14.12/4.47 MAYBE 14.27/4.48 proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl 14.27/4.48 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 14.27/4.48 14.27/4.48 14.27/4.48 Left Termination of the query pattern 14.27/4.48 14.27/4.48 ackermann(g,a,g) 14.27/4.48 14.27/4.48 w.r.t. the given Prolog program could not be shown: 14.27/4.48 14.27/4.48 (0) Prolog 14.27/4.48 (1) PrologToPiTRSProof [SOUND, 0 ms] 14.27/4.48 (2) PiTRS 14.27/4.48 (3) DependencyPairsProof [EQUIVALENT, 2 ms] 14.27/4.48 (4) PiDP 14.27/4.48 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 14.27/4.48 (6) AND 14.27/4.48 (7) PiDP 14.27/4.48 (8) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.48 (9) PiDP 14.27/4.48 (10) PiDPToQDPProof [SOUND, 13 ms] 14.27/4.48 (11) QDP 14.27/4.48 (12) QDPSizeChangeProof [EQUIVALENT, 2 ms] 14.27/4.48 (13) YES 14.27/4.48 (14) PiDP 14.27/4.48 (15) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.48 (16) PiDP 14.27/4.48 (17) PiDPToQDPProof [SOUND, 1 ms] 14.27/4.48 (18) QDP 14.27/4.48 (19) QDPOrderProof [EQUIVALENT, 56 ms] 14.27/4.48 (20) QDP 14.27/4.48 (21) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.48 (22) QDP 14.27/4.48 (23) QReductionProof [EQUIVALENT, 0 ms] 14.27/4.48 (24) QDP 14.27/4.48 (25) NonTerminationLoopProof [COMPLETE, 0 ms] 14.27/4.48 (26) NO 14.27/4.48 (27) PiDP 14.27/4.48 (28) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.48 (29) PiDP 14.27/4.48 (30) PiDPToQDPProof [SOUND, 0 ms] 14.27/4.48 (31) QDP 14.27/4.48 (32) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.27/4.48 (33) YES 14.27/4.48 (34) PiDP 14.27/4.48 (35) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.48 (36) PiDP 14.27/4.48 (37) PiDPToQDPProof [SOUND, 0 ms] 14.27/4.48 (38) QDP 14.27/4.48 (39) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.27/4.48 (40) YES 14.27/4.48 (41) PrologToPiTRSProof [SOUND, 0 ms] 14.27/4.48 (42) PiTRS 14.27/4.48 (43) DependencyPairsProof [EQUIVALENT, 5 ms] 14.27/4.48 (44) PiDP 14.27/4.48 (45) DependencyGraphProof [EQUIVALENT, 0 ms] 14.27/4.48 (46) AND 14.27/4.48 (47) PiDP 14.27/4.48 (48) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.48 (49) PiDP 14.27/4.48 (50) PiDPToQDPProof [SOUND, 0 ms] 14.27/4.48 (51) QDP 14.27/4.48 (52) QDPSizeChangeProof [EQUIVALENT, 1 ms] 14.27/4.48 (53) YES 14.27/4.48 (54) PiDP 14.27/4.48 (55) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.49 (56) PiDP 14.27/4.49 (57) PiDPToQDPProof [SOUND, 0 ms] 14.27/4.49 (58) QDP 14.27/4.49 (59) QDPOrderProof [EQUIVALENT, 17 ms] 14.27/4.49 (60) QDP 14.27/4.49 (61) DependencyGraphProof [EQUIVALENT, 0 ms] 14.27/4.49 (62) QDP 14.27/4.49 (63) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.49 (64) QDP 14.27/4.49 (65) QReductionProof [EQUIVALENT, 0 ms] 14.27/4.49 (66) QDP 14.27/4.49 (67) NonTerminationLoopProof [COMPLETE, 0 ms] 14.27/4.49 (68) NO 14.27/4.49 (69) PiDP 14.27/4.49 (70) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.49 (71) PiDP 14.27/4.49 (72) PiDPToQDPProof [SOUND, 0 ms] 14.27/4.49 (73) QDP 14.27/4.49 (74) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.27/4.49 (75) YES 14.27/4.49 (76) PiDP 14.27/4.49 (77) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.49 (78) PiDP 14.27/4.49 (79) PiDPToQDPProof [SOUND, 0 ms] 14.27/4.49 (80) QDP 14.27/4.49 (81) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.27/4.49 (82) YES 14.27/4.49 (83) PrologToDTProblemTransformerProof [SOUND, 54 ms] 14.27/4.49 (84) TRIPLES 14.27/4.49 (85) TriplesToPiDPProof [SOUND, 46 ms] 14.27/4.49 (86) PiDP 14.27/4.49 (87) DependencyGraphProof [EQUIVALENT, 0 ms] 14.27/4.49 (88) AND 14.27/4.49 (89) PiDP 14.27/4.49 (90) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.49 (91) PiDP 14.27/4.49 (92) PiDPToQDPProof [SOUND, 0 ms] 14.27/4.49 (93) QDP 14.27/4.49 (94) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.27/4.49 (95) YES 14.27/4.49 (96) PiDP 14.27/4.49 (97) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.49 (98) PiDP 14.27/4.49 (99) PiDPToQDPProof [SOUND, 0 ms] 14.27/4.49 (100) QDP 14.27/4.49 (101) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.27/4.49 (102) YES 14.27/4.49 (103) PiDP 14.27/4.49 (104) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.49 (105) PiDP 14.27/4.49 (106) PiDPToQDPProof [SOUND, 0 ms] 14.27/4.49 (107) QDP 14.27/4.49 (108) NonTerminationLoopProof [COMPLETE, 0 ms] 14.27/4.49 (109) NO 14.27/4.49 (110) PiDP 14.27/4.49 (111) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.49 (112) PiDP 14.27/4.49 (113) PiDPToQDPProof [SOUND, 0 ms] 14.27/4.49 (114) QDP 14.27/4.49 (115) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.27/4.49 (116) YES 14.27/4.49 (117) PrologToTRSTransformerProof [SOUND, 28 ms] 14.27/4.49 (118) QTRS 14.27/4.49 (119) DependencyPairsProof [EQUIVALENT, 0 ms] 14.27/4.49 (120) QDP 14.27/4.49 (121) DependencyGraphProof [EQUIVALENT, 0 ms] 14.27/4.49 (122) AND 14.27/4.49 (123) QDP 14.27/4.49 (124) QDPOrderProof [EQUIVALENT, 54 ms] 14.27/4.49 (125) QDP 14.27/4.49 (126) DependencyGraphProof [EQUIVALENT, 0 ms] 14.27/4.49 (127) QDP 14.27/4.49 (128) QDPOrderProof [EQUIVALENT, 32 ms] 14.27/4.49 (129) QDP 14.27/4.49 (130) DependencyGraphProof [EQUIVALENT, 0 ms] 14.27/4.49 (131) QDP 14.27/4.49 (132) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.49 (133) QDP 14.27/4.49 (134) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.27/4.49 (135) YES 14.27/4.49 (136) QDP 14.27/4.49 (137) QDPOrderProof [EQUIVALENT, 84 ms] 14.27/4.49 (138) QDP 14.27/4.49 (139) DependencyGraphProof [EQUIVALENT, 0 ms] 14.27/4.49 (140) QDP 14.27/4.49 (141) UsableRulesProof [EQUIVALENT, 0 ms] 14.27/4.49 (142) QDP 14.27/4.49 (143) NonTerminationLoopProof [COMPLETE, 0 ms] 14.27/4.49 (144) NO 14.27/4.49 (145) QDP 14.27/4.49 (146) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.27/4.49 (147) YES 14.27/4.49 (148) QDP 14.27/4.49 (149) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.27/4.49 (150) YES 14.27/4.49 (151) PrologToIRSwTTransformerProof [SOUND, 107 ms] 14.27/4.49 (152) AND 14.27/4.49 (153) IRSwT 14.27/4.49 (154) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 14.27/4.49 (155) IRSwT 14.27/4.49 (156) IntTRSCompressionProof [EQUIVALENT, 22 ms] 14.27/4.49 (157) IRSwT 14.27/4.49 (158) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 14.27/4.49 (159) IRSwT 14.27/4.49 (160) IRSwTTerminationDigraphProof [EQUIVALENT, 7 ms] 14.27/4.49 (161) IRSwT 14.27/4.49 (162) TempFilterProof [SOUND, 1 ms] 14.27/4.49 (163) IRSwT 14.27/4.49 (164) IRSwTToQDPProof [SOUND, 0 ms] 14.27/4.49 (165) QDP 14.27/4.49 (166) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.27/4.49 (167) YES 14.27/4.49 (168) IRSwT 14.27/4.49 (169) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 14.27/4.49 (170) IRSwT 14.27/4.49 (171) IntTRSCompressionProof [EQUIVALENT, 3 ms] 14.27/4.49 (172) IRSwT 14.27/4.49 (173) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 14.27/4.49 (174) IRSwT 14.27/4.49 (175) IRSwTTerminationDigraphProof [EQUIVALENT, 1 ms] 14.27/4.49 (176) IRSwT 14.27/4.49 (177) FilterProof [EQUIVALENT, 0 ms] 14.27/4.49 (178) IntTRS 14.27/4.49 (179) IntTRSNonPeriodicNontermProof [COMPLETE, 3 ms] 14.27/4.49 (180) NO 14.27/4.49 (181) IRSwT 14.27/4.49 (182) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 14.27/4.49 (183) IRSwT 14.27/4.49 (184) IRSwT 14.27/4.49 14.27/4.49 14.27/4.49 ---------------------------------------- 14.27/4.49 14.27/4.49 (0) 14.27/4.49 Obligation: 14.27/4.49 Clauses: 14.27/4.49 14.27/4.49 ackermann(0, N, s(N)). 14.27/4.49 ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val). 14.27/4.49 ackermann(s(M), s(N), Val) :- ','(ackermann(s(M), N, Val1), ackermann(M, Val1, Val)). 14.27/4.49 14.27/4.49 14.27/4.49 Query: ackermann(g,a,g) 14.27/4.49 ---------------------------------------- 14.27/4.49 14.27/4.49 (1) PrologToPiTRSProof (SOUND) 14.27/4.49 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 14.27/4.49 14.27/4.49 ackermann_in_3: (b,f,b) (b,b,b) (b,b,f) (b,f,f) 14.27/4.49 14.27/4.49 Transforming Prolog into the following Term Rewriting System: 14.27/4.49 14.27/4.49 Pi-finite rewrite system: 14.27/4.49 The TRS R consists of the following rules: 14.27/4.49 14.27/4.49 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.49 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.49 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.49 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.49 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.49 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.49 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.49 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.49 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.49 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.49 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.49 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.49 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.49 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.49 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.49 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.49 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.49 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.49 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.49 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.49 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.49 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.49 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.49 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.49 14.27/4.49 The argument filtering Pi contains the following mapping: 14.27/4.49 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.49 14.27/4.49 0 = 0 14.27/4.49 14.27/4.49 s(x1) = s(x1) 14.27/4.49 14.27/4.49 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag 14.27/4.49 14.27/4.49 U1_gag(x1, x2, x3) = U1_gag(x3) 14.27/4.49 14.27/4.49 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.49 14.27/4.49 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg 14.27/4.49 14.27/4.49 U1_ggg(x1, x2, x3) = U1_ggg(x3) 14.27/4.49 14.27/4.49 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x3, x4) 14.27/4.49 14.27/4.49 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.49 14.27/4.49 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) 14.27/4.49 14.27/4.49 U1_gga(x1, x2, x3) = U1_gga(x3) 14.27/4.49 14.27/4.49 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) 14.27/4.49 14.27/4.49 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 