_constant.)))) 312.36/171.93 down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) 312.36/171.93 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.93 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.93 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.93 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.93 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.93 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.93 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.93 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.93 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.93 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.93 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.93 down.0(a.) -> up.0(f.0(a.)) 312.36/171.93 down.0(a.) -> up.0(g.0(a.)) 312.36/171.93 312.36/171.93 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (106) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.93 TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.0(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.1(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.0(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.1(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.93 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.93 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.93 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.93 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.93 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.93 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.93 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.93 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.93 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.93 down.0(a.) -> up.0(f.0(a.)) 312.36/171.93 down.0(a.) -> up.0(g.0(a.)) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.93 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.93 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.93 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.93 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.93 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.93 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.93 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.93 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.93 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.93 down.0(g.0(b.)) -> g_flat.0(down.0(b.)) 312.36/171.93 down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.1(fresh_constant.))))) 312.36/171.93 down.0(g.0(g.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.1(fresh_constant.)))) 312.36/171.93 down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.93 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.93 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.93 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (107) UsableRulesReductionPairsProof (EQUIVALENT) 312.36/171.93 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. 312.36/171.93 312.36/171.93 No dependency pairs are removed. 312.36/171.93 312.36/171.93 The following rules are removed from R: 312.36/171.93 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.93 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.93 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.93 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.93 Used ordering: POLO with Polynomial interpretation [POLO]: 312.36/171.93 312.36/171.93 POL(TOP.0(x_1)) = x_1 312.36/171.93 POL(a.) = 0 312.36/171.93 POL(b.) = 0 312.36/171.93 POL(down.0(x_1)) = 1 + x_1 312.36/171.93 POL(f.0(x_1)) = x_1 312.36/171.93 POL(f.1(x_1)) = x_1 312.36/171.93 POL(f_flat.0(x_1)) = x_1 312.36/171.93 POL(fresh_constant.) = 0 312.36/171.93 POL(g.0(x_1)) = x_1 312.36/171.93 POL(g.1(x_1)) = x_1 312.36/171.93 POL(g_flat.0(x_1)) = x_1 312.36/171.93 POL(up.0(x_1)) = 1 + x_1 312.36/171.93 312.36/171.93 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (108) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.93 TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.0(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.1(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.0(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.1(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.93 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.93 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.93 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.93 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.93 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.93 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.93 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.93 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.93 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.93 down.0(a.) -> up.0(f.0(a.)) 312.36/171.93 down.0(a.) -> up.0(g.0(a.)) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.93 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.93 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.93 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.93 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.93 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.93 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.93 down.0(g.0(b.)) -> g_flat.0(down.0(b.)) 312.36/171.93 down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.1(fresh_constant.))))) 312.36/171.93 down.0(g.0(g.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.1(fresh_constant.)))) 312.36/171.93 down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.93 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.93 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.93 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (109) QDPOrderProof (EQUIVALENT) 312.36/171.93 We use the reduction pair processor [LPAR04,JAR06]. 312.36/171.93 312.36/171.93 312.36/171.93 The following pairs can be oriented strictly and are deleted. 312.36/171.93 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.1(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.1(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) 312.36/171.93 The remaining pairs can at least be oriented weakly. 