192.32/136.00 MAYBE 192.38/136.01 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 192.38/136.01 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 192.38/136.01 192.38/136.01 192.38/136.01 Outermost Termination of the given OTRS could not be shown: 192.38/136.01 192.38/136.01 (0) OTRS 192.38/136.01 (1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms] 192.38/136.01 (2) QTRS 192.38/136.01 (3) QTRSRRRProof [EQUIVALENT, 58 ms] 192.38/136.01 (4) QTRS 192.38/136.01 (5) DependencyPairsProof [EQUIVALENT, 0 ms] 192.38/136.01 (6) QDP 192.38/136.01 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 192.38/136.01 (8) AND 192.38/136.01 (9) QDP 192.38/136.01 (10) UsableRulesProof [EQUIVALENT, 0 ms] 192.38/136.01 (11) QDP 192.38/136.01 (12) QReductionProof [EQUIVALENT, 0 ms] 192.38/136.01 (13) QDP 192.38/136.01 (14) UsableRulesReductionPairsProof [EQUIVALENT, 6 ms] 192.38/136.01 (15) QDP 192.38/136.01 (16) DependencyGraphProof [EQUIVALENT, 0 ms] 192.38/136.01 (17) TRUE 192.38/136.01 (18) QDP 192.38/136.01 (19) UsableRulesProof [EQUIVALENT, 0 ms] 192.38/136.01 (20) QDP 192.38/136.01 (21) QReductionProof [EQUIVALENT, 0 ms] 192.38/136.01 (22) QDP 192.38/136.01 (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] 192.38/136.01 (24) YES 192.38/136.01 (25) QDP 192.38/136.01 (26) UsableRulesProof [EQUIVALENT, 0 ms] 192.38/136.01 (27) QDP 192.38/136.01 (28) QReductionProof [EQUIVALENT, 1 ms] 192.38/136.01 (29) QDP 192.38/136.01 (30) TransformationProof [SOUND, 0 ms] 192.38/136.01 (31) QDP 192.38/136.01 (32) TransformationProof [EQUIVALENT, 0 ms] 192.38/136.01 (33) QDP 192.38/136.01 (34) TransformationProof [EQUIVALENT, 0 ms] 192.38/136.01 (35) QDP 192.38/136.01 (36) TransformationProof [SOUND, 0 ms] 192.38/136.01 (37) QDP 192.38/136.01 (38) TransformationProof [EQUIVALENT, 0 ms] 192.38/136.01 (39) QDP 192.38/136.01 (40) QDPOrderProof [EQUIVALENT, 17 ms] 192.38/136.01 (41) QDP 192.38/136.01 (42) QDPOrderProof [EQUIVALENT, 0 ms] 192.38/136.01 (43) QDP 192.38/136.01 (44) QDPOrderProof [EQUIVALENT, 13 ms] 192.38/136.01 (45) QDP 192.38/136.01 (46) QDPOrderProof [EQUIVALENT, 306 ms] 192.38/136.01 (47) QDP 192.38/136.01 (48) UsableRulesProof [EQUIVALENT, 0 ms] 192.38/136.01 (49) QDP 192.38/136.01 (50) QReductionProof [EQUIVALENT, 0 ms] 192.38/136.01 (51) QDP 192.38/136.01 (52) Trivial-Transformation [SOUND, 0 ms] 192.38/136.01 (53) QTRS 192.38/136.01 (54) QTRSRRRProof [EQUIVALENT, 51 ms] 192.38/136.01 (55) QTRS 192.38/136.01 (56) Overlay + Local Confluence [EQUIVALENT, 0 ms] 192.38/136.01 (57) QTRS 192.38/136.01 (58) DependencyPairsProof [EQUIVALENT, 0 ms] 192.38/136.01 (59) QDP 192.38/136.01 (60) DependencyGraphProof [EQUIVALENT, 0 ms] 192.38/136.01 (61) QDP 192.38/136.01 (62) UsableRulesProof [EQUIVALENT, 0 ms] 192.38/136.01 (63) QDP 192.38/136.01 (64) QReductionProof [EQUIVALENT, 0 ms] 192.38/136.01 (65) QDP 192.38/136.01 (66) TransformationProof [EQUIVALENT, 0 ms] 192.38/136.01 (67) QDP 192.38/136.01 (68) NonTerminationLoopProof [COMPLETE, 0 ms] 192.38/136.01 (69) NO 192.38/136.01 (70) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] 192.38/136.01 (71) QTRS 192.38/136.01 (72) QTRSRRRProof [EQUIVALENT, 72 ms] 192.38/136.01 (73) QTRS 192.38/136.01 (74) QTRSRRRProof [EQUIVALENT, 41 ms] 192.38/136.01 (75) QTRS 192.38/136.01 (76) QTRSRRRProof [EQUIVALENT, 36 ms] 192.38/136.01 (77) QTRS 192.38/136.01 (78) QTRSRRRProof [EQUIVALENT, 14 ms] 192.38/136.01 (79) QTRS 192.38/136.01 (80) AAECC Innermost [EQUIVALENT, 0 ms] 192.38/136.01 (81) QTRS 192.38/136.01 (82) DependencyPairsProof [EQUIVALENT, 0 ms] 192.38/136.01 (83) QDP 192.38/136.01 (84) DependencyGraphProof [EQUIVALENT, 0 ms] 192.38/136.01 (85) AND 192.38/136.01 (86) QDP 192.38/136.01 (87) UsableRulesProof [EQUIVALENT, 0 ms] 192.38/136.01 (88) QDP 192.38/136.01 (89) QReductionProof [EQUIVALENT, 0 ms] 192.38/136.01 (90) QDP 192.38/136.01 (91) QDPSizeChangeProof [EQUIVALENT, 0 ms] 192.38/136.01 (92) YES 192.38/136.01 (93) QDP 192.38/136.01 (94) UsableRulesProof [EQUIVALENT, 0 ms] 192.38/136.01 (95) QDP 192.38/136.01 (96) QReductionProof [EQUIVALENT, 0 ms] 192.38/136.01 (97) QDP 192.38/136.01 (98) QDPSizeChangeProof [EQUIVALENT, 0 ms] 192.38/136.01 (99) YES 192.38/136.01 (100) QDP 192.38/136.01 (101) UsableRulesProof [EQUIVALENT, 0 ms] 192.38/136.01 (102) QDP 192.38/136.01 (103) QReductionProof [EQUIVALENT, 0 ms] 192.38/136.01 (104) QDP 192.38/136.01 (105) TransformationProof [EQUIVALENT, 0 ms] 192.38/136.01 (106) QDP 192.38/136.01 (107) DependencyGraphProof [EQUIVALENT, 0 ms] 192.38/136.01 (108) QDP 192.38/136.01 (109) TransformationProof [EQUIVALENT, 0 ms] 192.38/136.01 (110) QDP 192.38/136.01 (111) TransformationProof [EQUIVALENT, 0 ms] 192.38/136.01 (112) QDP 192.38/136.01 (113) TransformationProof [EQUIVALENT, 0 ms] 192.38/136.01 (114) QDP 192.38/136.01 (115) TransformationProof [EQUIVALENT, 0 ms] 192.38/136.01 (116) QDP 192.38/136.01 (117) QDPOrderProof [EQUIVALENT, 16 ms] 192.38/136.01 (118) QDP 192.38/136.01 (119) QDPOrderProof [EQUIVALENT, 11 ms] 192.38/136.01 (120) QDP 192.38/136.01 (121) DependencyGraphProof [EQUIVALENT, 0 ms] 192.38/136.01 (122) QDP 192.38/136.01 (123) QDPOrderProof [EQUIVALENT, 8 ms] 192.38/136.01 (124) QDP 192.38/136.01 (125) QDPOrderProof [EQUIVALENT, 0 ms] 192.38/136.01 (126) QDP 192.38/136.01 (127) QDPOrderProof [EQUIVALENT, 538 ms] 192.38/136.01 (128) QDP 192.38/136.01 (129) MNOCProof [EQUIVALENT, 0 ms] 192.38/136.01 (130) QDP 192.38/136.01 (131) SplitQDPProof [EQUIVALENT, 0 ms] 192.38/136.01 (132) AND 192.38/136.01 (133) QDP 192.38/136.01 (134) SemLabProof [SOUND, 0 ms] 192.38/136.01 (135) QDP 192.38/136.01 (136) DependencyGraphProof [EQUIVALENT, 0 ms] 192.38/136.01 (137) QDP 192.38/136.01 (138) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 192.38/136.01 (139) QDP 192.38/136.01 (140) UsableRulesReductionPairsProof [EQUIVALENT, 5 ms] 192.38/136.01 (141) QDP 192.38/136.01 (142) PisEmptyProof [SOUND, 0 ms] 192.38/136.01 (143) TRUE 192.38/136.01 (144) QDP 192.38/136.01 (145) QReductionProof [EQUIVALENT, 0 ms] 192.38/136.01 (146) QDP 192.38/136.01 (147) MNOCProof [EQUIVALENT, 0 ms] 192.38/136.01 (148) QDP 192.38/136.01 (149) SplitQDPProof [EQUIVALENT, 0 ms] 192.38/136.01 (150) AND 192.38/136.01 (151) QDP 192.38/136.01 (152) SemLabProof [SOUND, 0 ms] 192.38/136.01 (153) QDP 192.38/136.01 (154) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 192.38/136.01 (155) QDP 192.38/136.01 (156) MRRProof [EQUIVALENT, 0 ms] 192.38/136.01 (157) QDP 192.38/136.01 (158) QDPOrderProof [EQUIVALENT, 0 ms] 192.38/136.01 (159) QDP 192.38/136.01 (160) PisEmptyProof [SOUND, 0 ms] 192.38/136.01 (161) TRUE 192.38/136.01 (162) QDP 192.38/136.01 (163) QReductionProof [EQUIVALENT, 0 ms] 192.38/136.01 (164) QDP 192.38/136.01 (165) SplitQDPProof [EQUIVALENT, 0 ms] 192.38/136.01 (166) AND 192.38/136.01 (167) QDP 192.38/136.01 (168) SemLabProof [SOUND, 0 ms] 192.38/136.01 (169) QDP 192.38/136.01 (170) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 192.38/136.01 (171) QDP 192.38/136.01 (172) MRRProof [EQUIVALENT, 2 ms] 192.38/136.01 (173) QDP 192.38/136.01 (174) QDPOrderProof [EQUIVALENT, 0 ms] 192.38/136.01 (175) QDP 192.38/136.01 (176) PisEmptyProof [SOUND, 0 ms] 192.38/136.01 (177) TRUE 192.38/136.01 (178) QDP 192.38/136.01 (179) QReductionProof [EQUIVALENT, 0 ms] 192.38/136.01 (180) QDP 192.38/136.01 192.38/136.01 192.38/136.01 ---------------------------------------- 192.38/136.01 192.38/136.01 (0) 192.38/136.01 Obligation: 192.38/136.01 Term rewrite system R: 192.38/136.01 The TRS R consists of the following rules: 192.38/136.01 192.38/136.01 c -> f(n__g(n__c)) 192.38/136.01 f(n__g(X)) -> g(activate(X)) 192.38/136.01 g(X) -> n__g(X) 192.38/136.01 c -> n__c 192.38/136.01 activate(n__g(X)) -> g(X) 192.38/136.01 activate(n__c) -> c 192.38/136.01 activate(X) -> X 192.38/136.01 192.38/136.01 192.38/136.01 192.38/136.01 Outermost Strategy. 192.38/136.01 192.38/136.01 ---------------------------------------- 192.38/136.01 192.38/136.01 (1) Thiemann-SpecialC-Transformation (EQUIVALENT) 192.38/136.01 We applied the Thiemann-SpecialC transformation to transform the outermost TRS to an innermost TRS. 192.38/136.01 ---------------------------------------- 192.38/136.01 192.38/136.01 (2) 192.38/136.01 Obligation: 192.38/136.01 Q restricted rewrite system: 192.38/136.01 The TRS R consists of the following rules: 192.38/136.01 192.38/136.01 top(go_up(x)) -> top(reduce(x)) 192.38/136.01 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.01 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.01 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.01 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.01 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.01 redex_g(X) -> result_g(n__g(X)) 192.38/136.01 reduce(c) -> go_up(n__c) 192.38/136.01 redex_activate(n__g(X)) -> result_activate(g(X)) 192.38/136.01 redex_activate(n__c) -> result_activate(c) 192.38/136.01 redex_activate(X) -> result_activate(X) 192.38/136.01 check_f(result_f(x)) -> go_up(x) 192.38/136.01 check_g(result_g(x)) -> go_up(x) 192.38/136.01 check_activate(result_activate(x)) -> go_up(x) 192.38/136.01 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.01 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 192.38/136.01 check_activate(redex_activate(x_1)) -> in_activate_1(reduce(x_1)) 192.38/136.01 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.01 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.01 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.01 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 192.38/136.01 in_activate_1(go_up(x_1)) -> go_up(activate(x_1)) 192.38/136.01 192.38/136.01 The set Q consists of the following terms: 192.38/136.01 192.38/136.01 top(go_up(x0)) 192.38/136.01 reduce(f(x0)) 192.38/136.01 reduce(g(x0)) 192.38/136.01 reduce(activate(x0)) 192.38/136.01 reduce(c) 192.38/136.01 redex_f(n__g(x0)) 192.38/136.01 redex_g(x0) 192.38/136.01 redex_activate(x0) 192.38/136.01 check_f(result_f(x0)) 192.38/136.01 check_g(result_g(x0)) 192.38/136.01 check_activate(result_activate(x0)) 192.38/136.01 check_f(redex_f(x0)) 192.38/136.01 reduce(n__g(x0)) 192.38/136.01 in_f_1(go_up(x0)) 192.38/136.01 in_n__g_1(go_up(x0)) 192.38/136.01 in_g_1(go_up(x0)) 192.38/136.01 in_activate_1(go_up(x0)) 192.38/136.01 192.38/136.01 192.38/136.01 ---------------------------------------- 192.38/136.01 192.38/136.01 (3) QTRSRRRProof (EQUIVALENT) 192.38/136.01 Used ordering: 192.38/136.01 Polynomial interpretation [POLO]: 192.38/136.01 192.38/136.01 POL(activate(x_1)) = 1 + 2*x_1 192.38/136.01 POL(c) = 1 192.38/136.01 POL(check_activate(x_1)) = x_1 192.38/136.01 POL(check_f(x_1)) = x_1 192.38/136.01 POL(check_g(x_1)) = x_1 192.38/136.01 POL(f(x_1)) = 1 + 2*x_1 192.38/136.01 POL(g(x_1)) = x_1 192.38/136.01 POL(go_up(x_1)) = x_1 192.38/136.01 POL(in_activate_1(x_1)) = 1 + 2*x_1 192.38/136.01 POL(in_f_1(x_1)) = 1 + 2*x_1 192.38/136.01 POL(in_g_1(x_1)) = x_1 192.38/136.01 POL(in_n__g_1(x_1)) = x_1 192.38/136.01 POL(n__c) = 0 192.38/136.01 POL(n__g(x_1)) = x_1 192.38/136.01 POL(redex_activate(x_1)) = 1 + 2*x_1 192.38/136.01 POL(redex_f(x_1)) = 1 + 2*x_1 192.38/136.01 POL(redex_g(x_1)) = x_1 192.38/136.01 POL(reduce(x_1)) = x_1 192.38/136.01 POL(result_activate(x_1)) = x_1 192.38/136.01 POL(result_f(x_1)) = x_1 192.38/136.01 POL(result_g(x_1)) = x_1 192.38/136.01 POL(top(x_1)) = 2*x_1 192.38/136.01 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 192.38/136.01 192.38/136.01 reduce(c) -> go_up(n__c) 192.38/136.01 redex_activate(n__g(X)) -> result_activate(g(X)) 192.38/136.01 redex_activate(X) -> result_activate(X) 192.38/136.01 192.38/136.01 192.38/136.01 192.38/136.01 192.38/136.01 ---------------------------------------- 192.38/136.01 192.38/136.01 (4) 192.38/136.01 Obligation: 192.38/136.01 Q restricted rewrite system: 192.38/136.01 The TRS R consists of the following rules: 192.38/136.01 192.38/136.01 top(go_up(x)) -> top(reduce(x)) 192.38/136.01 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.01 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.01 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.01 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.01 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.01 redex_g(X) -> result_g(n__g(X)) 192.38/136.01 redex_activate(n__c) -> result_activate(c) 192.38/136.01 check_f(result_f(x)) -> go_up(x) 192.38/136.01 check_g(result_g(x)) -> go_up(x) 192.38/136.01 check_activate(result_activate(x)) -> go_up(x) 192.38/136.01 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.01 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 192.38/136.01 check_activate(redex_activate(x_1)) -> in_activate_1(reduce(x_1)) 192.38/136.01 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.01 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.01 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.01 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 192.38/136.01 in_activate_1(go_up(x_1)) -> go_up(activate(x_1)) 192.38/136.01 192.38/136.01 The set Q consists of the following terms: 192.38/136.01 192.38/136.01 top(go_up(x0)) 192.38/136.01 reduce(f(x0)) 192.38/136.01 reduce(g(x0)) 192.38/136.01 reduce(activate(x0)) 192.38/136.01 reduce(c) 192.38/136.01 redex_f(n__g(x0)) 192.38/136.01 redex_g(x0) 192.38/136.01 redex_activate(x0) 192.38/136.01 check_f(result_f(x0)) 192.38/136.01 check_g(result_g(x0)) 192.38/136.01 check_activate(result_activate(x0)) 192.38/136.01 check_f(redex_f(x0)) 192.38/136.01 reduce(n__g(x0)) 192.38/136.01 in_f_1(go_up(x0)) 192.38/136.01 in_n__g_1(go_up(x0)) 192.38/136.01 in_g_1(go_up(x0)) 192.38/136.01 in_activate_1(go_up(x0)) 192.38/136.01 192.38/136.01 192.38/136.01 ---------------------------------------- 192.38/136.01 192.38/136.01 (5) DependencyPairsProof (EQUIVALENT) 192.38/136.01 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 192.38/136.01 ---------------------------------------- 192.38/136.01 192.38/136.01 (6) 192.38/136.01 Obligation: 192.38/136.01 Q DP problem: 192.38/136.01 The TRS P consists of the following rules: 192.38/136.01 192.38/136.01 TOP(go_up(x)) -> TOP(reduce(x)) 192.38/136.01 TOP(go_up(x)) -> REDUCE(x) 192.38/136.01 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 192.38/136.01 REDUCE(f(x_1)) -> REDEX_F(x_1) 192.38/136.01 REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) 192.38/136.01 REDUCE(g(x_1)) -> REDEX_G(x_1) 192.38/136.01 REDUCE(activate(x_1)) -> CHECK_ACTIVATE(redex_activate(x_1)) 192.38/136.01 REDUCE(activate(x_1)) -> REDEX_ACTIVATE(x_1) 192.38/136.01 CHECK_F(redex_f(x_1)) -> IN_F_1(reduce(x_1)) 192.38/136.01 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 192.38/136.01 CHECK_G(redex_g(x_1)) -> IN_G_1(reduce(x_1)) 192.38/136.01 CHECK_G(redex_g(x_1)) -> REDUCE(x_1) 192.38/136.01 CHECK_ACTIVATE(redex_activate(x_1)) -> IN_ACTIVATE_1(reduce(x_1)) 192.38/136.01 CHECK_ACTIVATE(redex_activate(x_1)) -> REDUCE(x_1) 192.38/136.01 REDUCE(n__g(x_1)) -> IN_N__G_1(reduce(x_1)) 192.38/136.01 REDUCE(n__g(x_1)) -> REDUCE(x_1) 192.38/136.01 192.38/136.01 The TRS R consists of the following rules: 192.38/136.01 192.38/136.01 top(go_up(x)) -> top(reduce(x)) 192.38/136.01 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.01 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.01 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.01 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.01 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.01 redex_g(X) -> result_g(n__g(X)) 192.38/136.01 redex_activate(n__c) -> result_activate(c) 192.38/136.01 check_f(result_f(x)) -> go_up(x) 192.38/136.01 check_g(result_g(x)) -> go_up(x) 192.38/136.01 check_activate(result_activate(x)) -> go_up(x) 192.38/136.01 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.01 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 192.38/136.01 check_activate(redex_activate(x_1)) -> in_activate_1(reduce(x_1)) 192.38/136.01 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.01 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.01 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.01 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 192.38/136.01 in_activate_1(go_up(x_1)) -> go_up(activate(x_1)) 192.38/136.01 192.38/136.01 The set Q consists of the following terms: 192.38/136.01 192.38/136.01 top(go_up(x0)) 192.38/136.01 reduce(f(x0)) 192.38/136.01 reduce(g(x0)) 192.38/136.01 reduce(activate(x0)) 192.38/136.01 reduce(c) 192.38/136.01 redex_f(n__g(x0)) 192.38/136.01 redex_g(x0) 192.38/136.01 redex_activate(x0) 192.38/136.01 check_f(result_f(x0)) 192.38/136.01 check_g(result_g(x0)) 192.38/136.01 check_activate(result_activate(x0)) 192.38/136.01 check_f(redex_f(x0)) 192.38/136.01 reduce(n__g(x0)) 192.38/136.01 in_f_1(go_up(x0)) 192.38/136.01 in_n__g_1(go_up(x0)) 192.38/136.01 in_g_1(go_up(x0)) 192.38/136.01 in_activate_1(go_up(x0)) 192.38/136.01 192.38/136.01 We have to consider all minimal (P,Q,R)-chains. 