14.27/4.49 14.27/4.49 U3_ggg(x1, x2, x3, x4) = U3_ggg(x4) 14.27/4.49 14.27/4.49 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.49 14.27/4.49 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.49 14.27/4.49 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa 14.27/4.49 14.27/4.49 U1_gaa(x1, x2, x3) = U1_gaa(x3) 14.27/4.49 14.27/4.49 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.49 14.27/4.49 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 14.27/4.49 14.27/4.49 U3_gag(x1, x2, x3, x4) = U3_gag(x4) 14.27/4.49 14.27/4.49 14.27/4.49 14.27/4.49 14.27/4.49 14.27/4.49 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 14.27/4.49 14.27/4.49 14.27/4.49 14.27/4.49 ---------------------------------------- 14.27/4.49 14.27/4.49 (2) 14.27/4.49 Obligation: 14.27/4.49 Pi-finite rewrite system: 14.27/4.49 The TRS R consists of the following rules: 14.27/4.49 14.27/4.49 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.49 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.49 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.49 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.49 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.49 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.49 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.49 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.49 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.49 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.49 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.49 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.49 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.49 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.49 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.49 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.49 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.49 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.49 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.49 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.49 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.49 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.49 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.49 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.49 14.27/4.49 The argument filtering Pi contains the following mapping: 14.27/4.49 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.49 14.27/4.49 0 = 0 14.27/4.49 14.27/4.49 s(x1) = s(x1) 14.27/4.49 14.27/4.49 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag 14.27/4.49 14.27/4.49 U1_gag(x1, x2, x3) = U1_gag(x3) 14.27/4.49 14.27/4.49 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.49 14.27/4.49 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg 14.27/4.49 14.27/4.49 U1_ggg(x1, x2, x3) = U1_ggg(x3) 14.27/4.49 14.27/4.49 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x3, x4) 14.27/4.49 14.27/4.49 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.49 14.27/4.49 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) 14.27/4.49 14.27/4.49 U1_gga(x1, x2, x3) = U1_gga(x3) 14.27/4.49 14.27/4.49 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) 14.27/4.49 14.27/4.49 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 14.27/4.49 14.27/4.49 U3_ggg(x1, x2, x3, x4) = U3_ggg(x4) 14.27/4.49 14.27/4.49 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.49 14.27/4.49 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.49 14.27/4.49 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa 14.27/4.49 14.27/4.49 U1_gaa(x1, x2, x3) = U1_gaa(x3) 14.27/4.49 14.27/4.49 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.49 14.27/4.49 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 14.27/4.49 14.27/4.49 U3_gag(x1, x2, x3, x4) = U3_gag(x4) 14.27/4.49 14.27/4.49 14.27/4.49 14.27/4.49 ---------------------------------------- 14.27/4.49 14.27/4.49 (3) DependencyPairsProof (EQUIVALENT) 14.27/4.49 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 14.27/4.49 Pi DP problem: 14.27/4.49 The TRS P consists of the following rules: 14.27/4.49 14.27/4.49 ACKERMANN_IN_GAG(s(M), 0, Val) -> U1_GAG(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.49 ACKERMANN_IN_GAG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.49 ACKERMANN_IN_GGG(s(M), 0, Val) -> U1_GGG(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.49 ACKERMANN_IN_GGG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.49 ACKERMANN_IN_GGG(s(M), s(N), Val) -> U2_GGG(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.49 ACKERMANN_IN_GGG(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) 14.27/4.49 ACKERMANN_IN_GGA(s(M), 0, Val) -> U1_GGA(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.49 ACKERMANN_IN_GGA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) 14.27/4.49 ACKERMANN_IN_GGA(s(M), s(N), Val) -> U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.49 ACKERMANN_IN_GGA(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) 14.27/4.49 U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_GGA(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.49 U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGA(M, Val1, Val) 14.27/4.49 U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_GGG(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.49 