312.36/171.93 Used ordering: Polynomial interpretation [POLO]: 312.36/171.93 312.36/171.93 POL(TOP.0(x_1)) = x_1 312.36/171.93 POL(a.) = 1 312.36/171.93 POL(b.) = 0 312.36/171.93 POL(down.0(x_1)) = x_1 312.36/171.93 POL(f.0(x_1)) = 1 312.36/171.93 POL(f.1(x_1)) = 0 312.36/171.93 POL(f_flat.0(x_1)) = 1 312.36/171.93 POL(fresh_constant.) = 0 312.36/171.93 POL(g.0(x_1)) = x_1 312.36/171.93 POL(g.1(x_1)) = 0 312.36/171.93 POL(g_flat.0(x_1)) = x_1 312.36/171.93 POL(up.0(x_1)) = 1 312.36/171.93 312.36/171.93 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 312.36/171.93 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.93 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.93 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.93 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.93 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.93 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.93 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.93 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.93 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.93 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.93 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.93 down.0(a.) -> up.0(f.0(a.)) 312.36/171.93 down.0(a.) -> up.0(g.0(a.)) 312.36/171.93 312.36/171.93 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (110) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.93 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.93 TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(f.0(x0)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(f.0(x0))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.93 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.93 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.93 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.93 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.93 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.93 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.93 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.93 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.93 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.93 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.93 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.93 down.0(a.) -> up.0(f.0(a.)) 312.36/171.93 down.0(a.) -> up.0(g.0(a.)) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.93 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.93 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.93 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.93 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.93 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.93 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.93 down.0(g.0(b.)) -> g_flat.0(down.0(b.)) 312.36/171.93 down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.1(fresh_constant.))))) 312.36/171.93 down.0(g.0(g.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.1(fresh_constant.)))) 312.36/171.93 down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.93 down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) 312.36/171.93 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.93 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.93 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.93 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (111) PisEmptyProof (SOUND) 312.36/171.93 The TRS P is empty. Hence, there is no (P,Q,R) chain. 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (112) 312.36/171.93 TRUE 312.36/171.93 312.36/171.93 ---------------------------------------- 312.36/171.93 312.36/171.93 (113) 312.36/171.93 Obligation: 312.36/171.93 Q DP problem: 312.36/171.93 The TRS P consists of the following rules: 312.36/171.93 312.36/171.93 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.36/171.93 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.36/171.93 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.36/171.93 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.36/171.93 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.36/171.93 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.36/171.93 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.36/171.93 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.36/171.93 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.36/171.93 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.36/171.93 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.36/171.93 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.36/171.93 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.36/171.93 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.36/171.93 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.36/171.93 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.36/171.93 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.36/171.93 312.36/171.93 The TRS R consists of the following rules: 312.36/171.93 312.36/171.93 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.93 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.93 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.93 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.93 down(g(f(x))) -> up(b) 312.36/171.93 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.93 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.93 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.93 down(g(a)) -> g_flat(down(a)) 312.36/171.93 down(a) -> up(f(a)) 312.36/171.93 down(a) -> up(g(a)) 312.36/171.93 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.93 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.93 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.93 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.93 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.93 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.93 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.93 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.93 down(g(b)) -> g_flat(down(b)) 312.36/171.93 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.93 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.93 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.93 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.93 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.93 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.93 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.93 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.93 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.93 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.93 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.93 down(f(a)) -> f_flat(down(a)) 312.36/171.93 312.36/171.93 Q is empty. 312.36/171.93 We have to consider all minimal (P,Q,R)-chains. 312.36/171.93 ---------------------------------------- 312.36/171.94 312.36/171.94 (114) QDPOrderProof (EQUIVALENT) 312.36/171.94 We use the reduction pair processor [LPAR04,JAR06]. 312.36/171.94 312.36/171.94 312.36/171.94 The following pairs can be oriented strictly and are deleted. 312.36/171.94 312.36/171.94 TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) 312.36/171.94 The remaining pairs can at least be oriented weakly. 312.36/171.94 Used ordering: Matrix interpretation [MATRO]: 312.36/171.94 312.36/171.94 Non-tuple symbols: 312.36/171.94 <<< 312.36/171.94 M( a ) = [[1], [1]] 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( b ) = [[0], [0]] 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( down_1(x_1) ) = [[0], [1]] + [[0, 1], [0, 0]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( f_1(x_1) ) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( fresh_constant ) = [[0], [0]] 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( up_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( f_flat_1(x_1) ) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( g_1(x_1) ) = [[1], [0]] + [[0, 0], [1, 0]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( g_flat_1(x_1) ) = [[1], [0]] + [[0, 0], [1, 0]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 Tuple symbols: 312.36/171.94 <<< 312.36/171.94 M( TOP_1(x_1) ) = [[0]] + [[0, 1]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 312.36/171.94 312.36/171.94 Matrix type: 312.36/171.94 312.36/171.94 We used a basic matrix type which is not further parametrizeable. 