192.38/136.01 ---------------------------------------- 192.38/136.01 192.38/136.01 (7) DependencyGraphProof (EQUIVALENT) 192.38/136.01 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 12 less nodes. 192.38/136.01 ---------------------------------------- 192.38/136.01 192.38/136.01 (8) 192.38/136.01 Complex Obligation (AND) 192.38/136.01 192.38/136.01 ---------------------------------------- 192.38/136.01 192.38/136.01 (9) 192.38/136.01 Obligation: 192.38/136.01 Q DP problem: 192.38/136.01 The TRS P consists of the following rules: 192.38/136.01 192.38/136.01 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 192.38/136.01 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 192.38/136.01 192.38/136.01 The TRS R consists of the following rules: 192.38/136.01 192.38/136.01 top(go_up(x)) -> top(reduce(x)) 192.38/136.01 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.01 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.01 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.01 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.01 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.01 redex_g(X) -> result_g(n__g(X)) 192.38/136.01 redex_activate(n__c) -> result_activate(c) 192.38/136.01 check_f(result_f(x)) -> go_up(x) 192.38/136.01 check_g(result_g(x)) -> go_up(x) 192.38/136.01 check_activate(result_activate(x)) -> go_up(x) 192.38/136.01 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.01 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 192.38/136.01 check_activate(redex_activate(x_1)) -> in_activate_1(reduce(x_1)) 192.38/136.01 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.01 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.01 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.01 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 192.38/136.01 in_activate_1(go_up(x_1)) -> go_up(activate(x_1)) 192.38/136.01 192.38/136.01 The set Q consists of the following terms: 192.38/136.01 192.38/136.01 top(go_up(x0)) 192.38/136.01 reduce(f(x0)) 192.38/136.01 reduce(g(x0)) 192.38/136.01 reduce(activate(x0)) 192.38/136.01 reduce(c) 192.38/136.01 redex_f(n__g(x0)) 192.38/136.01 redex_g(x0) 192.38/136.01 redex_activate(x0) 192.38/136.01 check_f(result_f(x0)) 192.38/136.01 check_g(result_g(x0)) 192.38/136.01 check_activate(result_activate(x0)) 192.38/136.01 check_f(redex_f(x0)) 192.38/136.01 reduce(n__g(x0)) 192.38/136.01 in_f_1(go_up(x0)) 192.38/136.01 in_n__g_1(go_up(x0)) 192.38/136.01 in_g_1(go_up(x0)) 192.38/136.01 in_activate_1(go_up(x0)) 192.38/136.01 192.38/136.01 We have to consider all minimal (P,Q,R)-chains. 192.38/136.01 ---------------------------------------- 192.38/136.01 192.38/136.01 (10) UsableRulesProof (EQUIVALENT) 192.38/136.01 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. 192.38/136.01 ---------------------------------------- 192.38/136.01 192.38/136.01 (11) 192.38/136.01 Obligation: 192.38/136.01 Q DP problem: 192.38/136.01 The TRS P consists of the following rules: 192.38/136.01 192.38/136.01 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 192.38/136.01 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 192.38/136.01 192.38/136.01 The TRS R consists of the following rules: 192.38/136.01 192.38/136.01 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 top(go_up(x0)) 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 in_g_1(go_up(x0)) 192.38/136.02 in_activate_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (12) QReductionProof (EQUIVALENT) 192.38/136.02 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 192.38/136.02 192.38/136.02 top(go_up(x0)) 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 in_g_1(go_up(x0)) 192.38/136.02 in_activate_1(go_up(x0)) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (13) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 192.38/136.02 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (14) UsableRulesReductionPairsProof (EQUIVALENT) 192.38/136.02 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. 192.38/136.02 192.38/136.02 The following dependency pairs can be deleted: 192.38/136.02 192.38/136.02 REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) 192.38/136.02 The following rules are removed from R: 192.38/136.02 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 Used ordering: POLO with Polynomial interpretation [POLO]: 192.38/136.02 192.38/136.02 POL(CHECK_F(x_1)) = x_1 192.38/136.02 POL(REDUCE(x_1)) = 2*x_1 192.38/136.02 POL(activate(x_1)) = 2*x_1 192.38/136.02 POL(f(x_1)) = 2*x_1 192.38/136.02 POL(g(x_1)) = 2*x_1 192.38/136.02 POL(n__g(x_1)) = 2*x_1 192.38/136.02 POL(redex_f(x_1)) = 2*x_1 192.38/136.02 POL(result_f(x_1)) = x_1 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (15) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 CHECK_F(redex_f(x_1)) -> REDUCE(x_1) 192.38/136.02 192.38/136.02 R is empty. 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (16) DependencyGraphProof (EQUIVALENT) 192.38/136.02 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (17) 192.38/136.02 TRUE 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (18) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 REDUCE(n__g(x_1)) -> REDUCE(x_1) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 top(go_up(x)) -> top(reduce(x)) 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 192.38/136.02 check_activate(redex_activate(x_1)) -> in_activate_1(reduce(x_1)) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 192.38/136.02 in_activate_1(go_up(x_1)) -> go_up(activate(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 top(go_up(x0)) 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 in_g_1(go_up(x0)) 192.38/136.02 in_activate_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (19) UsableRulesProof (EQUIVALENT) 192.38/136.02 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. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (20) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 REDUCE(n__g(x_1)) -> REDUCE(x_1) 192.38/136.02 192.38/136.02 R is empty. 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 top(go_up(x0)) 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 in_g_1(go_up(x0)) 192.38/136.02 in_activate_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (21) QReductionProof (EQUIVALENT) 192.38/136.02 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 192.38/136.02 192.38/136.02 top(go_up(x0)) 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 in_g_1(go_up(x0)) 192.38/136.02 in_activate_1(go_up(x0)) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (22) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 REDUCE(n__g(x_1)) -> REDUCE(x_1) 192.38/136.02 192.38/136.02 R is empty. 192.38/136.02 Q is empty. 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (23) QDPSizeChangeProof (EQUIVALENT) 192.38/136.02 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 192.38/136.02 192.38/136.02 From the DPs we obtained the following set of size-change graphs: 192.38/136.02 *REDUCE(n__g(x_1)) -> REDUCE(x_1) 192.38/136.02 The graph contains the following edges 1 > 1 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (24) 192.38/136.02 YES 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (25) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(x)) -> TOP(reduce(x)) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 top(go_up(x)) -> top(reduce(x)) 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) 192.38/136.02 check_activate(redex_activate(x_1)) -> in_activate_1(reduce(x_1)) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 in_g_1(go_up(x_1)) -> go_up(g(x_1)) 192.38/136.02 in_activate_1(go_up(x_1)) -> go_up(activate(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 top(go_up(x0)) 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 in_g_1(go_up(x0)) 192.38/136.02 in_activate_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (26) UsableRulesProof (EQUIVALENT) 192.38/136.02 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. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (27) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(x)) -> TOP(reduce(x)) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 top(go_up(x0)) 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 in_g_1(go_up(x0)) 192.38/136.02 in_activate_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (28) QReductionProof (EQUIVALENT) 192.38/136.02 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 192.38/136.02 192.38/136.02 top(go_up(x0)) 192.38/136.02 in_g_1(go_up(x0)) 192.38/136.02 in_activate_1(go_up(x0)) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (29) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(x)) -> TOP(reduce(x)) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (30) TransformationProof (SOUND) 192.38/136.02 By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]: 192.38/136.02 192.38/136.02 (TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))),TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))) 192.38/136.02 (TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))),TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))) 192.38/136.02 (TOP(go_up(activate(x0))) -> TOP(check_activate(redex_activate(x0))),TOP(go_up(activate(x0))) -> TOP(check_activate(redex_activate(x0)))) 192.38/136.02 (TOP(go_up(c)) -> TOP(go_up(f(n__g(n__c)))),TOP(go_up(c)) -> TOP(go_up(f(n__g(n__c))))) 192.38/136.02 (TOP(go_up(n__g(x0))) -> TOP(in_n__g_1(reduce(x0))),TOP(go_up(n__g(x0))) -> TOP(in_n__g_1(reduce(x0)))) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (31) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 192.38/136.02 TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))) 192.38/136.02 TOP(go_up(activate(x0))) -> TOP(check_activate(redex_activate(x0))) 192.38/136.02 TOP(go_up(c)) -> TOP(go_up(f(n__g(n__c)))) 192.38/136.02 TOP(go_up(n__g(x0))) -> TOP(in_n__g_1(reduce(x0))) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (32) TransformationProof (EQUIVALENT) 192.38/136.02 By rewriting [LPAR04] the rule TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))) at position [0,0] we obtained the following new rules [LPAR04]: 192.38/136.02 192.38/136.02 (TOP(go_up(g(x0))) -> TOP(check_g(result_g(n__g(x0)))),TOP(go_up(g(x0))) -> TOP(check_g(result_g(n__g(x0))))) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (33) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 192.38/136.02 TOP(go_up(activate(x0))) -> TOP(check_activate(redex_activate(x0))) 192.38/136.02 TOP(go_up(c)) -> TOP(go_up(f(n__g(n__c)))) 192.38/136.02 TOP(go_up(n__g(x0))) -> TOP(in_n__g_1(reduce(x0))) 192.38/136.02 TOP(go_up(g(x0))) -> TOP(check_g(result_g(n__g(x0)))) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (34) TransformationProof (EQUIVALENT) 192.38/136.02 By rewriting [LPAR04] the rule TOP(go_up(g(x0))) -> TOP(check_g(result_g(n__g(x0)))) at position [0] we obtained the following new rules [LPAR04]: 192.38/136.02 192.38/136.02 (TOP(go_up(g(x0))) -> TOP(go_up(n__g(x0))),TOP(go_up(g(x0))) -> TOP(go_up(n__g(x0)))) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (35) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 192.38/136.02 TOP(go_up(activate(x0))) -> TOP(check_activate(redex_activate(x0))) 192.38/136.02 TOP(go_up(c)) -> TOP(go_up(f(n__g(n__c)))) 192.38/136.02 TOP(go_up(n__g(x0))) -> TOP(in_n__g_1(reduce(x0))) 192.38/136.02 TOP(go_up(g(x0))) -> TOP(go_up(n__g(x0))) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (36) TransformationProof (SOUND) 192.38/136.02 By narrowing [LPAR04] the rule TOP(go_up(activate(x0))) -> TOP(check_activate(redex_activate(x0))) at position [0,0] we obtained the following new rules [LPAR04]: 192.38/136.02 192.38/136.02 (TOP(go_up(activate(n__c))) -> TOP(check_activate(result_activate(c))),TOP(go_up(activate(n__c))) -> TOP(check_activate(result_activate(c)))) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (37) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 192.38/136.02 TOP(go_up(c)) -> TOP(go_up(f(n__g(n__c)))) 192.38/136.02 TOP(go_up(n__g(x0))) -> TOP(in_n__g_1(reduce(x0))) 192.38/136.02 TOP(go_up(g(x0))) -> TOP(go_up(n__g(x0))) 192.38/136.02 TOP(go_up(activate(n__c))) -> TOP(check_activate(result_activate(c))) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (38) TransformationProof (EQUIVALENT) 192.38/136.02 By rewriting [LPAR04] the rule TOP(go_up(activate(n__c))) -> TOP(check_activate(result_activate(c))) at position [0] we obtained the following new rules [LPAR04]: 192.38/136.02 192.38/136.02 (TOP(go_up(activate(n__c))) -> TOP(go_up(c)),TOP(go_up(activate(n__c))) -> TOP(go_up(c))) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (39) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 192.38/136.02 TOP(go_up(c)) -> TOP(go_up(f(n__g(n__c)))) 192.38/136.02 TOP(go_up(n__g(x0))) -> TOP(in_n__g_1(reduce(x0))) 192.38/136.02 TOP(go_up(g(x0))) -> TOP(go_up(n__g(x0))) 192.38/136.02 TOP(go_up(activate(n__c))) -> TOP(go_up(c)) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (40) QDPOrderProof (EQUIVALENT) 192.38/136.02 We use the reduction pair processor [LPAR04,JAR06]. 192.38/136.02 192.38/136.02 192.38/136.02 The following pairs can be oriented strictly and are deleted. 