U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGG(M, Val1, Val) 14.27/4.49 ACKERMANN_IN_GAG(s(M), s(N), Val) -> U2_GAG(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.49 ACKERMANN_IN_GAG(s(M), s(N), Val) -> ACKERMANN_IN_GAA(s(M), N, Val1) 14.27/4.49 ACKERMANN_IN_GAA(s(M), 0, Val) -> U1_GAA(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.49 ACKERMANN_IN_GAA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) 14.27/4.49 ACKERMANN_IN_GAA(s(M), s(N), Val) -> U2_GAA(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.49 ACKERMANN_IN_GAA(s(M), s(N), Val) -> ACKERMANN_IN_GAA(s(M), N, Val1) 14.27/4.49 U2_GAA(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_GAA(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.49 U2_GAA(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAA(M, Val1, Val) 14.27/4.49 U2_GAG(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_GAG(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.49 U2_GAG(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAG(M, Val1, Val) 14.27/4.49 14.27/4.49 The TRS R consists of the following rules: 14.27/4.49 14.27/4.49 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.49 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.49 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.49 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.49 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.49 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.49 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.49 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.49 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.49 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.49 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.49 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.49 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.49 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.49 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.49 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.49 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.49 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.49 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.49 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.49 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.49 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.49 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.49 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.49 14.27/4.49 The argument filtering Pi contains the following mapping: 14.27/4.49 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.49 14.27/4.49 0 = 0 14.27/4.49 14.27/4.49 s(x1) = s(x1) 14.27/4.49 14.27/4.49 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag 14.27/4.49 14.27/4.49 U1_gag(x1, x2, x3) = U1_gag(x3) 14.27/4.49 14.27/4.49 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.49 14.27/4.49 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg 14.27/4.49 14.27/4.49 U1_ggg(x1, x2, x3) = U1_ggg(x3) 14.27/4.49 14.27/4.49 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x3, x4) 14.27/4.49 14.27/4.49 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.49 14.27/4.49 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) 14.27/4.49 14.27/4.49 U1_gga(x1, x2, x3) = U1_gga(x3) 14.27/4.49 14.27/4.49 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) 14.27/4.49 14.27/4.49 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 14.27/4.49 14.27/4.49 U3_ggg(x1, x2, x3, x4) = U3_ggg(x4) 14.27/4.49 14.27/4.49 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.49 14.27/4.49 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.49 14.27/4.49 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa 14.27/4.49 14.27/4.49 U1_gaa(x1, x2, x3) = U1_gaa(x3) 14.27/4.49 14.27/4.49 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.49 14.27/4.49 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 14.27/4.49 14.27/4.49 U3_gag(x1, x2, x3, x4) = U3_gag(x4) 14.27/4.49 14.27/4.49 ACKERMANN_IN_GAG(x1, x2, x3) = ACKERMANN_IN_GAG(x1, x3) 14.27/4.49 14.27/4.49 U1_GAG(x1, x2, x3) = U1_GAG(x3) 14.27/4.49 14.27/4.49 ACKERMANN_IN_GGG(x1, x2, x3) = ACKERMANN_IN_GGG(x1, x2, x3) 14.27/4.49 14.27/4.49 U1_GGG(x1, x2, x3) = U1_GGG(x3) 14.27/4.49 14.27/4.49 U2_GGG(x1, x2, x3, x4) = U2_GGG(x1, x3, x4) 14.27/4.49 14.27/4.49 ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2) 14.27/4.49 14.27/4.49 U1_GGA(x1, x2, x3) = U1_GGA(x3) 14.27/4.49 14.27/4.49 U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x4) 14.27/4.49 14.27/4.49 U3_GGA(x1, x2, x3, x4) = U3_GGA(x4) 14.27/4.49 14.27/4.49 U3_GGG(x1, x2, x3, x4) = U3_GGG(x4) 14.27/4.49 14.27/4.49 U2_GAG(x1, x2, x3, x4) = U2_GAG(x1, x3, x4) 14.27/4.49 14.27/4.49 ACKERMANN_IN_GAA(x1, x2, x3) = ACKERMANN_IN_GAA(x1) 14.27/4.49 14.27/4.49 U1_GAA(x1, x2, x3) = U1_GAA(x3) 14.27/4.49 14.27/4.49 U2_GAA(x1, x2, x3, x4) = U2_GAA(x1, x4) 14.27/4.49 14.27/4.49 U3_GAA(x1, x2, x3, x4) = U3_GAA(x4) 14.27/4.49 14.27/4.49 U3_GAG(x1, x2, x3, x4) = U3_GAG(x4) 14.27/4.49 14.27/4.49 14.27/4.49 We have to consider all (P,R,Pi)-chains 14.27/4.49 ---------------------------------------- 14.27/4.49 14.27/4.49 (4) 