312.36/171.94 312.36/171.94 312.36/171.94 312.36/171.94 312.36/171.94 312.36/171.94 As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. 312.36/171.94 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 312.36/171.94 312.36/171.94 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.94 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.94 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.94 down(g(f(x))) -> up(b) 312.36/171.94 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.94 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.94 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.94 down(g(a)) -> g_flat(down(a)) 312.36/171.94 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.94 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.94 down(a) -> up(f(a)) 312.36/171.94 down(a) -> up(g(a)) 312.36/171.94 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (115) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.36/171.94 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.36/171.94 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.36/171.94 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.36/171.94 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.36/171.94 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.36/171.94 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.36/171.94 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.36/171.94 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.36/171.94 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.36/171.94 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.36/171.94 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.36/171.94 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.36/171.94 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.36/171.94 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.36/171.94 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.94 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.94 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.94 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.94 down(g(f(x))) -> up(b) 312.36/171.94 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.94 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.94 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.94 down(g(a)) -> g_flat(down(a)) 312.36/171.94 down(a) -> up(f(a)) 312.36/171.94 down(a) -> up(g(a)) 312.36/171.94 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.94 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.94 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.94 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.94 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.94 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.94 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.94 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.94 down(g(b)) -> g_flat(down(b)) 312.36/171.94 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.94 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.94 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.94 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.94 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.94 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.94 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.94 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.94 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.94 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.94 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.94 down(f(a)) -> f_flat(down(a)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (116) QDPOrderProof (EQUIVALENT) 312.36/171.94 We use the reduction pair processor [LPAR04,JAR06]. 312.36/171.94 312.36/171.94 312.36/171.94 The following pairs can be oriented strictly and are deleted. 312.36/171.94 312.36/171.94 TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) 312.36/171.94 TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) 312.36/171.94 TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) 312.36/171.94 The remaining pairs can at least be oriented weakly. 312.36/171.94 Used ordering: Matrix interpretation [MATRO]: 312.36/171.94 312.36/171.94 Non-tuple symbols: 312.36/171.94 <<< 312.36/171.94 M( a ) = [[1], [1]] 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( b ) = [[0], [0]] 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( down_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( f_1(x_1) ) = [[0], [1]] + [[0, 0], [0, 0]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( fresh_constant ) = [[0], [0]] 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( up_1(x_1) ) = [[0], [0]] + [[0, 1], [0, 0]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( f_flat_1(x_1) ) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( g_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 <<< 312.36/171.94 M( g_flat_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 Tuple symbols: 312.36/171.94 <<< 312.36/171.94 M( TOP_1(x_1) ) = [[0]] + [[1, 0]] * x_1 312.36/171.94 >>> 312.36/171.94 312.36/171.94 312.36/171.94 312.36/171.94 Matrix type: 312.36/171.94 312.36/171.94 We used a basic matrix type which is not further parametrizeable. 312.36/171.94 312.36/171.94 312.36/171.94 312.36/171.94 312.36/171.94 312.36/171.94 As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. 312.36/171.94 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 312.36/171.94 312.36/171.94 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.94 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.94 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.94 down(g(f(x))) -> up(b) 312.36/171.94 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.94 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.94 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.94 down(g(a)) -> g_flat(down(a)) 312.36/171.94 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.94 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.94 down(a) -> up(f(a)) 312.36/171.94 down(a) -> up(g(a)) 312.36/171.94 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (117) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.36/171.94 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.36/171.94 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.36/171.94 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.36/171.94 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.36/171.94 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.36/171.94 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.36/171.94 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.36/171.94 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.36/171.94 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.36/171.94 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.36/171.94 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.36/171.94 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.94 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.94 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.94 