192.38/136.02 192.38/136.02 TOP(go_up(activate(n__c))) -> TOP(go_up(c)) 192.38/136.02 The remaining pairs can at least be oriented weakly. 192.38/136.02 Used ordering: Polynomial interpretation [POLO]: 192.38/136.02 192.38/136.02 POL(TOP(x_1)) = x_1 192.38/136.02 POL(activate(x_1)) = x_1 192.38/136.02 POL(c) = 0 192.38/136.02 POL(check_activate(x_1)) = 1 192.38/136.02 POL(check_f(x_1)) = x_1 192.38/136.02 POL(check_g(x_1)) = 1 192.38/136.02 POL(f(x_1)) = 0 192.38/136.02 POL(g(x_1)) = 0 192.38/136.02 POL(go_up(x_1)) = x_1 192.38/136.02 POL(in_f_1(x_1)) = 0 192.38/136.02 POL(in_n__g_1(x_1)) = 0 192.38/136.02 POL(n__c) = 1 192.38/136.02 POL(n__g(x_1)) = 0 192.38/136.02 POL(redex_activate(x_1)) = x_1 192.38/136.02 POL(redex_f(x_1)) = 0 192.38/136.02 POL(redex_g(x_1)) = 1 + x_1 192.38/136.02 POL(reduce(x_1)) = 0 192.38/136.02 POL(result_activate(x_1)) = x_1 192.38/136.02 POL(result_f(x_1)) = x_1 192.38/136.02 POL(result_g(x_1)) = 1 + x_1 192.38/136.02 192.38/136.02 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 192.38/136.02 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (41) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 192.38/136.02 TOP(go_up(c)) -> TOP(go_up(f(n__g(n__c)))) 192.38/136.02 TOP(go_up(n__g(x0))) -> TOP(in_n__g_1(reduce(x0))) 192.38/136.02 TOP(go_up(g(x0))) -> TOP(go_up(n__g(x0))) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (42) QDPOrderProof (EQUIVALENT) 192.38/136.02 We use the reduction pair processor [LPAR04,JAR06]. 192.38/136.02 192.38/136.02 192.38/136.02 The following pairs can be oriented strictly and are deleted. 192.38/136.02 192.38/136.02 TOP(go_up(g(x0))) -> TOP(go_up(n__g(x0))) 192.38/136.02 The remaining pairs can at least be oriented weakly. 192.38/136.02 Used ordering: Polynomial interpretation [POLO]: 192.38/136.02 192.38/136.02 POL(TOP(x_1)) = x_1 192.38/136.02 POL(activate(x_1)) = x_1 192.38/136.02 POL(c) = 1 192.38/136.02 POL(check_activate(x_1)) = 1 192.38/136.02 POL(check_f(x_1)) = x_1 192.38/136.02 POL(check_g(x_1)) = 1 192.38/136.02 POL(f(x_1)) = 1 192.38/136.02 POL(g(x_1)) = 1 192.38/136.02 POL(go_up(x_1)) = x_1 192.38/136.02 POL(in_f_1(x_1)) = 1 192.38/136.02 POL(in_n__g_1(x_1)) = 0 192.38/136.02 POL(n__c) = 0 192.38/136.02 POL(n__g(x_1)) = 0 192.38/136.02 POL(redex_activate(x_1)) = 0 192.38/136.02 POL(redex_f(x_1)) = 1 192.38/136.02 POL(redex_g(x_1)) = 1 + x_1 192.38/136.02 POL(reduce(x_1)) = x_1 192.38/136.02 POL(result_activate(x_1)) = 1 + x_1 192.38/136.02 POL(result_f(x_1)) = x_1 192.38/136.02 POL(result_g(x_1)) = 1 + x_1 192.38/136.02 192.38/136.02 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 192.38/136.02 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (43) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 192.38/136.02 TOP(go_up(c)) -> TOP(go_up(f(n__g(n__c)))) 192.38/136.02 TOP(go_up(n__g(x0))) -> TOP(in_n__g_1(reduce(x0))) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (44) QDPOrderProof (EQUIVALENT) 192.38/136.02 We use the reduction pair processor [LPAR04,JAR06]. 192.38/136.02 192.38/136.02 192.38/136.02 The following pairs can be oriented strictly and are deleted. 192.38/136.02 192.38/136.02 TOP(go_up(c)) -> TOP(go_up(f(n__g(n__c)))) 192.38/136.02 The remaining pairs can at least be oriented weakly. 192.38/136.02 Used ordering: Polynomial interpretation [POLO]: 192.38/136.02 192.38/136.02 POL(TOP(x_1)) = x_1 192.38/136.02 POL(activate(x_1)) = x_1 192.38/136.02 POL(c) = 1 192.38/136.02 POL(check_activate(x_1)) = 1 192.38/136.02 POL(check_f(x_1)) = x_1 192.38/136.02 POL(check_g(x_1)) = 1 192.38/136.02 POL(f(x_1)) = 0 192.38/136.02 POL(g(x_1)) = 0 192.38/136.02 POL(go_up(x_1)) = x_1 192.38/136.02 POL(in_f_1(x_1)) = 0 192.38/136.02 POL(in_n__g_1(x_1)) = 0 192.38/136.02 POL(n__c) = 0 192.38/136.02 POL(n__g(x_1)) = 0 192.38/136.02 POL(redex_activate(x_1)) = 0 192.38/136.02 POL(redex_f(x_1)) = 0 192.38/136.02 POL(redex_g(x_1)) = 1 + x_1 192.38/136.02 POL(reduce(x_1)) = 0 192.38/136.02 POL(result_activate(x_1)) = x_1 192.38/136.02 POL(result_f(x_1)) = x_1 192.38/136.02 POL(result_g(x_1)) = 1 + x_1 192.38/136.02 192.38/136.02 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 192.38/136.02 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (45) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 192.38/136.02 TOP(go_up(n__g(x0))) -> TOP(in_n__g_1(reduce(x0))) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (46) QDPOrderProof (EQUIVALENT) 192.38/136.02 We use the reduction pair processor [LPAR04,JAR06]. 192.38/136.02 192.38/136.02 192.38/136.02 The following pairs can be oriented strictly and are deleted. 192.38/136.02 192.38/136.02 TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) 192.38/136.02 The remaining pairs can at least be oriented weakly. 192.38/136.02 Used ordering: Matrix interpretation [MATRO]: 192.38/136.02 192.38/136.02 Non-tuple symbols: 192.38/136.02 <<< 192.38/136.02 M( result_f_1(x_1) ) = [[0], [0]] + [[0, 1], [1, 1]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( reduce_1(x_1) ) = [[0], [0]] + [[0, 1], [1, 1]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( c ) = [[1], [1]] 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( in_n__g_1_1(x_1) ) = [[0], [1]] + [[0, 0], [0, 0]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( check_activate_1(x_1) ) = [[0], [1]] + [[0, 1], [1, 0]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( redex_f_1(x_1) ) = [[0], [1]] + [[1, 1], [0, 1]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( check_f_1(x_1) ) = [[0], [1]] + [[0, 1], [1, 0]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( activate_1(x_1) ) = [[0], [1]] + [[0, 0], [1, 0]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( g_1(x_1) ) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( redex_activate_1(x_1) ) = [[0], [1]] + [[1, 0], [1, 0]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( go_up_1(x_1) ) = [[0], [1]] + [[1, 1], [0, 1]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( result_g_1(x_1) ) = [[0], [0]] + [[1, 1], [0, 1]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( in_f_1_1(x_1) ) = [[1], [1]] + [[1, 0], [0, 1]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( result_activate_1(x_1) ) = [[0], [0]] + [[0, 1], [1, 1]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( f_1(x_1) ) = [[0], [1]] + [[1, 0], [0, 1]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( n__g_1(x_1) ) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( check_g_1(x_1) ) = [[0], [1]] + [[1, 0], [0, 1]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( n__c ) = [[1], [0]] 192.38/136.02 >>> 192.38/136.02 192.38/136.02 <<< 192.38/136.02 M( redex_g_1(x_1) ) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 Tuple symbols: 192.38/136.02 <<< 192.38/136.02 M( TOP_1(x_1) ) = [[0]] + [[1, 1]] * x_1 192.38/136.02 >>> 192.38/136.02 192.38/136.02 192.38/136.02 192.38/136.02 Matrix type: 192.38/136.02 192.38/136.02 We used a basic matrix type which is not further parametrizeable. 192.38/136.02 192.38/136.02 192.38/136.02 192.38/136.02 192.38/136.02 192.38/136.02 As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. 192.38/136.02 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 192.38/136.02 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (47) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(n__g(x0))) -> TOP(in_n__g_1(reduce(x0))) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (48) UsableRulesProof (EQUIVALENT) 192.38/136.02 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. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (49) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(x)) -> TOP(reduce(x)) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 top(go_up(x0)) 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 in_g_1(go_up(x0)) 192.38/136.02 in_activate_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (50) QReductionProof (EQUIVALENT) 192.38/136.02 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 192.38/136.02 192.38/136.02 top(go_up(x0)) 192.38/136.02 in_g_1(go_up(x0)) 192.38/136.02 in_activate_1(go_up(x0)) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (51) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 TOP(go_up(x)) -> TOP(reduce(x)) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 reduce(f(x_1)) -> check_f(redex_f(x_1)) 192.38/136.02 reduce(g(x_1)) -> check_g(redex_g(x_1)) 192.38/136.02 reduce(activate(x_1)) -> check_activate(redex_activate(x_1)) 192.38/136.02 reduce(c) -> go_up(f(n__g(n__c))) 192.38/136.02 reduce(n__g(x_1)) -> in_n__g_1(reduce(x_1)) 192.38/136.02 in_n__g_1(go_up(x_1)) -> go_up(n__g(x_1)) 192.38/136.02 redex_activate(n__c) -> result_activate(c) 192.38/136.02 check_activate(result_activate(x)) -> go_up(x) 192.38/136.02 redex_g(X) -> result_g(n__g(X)) 192.38/136.02 check_g(result_g(x)) -> go_up(x) 192.38/136.02 redex_f(n__g(X)) -> result_f(g(activate(X))) 192.38/136.02 check_f(result_f(x)) -> go_up(x) 192.38/136.02 check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) 192.38/136.02 in_f_1(go_up(x_1)) -> go_up(f(x_1)) 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 reduce(f(x0)) 192.38/136.02 reduce(g(x0)) 192.38/136.02 reduce(activate(x0)) 192.38/136.02 reduce(c) 192.38/136.02 redex_f(n__g(x0)) 192.38/136.02 redex_g(x0) 192.38/136.02 redex_activate(x0) 192.38/136.02 check_f(result_f(x0)) 192.38/136.02 check_g(result_g(x0)) 192.38/136.02 check_activate(result_activate(x0)) 192.38/136.02 check_f(redex_f(x0)) 192.38/136.02 reduce(n__g(x0)) 192.38/136.02 in_f_1(go_up(x0)) 192.38/136.02 in_n__g_1(go_up(x0)) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (52) Trivial-Transformation (SOUND) 192.38/136.02 We applied the Trivial transformation to transform the outermost TRS to a standard TRS. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (53) 192.38/136.02 Obligation: 192.38/136.02 Q restricted rewrite system: 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 c -> f(n__g(n__c)) 192.38/136.02 f(n__g(X)) -> g(activate(X)) 192.38/136.02 g(X) -> n__g(X) 192.38/136.02 c -> n__c 192.38/136.02 activate(n__g(X)) -> g(X) 192.38/136.02 activate(n__c) -> c 192.38/136.02 activate(X) -> X 192.38/136.02 192.38/136.02 Q is empty. 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (54) QTRSRRRProof (EQUIVALENT) 192.38/136.02 Used ordering: 192.38/136.02 Polynomial interpretation [POLO]: 192.38/136.02 192.38/136.02 POL(activate(x_1)) = 1 + 2*x_1 192.38/136.02 POL(c) = 1 192.38/136.02 POL(f(x_1)) = 1 + 2*x_1 192.38/136.02 POL(g(x_1)) = x_1 192.38/136.02 POL(n__c) = 0 192.38/136.02 POL(n__g(x_1)) = x_1 192.38/136.02 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 192.38/136.02 192.38/136.02 c -> n__c 192.38/136.02 activate(n__g(X)) -> g(X) 192.38/136.02 activate(X) -> X 192.38/136.02 192.38/136.02 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (55) 192.38/136.02 Obligation: 192.38/136.02 Q restricted rewrite system: 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 c -> f(n__g(n__c)) 192.38/136.02 f(n__g(X)) -> g(activate(X)) 192.38/136.02 g(X) -> n__g(X) 192.38/136.02 activate(n__c) -> c 192.38/136.02 192.38/136.02 Q is empty. 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (56) Overlay + Local Confluence (EQUIVALENT) 192.38/136.02 The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (57) 192.38/136.02 Obligation: 192.38/136.02 Q restricted rewrite system: 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 c -> f(n__g(n__c)) 192.38/136.02 f(n__g(X)) -> g(activate(X)) 192.38/136.02 g(X) -> n__g(X) 192.38/136.02 activate(n__c) -> c 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 c 192.38/136.02 f(n__g(x0)) 192.38/136.02 g(x0) 192.38/136.02 activate(n__c) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (58) DependencyPairsProof (EQUIVALENT) 192.38/136.02 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (59) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 C -> F(n__g(n__c)) 192.38/136.02 F(n__g(X)) -> G(activate(X)) 192.38/136.02 F(n__g(X)) -> ACTIVATE(X) 192.38/136.02 ACTIVATE(n__c) -> C 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 c -> f(n__g(n__c)) 192.38/136.02 f(n__g(X)) -> g(activate(X)) 192.38/136.02 g(X) -> n__g(X) 192.38/136.02 activate(n__c) -> c 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 c 192.38/136.02 f(n__g(x0)) 192.38/136.02 g(x0) 192.38/136.02 activate(n__c) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (60) DependencyGraphProof (EQUIVALENT) 192.38/136.02 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (61) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 F(n__g(X)) -> ACTIVATE(X) 192.38/136.02 ACTIVATE(n__c) -> C 192.38/136.02 C -> F(n__g(n__c)) 192.38/136.02 192.38/136.02 The TRS R consists of the following rules: 192.38/136.02 192.38/136.02 c -> f(n__g(n__c)) 192.38/136.02 f(n__g(X)) -> g(activate(X)) 192.38/136.02 g(X) -> n__g(X) 192.38/136.02 activate(n__c) -> c 192.38/136.02 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 c 192.38/136.02 f(n__g(x0)) 192.38/136.02 g(x0) 192.38/136.02 activate(n__c) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (62) UsableRulesProof (EQUIVALENT) 192.38/136.02 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. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (63) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 F(n__g(X)) -> ACTIVATE(X) 192.38/136.02 ACTIVATE(n__c) -> C 192.38/136.02 C -> F(n__g(n__c)) 192.38/136.02 192.38/136.02 R is empty. 192.38/136.02 The set Q consists of the following terms: 192.38/136.02 192.38/136.02 c 192.38/136.02 f(n__g(x0)) 192.38/136.02 g(x0) 192.38/136.02 activate(n__c) 192.38/136.02 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (64) QReductionProof (EQUIVALENT) 192.38/136.02 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 192.38/136.02 192.38/136.02 c 192.38/136.02 f(n__g(x0)) 192.38/136.02 g(x0) 192.38/136.02 activate(n__c) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (65) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 F(n__g(X)) -> ACTIVATE(X) 192.38/136.02 ACTIVATE(n__c) -> C 192.38/136.02 C -> F(n__g(n__c)) 192.38/136.02 192.38/136.02 R is empty. 192.38/136.02 Q is empty. 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (66) TransformationProof (EQUIVALENT) 192.38/136.02 By instantiating [LPAR04] the rule F(n__g(X)) -> ACTIVATE(X) we obtained the following new rules [LPAR04]: 192.38/136.02 192.38/136.02 (F(n__g(n__c)) -> ACTIVATE(n__c),F(n__g(n__c)) -> ACTIVATE(n__c)) 192.38/136.02 192.38/136.02 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (67) 192.38/136.02 Obligation: 192.38/136.02 Q DP problem: 192.38/136.02 The TRS P consists of the following rules: 192.38/136.02 192.38/136.02 ACTIVATE(n__c) -> C 192.38/136.02 C -> F(n__g(n__c)) 192.38/136.02 F(n__g(n__c)) -> ACTIVATE(n__c) 192.38/136.02 192.38/136.02 R is empty. 192.38/136.02 Q is empty. 192.38/136.02 We have to consider all minimal (P,Q,R)-chains. 