14.27/4.49 Obligation: 14.27/4.49 Pi DP problem: 14.27/4.49 The TRS P consists of the following rules: 14.27/4.49 14.27/4.49 ACKERMANN_IN_GAG(s(M), 0, Val) -> U1_GAG(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.49 ACKERMANN_IN_GAG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.49 ACKERMANN_IN_GGG(s(M), 0, Val) -> U1_GGG(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.49 ACKERMANN_IN_GGG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.49 ACKERMANN_IN_GGG(s(M), s(N), Val) -> U2_GGG(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.49 ACKERMANN_IN_GGG(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) 14.27/4.49 ACKERMANN_IN_GGA(s(M), 0, Val) -> U1_GGA(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.49 ACKERMANN_IN_GGA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) 14.27/4.49 ACKERMANN_IN_GGA(s(M), s(N), Val) -> U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.49 ACKERMANN_IN_GGA(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) 14.27/4.49 U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_GGA(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.49 U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGA(M, Val1, Val) 14.27/4.49 U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_GGG(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.49 U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGG(M, Val1, Val) 14.27/4.49 ACKERMANN_IN_GAG(s(M), s(N), Val) -> U2_GAG(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.49 ACKERMANN_IN_GAG(s(M), s(N), Val) -> ACKERMANN_IN_GAA(s(M), N, Val1) 14.27/4.49 ACKERMANN_IN_GAA(s(M), 0, Val) -> U1_GAA(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.49 ACKERMANN_IN_GAA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) 14.27/4.49 ACKERMANN_IN_GAA(s(M), s(N), Val) -> U2_GAA(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.49 ACKERMANN_IN_GAA(s(M), s(N), Val) -> ACKERMANN_IN_GAA(s(M), N, Val1) 14.27/4.49 U2_GAA(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_GAA(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.49 U2_GAA(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAA(M, Val1, Val) 14.27/4.49 U2_GAG(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_GAG(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.49 U2_GAG(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAG(M, Val1, Val) 14.27/4.49 14.27/4.49 The TRS R consists of the following rules: 14.27/4.49 14.27/4.49 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.49 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.49 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.49 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.49 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.49 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.49 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.49 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.49 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.49 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.49 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.49 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.49 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.49 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.49 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.49 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.49 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.49 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.49 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.49 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.49 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.49 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.49 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.49 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.49 14.27/4.49 The argument filtering Pi contains the following mapping: 14.27/4.49 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.49 14.27/4.49 0 = 0 14.27/4.49 14.27/4.49 s(x1) = s(x1) 14.27/4.49 14.27/4.49 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag 14.27/4.49 14.27/4.49 U1_gag(x1, x2, x3) = U1_gag(x3) 14.27/4.49 14.27/4.49 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.49 14.27/4.49 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg 14.27/4.49 14.27/4.49 U1_ggg(x1, x2, x3) = U1_ggg(x3) 14.27/4.49 14.27/4.49 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x3, x4) 14.27/4.50 14.27/4.50 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.50 14.27/4.50 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) 14.27/4.50 14.27/4.50 U1_gga(x1, x2, x3) = U1_gga(x3) 14.27/4.50 14.27/4.50 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) 14.27/4.50 14.27/4.50 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 14.27/4.50 14.27/4.50 U3_ggg(x1, x2, x3, x4) = U3_ggg(x4) 14.27/4.50 14.27/4.50 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.50 14.27/4.50 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.50 14.27/4.50 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa 14.27/4.50 14.27/4.50 U1_gaa(x1, x2, x3) = U1_gaa(x3) 14.27/4.50 14.27/4.50 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.50 14.27/4.50 