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.94 down(g(f(x))) -> up(b) 312.36/171.94 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.94 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.94 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.94 down(g(a)) -> g_flat(down(a)) 312.36/171.94 down(a) -> up(f(a)) 312.36/171.94 down(a) -> up(g(a)) 312.36/171.94 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.94 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.94 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.94 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.94 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.94 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.94 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.94 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.94 down(g(b)) -> g_flat(down(b)) 312.36/171.94 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.94 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.94 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.94 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.94 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.94 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.94 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.94 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.94 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.94 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.94 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.94 down(f(a)) -> f_flat(down(a)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (118) SplitQDPProof (EQUIVALENT) 312.36/171.94 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (119) 312.36/171.94 Complex Obligation (AND) 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (120) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.36/171.94 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.36/171.94 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.36/171.94 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.36/171.94 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.36/171.94 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.36/171.94 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.36/171.94 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.36/171.94 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.36/171.94 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.36/171.94 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.36/171.94 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.36/171.94 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.94 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.94 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.94 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.94 down(g(f(x))) -> up(b) 312.36/171.94 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.94 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.94 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.94 down(g(a)) -> g_flat(down(a)) 312.36/171.94 down(a) -> up(f(a)) 312.36/171.94 down(a) -> up(g(a)) 312.36/171.94 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.94 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.94 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.94 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.94 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.94 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.94 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.94 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.94 down(g(b)) -> g_flat(down(b)) 312.36/171.94 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.94 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.94 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.94 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.94 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.94 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.94 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.94 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.94 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.94 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.94 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.94 down(f(a)) -> f_flat(down(a)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (121) SemLabProof (SOUND) 312.36/171.94 We found the following model for the rules of the TRSs R and P. 312.36/171.94 Interpretation over the domain with elements from 0 to 1. 312.36/171.94 a: 0 312.36/171.94 b: 1 312.36/171.94 down: 0 312.36/171.94 f: 0 312.36/171.94 fresh_constant: 0 312.36/171.94 up: 0 312.36/171.94 f_flat: 0 312.36/171.94 TOP: 0 312.36/171.94 g_flat: 0 312.36/171.94 g: 0 312.36/171.94 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (122) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.94 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.1(x0)))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.1(x0))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.1(b.) 312.36/171.94 down.0(g.0(f.1(x))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.94 down.0(g.0(g.0(g.0(g.1(b.))))) -> g_flat.0(down.0(g.0(g.0(g.1(b.))))) 312.36/171.94 down.0(g.0(g.0(g.1(b.)))) -> g_flat.0(down.0(g.0(g.1(b.)))) 312.36/171.94 down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.))) 312.36/171.94 down.0(g.1(b.)) -> g_flat.0(down.1(b.)) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) 312.36/171.94 down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) 312.36/171.94 down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(b.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (123) MRRProof (EQUIVALENT) 312.36/171.94 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. 312.36/171.94 312.36/171.94 312.36/171.94 Strictly oriented rules of the TRS R: 312.36/171.94 312.36/171.94 down.0(g.1(b.)) -> g_flat.0(down.1(b.)) 312.36/171.94 312.36/171.94 Used ordering: Polynomial interpretation [POLO]: 312.36/171.94 312.36/171.94 POL(TOP.0(x_1)) = x_1 312.36/171.94 POL(a.) = 0 312.36/171.94 POL(b.) = 0 312.36/171.94 POL(down.0(x_1)) = 1 + x_1 312.36/171.94 POL(down.1(x_1)) = x_1 312.36/171.94 POL(f.0(x_1)) = x_1 312.36/171.94 POL(f.1(x_1)) = x_1 312.36/171.94 POL(f_flat.0(x_1)) = x_1 312.36/171.94 POL(fresh_constant.) = 0 312.36/171.94 POL(g.0(x_1)) = x_1 312.36/171.94 POL(g.1(x_1)) = x_1 312.36/171.94 POL(g_flat.0(x_1)) = x_1 312.36/171.94 POL(up.0(x_1)) = 1 + x_1 312.36/171.94 POL(up.1(x_1)) = 1 + x_1 312.36/171.94 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (124) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.94 TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.1(x0)))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.1(x0))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.1(b.) 312.36/171.94 down.0(g.0(f.1(x))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.94 down.0(g.0(g.0(g.0(g.1(b.))))) -> g_flat.0(down.0(g.0(g.0(g.1(b.))))) 312.36/171.94 down.0(g.0(g.0(g.1(b.)))) -> g_flat.0(down.0(g.0(g.1(b.)))) 312.36/171.94 down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) 312.36/171.94 down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) 312.36/171.94 down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(b.