192.38/136.02 ---------------------------------------- 192.38/136.02 192.38/136.02 (68) NonTerminationLoopProof (COMPLETE) 192.38/136.02 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 192.38/136.02 Found a loop by narrowing to the right: 192.38/136.02 192.38/136.02 s = ACTIVATE(n__c) evaluates to t =ACTIVATE(n__c) 192.38/136.02 192.38/136.02 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 192.38/136.03 * Matcher: [ ] 192.38/136.03 * Semiunifier: [ ] 192.38/136.03 192.38/136.03 -------------------------------------------------------------------------------- 192.38/136.03 Rewriting sequence 192.38/136.03 192.38/136.03 ACTIVATE(n__c) -> C 192.38/136.03 with rule ACTIVATE(n__c) -> C and matcher [ ]. 192.38/136.03 192.38/136.03 C -> F(n__g(n__c)) 192.38/136.03 with rule C -> F(n__g(n__c)) at position [] and matcher [ ] 192.38/136.03 192.38/136.03 F(n__g(n__c)) -> ACTIVATE(n__c) 192.38/136.03 with rule F(n__g(n__c)) -> ACTIVATE(n__c) at position [] and matcher [ ] 192.38/136.03 192.38/136.03 Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence 192.38/136.03 192.38/136.03 192.38/136.03 All these steps are and every following step will be a correct step w.r.t to Q. 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (69) 192.38/136.03 NO 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (70) Raffelsieper-Zantema-Transformation (SOUND) 192.38/136.03 We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (71) 192.38/136.03 Obligation: 192.38/136.03 Q restricted rewrite system: 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(c) -> up(n__c) 192.38/136.03 down(activate(n__g(X))) -> up(g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(activate(X)) -> up(X) 192.38/136.03 top(up(x)) -> top(down(x)) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 g_flat(up(x_1)) -> up(g(x_1)) 192.38/136.03 activate_flat(up(x_1)) -> up(activate(x_1)) 192.38/136.03 192.38/136.03 Q is empty. 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (72) QTRSRRRProof (EQUIVALENT) 192.38/136.03 Used ordering: 192.38/136.03 Polynomial interpretation [POLO]: 192.38/136.03 192.38/136.03 POL(activate(x_1)) = x_1 192.38/136.03 POL(activate_flat(x_1)) = x_1 192.38/136.03 POL(c) = 0 192.38/136.03 POL(down(x_1)) = 2*x_1 192.38/136.03 POL(f(x_1)) = 2*x_1 192.38/136.03 POL(f_flat(x_1)) = 2*x_1 192.38/136.03 POL(fresh_constant) = 0 192.38/136.03 POL(g(x_1)) = 2*x_1 192.38/136.03 POL(g_flat(x_1)) = 2 + 2*x_1 192.38/136.03 POL(n__c) = 0 192.38/136.03 POL(n__g(x_1)) = 2*x_1 192.38/136.03 POL(n__g_flat(x_1)) = 2*x_1 192.38/136.03 POL(top(x_1)) = 2*x_1 192.38/136.03 POL(up(x_1)) = 2*x_1 192.38/136.03 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 192.38/136.03 192.38/136.03 g_flat(up(x_1)) -> up(g(x_1)) 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (73) 192.38/136.03 Obligation: 192.38/136.03 Q restricted rewrite system: 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(c) -> up(n__c) 192.38/136.03 down(activate(n__g(X))) -> up(g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(activate(X)) -> up(X) 192.38/136.03 top(up(x)) -> top(down(x)) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 activate_flat(up(x_1)) -> up(activate(x_1)) 192.38/136.03 192.38/136.03 Q is empty. 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (74) QTRSRRRProof (EQUIVALENT) 192.38/136.03 Used ordering: 192.38/136.03 Polynomial interpretation [POLO]: 192.38/136.03 192.38/136.03 POL(activate(x_1)) = x_1 192.38/136.03 POL(activate_flat(x_1)) = 1 + x_1 192.38/136.03 POL(c) = 0 192.38/136.03 POL(down(x_1)) = 2*x_1 192.38/136.03 POL(f(x_1)) = x_1 192.38/136.03 POL(f_flat(x_1)) = x_1 192.38/136.03 POL(fresh_constant) = 0 192.38/136.03 POL(g(x_1)) = 2*x_1 192.38/136.03 POL(n__c) = 0 192.38/136.03 POL(n__g(x_1)) = 2*x_1 192.38/136.03 POL(n__g_flat(x_1)) = 2*x_1 192.38/136.03 POL(top(x_1)) = x_1 192.38/136.03 POL(up(x_1)) = 2*x_1 192.38/136.03 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 192.38/136.03 192.38/136.03 activate_flat(up(x_1)) -> up(activate(x_1)) 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (75) 192.38/136.03 Obligation: 192.38/136.03 Q restricted rewrite system: 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(c) -> up(n__c) 192.38/136.03 down(activate(n__g(X))) -> up(g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(activate(X)) -> up(X) 192.38/136.03 top(up(x)) -> top(down(x)) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 Q is empty. 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (76) QTRSRRRProof (EQUIVALENT) 192.38/136.03 Used ordering: 192.38/136.03 Polynomial interpretation [POLO]: 192.38/136.03 192.38/136.03 POL(activate(x_1)) = 2*x_1 192.38/136.03 POL(c) = 2 192.38/136.03 POL(down(x_1)) = 2*x_1 192.38/136.03 POL(f(x_1)) = 2*x_1 192.38/136.03 POL(f_flat(x_1)) = 2*x_1 192.38/136.03 POL(fresh_constant) = 0 192.38/136.03 POL(g(x_1)) = x_1 192.38/136.03 POL(n__c) = 1 192.38/136.03 POL(n__g(x_1)) = x_1 192.38/136.03 POL(n__g_flat(x_1)) = x_1 192.38/136.03 POL(top(x_1)) = 2*x_1 192.38/136.03 POL(up(x_1)) = 2*x_1 192.38/136.03 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 192.38/136.03 192.38/136.03 down(c) -> up(n__c) 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (77) 192.38/136.03 Obligation: 192.38/136.03 Q restricted rewrite system: 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__g(X))) -> up(g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(activate(X)) -> up(X) 192.38/136.03 top(up(x)) -> top(down(x)) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 Q is empty. 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (78) QTRSRRRProof (EQUIVALENT) 192.38/136.03 Used ordering: 192.38/136.03 Polynomial interpretation [POLO]: 192.38/136.03 192.38/136.03 POL(activate(x_1)) = 1 + x_1 192.38/136.03 POL(c) = 1 192.38/136.03 POL(down(x_1)) = 2*x_1 192.38/136.03 POL(f(x_1)) = 1 + 2*x_1 192.38/136.03 POL(f_flat(x_1)) = 2 + 2*x_1 192.38/136.03 POL(fresh_constant) = 0 192.38/136.03 POL(g(x_1)) = x_1 192.38/136.03 POL(n__c) = 0 192.38/136.03 POL(n__g(x_1)) = x_1 192.38/136.03 POL(n__g_flat(x_1)) = x_1 192.38/136.03 POL(top(x_1)) = 2*x_1 192.38/136.03 POL(up(x_1)) = 2*x_1 192.38/136.03 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 192.38/136.03 192.38/136.03 down(activate(n__g(X))) -> up(g(X)) 192.38/136.03 down(activate(X)) -> up(X) 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (79) 192.38/136.03 Obligation: 192.38/136.03 Q restricted rewrite system: 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 top(up(x)) -> top(down(x)) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 Q is empty. 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (80) AAECC Innermost (EQUIVALENT) 192.38/136.03 We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 192.38/136.03 The TRS R 2 is 192.38/136.03 top(up(x)) -> top(down(x)) 192.38/136.03 192.38/136.03 The signature Sigma is {top_1} 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (81) 192.38/136.03 Obligation: 192.38/136.03 Q restricted rewrite system: 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 top(up(x)) -> top(down(x)) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 top(up(x0)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (82) DependencyPairsProof (EQUIVALENT) 192.38/136.03 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (83) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(x)) -> TOP(down(x)) 192.38/136.03 TOP(up(x)) -> DOWN(x) 192.38/136.03 DOWN(n__g(y1)) -> N__G_FLAT(down(y1)) 192.38/136.03 DOWN(n__g(y1)) -> DOWN(y1) 192.38/136.03 DOWN(f(c)) -> F_FLAT(down(c)) 192.38/136.03 DOWN(f(c)) -> DOWN(c) 192.38/136.03 DOWN(f(f(y5))) -> F_FLAT(down(f(y5))) 192.38/136.03 DOWN(f(f(y5))) -> DOWN(f(y5)) 192.38/136.03 DOWN(f(n__c)) -> F_FLAT(down(n__c)) 192.38/136.03 DOWN(f(n__c)) -> DOWN(n__c) 192.38/136.03 DOWN(f(g(y7))) -> F_FLAT(down(g(y7))) 192.38/136.03 DOWN(f(g(y7))) -> DOWN(g(y7)) 192.38/136.03 DOWN(f(activate(y8))) -> F_FLAT(down(activate(y8))) 192.38/136.03 DOWN(f(activate(y8))) -> DOWN(activate(y8)) 192.38/136.03 DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant)) 192.38/136.03 DOWN(f(fresh_constant)) -> DOWN(fresh_constant) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 top(up(x)) -> top(down(x)) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 top(up(x0)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (84) DependencyGraphProof (EQUIVALENT) 192.38/136.03 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 13 less nodes. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (85) 192.38/136.03 Complex Obligation (AND) 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (86) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 DOWN(f(f(y5))) -> DOWN(f(y5)) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 top(up(x)) -> top(down(x)) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 top(up(x0)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (87) UsableRulesProof (EQUIVALENT) 192.38/136.03 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. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (88) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 DOWN(f(f(y5))) -> DOWN(f(y5)) 192.38/136.03 192.38/136.03 R is empty. 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 top(up(x0)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (89) QReductionProof (EQUIVALENT) 192.38/136.03 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 top(up(x0)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (90) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 DOWN(f(f(y5))) -> DOWN(f(y5)) 192.38/136.03 192.38/136.03 R is empty. 192.38/136.03 Q is empty. 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (91) QDPSizeChangeProof (EQUIVALENT) 192.38/136.03 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 192.38/136.03 192.38/136.03 From the DPs we obtained the following set of size-change graphs: 192.38/136.03 *DOWN(f(f(y5))) -> DOWN(f(y5)) 192.38/136.03 The graph contains the following edges 1 > 1 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (92) 192.38/136.03 YES 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (93) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 DOWN(n__g(y1)) -> DOWN(y1) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 top(up(x)) -> top(down(x)) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 top(up(x0)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (94) UsableRulesProof (EQUIVALENT) 192.38/136.03 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. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (95) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 DOWN(n__g(y1)) -> DOWN(y1) 192.38/136.03 192.38/136.03 R is empty. 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 top(up(x0)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (96) QReductionProof (EQUIVALENT) 192.38/136.03 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 top(up(x0)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (97) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 DOWN(n__g(y1)) -> DOWN(y1) 192.38/136.03 192.38/136.03 R is empty. 192.38/136.03 Q is empty. 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (98) QDPSizeChangeProof (EQUIVALENT) 192.38/136.03 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 192.38/136.03 192.38/136.03 From the DPs we obtained the following set of size-change graphs: 192.38/136.03 *DOWN(n__g(y1)) -> DOWN(y1) 192.38/136.03 The graph contains the following edges 1 > 1 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (99) 192.38/136.03 YES 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (100) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(x)) -> TOP(down(x)) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 top(up(x)) -> top(down(x)) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 top(up(x0)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (101) UsableRulesProof (EQUIVALENT) 192.38/136.03 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. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (102) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(x)) -> TOP(down(x)) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 top(up(x0)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (103) QReductionProof (EQUIVALENT) 192.38/136.03 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 192.38/136.03 192.38/136.03 top(up(x0)) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (104) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(x)) -> TOP(down(x)) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (105) TransformationProof (EQUIVALENT) 192.38/136.03 By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]: 192.38/136.03 192.38/136.03 (TOP(up(c)) -> TOP(up(f(n__g(n__c)))),TOP(up(c)) -> TOP(up(f(n__g(n__c))))) 192.38/136.03 (TOP(up(f(n__g(x0)))) -> TOP(up(g(activate(x0)))),TOP(up(f(n__g(x0)))) -> TOP(up(g(activate(x0))))) 192.38/136.03 (TOP(up(g(x0))) -> TOP(up(n__g(x0))),TOP(up(g(x0))) -> TOP(up(n__g(x0)))) 192.38/136.03 (TOP(up(activate(n__c))) -> TOP(up(c)),TOP(up(activate(n__c))) -> TOP(up(c))) 192.38/136.03 (TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))),TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0)))) 192.38/136.03 (TOP(up(f(c))) -> TOP(f_flat(down(c))),TOP(up(f(c))) -> TOP(f_flat(down(c)))) 192.38/136.03 (TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))),TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))) 192.38/136.03 (TOP(up(f(n__c))) -> TOP(f_flat(down(n__c))),TOP(up(f(n__c))) -> TOP(f_flat(down(n__c)))) 192.38/136.03 (TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))),TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))) 192.38/136.03 (TOP(up(f(activate(x0)))) -> TOP(f_flat(down(activate(x0)))),TOP(up(f(activate(x0)))) -> TOP(f_flat(down(activate(x0))))) 192.38/136.03 (TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))),TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant)))) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (106) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(c)) -> TOP(up(f(n__g(n__c)))) 192.38/136.03 TOP(up(f(n__g(x0)))) -> TOP(up(g(activate(x0)))) 192.38/136.03 TOP(up(g(x0))) -> TOP(up(n__g(x0))) 192.38/136.03 TOP(up(activate(n__c))) -> TOP(up(c)) 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 TOP(up(f(c))) -> TOP(f_flat(down(c))) 192.38/136.03 TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))) 192.38/136.03 TOP(up(f(n__c))) -> TOP(f_flat(down(n__c))) 192.38/136.03 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 192.38/136.03 TOP(up(f(activate(x0)))) -> TOP(f_flat(down(activate(x0)))) 192.38/136.03 TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (107) DependencyGraphProof (EQUIVALENT) 192.38/136.03 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (108) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(f(n__g(x0)))) -> TOP(up(g(activate(x0)))) 192.38/136.03 TOP(up(g(x0))) -> TOP(up(n__g(x0))) 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 TOP(up(c)) -> TOP(up(f(n__g(n__c)))) 192.38/136.03 TOP(up(f(c))) -> TOP(f_flat(down(c))) 192.38/136.03 TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))) 192.38/136.03 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 192.38/136.03 TOP(up(f(activate(x0)))) -> TOP(f_flat(down(activate(x0)))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (109) TransformationProof (EQUIVALENT) 192.38/136.03 By rewriting [LPAR04] the rule TOP(up(f(c))) -> TOP(f_flat(down(c))) at position [0,0] we obtained the following new rules [LPAR04]: 192.38/136.03 192.38/136.03 (TOP(up(f(c))) -> TOP(f_flat(up(f(n__g(n__c))))),TOP(up(f(c))) -> TOP(f_flat(up(f(n__g(n__c)))))) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (110) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(f(n__g(x0)))) -> TOP(up(g(activate(x0)))) 192.38/136.03 TOP(up(g(x0))) -> TOP(up(n__g(x0))) 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 TOP(up(c)) -> TOP(up(f(n__g(n__c)))) 192.38/136.03 TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))) 192.38/136.03 TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) 192.38/136.03 TOP(up(f(activate(x0)))) -> TOP(f_flat(down(activate(x0)))) 192.38/136.03 TOP(up(f(c))) -> TOP(f_flat(up(f(n__g(n__c))))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (111) TransformationProof (EQUIVALENT) 192.38/136.03 By rewriting [LPAR04] the rule TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) at position [0,0] we obtained the following new rules [LPAR04]: 192.38/136.03 192.38/136.03 (TOP(up(f(g(x0)))) -> TOP(f_flat(up(n__g(x0)))),TOP(up(f(g(x0)))) -> TOP(f_flat(up(n__g(x0))))) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (112) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(f(n__g(x0)))) -> TOP(up(g(activate(x0)))) 192.38/136.03 TOP(up(g(x0))) -> TOP(up(n__g(x0))) 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 TOP(up(c)) -> TOP(up(f(n__g(n__c)))) 192.38/136.03 TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))) 192.38/136.03 TOP(up(f(activate(x0)))) -> TOP(f_flat(down(activate(x0)))) 192.38/136.03 TOP(up(f(c))) -> TOP(f_flat(up(f(n__g(n__c))))) 192.38/136.03 TOP(up(f(g(x0)))) -> TOP(f_flat(up(n__g(x0)))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (113) TransformationProof (EQUIVALENT) 192.38/136.03 By rewriting [LPAR04] the rule TOP(up(f(c))) -> TOP(f_flat(up(f(n__g(n__c))))) at position [0] we obtained the following new rules [LPAR04]: 192.38/136.03 192.38/136.03 (TOP(up(f(c))) -> TOP(up(f(f(n__g(n__c))))),TOP(up(f(c))) -> TOP(up(f(f(n__g(n__c)))))) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (114) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(f(n__g(x0)))) -> TOP(up(g(activate(x0)))) 192.38/136.03 TOP(up(g(x0))) -> TOP(up(n__g(x0))) 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 TOP(up(c)) -> TOP(up(f(n__g(n__c)))) 192.38/136.03 TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))) 192.38/136.03 TOP(up(f(activate(x0)))) -> TOP(f_flat(down(activate(x0)))) 192.38/136.03 TOP(up(f(g(x0)))) -> TOP(f_flat(up(n__g(x0)))) 192.38/136.03 TOP(up(f(c))) -> TOP(up(f(f(n__g(n__c))))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (115) TransformationProof (EQUIVALENT) 192.38/136.03 By rewriting [LPAR04] the rule TOP(up(f(g(x0)))) -> TOP(f_flat(up(n__g(x0)))) at position [0] we obtained the following new rules [LPAR04]: 192.38/136.03 192.38/136.03 (TOP(up(f(g(x0)))) -> TOP(up(f(n__g(x0)))),TOP(up(f(g(x0)))) -> TOP(up(f(n__g(x0))))) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (116) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(f(n__g(x0)))) -> TOP(up(g(activate(x0)))) 192.38/136.03 TOP(up(g(x0))) -> TOP(up(n__g(x0))) 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 TOP(up(c)) -> TOP(up(f(n__g(n__c)))) 192.38/136.03 TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))) 192.38/136.03 TOP(up(f(activate(x0)))) -> TOP(f_flat(down(activate(x0)))) 192.38/136.03 TOP(up(f(c))) -> TOP(up(f(f(n__g(n__c))))) 192.38/136.03 TOP(up(f(g(x0)))) -> TOP(up(f(n__g(x0)))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (117) QDPOrderProof (EQUIVALENT) 192.38/136.03 We use the reduction pair processor [LPAR04,JAR06]. 192.38/136.03 192.38/136.03 192.38/136.03 The following pairs can be oriented strictly and are deleted. 192.38/136.03 192.38/136.03 TOP(up(c)) -> TOP(up(f(n__g(n__c)))) 192.38/136.03 The remaining pairs can at least be oriented weakly. 192.38/136.03 Used ordering: Polynomial interpretation [POLO]: 192.38/136.03 192.38/136.03 POL(TOP(x_1)) = x_1 192.38/136.03 POL(activate(x_1)) = 0 192.38/136.03 POL(c) = 1 192.38/136.03 POL(down(x_1)) = 0 192.38/136.03 POL(f(x_1)) = 0 192.38/136.03 POL(f_flat(x_1)) = 0 192.38/136.03 POL(fresh_constant) = 0 192.38/136.03 POL(g(x_1)) = 0 192.38/136.03 POL(n__c) = 0 192.38/136.03 POL(n__g(x_1)) = 0 192.38/136.03 POL(n__g_flat(x_1)) = 0 192.38/136.03 POL(up(x_1)) = x_1 192.38/136.03 192.38/136.03 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 192.38/136.03 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (118) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(f(n__g(x0)))) -> TOP(up(g(activate(x0)))) 192.38/136.03 TOP(up(g(x0))) -> TOP(up(n__g(x0))) 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))) 192.38/136.03 TOP(up(f(activate(x0)))) -> TOP(f_flat(down(activate(x0)))) 192.38/136.03 TOP(up(f(c))) -> TOP(up(f(f(n__g(n__c))))) 192.38/136.03 TOP(up(f(g(x0)))) -> TOP(up(f(n__g(x0)))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (119) QDPOrderProof (EQUIVALENT) 192.38/136.03 We use the reduction pair processor [LPAR04,JAR06]. 192.38/136.03 192.38/136.03 192.38/136.03 The following pairs can be oriented strictly and are deleted. 192.38/136.03 192.38/136.03 TOP(up(g(x0))) -> TOP(up(n__g(x0))) 192.38/136.03 The remaining pairs can at least be oriented weakly. 192.38/136.03 Used ordering: Polynomial interpretation [POLO]: 192.38/136.03 192.38/136.03 POL(TOP(x_1)) = x_1 192.38/136.03 POL(activate(x_1)) = 0 192.38/136.03 POL(c) = 0 192.38/136.03 POL(down(x_1)) = 0 192.38/136.03 POL(f(x_1)) = 1 192.38/136.03 POL(f_flat(x_1)) = 1 192.38/136.03 POL(fresh_constant) = 0 192.38/136.03 POL(g(x_1)) = 1 192.38/136.03 POL(n__c) = 0 192.38/136.03 POL(n__g(x_1)) = 0 192.38/136.03 POL(n__g_flat(x_1)) = 0 192.38/136.03 POL(up(x_1)) = x_1 192.38/136.03 192.38/136.03 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 192.38/136.03 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (120) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(f(n__g(x0)))) -> TOP(up(g(activate(x0)))) 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))) 192.38/136.03 TOP(up(f(activate(x0)))) -> TOP(f_flat(down(activate(x0)))) 192.38/136.03 TOP(up(f(c))) -> TOP(up(f(f(n__g(n__c))))) 192.38/136.03 TOP(up(f(g(x0)))) -> TOP(up(f(n__g(x0)))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (121) DependencyGraphProof (EQUIVALENT) 192.38/136.03 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (122) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))) 192.38/136.03 TOP(up(f(activate(x0)))) -> TOP(f_flat(down(activate(x0)))) 192.38/136.03 TOP(up(f(c))) -> TOP(up(f(f(n__g(n__c))))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (123) QDPOrderProof (EQUIVALENT) 192.38/136.03 We use the reduction pair processor [LPAR04,JAR06]. 192.38/136.03 192.38/136.03 192.38/136.03 The following pairs can be oriented strictly and are deleted. 192.38/136.03 192.38/136.03 TOP(up(f(c))) -> TOP(up(f(f(n__g(n__c))))) 192.38/136.03 The remaining pairs can at least be oriented weakly. 192.38/136.03 Used ordering: Polynomial interpretation [POLO]: 192.38/136.03 192.38/136.03 POL(TOP(x_1)) = x_1 192.38/136.03 POL(activate(x_1)) = x_1 192.38/136.03 POL(c) = 1 192.38/136.03 POL(down(x_1)) = x_1 192.38/136.03 POL(f(x_1)) = x_1 192.38/136.03 POL(f_flat(x_1)) = x_1 192.38/136.03 POL(fresh_constant) = 0 192.38/136.03 POL(g(x_1)) = 0 192.38/136.03 POL(n__c) = 1 192.38/136.03 POL(n__g(x_1)) = 0 192.38/136.03 POL(n__g_flat(x_1)) = 0 192.38/136.03 POL(up(x_1)) = x_1 192.38/136.03 192.38/136.03 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (124) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))) 192.38/136.03 TOP(up(f(activate(x0)))) -> TOP(f_flat(down(activate(x0)))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (125) QDPOrderProof (EQUIVALENT) 192.38/136.03 We use the reduction pair processor [LPAR04,JAR06]. 192.38/136.03 192.38/136.03 192.38/136.03 The following pairs can be oriented strictly and are deleted. 192.38/136.03 192.38/136.03 TOP(up(f(activate(x0)))) -> TOP(f_flat(down(activate(x0)))) 192.38/136.03 The remaining pairs can at least be oriented weakly. 192.38/136.03 Used ordering: Polynomial interpretation [POLO]: 192.38/136.03 192.38/136.03 POL(TOP(x_1)) = x_1 192.38/136.03 POL(activate(x_1)) = 1 192.38/136.03 POL(c) = 0 192.38/136.03 POL(down(x_1)) = 0 192.38/136.03 POL(f(x_1)) = x_1 192.38/136.03 POL(f_flat(x_1)) = x_1 192.38/136.03 POL(fresh_constant) = 0 192.38/136.03 POL(g(x_1)) = 0 192.38/136.03 POL(n__c) = 0 192.38/136.03 POL(n__g(x_1)) = 0 192.38/136.03 POL(n__g_flat(x_1)) = 0 192.38/136.03 POL(up(x_1)) = x_1 192.38/136.03 192.38/136.03 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (126) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (127) QDPOrderProof (EQUIVALENT) 192.38/136.03 We use the reduction pair processor [LPAR04,JAR06]. 192.38/136.03 192.38/136.03 192.38/136.03 The following pairs can be oriented strictly and are deleted. 192.38/136.03 192.38/136.03 TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))) 192.38/136.03 The remaining pairs can at least be oriented weakly. 192.38/136.03 Used ordering: Matrix interpretation [MATRO]: 192.38/136.03 192.38/136.03 Non-tuple symbols: 192.38/136.03 <<< 192.38/136.03 M( c ) = [[1], [1]] 192.38/136.03 >>> 192.38/136.03 192.38/136.03 <<< 192.38/136.03 M( down_1(x_1) ) = [[0], [0]] + [[1, 0], [1, 1]] * x_1 192.38/136.03 >>> 192.38/136.03 192.38/136.03 <<< 192.38/136.03 M( f_1(x_1) ) = [[1], [0]] + [[1, 0], [0, 1]] * x_1 192.38/136.03 >>> 192.38/136.03 192.38/136.03 <<< 192.38/136.03 M( n__g_1(x_1) ) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 192.38/136.03 >>> 192.38/136.03 192.38/136.03 <<< 192.38/136.03 M( fresh_constant ) = [[0], [0]] 192.38/136.03 >>> 192.38/136.03 192.38/136.03 <<< 192.38/136.03 M( up_1(x_1) ) = [[0], [1]] + [[1, 1], [1, 0]] * x_1 192.38/136.03 >>> 192.38/136.03 192.38/136.03 <<< 192.38/136.03 M( n__g_flat_1(x_1) ) = [[0], [1]] + [[0, 0], [0, 0]] * x_1 192.38/136.03 >>> 192.38/136.03 192.38/136.03 <<< 192.38/136.03 M( f_flat_1(x_1) ) = [[1], [1]] + [[1, 0], [0, 1]] * x_1 192.38/136.03 >>> 192.38/136.03 192.38/136.03 <<< 192.38/136.03 M( n__c ) = [[1], [0]] 192.38/136.03 >>> 192.38/136.03 192.38/136.03 <<< 192.38/136.03 M( activate_1(x_1) ) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 192.38/136.03 >>> 192.38/136.03 192.38/136.03 <<< 192.38/136.03 M( g_1(x_1) ) = [[0], [1]] + [[0, 0], [0, 0]] * x_1 192.38/136.03 >>> 192.38/136.03 192.38/136.03 Tuple symbols: 192.38/136.03 <<< 192.38/136.03 M( TOP_1(x_1) ) = [[0]] + [[1, 1]] * x_1 192.38/136.03 >>> 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 Matrix type: 192.38/136.03 192.38/136.03 We used a basic matrix type which is not further parametrizeable. 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 192.38/136.03 As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. 192.38/136.03 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (128) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (129) MNOCProof (EQUIVALENT) 192.38/136.03 We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (130) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 Q is empty. 192.38/136.03 We have to consider all (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (131) SplitQDPProof (EQUIVALENT) 192.38/136.03 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (132) 192.38/136.03 Complex Obligation (AND) 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (133) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.03 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.03 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.03 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.03 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.03 down(f(fresh_constant)) -> f_flat(down(fresh_constant)) 192.38/136.03 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.03 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down(c) 192.38/136.03 down(f(n__g(x0))) 192.38/136.03 down(g(x0)) 192.38/136.03 down(activate(n__c)) 192.38/136.03 down(n__g(x0)) 192.38/136.03 down(f(c)) 192.38/136.03 down(f(f(x0))) 192.38/136.03 down(f(n__c)) 192.38/136.03 down(f(g(x0))) 192.38/136.03 down(f(activate(x0))) 192.38/136.03 down(f(fresh_constant)) 192.38/136.03 f_flat(up(x0)) 192.38/136.03 n__g_flat(up(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (134) SemLabProof (SOUND) 192.38/136.03 We found the following model for the rules of the TRSs R and P. 192.38/136.03 Interpretation over the domain with elements from 0 to 1. 192.38/136.03 c: 0 192.38/136.03 down: 0 192.38/136.03 f: 0 192.38/136.03 n__g: 0 192.38/136.03 fresh_constant: 1 192.38/136.03 up: 0 192.38/136.03 n__g_flat: 0 192.38/136.03 f_flat: 0 192.38/136.03 n__c: 0 192.38/136.03 activate: 0 192.38/136.03 TOP: 0 192.38/136.03 g: 0 192.38/136.03 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (135) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP.0(up.0(n__g.0(x0))) -> TOP.0(n__g_flat.0(down.0(x0))) 192.38/136.03 TOP.0(up.0(n__g.1(x0))) -> TOP.0(n__g_flat.0(down.1(x0))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down.0(c.) -> up.0(f.0(n__g.0(n__c.))) 192.38/136.03 down.0(f.0(n__g.0(X))) -> up.0(g.0(activate.0(X))) 192.38/136.03 down.0(f.0(n__g.1(X))) -> up.0(g.0(activate.1(X))) 192.38/136.03 down.0(g.0(X)) -> up.0(n__g.0(X)) 192.38/136.03 down.0(g.1(X)) -> up.0(n__g.1(X)) 192.38/136.03 down.0(activate.0(n__c.)) -> up.0(c.) 192.38/136.03 down.0(n__g.0(y1)) -> n__g_flat.0(down.0(y1)) 192.38/136.03 down.0(n__g.1(y1)) -> n__g_flat.0(down.1(y1)) 192.38/136.03 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 192.38/136.03 down.0(f.0(f.0(y5))) -> f_flat.0(down.0(f.0(y5))) 192.38/136.03 down.0(f.0(f.1(y5))) -> f_flat.0(down.0(f.1(y5))) 192.38/136.03 down.0(f.0(n__c.)) -> f_flat.0(down.0(n__c.)) 192.38/136.03 down.0(f.0(g.0(y7))) -> f_flat.0(down.0(g.0(y7))) 192.38/136.03 down.0(f.0(g.1(y7))) -> f_flat.0(down.0(g.1(y7))) 192.38/136.03 down.0(f.0(activate.0(y8))) -> f_flat.0(down.0(activate.0(y8))) 192.38/136.03 down.0(f.0(activate.1(y8))) -> f_flat.0(down.0(activate.1(y8))) 192.38/136.03 down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.)) 192.38/136.03 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 192.38/136.03 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 192.38/136.03 n__g_flat.0(up.0(x_1)) -> up.0(n__g.0(x_1)) 192.38/136.03 n__g_flat.0(up.1(x_1)) -> up.0(n__g.1(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down.0(c.) 192.38/136.03 down.0(f.0(n__g.0(x0))) 192.38/136.03 down.0(f.0(n__g.1(x0))) 192.38/136.03 down.0(g.0(x0)) 192.38/136.03 down.0(g.1(x0)) 192.38/136.03 down.0(activate.0(n__c.)) 192.38/136.03 down.0(n__g.0(x0)) 192.38/136.03 down.0(n__g.1(x0)) 192.38/136.03 down.0(f.0(c.)) 192.38/136.03 down.0(f.0(f.0(x0))) 192.38/136.03 down.0(f.0(f.1(x0))) 192.38/136.03 down.0(f.0(n__c.)) 192.38/136.03 down.0(f.0(g.0(x0))) 192.38/136.03 down.0(f.0(g.1(x0))) 192.38/136.03 down.0(f.0(activate.0(x0))) 192.38/136.03 down.0(f.0(activate.1(x0))) 192.38/136.03 down.0(f.1(fresh_constant.)) 