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 14.27/4.50 14.27/4.50 U3_gag(x1, x2, x3, x4) = U3_gag(x4) 14.27/4.50 14.27/4.50 ACKERMANN_IN_GAG(x1, x2, x3) = ACKERMANN_IN_GAG(x1, x3) 14.27/4.50 14.27/4.50 U1_GAG(x1, x2, x3) = U1_GAG(x3) 14.27/4.50 14.27/4.50 ACKERMANN_IN_GGG(x1, x2, x3) = ACKERMANN_IN_GGG(x1, x2, x3) 14.27/4.50 14.27/4.50 U1_GGG(x1, x2, x3) = U1_GGG(x3) 14.27/4.50 14.27/4.50 U2_GGG(x1, x2, x3, x4) = U2_GGG(x1, x3, x4) 14.27/4.50 14.27/4.50 ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2) 14.27/4.50 14.27/4.50 U1_GGA(x1, x2, x3) = U1_GGA(x3) 14.27/4.50 14.27/4.50 U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x4) 14.27/4.50 14.27/4.50 U3_GGA(x1, x2, x3, x4) = U3_GGA(x4) 14.27/4.50 14.27/4.50 U3_GGG(x1, x2, x3, x4) = U3_GGG(x4) 14.27/4.50 14.27/4.50 U2_GAG(x1, x2, x3, x4) = U2_GAG(x1, x3, x4) 14.27/4.50 14.27/4.50 ACKERMANN_IN_GAA(x1, x2, x3) = ACKERMANN_IN_GAA(x1) 14.27/4.50 14.27/4.50 U1_GAA(x1, x2, x3) = U1_GAA(x3) 14.27/4.50 14.27/4.50 U2_GAA(x1, x2, x3, x4) = U2_GAA(x1, x4) 14.27/4.50 14.27/4.50 U3_GAA(x1, x2, x3, x4) = U3_GAA(x4) 14.27/4.50 14.27/4.50 U3_GAG(x1, x2, x3, x4) = U3_GAG(x4) 14.27/4.50 14.27/4.50 14.27/4.50 We have to consider all (P,R,Pi)-chains 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (5) DependencyGraphProof (EQUIVALENT) 14.27/4.50 The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 12 less nodes. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (6) 14.27/4.50 Complex Obligation (AND) 14.27/4.50 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (7) 14.27/4.50 Obligation: 14.27/4.50 Pi DP problem: 14.27/4.50 The TRS P consists of the following rules: 14.27/4.50 14.27/4.50 ACKERMANN_IN_GGA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) 14.27/4.50 ACKERMANN_IN_GGA(s(M), s(N), Val) -> U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGA(M, Val1, Val) 14.27/4.50 ACKERMANN_IN_GGA(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) 14.27/4.50 14.27/4.50 The TRS R consists of the following rules: 14.27/4.50 14.27/4.50 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.50 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.50 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.50 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.50 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.50 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.50 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.50 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.50 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.50 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.50 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.50 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.50 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.50 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.50 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.50 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.50 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.50 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.50 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.50 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.50 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.50 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.50 14.27/4.50 The argument filtering Pi contains the following mapping: 14.27/4.50 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.50 14.27/4.50 0 = 0 14.27/4.50 14.27/4.50 s(x1) = s(x1) 14.27/4.50 14.27/4.50 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag 14.27/4.50 14.27/4.50 U1_gag(x1, x2, x3) = U1_gag(x3) 14.27/4.50 14.27/4.50 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.50 14.27/4.50 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg 14.27/4.50 14.27/4.50 U1_ggg(x1, x2, x3) = U1_ggg(x3) 14.27/4.50 14.27/4.50 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x3, x4) 14.27/4.50 14.27/4.50 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.50 14.27/4.50 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) 14.27/4.50 14.27/4.50 U1_gga(x1, x2, x3) = U1_gga(x3) 14.27/4.50 14.27/4.50 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) 14.27/4.50 14.27/4.50 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 14.27/4.50 14.27/4.50 U3_ggg(x1, x2, x3, x4) = U3_ggg(x4) 14.27/4.50 14.27/4.50 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.50 14.27/4.50 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.50 14.27/4.50 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa 14.27/4.50 14.27/4.50 U1_gaa(x1, x2, x3) = U1_gaa(x3) 14.27/4.50 14.27/4.50 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.50 14.27/4.50 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 14.27/4.50 14.27/4.50 U3_gag(x1, x2, x3, x4) = U3_gag(x4) 14.27/4.50 14.27/4.50 ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2) 14.27/4.50 14.27/4.50 U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x4) 14.27/4.50 14.27/4.50 14.27/4.50 We have to consider all (P,R,Pi)-chains 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (8) UsableRulesProof (EQUIVALENT) 14.27/4.50 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (9) 14.27/4.50 Obligation: 14.27/4.50 Pi DP problem: 14.27/4.50 The