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (125) DependencyGraphProof (EQUIVALENT) 312.36/171.94 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (126) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.1(x0)))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.1(x0))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.1(b.) 312.36/171.94 down.0(g.0(f.1(x))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.94 down.0(g.0(g.0(g.0(g.1(b.))))) -> g_flat.0(down.0(g.0(g.0(g.1(b.))))) 312.36/171.94 down.0(g.0(g.0(g.1(b.)))) -> g_flat.0(down.0(g.0(g.1(b.)))) 312.36/171.94 down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) 312.36/171.94 down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) 312.36/171.94 down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(b.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (127) QDPOrderProof (EQUIVALENT) 312.36/171.94 We use the reduction pair processor [LPAR04,JAR06]. 312.36/171.94 312.36/171.94 312.36/171.94 The following pairs can be oriented strictly and are deleted. 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.1(x0)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.1(x0))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) 312.36/171.94 The remaining pairs can at least be oriented weakly. 312.36/171.94 Used ordering: Polynomial interpretation [POLO]: 312.36/171.94 312.36/171.94 POL(TOP.0(x_1)) = x_1 312.36/171.94 POL(a.) = 1 312.36/171.94 POL(b.) = 0 312.36/171.94 POL(down.0(x_1)) = x_1 312.36/171.94 POL(f.0(x_1)) = x_1 312.36/171.94 POL(f.1(x_1)) = x_1 312.36/171.94 POL(f_flat.0(x_1)) = x_1 312.36/171.94 POL(fresh_constant.) = 0 312.36/171.94 POL(g.0(x_1)) = 1 312.36/171.94 POL(g.1(x_1)) = 0 312.36/171.94 POL(g_flat.0(x_1)) = 1 312.36/171.94 POL(up.0(x_1)) = 1 312.36/171.94 POL(up.1(x_1)) = 1 + x_1 312.36/171.94 312.36/171.94 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.1(b.) 312.36/171.94 down.0(g.0(f.1(x))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(g.0(g.0(g.0(g.1(b.))))) -> g_flat.0(down.0(g.0(g.0(g.1(b.))))) 312.36/171.94 down.0(g.0(g.0(g.1(b.)))) -> g_flat.0(down.0(g.0(g.1(b.)))) 312.36/171.94 down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) 312.36/171.94 down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) 312.36/171.94 down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(b.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (128) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.1(b.) 312.36/171.94 down.0(g.0(f.1(x))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.94 down.0(g.0(g.0(g.0(g.1(b.))))) -> g_flat.0(down.0(g.0(g.0(g.1(b.))))) 312.36/171.94 down.0(g.0(g.0(g.1(b.)))) -> g_flat.0(down.0(g.0(g.1(b.)))) 312.36/171.94 down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) 312.36/171.94 down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) 312.36/171.94 down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(b.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (129) UsableRulesReductionPairsProof (EQUIVALENT) 312.36/171.94 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. 312.36/171.94 312.36/171.94 No dependency pairs are removed. 312.36/171.94 312.36/171.94 The following rules are removed from R: 312.36/171.94 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) 312.36/171.94 down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) 312.36/171.94 down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) 312.36/171.94 Used ordering: POLO with Polynomial interpretation [POLO]: 312.36/171.94 312.36/171.94 POL(TOP.0(x_1)) = x_1 312.36/171.94 POL(a.) = 0 312.36/171.94 POL(b.) = 0 312.36/171.94 POL(down.0(x_1)) = 1 + x_1 312.36/171.94 POL(f.0(x_1)) = x_1 312.36/171.94 POL(f.1(x_1)) = x_1 312.36/171.94 POL(f_flat.0(x_1)) = x_1 312.36/171.94 POL(fresh_constant.) = 0 312.36/171.94 POL(g.0(x_1)) = x_1 312.36/171.94 POL(g.1(x_1)) = x_1 312.36/171.94 POL(g_flat.0(x_1)) = x_1 312.36/171.94 POL(up.0(x_1)) = 1 + x_1 312.36/171.94 POL(up.1(x_1)) = 1 + x_1 312.36/171.94 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (130) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.1(b.) 312.36/171.94 down.0(g.0(f.1(x))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.1(b.))))) -> g_flat.0(down.0(g.0(g.0(g.1(b.))))) 312.36/171.94 down.0(g.0(g.0(g.1(b.)))) -> g_flat.0(down.0(g.0(g.1(b.)))) 312.36/171.94 down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) 312.36/171.94 down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) 312.36/171.94 down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))) -> up.1(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(b.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (131) PisEmptyProof (SOUND) 312.36/171.94 The TRS P is empty. Hence, there is no (P,Q,R) chain. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (132) 312.36/171.94 TRUE 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (133) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.36/171.94 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.36/171.94 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.36/171.94 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.36/171.94 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.36/171.94 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.36/171.94 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.36/171.94 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.36/171.94 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.36/171.94 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.36/171.94 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.36/171.94 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.36/171.94 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.94 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.94 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.94 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.94 down(g(a)) -> g_flat(down(a)) 312.36/171.94 down(a) -> up(f(a)) 312.36/171.94 down(a) -> up(g(a)) 312.36/171.94 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.94 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.94 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.94 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.94 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.94 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.94 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.94 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.94 down(g(f(x))) -> up(b) 312.36/171.94 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.94 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.94 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.94 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.94 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.94 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.94 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.94 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.94 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.94 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.94 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.94 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.94 