192.38/136.03 f_flat.0(up.0(x0)) 192.38/136.03 f_flat.0(up.1(x0)) 192.38/136.03 n__g_flat.0(up.0(x0)) 192.38/136.03 n__g_flat.0(up.1(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (136) DependencyGraphProof (EQUIVALENT) 192.38/136.03 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (137) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP.0(up.0(n__g.0(x0))) -> TOP.0(n__g_flat.0(down.0(x0))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down.0(c.) -> up.0(f.0(n__g.0(n__c.))) 192.38/136.03 down.0(f.0(n__g.0(X))) -> up.0(g.0(activate.0(X))) 192.38/136.03 down.0(f.0(n__g.1(X))) -> up.0(g.0(activate.1(X))) 192.38/136.03 down.0(g.0(X)) -> up.0(n__g.0(X)) 192.38/136.03 down.0(g.1(X)) -> up.0(n__g.1(X)) 192.38/136.03 down.0(activate.0(n__c.)) -> up.0(c.) 192.38/136.03 down.0(n__g.0(y1)) -> n__g_flat.0(down.0(y1)) 192.38/136.03 down.0(n__g.1(y1)) -> n__g_flat.0(down.1(y1)) 192.38/136.03 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 192.38/136.03 down.0(f.0(f.0(y5))) -> f_flat.0(down.0(f.0(y5))) 192.38/136.03 down.0(f.0(f.1(y5))) -> f_flat.0(down.0(f.1(y5))) 192.38/136.03 down.0(f.0(n__c.)) -> f_flat.0(down.0(n__c.)) 192.38/136.03 down.0(f.0(g.0(y7))) -> f_flat.0(down.0(g.0(y7))) 192.38/136.03 down.0(f.0(g.1(y7))) -> f_flat.0(down.0(g.1(y7))) 192.38/136.03 down.0(f.0(activate.0(y8))) -> f_flat.0(down.0(activate.0(y8))) 192.38/136.03 down.0(f.0(activate.1(y8))) -> f_flat.0(down.0(activate.1(y8))) 192.38/136.03 down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.)) 192.38/136.03 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 192.38/136.03 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 192.38/136.03 n__g_flat.0(up.0(x_1)) -> up.0(n__g.0(x_1)) 192.38/136.03 n__g_flat.0(up.1(x_1)) -> up.0(n__g.1(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down.0(c.) 192.38/136.03 down.0(f.0(n__g.0(x0))) 192.38/136.03 down.0(f.0(n__g.1(x0))) 192.38/136.03 down.0(g.0(x0)) 192.38/136.03 down.0(g.1(x0)) 192.38/136.03 down.0(activate.0(n__c.)) 192.38/136.03 down.0(n__g.0(x0)) 192.38/136.03 down.0(n__g.1(x0)) 192.38/136.03 down.0(f.0(c.)) 192.38/136.03 down.0(f.0(f.0(x0))) 192.38/136.03 down.0(f.0(f.1(x0))) 192.38/136.03 down.0(f.0(n__c.)) 192.38/136.03 down.0(f.0(g.0(x0))) 192.38/136.03 down.0(f.0(g.1(x0))) 192.38/136.03 down.0(f.0(activate.0(x0))) 192.38/136.03 down.0(f.0(activate.1(x0))) 192.38/136.03 down.0(f.1(fresh_constant.)) 192.38/136.03 f_flat.0(up.0(x0)) 192.38/136.03 f_flat.0(up.1(x0)) 192.38/136.03 n__g_flat.0(up.0(x0)) 192.38/136.03 n__g_flat.0(up.1(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (138) UsableRulesReductionPairsProof (EQUIVALENT) 192.38/136.03 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. 192.38/136.03 192.38/136.03 No dependency pairs are removed. 192.38/136.03 192.38/136.03 The following rules are removed from R: 192.38/136.03 192.38/136.03 down.0(g.1(X)) -> up.0(n__g.1(X)) 192.38/136.03 down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.)) 192.38/136.03 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 192.38/136.03 n__g_flat.0(up.1(x_1)) -> up.0(n__g.1(x_1)) 192.38/136.03 Used ordering: POLO with Polynomial interpretation [POLO]: 192.38/136.03 192.38/136.03 POL(TOP.0(x_1)) = x_1 192.38/136.03 POL(activate.0(x_1)) = x_1 192.38/136.03 POL(activate.1(x_1)) = x_1 192.38/136.03 POL(c.) = 0 192.38/136.03 POL(down.0(x_1)) = x_1 192.38/136.03 POL(down.1(x_1)) = x_1 192.38/136.03 POL(f.0(x_1)) = x_1 192.38/136.03 POL(f.1(x_1)) = 1 + x_1 192.38/136.03 POL(f_flat.0(x_1)) = x_1 192.38/136.03 POL(fresh_constant.) = 0 192.38/136.03 POL(g.0(x_1)) = x_1 192.38/136.03 POL(g.1(x_1)) = 1 + x_1 192.38/136.03 POL(n__c.) = 0 192.38/136.03 POL(n__g.0(x_1)) = x_1 192.38/136.03 POL(n__g.1(x_1)) = x_1 192.38/136.03 POL(n__g_flat.0(x_1)) = x_1 192.38/136.03 POL(up.0(x_1)) = x_1 192.38/136.03 POL(up.1(x_1)) = 1 + x_1 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (139) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP.0(up.0(n__g.0(x0))) -> TOP.0(n__g_flat.0(down.0(x0))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down.0(c.) -> up.0(f.0(n__g.0(n__c.))) 192.38/136.03 down.0(f.0(n__g.0(X))) -> up.0(g.0(activate.0(X))) 192.38/136.03 down.0(f.0(n__g.1(X))) -> up.0(g.0(activate.1(X))) 192.38/136.03 down.0(g.0(X)) -> up.0(n__g.0(X)) 192.38/136.03 down.0(activate.0(n__c.)) -> up.0(c.) 192.38/136.03 down.0(n__g.0(y1)) -> n__g_flat.0(down.0(y1)) 192.38/136.03 down.0(n__g.1(y1)) -> n__g_flat.0(down.1(y1)) 192.38/136.03 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 192.38/136.03 down.0(f.0(f.0(y5))) -> f_flat.0(down.0(f.0(y5))) 192.38/136.03 down.0(f.0(f.1(y5))) -> f_flat.0(down.0(f.1(y5))) 192.38/136.03 down.0(f.0(n__c.)) -> f_flat.0(down.0(n__c.)) 192.38/136.03 down.0(f.0(g.0(y7))) -> f_flat.0(down.0(g.0(y7))) 192.38/136.03 down.0(f.0(g.1(y7))) -> f_flat.0(down.0(g.1(y7))) 192.38/136.03 down.0(f.0(activate.0(y8))) -> f_flat.0(down.0(activate.0(y8))) 192.38/136.03 down.0(f.0(activate.1(y8))) -> f_flat.0(down.0(activate.1(y8))) 192.38/136.03 n__g_flat.0(up.0(x_1)) -> up.0(n__g.0(x_1)) 192.38/136.03 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down.0(c.) 192.38/136.03 down.0(f.0(n__g.0(x0))) 192.38/136.03 down.0(f.0(n__g.1(x0))) 192.38/136.03 down.0(g.0(x0)) 192.38/136.03 down.0(g.1(x0)) 192.38/136.03 down.0(activate.0(n__c.)) 192.38/136.03 down.0(n__g.0(x0)) 192.38/136.03 down.0(n__g.1(x0)) 192.38/136.03 down.0(f.0(c.)) 192.38/136.03 down.0(f.0(f.0(x0))) 192.38/136.03 down.0(f.0(f.1(x0))) 192.38/136.03 down.0(f.0(n__c.)) 192.38/136.03 down.0(f.0(g.0(x0))) 192.38/136.03 down.0(f.0(g.1(x0))) 192.38/136.03 down.0(f.0(activate.0(x0))) 192.38/136.03 down.0(f.0(activate.1(x0))) 192.38/136.03 down.0(f.1(fresh_constant.)) 192.38/136.03 f_flat.0(up.0(x0)) 192.38/136.03 f_flat.0(up.1(x0)) 192.38/136.03 n__g_flat.0(up.0(x0)) 192.38/136.03 n__g_flat.0(up.1(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (140) UsableRulesReductionPairsProof (EQUIVALENT) 192.38/136.03 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. 192.38/136.03 192.38/136.03 No dependency pairs are removed. 192.38/136.03 192.38/136.03 The following rules are removed from R: 192.38/136.03 192.38/136.03 down.0(f.0(n__g.1(X))) -> up.0(g.0(activate.1(X))) 192.38/136.03 down.0(n__g.1(y1)) -> n__g_flat.0(down.1(y1)) 192.38/136.03 Used ordering: POLO with Polynomial interpretation [POLO]: 192.38/136.03 192.38/136.03 POL(TOP.0(x_1)) = x_1 192.38/136.03 POL(activate.0(x_1)) = 1 + x_1 192.38/136.03 POL(activate.1(x_1)) = 1 + x_1 192.38/136.03 POL(c.) = 1 192.38/136.03 POL(down.0(x_1)) = 1 + x_1 192.38/136.03 POL(down.1(x_1)) = 1 + x_1 192.38/136.03 POL(f.0(x_1)) = 1 + x_1 192.38/136.03 POL(f.1(x_1)) = x_1 192.38/136.03 POL(f_flat.0(x_1)) = 1 + x_1 192.38/136.03 POL(g.0(x_1)) = x_1 192.38/136.03 POL(g.1(x_1)) = x_1 192.38/136.03 POL(n__c.) = 0 192.38/136.03 POL(n__g.0(x_1)) = x_1 192.38/136.03 POL(n__g.1(x_1)) = 1 + x_1 192.38/136.03 POL(n__g_flat.0(x_1)) = x_1 192.38/136.03 POL(up.0(x_1)) = 1 + x_1 192.38/136.03 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (141) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP.0(up.0(n__g.0(x0))) -> TOP.0(n__g_flat.0(down.0(x0))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down.0(c.) -> up.0(f.0(n__g.0(n__c.))) 192.38/136.03 down.0(f.0(n__g.0(X))) -> up.0(g.0(activate.0(X))) 192.38/136.03 down.0(g.0(X)) -> up.0(n__g.0(X)) 192.38/136.03 down.0(activate.0(n__c.)) -> up.0(c.) 192.38/136.03 down.0(n__g.0(y1)) -> n__g_flat.0(down.0(y1)) 192.38/136.03 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 192.38/136.03 down.0(f.0(f.0(y5))) -> f_flat.0(down.0(f.0(y5))) 192.38/136.03 down.0(f.0(f.1(y5))) -> f_flat.0(down.0(f.1(y5))) 192.38/136.03 down.0(f.0(n__c.)) -> f_flat.0(down.0(n__c.)) 192.38/136.03 down.0(f.0(g.0(y7))) -> f_flat.0(down.0(g.0(y7))) 192.38/136.03 down.0(f.0(g.1(y7))) -> f_flat.0(down.0(g.1(y7))) 192.38/136.03 down.0(f.0(activate.0(y8))) -> f_flat.0(down.0(activate.0(y8))) 192.38/136.03 down.0(f.0(activate.1(y8))) -> f_flat.0(down.0(activate.1(y8))) 192.38/136.03 n__g_flat.0(up.0(x_1)) -> up.0(n__g.0(x_1)) 192.38/136.03 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 192.38/136.03 192.38/136.03 The set Q consists of the following terms: 192.38/136.03 192.38/136.03 down.0(c.) 192.38/136.03 down.0(f.0(n__g.0(x0))) 192.38/136.03 down.0(f.0(n__g.1(x0))) 192.38/136.03 down.0(g.0(x0)) 192.38/136.03 down.0(g.1(x0)) 192.38/136.03 down.0(activate.0(n__c.)) 192.38/136.03 down.0(n__g.0(x0)) 192.38/136.03 down.0(n__g.1(x0)) 192.38/136.03 down.0(f.0(c.)) 192.38/136.03 down.0(f.0(f.0(x0))) 192.38/136.03 down.0(f.0(f.1(x0))) 192.38/136.03 down.0(f.0(n__c.)) 192.38/136.03 down.0(f.0(g.0(x0))) 192.38/136.03 down.0(f.0(g.1(x0))) 192.38/136.03 down.0(f.0(activate.0(x0))) 192.38/136.03 down.0(f.0(activate.1(x0))) 192.38/136.03 down.0(f.1(fresh_constant.)) 192.38/136.03 f_flat.0(up.0(x0)) 192.38/136.03 f_flat.0(up.1(x0)) 192.38/136.03 n__g_flat.0(up.0(x0)) 192.38/136.03 n__g_flat.0(up.1(x0)) 192.38/136.03 192.38/136.03 We have to consider all minimal (P,Q,R)-chains. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (142) PisEmptyProof (SOUND) 192.38/136.03 The TRS P is empty. Hence, there is no (P,Q,R) chain. 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (143) 192.38/136.03 TRUE 192.38/136.03 192.38/136.03 ---------------------------------------- 192.38/136.03 192.38/136.03 (144) 192.38/136.03 Obligation: 192.38/136.03 Q DP problem: 192.38/136.03 The TRS P consists of the following rules: 192.38/136.03 192.38/136.03 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.03 192.38/136.03 The TRS R consists of the following rules: 192.38/136.03 192.38/136.03 down(c) -> up(f(n__g(n__c))) 192.38/136.03 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.03 down(g(X)) -> up(n__g(X)) 192.38/136.03 down(activate(n__c)) -> up(c) 192.38/136.03 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.03 down(f(c)) -> f_flat(down(c)) 192.38/136.04 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.04 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.04 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.04 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.04 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.04 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down(c) 192.38/136.04 down(f(n__g(x0))) 192.38/136.04 down(g(x0)) 192.38/136.04 down(activate(n__c)) 192.38/136.04 down(n__g(x0)) 192.38/136.04 down(f(c)) 192.38/136.04 down(f(f(x0))) 192.38/136.04 down(f(n__c)) 192.38/136.04 down(f(g(x0))) 192.38/136.04 down(f(activate(x0))) 192.38/136.04 down(f(fresh_constant)) 192.38/136.04 f_flat(up(x0)) 192.38/136.04 n__g_flat(up(x0)) 192.38/136.04 192.38/136.04 We have to consider all minimal (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (145) QReductionProof (EQUIVALENT) 192.38/136.04 We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN]. 192.38/136.04 192.38/136.04 down(f(fresh_constant)) 192.38/136.04 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (146) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down(c) -> up(f(n__g(n__c))) 192.38/136.04 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.04 down(g(X)) -> up(n__g(X)) 192.38/136.04 down(activate(n__c)) -> up(c) 192.38/136.04 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.04 down(f(c)) -> f_flat(down(c)) 192.38/136.04 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.04 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.04 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.04 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.04 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.04 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down(c) 192.38/136.04 down(f(n__g(x0))) 192.38/136.04 down(g(x0)) 192.38/136.04 down(activate(n__c)) 192.38/136.04 down(n__g(x0)) 192.38/136.04 down(f(c)) 192.38/136.04 down(f(f(x0))) 192.38/136.04 down(f(n__c)) 192.38/136.04 down(f(g(x0))) 192.38/136.04 down(f(activate(x0))) 192.38/136.04 f_flat(up(x0)) 192.38/136.04 n__g_flat(up(x0)) 192.38/136.04 192.38/136.04 We have to consider all (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (147) MNOCProof (EQUIVALENT) 192.38/136.04 We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (148) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down(c) -> up(f(n__g(n__c))) 192.38/136.04 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.04 down(g(X)) -> up(n__g(X)) 192.38/136.04 down(activate(n__c)) -> up(c) 192.38/136.04 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.04 down(f(c)) -> f_flat(down(c)) 192.38/136.04 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.04 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.04 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.04 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.04 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.04 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.04 192.38/136.04 Q is empty. 192.38/136.04 We have to consider all (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (149) SplitQDPProof (EQUIVALENT) 192.38/136.04 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (150) 192.38/136.04 Complex Obligation (AND) 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (151) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down(c) -> up(f(n__g(n__c))) 192.38/136.04 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.04 down(g(X)) -> up(n__g(X)) 192.38/136.04 down(activate(n__c)) -> up(c) 192.38/136.04 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.04 down(f(c)) -> f_flat(down(c)) 192.38/136.04 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.04 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.04 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.04 down(f(activate(y8))) -> f_flat(down(activate(y8))) 192.38/136.04 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.04 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down(c) 192.38/136.04 down(f(n__g(x0))) 192.38/136.04 down(g(x0)) 192.38/136.04 down(activate(n__c)) 192.38/136.04 down(n__g(x0)) 192.38/136.04 down(f(c)) 192.38/136.04 down(f(f(x0))) 192.38/136.04 down(f(n__c)) 192.38/136.04 down(f(g(x0))) 192.38/136.04 down(f(activate(x0))) 192.38/136.04 down(f(fresh_constant)) 192.38/136.04 f_flat(up(x0)) 192.38/136.04 n__g_flat(up(x0)) 192.38/136.04 192.38/136.04 We have to consider all minimal (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (152) SemLabProof (SOUND) 192.38/136.04 We found the following model for the rules of the TRSs R and P. 192.38/136.04 Interpretation over the domain with elements from 0 to 1. 