TRS P consists of the following rules: 14.27/4.50 14.27/4.50 ACKERMANN_IN_GGA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) 14.27/4.50 ACKERMANN_IN_GGA(s(M), s(N), Val) -> U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGA(M, Val1, Val) 14.27/4.50 ACKERMANN_IN_GGA(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) 14.27/4.50 14.27/4.50 The TRS R consists of the following rules: 14.27/4.50 14.27/4.50 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.50 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.50 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.50 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.50 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.50 14.27/4.50 The argument filtering Pi contains the following mapping: 14.27/4.50 0 = 0 14.27/4.50 14.27/4.50 s(x1) = s(x1) 14.27/4.50 14.27/4.50 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.50 14.27/4.50 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) 14.27/4.50 14.27/4.50 U1_gga(x1, x2, x3) = U1_gga(x3) 14.27/4.50 14.27/4.50 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) 14.27/4.50 14.27/4.50 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 14.27/4.50 14.27/4.50 ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2) 14.27/4.50 14.27/4.50 U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x4) 14.27/4.50 14.27/4.50 14.27/4.50 We have to consider all (P,R,Pi)-chains 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (10) PiDPToQDPProof (SOUND) 14.27/4.50 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (11) 14.27/4.50 Obligation: 14.27/4.50 Q DP problem: 14.27/4.50 The TRS P consists of the following rules: 14.27/4.50 14.27/4.50 ACKERMANN_IN_GGA(s(M), 0) -> ACKERMANN_IN_GGA(M, s(0)) 14.27/4.50 ACKERMANN_IN_GGA(s(M), s(N)) -> U2_GGA(M, ackermann_in_gga(s(M), N)) 14.27/4.50 U2_GGA(M, ackermann_out_gga(Val1)) -> ACKERMANN_IN_GGA(M, Val1) 14.27/4.50 ACKERMANN_IN_GGA(s(M), s(N)) -> ACKERMANN_IN_GGA(s(M), N) 14.27/4.50 14.27/4.50 The TRS R consists of the following rules: 14.27/4.50 14.27/4.50 ackermann_in_gga(s(M), 0) -> U1_gga(ackermann_in_gga(M, s(0))) 14.27/4.50 ackermann_in_gga(s(M), s(N)) -> U2_gga(M, ackermann_in_gga(s(M), N)) 14.27/4.50 U1_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) 14.27/4.50 U2_gga(M, ackermann_out_gga(Val1)) -> U3_gga(ackermann_in_gga(M, Val1)) 14.27/4.50 ackermann_in_gga(0, N) -> ackermann_out_gga(s(N)) 14.27/4.50 U3_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) 14.27/4.50 14.27/4.50 The set Q consists of the following terms: 14.27/4.50 14.27/4.50 ackermann_in_gga(x0, x1) 14.27/4.50 U1_gga(x0) 14.27/4.50 U2_gga(x0, x1) 14.27/4.50 U3_gga(x0) 14.27/4.50 14.27/4.50 We have to consider all (P,Q,R)-chains. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (12) QDPSizeChangeProof (EQUIVALENT) 14.27/4.50 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. 14.27/4.50 14.27/4.50 From the DPs we obtained the following set of size-change graphs: 14.27/4.50 *ACKERMANN_IN_GGA(s(M), s(N)) -> ACKERMANN_IN_GGA(s(M), N) 14.27/4.50 The graph contains the following edges 1 >= 1, 2 > 2 14.27/4.50 14.27/4.50 14.27/4.50 *ACKERMANN_IN_GGA(s(M), s(N)) -> U2_GGA(M, ackermann_in_gga(s(M), N)) 14.27/4.50 The graph contains the following edges 1 > 1 14.27/4.50 14.27/4.50 14.27/4.50 *U2_GGA(M, ackermann_out_gga(Val1)) -> ACKERMANN_IN_GGA(M, Val1) 14.27/4.50 The graph contains the following edges 1 >= 1, 2 > 2 14.27/4.50 14.27/4.50 14.27/4.50 *ACKERMANN_IN_GGA(s(M), 0) -> ACKERMANN_IN_GGA(M, s(0)) 14.27/4.50 The graph contains the following edges 1 > 1 14.27/4.50 14.27/4.50 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (13) 14.27/4.50 YES 14.27/4.50 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (14) 14.27/4.50 Obligation: 14.27/4.50 Pi DP problem: 14.27/4.50 The TRS P consists of the following rules: 14.27/4.50 14.27/4.50 ACKERMANN_IN_GAA(s(M), s(N), Val) -> U2_GAA(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.50 U2_GAA(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAA(M, Val1, Val) 14.27/4.50 ACKERMANN_IN_GAA(s(M), s(N), Val) -> ACKERMANN_IN_GAA(s(M), N, Val1) 14.27/4.50 14.27/4.50 The TRS R consists of the following rules: 14.27/4.50 14.27/4.50 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.50 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.50 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.50 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.50 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.50 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.50 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.50 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.50 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.50 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.50 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.50 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.50 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.50 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.50 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.50 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.50 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.50 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.50 