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.94 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.94 down(f(a)) -> f_flat(down(a)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (134) SplitQDPProof (EQUIVALENT) 312.36/171.94 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (135) 312.36/171.94 Complex Obligation (AND) 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (136) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.36/171.94 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.36/171.94 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.36/171.94 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.36/171.94 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.36/171.94 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.36/171.94 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.36/171.94 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.36/171.94 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.36/171.94 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.36/171.94 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.36/171.94 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.36/171.94 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.94 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.94 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.94 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.94 down(g(a)) -> g_flat(down(a)) 312.36/171.94 down(a) -> up(f(a)) 312.36/171.94 down(a) -> up(g(a)) 312.36/171.94 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.94 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.94 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.94 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.94 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.94 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.94 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.94 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.94 down(g(f(x))) -> up(b) 312.36/171.94 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.94 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.94 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.94 down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) 312.36/171.94 down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) 312.36/171.94 down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) 312.36/171.94 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.94 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.94 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.94 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.94 down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) 312.36/171.94 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.94 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.94 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.94 down(f(a)) -> f_flat(down(a)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (137) SemLabProof (SOUND) 312.36/171.94 We found the following model for the rules of the TRSs R and P. 312.36/171.94 Interpretation over the domain with elements from 0 to 1. 312.36/171.94 a: 0 312.36/171.94 b: 0 312.36/171.94 down: 0 312.36/171.94 f: 0 312.36/171.94 fresh_constant: 1 312.36/171.94 up: 0 312.36/171.94 f_flat: 0 312.36/171.94 TOP: 0 312.36/171.94 g_flat: 0 312.36/171.94 g: x0 312.36/171.94 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (138) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.94 TOP.0(up.0(f.1(g.1(x0)))) -> TOP.0(f_flat.0(down.1(g.1(x0)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.94 TOP.0(up.0(f.0(f.1(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.1(g.1(x0))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.1(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(g.1(x0)))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.1(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.1(g.1(x0))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.1(g.1(x_1)) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.1(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.1(g.1(y22)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(f.1(g.1(y16))))) -> f_flat.0(down.0(f.0(f.1(g.1(y16))))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(f.1(g.1(y10)))) -> f_flat.0(down.0(f.1(g.1(y10)))) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 down.0(f.1(g.1(y4))) -> f_flat.0(down.1(g.1(y4))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.94 down.0(g.0(f.1(x))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.94 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.94 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.94 down.1(g.1(g.1(g.1(g.1(fresh_constant.))))) -> g_flat.0(down.1(g.1(g.1(g.1(fresh_constant.))))) 312.36/171.94 down.1(g.1(g.1(g.1(fresh_constant.)))) -> g_flat.0(down.1(g.1(g.1(fresh_constant.)))) 312.36/171.94 down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.94 down.1(g.1(g.1(g.1(g.1(g.1(g.1(x))))))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.94 down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (139) MRRProof (EQUIVALENT) 312.36/171.94 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. 312.36/171.94 312.36/171.94 Strictly oriented dependency pairs: 312.36/171.94 312.36/171.94 TOP.0(up.0(f.1(g.1(x0)))) -> TOP.0(f_flat.0(down.1(g.1(x0)))) 312.36/171.94 312.36/171.94 Strictly oriented rules of the TRS R: 312.36/171.94 312.36/171.94 down.0(f.1(g.1(y4))) -> f_flat.0(down.1(g.1(y4))) 312.36/171.94 down.0(g.0(f.1(x))) -> up.0(b.) 312.36/171.94 312.36/171.94 Used ordering: Polynomial interpretation [POLO]: 312.36/171.94 312.36/171.94 POL(TOP.0(x_1)) = x_1 312.36/171.94 POL(a.) = 0 312.36/171.94 POL(b.) = 0 312.36/171.94 POL(down.0(x_1)) = x_1 312.36/171.94 POL(down.1(x_1)) = x_1 312.36/171.94 POL(f.0(x_1)) = x_1 312.36/171.94 POL(f.1(x_1)) = 1 + x_1 312.36/171.94 POL(f_flat.0(x_1)) = x_1 312.36/171.94 POL(fresh_constant.) = 0 312.36/171.94 POL(g.0(x_1)) = x_1 312.36/171.94 POL(g.1(x_1)) = x_1 312.36/171.94 POL(g_flat.0(x_1)) = x_1 312.36/171.94 POL(up.0(x_1)) = x_1 312.36/171.94 POL(up.1(x_1)) = 1 + x_1 312.36/171.94 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (140) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.94 TOP.0(up.0(f.0(f.1(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.1(g.1(x0))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.1(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(g.1(x0)))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.1(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.1(g.1(x0))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.1(g.1(x_1)) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.1(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.1(g.1(y22)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(f.1(g.1(y16))))) -> f_flat.0(down.0(f.0(f.1(g.1(y16))))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(f.1(g.1(y10)))) -> f_flat.0(down.0(f.1(g.1(y10)))) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.94 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.94 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.94 down.1(g.1(g.1(g.1(g.1(fresh_constant.))))) -> g_flat.0(down.1(g.1(g.1(g.1(fresh_constant.))))) 