192.38/136.04 c: 0 192.38/136.04 down: 0 192.38/136.04 f: 0 192.38/136.04 fresh_constant: 0 192.38/136.04 n__g: 0 192.38/136.04 up: 0 192.38/136.04 n__g_flat: 0 192.38/136.04 f_flat: 0 192.38/136.04 n__c: 0 192.38/136.04 activate: 1 192.38/136.04 TOP: 0 192.38/136.04 g: 0 192.38/136.04 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (153) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP.0(up.0(n__g.0(x0))) -> TOP.0(n__g_flat.0(down.0(x0))) 192.38/136.04 TOP.0(up.0(n__g.1(x0))) -> TOP.0(n__g_flat.0(down.1(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down.0(c.) -> up.0(f.0(n__g.0(n__c.))) 192.38/136.04 down.0(f.0(n__g.0(X))) -> up.0(g.1(activate.0(X))) 192.38/136.04 down.0(f.0(n__g.1(X))) -> up.0(g.1(activate.1(X))) 192.38/136.04 down.0(g.0(X)) -> up.0(n__g.0(X)) 192.38/136.04 down.0(g.1(X)) -> up.0(n__g.1(X)) 192.38/136.04 down.1(activate.0(n__c.)) -> up.0(c.) 192.38/136.04 down.0(n__g.0(y1)) -> n__g_flat.0(down.0(y1)) 192.38/136.04 down.0(n__g.1(y1)) -> n__g_flat.0(down.1(y1)) 192.38/136.04 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 192.38/136.04 down.0(f.0(f.0(y5))) -> f_flat.0(down.0(f.0(y5))) 192.38/136.04 down.0(f.0(f.1(y5))) -> f_flat.0(down.0(f.1(y5))) 192.38/136.04 down.0(f.0(n__c.)) -> f_flat.0(down.0(n__c.)) 192.38/136.04 down.0(f.0(g.0(y7))) -> f_flat.0(down.0(g.0(y7))) 192.38/136.04 down.0(f.0(g.1(y7))) -> f_flat.0(down.0(g.1(y7))) 192.38/136.04 down.0(f.1(activate.0(y8))) -> f_flat.0(down.1(activate.0(y8))) 192.38/136.04 down.0(f.1(activate.1(y8))) -> f_flat.0(down.1(activate.1(y8))) 192.38/136.04 n__g_flat.0(up.0(x_1)) -> up.0(n__g.0(x_1)) 192.38/136.04 n__g_flat.0(up.1(x_1)) -> up.0(n__g.1(x_1)) 192.38/136.04 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 192.38/136.04 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down.0(c.) 192.38/136.04 down.0(f.0(n__g.0(x0))) 192.38/136.04 down.0(f.0(n__g.1(x0))) 192.38/136.04 down.0(g.0(x0)) 192.38/136.04 down.0(g.1(x0)) 192.38/136.04 down.1(activate.0(n__c.)) 192.38/136.04 down.0(n__g.0(x0)) 192.38/136.04 down.0(n__g.1(x0)) 192.38/136.04 down.0(f.0(c.)) 192.38/136.04 down.0(f.0(f.0(x0))) 192.38/136.04 down.0(f.0(f.1(x0))) 192.38/136.04 down.0(f.0(n__c.)) 192.38/136.04 down.0(f.0(g.0(x0))) 192.38/136.04 down.0(f.0(g.1(x0))) 192.38/136.04 down.0(f.1(activate.0(x0))) 192.38/136.04 down.0(f.1(activate.1(x0))) 192.38/136.04 down.0(f.0(fresh_constant.)) 192.38/136.04 f_flat.0(up.0(x0)) 192.38/136.04 f_flat.0(up.1(x0)) 192.38/136.04 n__g_flat.0(up.0(x0)) 192.38/136.04 n__g_flat.0(up.1(x0)) 192.38/136.04 192.38/136.04 We have to consider all minimal (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (154) UsableRulesReductionPairsProof (EQUIVALENT) 192.38/136.04 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. 192.38/136.04 192.38/136.04 No dependency pairs are removed. 192.38/136.04 192.38/136.04 The following rules are removed from R: 192.38/136.04 192.38/136.04 down.0(g.0(X)) -> up.0(n__g.0(X)) 192.38/136.04 down.0(f.1(activate.0(y8))) -> f_flat.0(down.1(activate.0(y8))) 192.38/136.04 down.0(f.1(activate.1(y8))) -> f_flat.0(down.1(activate.1(y8))) 192.38/136.04 n__g_flat.0(up.1(x_1)) -> up.0(n__g.1(x_1)) 192.38/136.04 f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) 192.38/136.04 Used ordering: POLO with Polynomial interpretation [POLO]: 192.38/136.04 192.38/136.04 POL(TOP.0(x_1)) = x_1 192.38/136.04 POL(activate.0(x_1)) = x_1 192.38/136.04 POL(activate.1(x_1)) = x_1 192.38/136.04 POL(c.) = 0 192.38/136.04 POL(down.0(x_1)) = x_1 192.38/136.04 POL(down.1(x_1)) = x_1 192.38/136.04 POL(f.0(x_1)) = x_1 192.38/136.04 POL(f.1(x_1)) = 1 + x_1 192.38/136.04 POL(f_flat.0(x_1)) = x_1 192.38/136.04 POL(g.0(x_1)) = 1 + x_1 192.38/136.04 POL(g.1(x_1)) = x_1 192.38/136.04 POL(n__c.) = 0 192.38/136.04 POL(n__g.0(x_1)) = x_1 192.38/136.04 POL(n__g.1(x_1)) = x_1 192.38/136.04 POL(n__g_flat.0(x_1)) = x_1 192.38/136.04 POL(up.0(x_1)) = x_1 192.38/136.04 POL(up.1(x_1)) = 1 + x_1 192.38/136.04 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (155) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP.0(up.0(n__g.0(x0))) -> TOP.0(n__g_flat.0(down.0(x0))) 192.38/136.04 TOP.0(up.0(n__g.1(x0))) -> TOP.0(n__g_flat.0(down.1(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down.1(activate.0(n__c.)) -> up.0(c.) 192.38/136.04 n__g_flat.0(up.0(x_1)) -> up.0(n__g.0(x_1)) 192.38/136.04 down.0(c.) -> up.0(f.0(n__g.0(n__c.))) 192.38/136.04 down.0(f.0(n__g.0(X))) -> up.0(g.1(activate.0(X))) 192.38/136.04 down.0(f.0(n__g.1(X))) -> up.0(g.1(activate.1(X))) 192.38/136.04 down.0(g.1(X)) -> up.0(n__g.1(X)) 192.38/136.04 down.0(n__g.0(y1)) -> n__g_flat.0(down.0(y1)) 192.38/136.04 down.0(n__g.1(y1)) -> n__g_flat.0(down.1(y1)) 192.38/136.04 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 192.38/136.04 down.0(f.0(f.0(y5))) -> f_flat.0(down.0(f.0(y5))) 192.38/136.04 down.0(f.0(f.1(y5))) -> f_flat.0(down.0(f.1(y5))) 192.38/136.04 down.0(f.0(n__c.)) -> f_flat.0(down.0(n__c.)) 192.38/136.04 down.0(f.0(g.0(y7))) -> f_flat.0(down.0(g.0(y7))) 192.38/136.04 down.0(f.0(g.1(y7))) -> f_flat.0(down.0(g.1(y7))) 192.38/136.04 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down.0(c.) 192.38/136.04 down.0(f.0(n__g.0(x0))) 192.38/136.04 down.0(f.0(n__g.1(x0))) 192.38/136.04 down.0(g.0(x0)) 192.38/136.04 down.0(g.1(x0)) 192.38/136.04 down.1(activate.0(n__c.)) 192.38/136.04 down.0(n__g.0(x0)) 192.38/136.04 down.0(n__g.1(x0)) 192.38/136.04 down.0(f.0(c.)) 192.38/136.04 down.0(f.0(f.0(x0))) 192.38/136.04 down.0(f.0(f.1(x0))) 192.38/136.04 down.0(f.0(n__c.)) 192.38/136.04 down.0(f.0(g.0(x0))) 192.38/136.04 down.0(f.0(g.1(x0))) 192.38/136.04 down.0(f.1(activate.0(x0))) 192.38/136.04 down.0(f.1(activate.1(x0))) 192.38/136.04 down.0(f.0(fresh_constant.)) 192.38/136.04 f_flat.0(up.0(x0)) 192.38/136.04 f_flat.0(up.1(x0)) 192.38/136.04 n__g_flat.0(up.0(x0)) 192.38/136.04 n__g_flat.0(up.1(x0)) 192.38/136.04 192.38/136.04 We have to consider all minimal (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (156) MRRProof (EQUIVALENT) 192.38/136.04 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. 192.38/136.04 192.38/136.04 192.38/136.04 Strictly oriented rules of the TRS R: 192.38/136.04 192.38/136.04 down.0(f.0(n__g.1(X))) -> up.0(g.1(activate.1(X))) 192.38/136.04 192.38/136.04 Used ordering: Polynomial interpretation [POLO]: 192.38/136.04 192.38/136.04 POL(TOP.0(x_1)) = x_1 192.38/136.04 POL(activate.0(x_1)) = 1 + x_1 192.38/136.04 POL(activate.1(x_1)) = x_1 192.38/136.04 POL(c.) = 1 192.38/136.04 POL(down.0(x_1)) = x_1 192.38/136.04 POL(down.1(x_1)) = x_1 192.38/136.04 POL(f.0(x_1)) = 1 + x_1 192.38/136.04 POL(f.1(x_1)) = x_1 192.38/136.04 POL(f_flat.0(x_1)) = 1 + x_1 192.38/136.04 POL(g.0(x_1)) = x_1 192.38/136.04 POL(g.1(x_1)) = x_1 192.38/136.04 POL(n__c.) = 0 192.38/136.04 POL(n__g.0(x_1)) = x_1 192.38/136.04 POL(n__g.1(x_1)) = x_1 192.38/136.04 POL(n__g_flat.0(x_1)) = x_1 192.38/136.04 POL(up.0(x_1)) = x_1 192.38/136.04 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (157) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP.0(up.0(n__g.0(x0))) -> TOP.0(n__g_flat.0(down.0(x0))) 192.38/136.04 TOP.0(up.0(n__g.1(x0))) -> TOP.0(n__g_flat.0(down.1(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down.1(activate.0(n__c.)) -> up.0(c.) 192.38/136.04 n__g_flat.0(up.0(x_1)) -> up.0(n__g.0(x_1)) 192.38/136.04 down.0(c.) -> up.0(f.0(n__g.0(n__c.))) 192.38/136.04 down.0(f.0(n__g.0(X))) -> up.0(g.1(activate.0(X))) 192.38/136.04 down.0(g.1(X)) -> up.0(n__g.1(X)) 192.38/136.04 down.0(n__g.0(y1)) -> n__g_flat.0(down.0(y1)) 192.38/136.04 down.0(n__g.1(y1)) -> n__g_flat.0(down.1(y1)) 192.38/136.04 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 192.38/136.04 down.0(f.0(f.0(y5))) -> f_flat.0(down.0(f.0(y5))) 192.38/136.04 down.0(f.0(f.1(y5))) -> f_flat.0(down.0(f.1(y5))) 192.38/136.04 down.0(f.0(n__c.)) -> f_flat.0(down.0(n__c.)) 192.38/136.04 down.0(f.0(g.0(y7))) -> f_flat.0(down.0(g.0(y7))) 192.38/136.04 down.0(f.0(g.1(y7))) -> f_flat.0(down.0(g.1(y7))) 192.38/136.04 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down.0(c.) 192.38/136.04 down.0(f.0(n__g.0(x0))) 192.38/136.04 down.0(f.0(n__g.1(x0))) 192.38/136.04 down.0(g.0(x0)) 192.38/136.04 down.0(g.1(x0)) 192.38/136.04 down.1(activate.0(n__c.)) 192.38/136.04 down.0(n__g.0(x0)) 192.38/136.04 down.0(n__g.1(x0)) 192.38/136.04 down.0(f.0(c.)) 192.38/136.04 down.0(f.0(f.0(x0))) 192.38/136.04 down.0(f.0(f.1(x0))) 192.38/136.04 down.0(f.0(n__c.)) 192.38/136.04 down.0(f.0(g.0(x0))) 192.38/136.04 down.0(f.0(g.1(x0))) 192.38/136.04 down.0(f.1(activate.0(x0))) 192.38/136.04 down.0(f.1(activate.1(x0))) 192.38/136.04 down.0(f.0(fresh_constant.)) 192.38/136.04 f_flat.0(up.0(x0)) 192.38/136.04 f_flat.0(up.1(x0)) 192.38/136.04 n__g_flat.0(up.0(x0)) 192.38/136.04 n__g_flat.0(up.1(x0)) 192.38/136.04 192.38/136.04 We have to consider all minimal (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (158) QDPOrderProof (EQUIVALENT) 192.38/136.04 We use the reduction pair processor [LPAR04,JAR06]. 192.38/136.04 192.38/136.04 192.38/136.04 The following pairs can be oriented strictly and are deleted. 192.38/136.04 192.38/136.04 TOP.0(up.0(n__g.1(x0))) -> TOP.0(n__g_flat.0(down.1(x0))) 192.38/136.04 The remaining pairs can at least be oriented weakly. 192.38/136.04 Used ordering: Polynomial interpretation [POLO]: 192.38/136.04 192.38/136.04 POL(TOP.0(x_1)) = x_1 192.38/136.04 POL(activate.0(x_1)) = x_1 192.38/136.04 POL(c.) = 0 192.38/136.04 POL(down.0(x_1)) = 0 192.38/136.04 POL(down.1(x_1)) = 0 192.38/136.04 POL(f.0(x_1)) = 0 192.38/136.04 POL(f.1(x_1)) = x_1 192.38/136.04 POL(f_flat.0(x_1)) = 0 192.38/136.04 POL(g.0(x_1)) = x_1 192.38/136.04 POL(g.1(x_1)) = 0 192.38/136.04 POL(n__c.) = 0 192.38/136.04 POL(n__g.0(x_1)) = 0 192.38/136.04 POL(n__g.1(x_1)) = 1 + x_1 192.38/136.04 POL(n__g_flat.0(x_1)) = 0 192.38/136.04 POL(up.0(x_1)) = x_1 192.38/136.04 192.38/136.04 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 192.38/136.04 192.38/136.04 n__g_flat.0(up.0(x_1)) -> up.0(n__g.0(x_1)) 192.38/136.04 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (159) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP.0(up.0(n__g.0(x0))) -> TOP.0(n__g_flat.0(down.0(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down.1(activate.0(n__c.)) -> up.0(c.) 192.38/136.04 n__g_flat.0(up.0(x_1)) -> up.0(n__g.0(x_1)) 192.38/136.04 down.0(c.) -> up.0(f.0(n__g.0(n__c.))) 192.38/136.04 down.0(f.0(n__g.0(X))) -> up.0(g.1(activate.0(X))) 192.38/136.04 down.0(g.1(X)) -> up.0(n__g.1(X)) 192.38/136.04 down.0(n__g.0(y1)) -> n__g_flat.0(down.0(y1)) 192.38/136.04 down.0(n__g.1(y1)) -> n__g_flat.0(down.1(y1)) 192.38/136.04 down.0(f.0(c.)) -> f_flat.0(down.0(c.)) 192.38/136.04 down.0(f.0(f.0(y5))) -> f_flat.0(down.0(f.0(y5))) 192.38/136.04 down.0(f.0(f.1(y5))) -> f_flat.0(down.0(f.1(y5))) 192.38/136.04 down.0(f.0(n__c.)) -> f_flat.0(down.0(n__c.)) 192.38/136.04 down.0(f.0(g.0(y7))) -> f_flat.0(down.0(g.0(y7))) 192.38/136.04 down.0(f.0(g.1(y7))) -> f_flat.0(down.0(g.1(y7))) 192.38/136.04 f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down.0(c.) 192.38/136.04 down.0(f.0(n__g.0(x0))) 192.38/136.04 down.0(f.0(n__g.1(x0))) 192.38/136.04 down.0(g.0(x0)) 192.38/136.04 down.0(g.1(x0)) 192.38/136.04 down.1(activate.0(n__c.)) 192.38/136.04 down.0(n__g.0(x0)) 192.38/136.04 down.0(n__g.1(x0)) 192.38/136.04 down.0(f.0(c.)) 192.38/136.04 down.0(f.0(f.0(x0))) 192.38/136.04 down.0(f.0(f.1(x0))) 192.38/136.04 down.0(f.0(n__c.)) 192.38/136.04 down.0(f.0(g.0(x0))) 192.38/136.04 down.0(f.0(g.1(x0))) 192.38/136.04 down.0(f.1(activate.0(x0))) 192.38/136.04 down.0(f.1(activate.1(x0))) 192.38/136.04 down.0(f.0(fresh_constant.)) 192.38/136.04 f_flat.0(up.0(x0)) 192.38/136.04 f_flat.0(up.1(x0)) 192.38/136.04 n__g_flat.0(up.0(x0)) 192.38/136.04 n__g_flat.0(up.1(x0)) 192.38/136.04 192.38/136.04 We have to consider all minimal (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (160) PisEmptyProof (SOUND) 192.38/136.04 The TRS P is empty. Hence, there is no (P,Q,R) chain. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (161) 192.38/136.04 TRUE 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (162) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down(activate(n__c)) -> up(c) 192.38/136.04 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.04 down(c) -> up(f(n__g(n__c))) 192.38/136.04 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.04 down(g(X)) -> up(n__g(X)) 192.38/136.04 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.04 down(f(c)) -> f_flat(down(c)) 192.38/136.04 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.04 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.04 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.04 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down(c) 192.38/136.04 down(f(n__g(x0))) 192.38/136.04 down(g(x0)) 192.38/136.04 down(activate(n__c)) 192.38/136.04 down(n__g(x0)) 192.38/136.04 down(f(c)) 192.38/136.04 down(f(f(x0))) 192.38/136.04 down(f(n__c)) 192.38/136.04 down(f(g(x0))) 192.38/136.04 down(f(activate(x0))) 192.38/136.04 down(f(fresh_constant)) 192.38/136.04 f_flat(up(x0)) 192.38/136.04 n__g_flat(up(x0)) 192.38/136.04 192.38/136.04 We have to consider all minimal (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (163) QReductionProof (EQUIVALENT) 192.38/136.04 We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN]. 192.38/136.04 192.38/136.04 down(f(fresh_constant)) 192.38/136.04 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (164) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down(activate(n__c)) -> up(c) 192.38/136.04 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.04 down(c) -> up(f(n__g(n__c))) 192.38/136.04 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.04 down(g(X)) -> up(n__g(X)) 192.38/136.04 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.04 down(f(c)) -> f_flat(down(c)) 192.38/136.04 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.04 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.04 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.04 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down(c) 192.38/136.04 down(f(n__g(x0))) 192.38/136.04 down(g(x0)) 192.38/136.04 down(activate(n__c)) 192.38/136.04 down(n__g(x0)) 192.38/136.04 down(f(c)) 192.38/136.04 down(f(f(x0))) 192.38/136.04 down(f(n__c)) 192.38/136.04 down(f(g(x0))) 192.38/136.04 down(f(activate(x0))) 192.38/136.04 f_flat(up(x0)) 192.38/136.04 n__g_flat(up(x0)) 192.38/136.04 192.38/136.04 We have to consider all (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (165) SplitQDPProof (EQUIVALENT) 192.38/136.04 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (166) 192.38/136.04 Complex Obligation (AND) 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (167) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down(activate(n__c)) -> up(c) 192.38/136.04 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.04 down(c) -> up(f(n__g(n__c))) 192.38/136.04 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.04 down(g(X)) -> up(n__g(X)) 192.38/136.04 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.04 down(f(c)) -> f_flat(down(c)) 192.38/136.04 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.04 down(f(n__c)) -> f_flat(down(n__c)) 192.38/136.04 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.04 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down(c) 192.38/136.04 down(f(n__g(x0))) 192.38/136.04 down(g(x0)) 192.38/136.04 down(activate(n__c)) 192.38/136.04 down(n__g(x0)) 192.38/136.04 down(f(c)) 192.38/136.04 down(f(f(x0))) 192.38/136.04 down(f(n__c)) 192.38/136.04 down(f(g(x0))) 192.38/136.04 down(f(activate(x0))) 192.38/136.04 down(f(fresh_constant)) 192.38/136.04 f_flat(up(x0)) 192.38/136.04 n__g_flat(up(x0)) 192.38/136.04 192.38/136.04 We have to consider all minimal (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (168) SemLabProof (SOUND) 192.38/136.04 We found the following model for the rules of the TRSs R and P. 192.38/136.04 Interpretation over the domain with elements from 0 to 1. 192.38/136.04 c: 1 192.38/136.04 down: 0 192.38/136.04 f: 1 192.38/136.04 fresh_constant: 0 192.38/136.04 n__g: 1 192.38/136.04 up: 0 192.38/136.04 n__g_flat: 0 192.38/136.04 f_flat: 0 192.38/136.04 activate: 0 192.38/136.04 n__c: 0 192.38/136.04 TOP: 0 192.38/136.04 g: 1 192.38/136.04 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (169) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP.0(up.1(n__g.0(x0))) -> TOP.0(n__g_flat.0(down.0(x0))) 192.38/136.04 TOP.0(up.1(n__g.1(x0))) -> TOP.0(n__g_flat.0(down.1(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down.0(activate.0(n__c.)) -> up.1(c.) 192.38/136.04 n__g_flat.0(up.0(x_1)) -> up.1(n__g.0(x_1)) 192.38/136.04 n__g_flat.0(up.1(x_1)) -> up.1(n__g.1(x_1)) 192.38/136.04 down.1(c.) -> up.1(f.1(n__g.0(n__c.))) 192.38/136.04 down.1(f.1(n__g.0(X))) -> up.1(g.0(activate.0(X))) 192.38/136.04 down.1(f.1(n__g.1(X))) -> up.1(g.0(activate.1(X))) 192.38/136.04 down.1(g.0(X)) -> up.1(n__g.0(X)) 192.38/136.04 down.1(g.1(X)) -> up.1(n__g.1(X)) 192.38/136.04 down.1(n__g.0(y1)) -> n__g_flat.0(down.0(y1)) 192.38/136.04 down.1(n__g.1(y1)) -> n__g_flat.0(down.1(y1)) 192.38/136.04 down.1(f.1(c.)) -> f_flat.0(down.1(c.)) 192.38/136.04 down.1(f.1(f.0(y5))) -> f_flat.0(down.1(f.0(y5))) 192.38/136.04 down.1(f.1(f.1(y5))) -> f_flat.0(down.1(f.1(y5))) 192.38/136.04 down.1(f.0(n__c.)) -> f_flat.0(down.0(n__c.)) 192.38/136.04 down.1(f.1(g.0(y7))) -> f_flat.0(down.1(g.0(y7))) 192.38/136.04 down.1(f.1(g.1(y7))) -> f_flat.0(down.1(g.1(y7))) 192.38/136.04 f_flat.0(up.0(x_1)) -> up.1(f.0(x_1)) 192.38/136.04 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down.1(c.) 192.38/136.04 down.1(f.1(n__g.0(x0))) 192.38/136.04 down.1(f.1(n__g.1(x0))) 192.38/136.04 down.1(g.0(x0)) 192.38/136.04 down.1(g.1(x0)) 192.38/136.04 down.0(activate.0(n__c.)) 192.38/136.04 down.1(n__g.0(x0)) 192.38/136.04 down.1(n__g.1(x0)) 192.38/136.04 down.1(f.1(c.)) 192.38/136.04 down.1(f.1(f.0(x0))) 192.38/136.04 down.1(f.1(f.1(x0))) 192.38/136.04 down.1(f.0(n__c.)) 192.38/136.04 down.1(f.1(g.0(x0))) 192.38/136.04 down.1(f.1(g.1(x0))) 192.38/136.04 down.1(f.0(activate.0(x0))) 192.38/136.04 down.1(f.0(activate.1(x0))) 192.38/136.04 down.1(f.0(fresh_constant.)) 192.38/136.04 f_flat.0(up.0(x0)) 192.38/136.04 f_flat.0(up.1(x0)) 192.38/136.04 n__g_flat.0(up.0(x0)) 192.38/136.04 n__g_flat.0(up.1(x0)) 192.38/136.04 192.38/136.04 We have to consider all minimal (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (170) UsableRulesReductionPairsProof (EQUIVALENT) 192.38/136.04 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. 192.38/136.04 192.38/136.04 No dependency pairs are removed. 192.38/136.04 192.38/136.04 The following rules are removed from R: 192.38/136.04 192.38/136.04 n__g_flat.0(up.0(x_1)) -> up.1(n__g.0(x_1)) 192.38/136.04 down.1(g.1(X)) -> up.1(n__g.1(X)) 192.38/136.04 down.1(f.0(n__c.)) -> f_flat.0(down.0(n__c.)) 192.38/136.04 f_flat.0(up.0(x_1)) -> up.1(f.0(x_1)) 192.38/136.04 Used ordering: POLO with Polynomial interpretation [POLO]: 192.38/136.04 192.38/136.04 POL(TOP.0(x_1)) = x_1 192.38/136.04 POL(activate.0(x_1)) = x_1 192.38/136.04 POL(activate.1(x_1)) = x_1 192.38/136.04 POL(c.) = 0 192.38/136.04 POL(down.0(x_1)) = x_1 192.38/136.04 POL(down.1(x_1)) = x_1 192.38/136.04 POL(f.0(x_1)) = 1 + x_1 192.38/136.04 POL(f.1(x_1)) = x_1 192.38/136.04 POL(f_flat.0(x_1)) = x_1 192.38/136.04 POL(g.0(x_1)) = x_1 192.38/136.04 POL(g.1(x_1)) = 1 + x_1 192.38/136.04 POL(n__c.) = 0 192.38/136.04 POL(n__g.0(x_1)) = x_1 192.38/136.04 POL(n__g.1(x_1)) = x_1 192.38/136.04 POL(n__g_flat.0(x_1)) = x_1 192.38/136.04 POL(up.0(x_1)) = 1 + x_1 192.38/136.04 POL(up.1(x_1)) = x_1 192.38/136.04 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (171) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP.0(up.1(n__g.0(x0))) -> TOP.0(n__g_flat.0(down.0(x0))) 192.38/136.04 TOP.0(up.1(n__g.1(x0))) -> TOP.0(n__g_flat.0(down.1(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down.1(c.) -> up.1(f.1(n__g.0(n__c.))) 192.38/136.04 down.1(f.1(n__g.0(X))) -> up.1(g.0(activate.0(X))) 192.38/136.04 down.1(f.1(n__g.1(X))) -> up.1(g.0(activate.1(X))) 192.38/136.04 down.1(g.0(X)) -> up.1(n__g.0(X)) 192.38/136.04 down.1(n__g.0(y1)) -> n__g_flat.0(down.0(y1)) 192.38/136.04 down.1(n__g.1(y1)) -> n__g_flat.0(down.1(y1)) 192.38/136.04 down.1(f.1(c.)) -> f_flat.0(down.1(c.)) 192.38/136.04 down.1(f.1(f.0(y5))) -> f_flat.0(down.1(f.0(y5))) 192.38/136.04 down.1(f.1(f.1(y5))) -> f_flat.0(down.1(f.1(y5))) 192.38/136.04 down.1(f.1(g.0(y7))) -> f_flat.0(down.1(g.0(y7))) 192.38/136.04 down.1(f.1(g.1(y7))) -> f_flat.0(down.1(g.1(y7))) 192.38/136.04 n__g_flat.0(up.1(x_1)) -> up.1(n__g.1(x_1)) 192.38/136.04 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 192.38/136.04 down.0(activate.0(n__c.)) -> up.1(c.) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down.1(c.) 192.38/136.04 down.1(f.1(n__g.0(x0))) 192.38/136.04 down.1(f.1(n__g.1(x0))) 192.38/136.04 down.1(g.0(x0)) 192.38/136.04 down.1(g.1(x0)) 192.38/136.04 down.0(activate.0(n__c.)) 192.38/136.04 down.1(n__g.0(x0)) 192.38/136.04 down.1(n__g.1(x0)) 192.38/136.04 down.1(f.1(c.)) 192.38/136.04 down.1(f.1(f.0(x0))) 192.38/136.04 down.1(f.1(f.1(x0))) 192.38/136.04 down.1(f.0(n__c.)) 192.38/136.04 down.1(f.1(g.0(x0))) 192.38/136.04 down.1(f.1(g.1(x0))) 192.38/136.04 down.1(f.0(activate.0(x0))) 192.38/136.04 down.1(f.0(activate.1(x0))) 192.38/136.04 down.1(f.0(fresh_constant.)) 192.38/136.04 f_flat.0(up.0(x0)) 192.38/136.04 f_flat.0(up.1(x0)) 192.38/136.04 n__g_flat.0(up.0(x0)) 192.38/136.04 n__g_flat.0(up.1(x0)) 192.38/136.04 192.38/136.04 We have to consider all minimal (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (172) MRRProof (EQUIVALENT) 192.38/136.04 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. 192.38/136.04 192.38/136.04 192.38/136.04 Strictly oriented rules of the TRS R: 192.38/136.04 192.38/136.04 down.1(f.1(n__g.1(X))) -> up.1(g.0(activate.1(X))) 192.38/136.04 192.38/136.04 Used ordering: Polynomial interpretation [POLO]: 192.38/136.04 192.38/136.04 POL(TOP.0(x_1)) = x_1 192.38/136.04 POL(activate.0(x_1)) = 1 + x_1 192.38/136.04 POL(activate.1(x_1)) = x_1 192.38/136.04 POL(c.) = 1 192.38/136.04 POL(down.0(x_1)) = x_1 192.38/136.04 POL(down.1(x_1)) = x_1 192.38/136.04 POL(f.0(x_1)) = x_1 192.38/136.04 POL(f.1(x_1)) = 1 + x_1 192.38/136.04 POL(f_flat.0(x_1)) = 1 + x_1 192.38/136.04 POL(g.0(x_1)) = x_1 192.38/136.04 POL(g.1(x_1)) = x_1 192.38/136.04 POL(n__c.) = 0 192.38/136.04 POL(n__g.0(x_1)) = x_1 192.38/136.04 POL(n__g.1(x_1)) = x_1 192.38/136.04 POL(n__g_flat.0(x_1)) = x_1 192.38/136.04 POL(up.1(x_1)) = x_1 192.38/136.04 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (173) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP.0(up.1(n__g.0(x0))) -> TOP.0(n__g_flat.0(down.0(x0))) 192.38/136.04 TOP.0(up.1(n__g.1(x0))) -> TOP.0(n__g_flat.0(down.1(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down.1(c.) -> up.1(f.1(n__g.0(n__c.))) 192.38/136.04 down.1(f.1(n__g.0(X))) -> up.1(g.0(activate.0(X))) 192.38/136.04 down.1(g.0(X)) -> up.1(n__g.0(X)) 192.38/136.04 down.1(n__g.0(y1)) -> n__g_flat.0(down.0(y1)) 192.38/136.04 down.1(n__g.1(y1)) -> n__g_flat.0(down.1(y1)) 192.38/136.04 down.1(f.1(c.)) -> f_flat.0(down.1(c.)) 192.38/136.04 down.1(f.1(f.0(y5))) -> f_flat.0(down.1(f.0(y5))) 192.38/136.04 down.1(f.1(f.1(y5))) -> f_flat.0(down.1(f.1(y5))) 192.38/136.04 down.1(f.1(g.0(y7))) -> f_flat.0(down.1(g.0(y7))) 192.38/136.04 down.1(f.1(g.1(y7))) -> f_flat.0(down.1(g.1(y7))) 192.38/136.04 n__g_flat.0(up.1(x_1)) -> up.1(n__g.1(x_1)) 192.38/136.04 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 192.38/136.04 down.0(activate.0(n__c.)) -> up.1(c.) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down.1(c.) 192.38/136.04 down.1(f.1(n__g.0(x0))) 192.38/136.04 down.1(f.1(n__g.1(x0))) 192.38/136.04 down.1(g.0(x0)) 192.38/136.04 down.1(g.1(x0)) 192.38/136.04 down.0(activate.0(n__c.)) 192.38/136.04 down.1(n__g.0(x0)) 192.38/136.04 down.1(n__g.1(x0)) 192.38/136.04 down.1(f.1(c.)) 192.38/136.04 down.1(f.1(f.0(x0))) 192.38/136.04 down.1(f.1(f.1(x0))) 192.38/136.04 down.1(f.0(n__c.)) 192.38/136.04 down.1(f.1(g.0(x0))) 192.38/136.04 down.1(f.1(g.1(x0))) 192.38/136.04 down.1(f.0(activate.0(x0))) 192.38/136.04 down.1(f.0(activate.1(x0))) 192.38/136.04 down.1(f.0(fresh_constant.)) 192.38/136.04 f_flat.0(up.0(x0)) 192.38/136.04 f_flat.0(up.1(x0)) 192.38/136.04 n__g_flat.0(up.0(x0)) 192.38/136.04 n__g_flat.0(up.1(x0)) 192.38/136.04 192.38/136.04 We have to consider all minimal (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (174) QDPOrderProof (EQUIVALENT) 192.38/136.04 We use the reduction pair processor [LPAR04,JAR06]. 192.38/136.04 192.38/136.04 192.38/136.04 The following pairs can be oriented strictly and are deleted. 192.38/136.04 192.38/136.04 TOP.0(up.1(n__g.0(x0))) -> TOP.0(n__g_flat.0(down.0(x0))) 192.38/136.04 The remaining pairs can at least be oriented weakly. 192.38/136.04 Used ordering: Polynomial interpretation [POLO]: 192.38/136.04 192.38/136.04 POL(TOP.0(x_1)) = x_1 192.38/136.04 POL(activate.0(x_1)) = x_1 192.38/136.04 POL(c.) = 0 192.38/136.04 POL(down.0(x_1)) = 0 192.38/136.04 POL(down.1(x_1)) = 0 192.38/136.04 POL(f.0(x_1)) = x_1 192.38/136.04 POL(f.1(x_1)) = 0 192.38/136.04 POL(f_flat.0(x_1)) = 0 192.38/136.04 POL(g.0(x_1)) = 0 192.38/136.04 POL(g.1(x_1)) = x_1 192.38/136.04 POL(n__c.) = 1 192.38/136.04 POL(n__g.0(x_1)) = 1 192.38/136.04 POL(n__g.1(x_1)) = 0 192.38/136.04 POL(n__g_flat.0(x_1)) = 0 192.38/136.04 POL(up.1(x_1)) = x_1 192.38/136.04 192.38/136.04 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 192.38/136.04 192.38/136.04 n__g_flat.0(up.1(x_1)) -> up.1(n__g.1(x_1)) 192.38/136.04 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (175) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP.0(up.1(n__g.1(x0))) -> TOP.0(n__g_flat.0(down.1(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down.1(c.) -> up.1(f.1(n__g.0(n__c.))) 192.38/136.04 down.1(f.1(n__g.0(X))) -> up.1(g.0(activate.0(X))) 192.38/136.04 down.1(g.0(X)) -> up.1(n__g.0(X)) 192.38/136.04 down.1(n__g.0(y1)) -> n__g_flat.0(down.0(y1)) 192.38/136.04 down.1(n__g.1(y1)) -> n__g_flat.0(down.1(y1)) 192.38/136.04 down.1(f.1(c.)) -> f_flat.0(down.1(c.)) 192.38/136.04 down.1(f.1(f.0(y5))) -> f_flat.0(down.1(f.0(y5))) 192.38/136.04 down.1(f.1(f.1(y5))) -> f_flat.0(down.1(f.1(y5))) 192.38/136.04 down.1(f.1(g.0(y7))) -> f_flat.0(down.1(g.0(y7))) 192.38/136.04 down.1(f.1(g.1(y7))) -> f_flat.0(down.1(g.1(y7))) 192.38/136.04 n__g_flat.0(up.1(x_1)) -> up.1(n__g.1(x_1)) 192.38/136.04 f_flat.0(up.1(x_1)) -> up.1(f.1(x_1)) 192.38/136.04 down.0(activate.0(n__c.)) -> up.1(c.) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down.1(c.) 192.38/136.04 down.1(f.1(n__g.0(x0))) 192.38/136.04 down.1(f.1(n__g.1(x0))) 192.38/136.04 down.1(g.0(x0)) 192.38/136.04 down.1(g.1(x0)) 192.38/136.04 down.0(activate.0(n__c.)) 192.38/136.04 down.1(n__g.0(x0)) 192.38/136.04 down.1(n__g.1(x0)) 192.38/136.04 down.1(f.1(c.)) 192.38/136.04 down.1(f.1(f.0(x0))) 192.38/136.04 down.1(f.1(f.1(x0))) 192.38/136.04 down.1(f.0(n__c.)) 192.38/136.04 down.1(f.1(g.0(x0))) 192.38/136.04 down.1(f.1(g.1(x0))) 192.38/136.04 down.1(f.0(activate.0(x0))) 192.38/136.04 down.1(f.0(activate.1(x0))) 192.38/136.04 down.1(f.0(fresh_constant.)) 192.38/136.04 f_flat.0(up.0(x0)) 192.38/136.04 f_flat.0(up.1(x0)) 192.38/136.04 n__g_flat.0(up.0(x0)) 192.38/136.04 n__g_flat.0(up.1(x0)) 192.38/136.04 192.38/136.04 We have to consider all minimal (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (176) PisEmptyProof (SOUND) 192.38/136.04 The TRS P is empty. Hence, there is no (P,Q,R) chain. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (177) 192.38/136.04 TRUE 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (178) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down(c) -> up(f(n__g(n__c))) 192.38/136.04 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.04 down(g(X)) -> up(n__g(X)) 192.38/136.04 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.04 down(f(c)) -> f_flat(down(c)) 192.38/136.04 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.04 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.04 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.04 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.04 down(activate(n__c)) -> up(c) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down(c) 192.38/136.04 down(f(n__g(x0))) 192.38/136.04 down(g(x0)) 192.38/136.04 down(activate(n__c)) 192.38/136.04 down(n__g(x0)) 192.38/136.04 down(f(c)) 192.38/136.04 down(f(f(x0))) 192.38/136.04 down(f(n__c)) 192.38/136.04 down(f(g(x0))) 192.38/136.04 down(f(activate(x0))) 192.38/136.04 down(f(fresh_constant)) 192.38/136.04 f_flat(up(x0)) 192.38/136.04 n__g_flat(up(x0)) 192.38/136.04 192.38/136.04 We have to consider all minimal (P,Q,R)-chains. 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (179) QReductionProof (EQUIVALENT) 192.38/136.04 We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN]. 192.38/136.04 192.38/136.04 down(f(fresh_constant)) 192.38/136.04 192.38/136.04 192.38/136.04 ---------------------------------------- 192.38/136.04 192.38/136.04 (180) 192.38/136.04 Obligation: 192.38/136.04 Q DP problem: 192.38/136.04 The TRS P consists of the following rules: 192.38/136.04 192.38/136.04 TOP(up(n__g(x0))) -> TOP(n__g_flat(down(x0))) 192.38/136.04 192.38/136.04 The TRS R consists of the following rules: 192.38/136.04 192.38/136.04 down(c) -> up(f(n__g(n__c))) 192.38/136.04 down(f(n__g(X))) -> up(g(activate(X))) 192.38/136.04 down(g(X)) -> up(n__g(X)) 192.38/136.04 down(n__g(y1)) -> n__g_flat(down(y1)) 192.38/136.04 down(f(c)) -> f_flat(down(c)) 192.38/136.04 down(f(f(y5))) -> f_flat(down(f(y5))) 192.38/136.04 down(f(g(y7))) -> f_flat(down(g(y7))) 192.38/136.04 n__g_flat(up(x_1)) -> up(n__g(x_1)) 192.38/136.04 f_flat(up(x_1)) -> up(f(x_1)) 192.38/136.04 down(activate(n__c)) -> up(c) 192.38/136.04 192.38/136.04 The set Q consists of the following terms: 192.38/136.04 192.38/136.04 down(c) 192.38/136.04 down(f(n__g(x0))) 192.38/136.04 down(g(x0)) 192.38/136.04 down(activate(n__c)) 192.38/136.04 down(n__g(x0)) 192.38/136.04 down(f(c)) 192.38/136.04 down(f(f(x0))) 192.38/136.04 down(f(n__c)) 192.38/136.04 down(f(g(x0))) 192.38/136.04 down(f(activate(x0))) 192.38/136.04 f_flat(up(x0)) 192.38/136.04 n__g_flat(up(x0)) 192.38/136.04 192.38/136.04 We have to consider all (P,Q,R)-chains. 192.61/136.14 EOF