312.36/171.94 down.1(g.1(g.1(g.1(fresh_constant.)))) -> g_flat.0(down.1(g.1(g.1(fresh_constant.)))) 312.36/171.94 down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.94 down.1(g.1(g.1(g.1(g.1(g.1(g.1(x))))))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.94 down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (141) DependencyGraphProof (EQUIVALENT) 312.36/171.94 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (142) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.1(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(g.1(x0)))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.1(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.1(g.1(x0))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.1(g.1(x_1)) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.1(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.1(g.1(y22)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(f.1(g.1(y16))))) -> f_flat.0(down.0(f.0(f.1(g.1(y16))))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(f.1(g.1(y10)))) -> f_flat.0(down.0(f.1(g.1(y10)))) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.94 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.94 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.94 down.1(g.1(g.1(g.1(g.1(fresh_constant.))))) -> g_flat.0(down.1(g.1(g.1(g.1(fresh_constant.))))) 312.36/171.94 down.1(g.1(g.1(g.1(fresh_constant.)))) -> g_flat.0(down.1(g.1(g.1(fresh_constant.)))) 312.36/171.94 down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.94 down.1(g.1(g.1(g.1(g.1(g.1(g.1(x))))))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.94 down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (143) UsableRulesReductionPairsProof (EQUIVALENT) 312.36/171.94 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. 312.36/171.94 312.36/171.94 No dependency pairs are removed. 312.36/171.94 312.36/171.94 The following rules are removed from R: 312.36/171.94 312.36/171.94 down.1(g.1(g.1(g.1(g.1(fresh_constant.))))) -> g_flat.0(down.1(g.1(g.1(g.1(fresh_constant.))))) 312.36/171.94 down.1(g.1(g.1(g.1(fresh_constant.)))) -> g_flat.0(down.1(g.1(g.1(fresh_constant.)))) 312.36/171.94 down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.))) 312.36/171.94 down.1(g.1(g.1(g.1(g.1(g.1(g.1(x))))))) -> up.0(b.) 312.36/171.94 down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) 312.36/171.94 Used ordering: POLO with Polynomial interpretation [POLO]: 312.36/171.94 312.36/171.94 POL(TOP.0(x_1)) = x_1 312.36/171.94 POL(a.) = 0 312.36/171.94 POL(b.) = 0 312.36/171.94 POL(down.0(x_1)) = 1 + x_1 312.36/171.94 POL(f.0(x_1)) = x_1 312.36/171.94 POL(f.1(x_1)) = x_1 312.36/171.94 POL(f_flat.0(x_1)) = x_1 312.36/171.94 POL(g.0(x_1)) = x_1 312.36/171.94 POL(g.1(x_1)) = x_1 312.36/171.94 POL(g_flat.0(x_1)) = x_1 312.36/171.94 POL(up.0(x_1)) = 1 + x_1 312.36/171.94 POL(up.1(x_1)) = 1 + x_1 312.36/171.94 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (144) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.1(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(g.1(x0)))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.1(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.1(g.1(x0))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.1(g.1(x_1)) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 down.0(f.0(f.0(f.0(f.1(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.1(g.1(y22)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 312.36/171.94 down.0(f.0(f.0(f.1(g.1(y16))))) -> f_flat.0(down.0(f.0(f.1(g.1(y16))))) 312.36/171.94 down.0(f.0(f.1(g.1(y10)))) -> f_flat.0(down.0(f.1(g.1(y10)))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.94 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.94 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (145) MRRProof (EQUIVALENT) 312.36/171.94 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. 312.36/171.94 312.36/171.94 312.36/171.94 Strictly oriented rules of the TRS R: 312.36/171.94 312.36/171.94 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 312.36/171.94 312.36/171.94 Used ordering: Polynomial interpretation [POLO]: 312.36/171.94 312.36/171.94 POL(TOP.0(x_1)) = x_1 312.36/171.94 POL(a.) = 0 312.36/171.94 POL(b.) = 0 312.36/171.94 POL(down.0(x_1)) = x_1 312.36/171.94 POL(f.0(x_1)) = x_1 312.36/171.94 POL(f.1(x_1)) = x_1 312.36/171.94 POL(f_flat.0(x_1)) = x_1 312.36/171.94 POL(g.0(x_1)) = x_1 312.36/171.94 POL(g.1(x_1)) = x_1 312.36/171.94 POL(g_flat.0(x_1)) = x_1 312.36/171.94 POL(up.0(x_1)) = x_1 312.36/171.94 POL(up.1(x_1)) = 1 + x_1 312.36/171.94 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (146) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.1(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(g.1(x0)))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.1(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.1(g.1(x0))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.1(g.1(x_1)) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 down.0(f.0(f.0(f.0(f.1(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.1(g.1(y22)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 down.0(f.0(f.0(f.1(g.1(y16))))) -> f_flat.0(down.0(f.0(f.1(g.1(y16))))) 312.36/171.94 down.0(f.0(f.1(g.1(y10)))) -> f_flat.0(down.0(f.1(g.1(y10)))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.94 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.94 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (147) QDPOrderProof (EQUIVALENT) 312.36/171.94 We use the reduction pair processor [LPAR04,JAR06]. 312.36/171.94 312.36/171.94 312.36/171.94 The following pairs can be oriented strictly and are deleted. 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(f.0(f.1(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.1(g.1(x0)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.1(g.1(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.1(g.1(x0))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) 312.36/171.94 The remaining pairs can at least be oriented weakly. 312.36/171.94 Used ordering: Polynomial interpretation [POLO]: 312.36/171.94 312.36/171.94 POL(TOP.0(x_1)) = x_1 312.36/171.94 POL(a.) = 1 312.36/171.94 POL(b.) = 0 312.36/171.94 POL(down.0(x_1)) = x_1 312.36/171.94 POL(f.0(x_1)) = x_1 312.36/171.94 POL(f.1(x_1)) = 0 312.36/171.94 POL(f_flat.0(x_1)) = x_1 312.36/171.94 POL(g.0(x_1)) = 1 312.36/171.94 POL(g.1(x_1)) = x_1 312.36/171.94 POL(g_flat.0(x_1)) = 1 312.36/171.94 POL(up.0(x_1)) = 1 312.36/171.94 POL(up.1(x_1)) = 1 312.36/171.94 312.36/171.94 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.94 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.94 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.1(g.1(x_1)) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(f.1(g.1(y10)))) -> f_flat.0(down.0(f.1(g.1(y10)))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(f.1(g.1(y16))))) -> f_flat.0(down.0(f.0(f.1(g.1(y16))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.1(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.1(g.1(y22)))))) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (148) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.1(g.1(x_1)) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 down.0(f.0(f.0(f.0(f.1(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.1(g.1(y22)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 down.0(f.0(f.0(f.1(g.1(y16))))) -> f_flat.0(down.0(f.0(f.1(g.1(y16))))) 312.36/171.94 down.0(f.0(f.1(g.1(y10)))) -> f_flat.0(down.0(f.1(g.1(y10)))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.94 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.94 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (149) UsableRulesReductionPairsProof (EQUIVALENT) 312.36/171.94 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. 312.36/171.94 312.36/171.94 No dependency pairs are removed. 312.36/171.94 312.36/171.94 The following rules are removed from R: 312.36/171.94 312.36/171.94 down.0(f.0(f.0(f.0(f.1(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.1(g.1(y22)))))) 312.36/171.94 down.0(f.0(f.0(f.1(g.1(y16))))) -> f_flat.0(down.0(f.0(f.1(g.1(y16))))) 312.36/171.94 down.0(f.0(f.1(g.1(y10)))) -> f_flat.0(down.0(f.1(g.1(y10)))) 312.36/171.94 Used ordering: POLO with Polynomial interpretation [POLO]: 312.36/171.94 312.36/171.94 POL(TOP.0(x_1)) = x_1 312.36/171.94 POL(a.) = 0 312.36/171.94 POL(b.) = 0 312.36/171.94 POL(down.0(x_1)) = 1 + x_1 312.36/171.94 POL(f.0(x_1)) = x_1 312.36/171.94 POL(f.1(x_1)) = x_1 312.36/171.94 POL(f_flat.0(x_1)) = x_1 312.36/171.94 POL(g.0(x_1)) = x_1 312.36/171.94 POL(g.1(x_1)) = x_1 312.36/171.94 POL(g_flat.0(x_1)) = x_1 312.36/171.94 POL(up.0(x_1)) = 1 + x_1 312.36/171.94 POL(up.1(x_1)) = x_1 312.36/171.94 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (150) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(a.)))) -> TOP.0(f_flat.0(down.0(f.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(a.)))) -> TOP.0(g_flat.0(down.0(g.0(a.)))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(a.))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(g.0(x0)))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(a.))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(a.)))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(g.0(x0))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(a.)))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(a.))))))) 312.36/171.94 TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) 312.36/171.94 TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> TOP.0(g_flat.0(down.0(g.0(g.0(g.0(g.0(a.))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) 312.36/171.94 g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) 312.36/171.94 g_flat.0(up.1(x_1)) -> up.1(g.1(x_1)) 312.36/171.94 down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) 312.36/171.94 down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) 312.36/171.94 down.0(g.0(a.)) -> g_flat.0(down.0(a.)) 312.36/171.94 down.0(a.) -> up.0(f.0(a.)) 312.36/171.94 down.0(a.) -> up.0(g.0(a.)) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) 312.36/171.94 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 312.36/171.94 down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) 312.36/171.94 down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) 312.36/171.94 down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) 312.36/171.94 down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) 312.36/171.94 down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) 312.36/171.94 down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) 312.36/171.94 down.0(g.0(f.0(x))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) 312.36/171.94 down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) 312.36/171.94 down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))) -> up.0(b.) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) 312.36/171.94 down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) 312.36/171.94 down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) 312.36/171.94 down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) 312.36/171.94 down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) 312.36/171.94 down.0(f.0(a.)) -> f_flat.0(down.0(a.)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (151) PisEmptyProof (SOUND) 312.36/171.94 The TRS P is empty. Hence, there is no (P,Q,R) chain. 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (152) 312.36/171.94 TRUE 312.36/171.94 312.36/171.94 ---------------------------------------- 312.36/171.94 312.36/171.94 (153) 312.36/171.94 Obligation: 312.36/171.94 Q DP problem: 312.36/171.94 The TRS P consists of the following rules: 312.36/171.94 312.36/171.94 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 312.36/171.94 TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) 312.36/171.94 TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) 312.36/171.94 TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) 312.36/171.94 TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) 312.36/171.94 TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) 312.36/171.94 TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) 312.36/171.94 TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) 312.36/171.94 TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) 312.36/171.94 TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) 312.36/171.94 TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) 312.36/171.94 TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) 312.36/171.94 TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) 312.36/171.94 312.36/171.94 The TRS R consists of the following rules: 312.36/171.94 312.36/171.94 down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) 312.36/171.94 g_flat(up(x_1)) -> up(g(x_1)) 312.36/171.94 down(g(g(g(a)))) -> g_flat(down(g(g(a)))) 312.36/171.94 down(g(g(a))) -> g_flat(down(g(a))) 312.36/171.94 down(g(a)) -> g_flat(down(a)) 312.36/171.94 down(a) -> up(f(a)) 312.36/171.94 down(a) -> up(g(a)) 312.36/171.94 down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) 312.36/171.94 f_flat(up(x_1)) -> up(f(x_1)) 312.36/171.94 down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) 312.36/171.94 down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) 312.36/171.94 down(f(g(y4))) -> f_flat(down(g(y4))) 312.36/171.94 down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) 312.36/171.94 down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) 312.36/171.94 down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) 312.36/171.94 down(g(f(x))) -> up(b) 312.36/171.94 down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) 312.36/171.94 down(g(g(g(b)))) -> g_flat(down(g(g(b)))) 312.36/171.94 down(g(g(b))) -> g_flat(down(g(b))) 312.36/171.94 down(g(g(g(g(g(g(x))))))) -> up(b) 312.36/171.94 down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) 312.36/171.94 down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) 312.36/171.94 down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) 312.36/171.94 down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) 312.36/171.94 down(f(f(f(a)))) -> f_flat(down(f(f(a)))) 312.36/171.94 down(f(f(a))) -> f_flat(down(f(a))) 312.36/171.94 down(f(a)) -> f_flat(down(a)) 312.36/171.94 312.36/171.94 Q is empty. 312.36/171.94 We have to consider all minimal (P,Q,R)-chains. 312.53/172.03 EOF