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.50 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.50 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.50 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.50 14.27/4.50 The argument filtering Pi contains the following mapping: 14.27/4.50 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.50 14.27/4.50 0 = 0 14.27/4.50 14.27/4.50 s(x1) = s(x1) 14.27/4.50 14.27/4.50 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag 14.27/4.50 14.27/4.50 U1_gag(x1, x2, x3) = U1_gag(x3) 14.27/4.50 14.27/4.50 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.50 14.27/4.50 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg 14.27/4.50 14.27/4.50 U1_ggg(x1, x2, x3) = U1_ggg(x3) 14.27/4.50 14.27/4.50 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x3, x4) 14.27/4.50 14.27/4.50 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.50 14.27/4.50 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) 14.27/4.50 14.27/4.50 U1_gga(x1, x2, x3) = U1_gga(x3) 14.27/4.50 14.27/4.50 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) 14.27/4.50 14.27/4.50 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 14.27/4.50 14.27/4.50 U3_ggg(x1, x2, x3, x4) = U3_ggg(x4) 14.27/4.50 14.27/4.50 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.50 14.27/4.50 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.50 14.27/4.50 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa 14.27/4.50 14.27/4.50 U1_gaa(x1, x2, x3) = U1_gaa(x3) 14.27/4.50 14.27/4.50 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.50 14.27/4.50 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 14.27/4.50 14.27/4.50 U3_gag(x1, x2, x3, x4) = U3_gag(x4) 14.27/4.50 14.27/4.50 ACKERMANN_IN_GAA(x1, x2, x3) = ACKERMANN_IN_GAA(x1) 14.27/4.50 14.27/4.50 U2_GAA(x1, x2, x3, x4) = U2_GAA(x1, x4) 14.27/4.50 14.27/4.50 14.27/4.50 We have to consider all (P,R,Pi)-chains 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (15) UsableRulesProof (EQUIVALENT) 14.27/4.50 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (16) 14.27/4.50 Obligation: 14.27/4.50 Pi DP problem: 14.27/4.50 The TRS P consists of the following rules: 14.27/4.50 14.27/4.50 ACKERMANN_IN_GAA(s(M), s(N), Val) -> U2_GAA(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.50 U2_GAA(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAA(M, Val1, Val) 14.27/4.50 ACKERMANN_IN_GAA(s(M), s(N), Val) -> ACKERMANN_IN_GAA(s(M), N, Val1) 14.27/4.50 14.27/4.50 The TRS R consists of the following rules: 14.27/4.50 14.27/4.50 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.50 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.50 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.50 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.50 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.50 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.50 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.50 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.50 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.50 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.50 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.50 14.27/4.50 The argument filtering Pi contains the following mapping: 14.27/4.50 0 = 0 14.27/4.50 14.27/4.50 s(x1) = s(x1) 14.27/4.50 14.27/4.50 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.50 14.27/4.50 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) 14.27/4.50 14.27/4.50 U1_gga(x1, x2, x3) = U1_gga(x3) 14.27/4.50 14.27/4.50 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) 14.27/4.50 14.27/4.50 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 14.27/4.50 14.27/4.50 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.50 14.27/4.50 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa 14.27/4.50 14.27/4.50 U1_gaa(x1, x2, x3) = U1_gaa(x3) 14.27/4.50 14.27/4.50 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.50 14.27/4.50 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 14.27/4.50 14.27/4.50 ACKERMANN_IN_GAA(x1, x2, x3) = ACKERMANN_IN_GAA(x1) 14.27/4.50 14.27/4.50 U2_GAA(x1, x2, x3, x4) = U2_GAA(x1, x4) 14.27/4.50 14.27/4.50 14.27/4.50 We have to consider all (P,R,Pi)-chains 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (17) PiDPToQDPProof (SOUND) 14.27/4.50 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (18) 14.27/4.50 Obligation: 14.27/4.50 Q DP problem: 14.27/4.50 The TRS P consists of the following rules: 14.27/4.50 14.27/4.50 ACKERMANN_IN_GAA(s(M)) -> U2_GAA(M, ackermann_in_gaa(s(M))) 14.27/4.50 U2_GAA(M, ackermann_out_gaa) -> ACKERMANN_IN_GAA(M) 14.27/4.50 ACKERMANN_IN_GAA(s(M)) -> ACKERMANN_IN_GAA(s(M)) 14.27/4.50 14.27/4.50 The TRS R consists of the following rules: 14.27/4.50 14.27/4.50 ackermann_in_gaa(s(M)) -> U1_gaa(ackermann_in_gga(M, s(0))) 14.27/4.50 ackermann_in_gaa(s(M)) -> U2_gaa(M, ackermann_in_gaa(s(M))) 14.27/4.50 U1_gaa(ackermann_out_gga(Val)) -> ackermann_out_gaa 14.27/4.50 U2_gaa(M, ackermann_out_gaa) -> U3_gaa(ackermann_in_gaa(M)) 14.27/4.50 ackermann_in_gga(0, N) -> ackermann_out_gga(s(N)) 14.27/4.50 ackermann_in_gga(s(M), s(N)) -> U2_gga(M, ackermann_in_gga(s(M), N)) 14.27/4.50 U3_gaa(ackermann_out_gaa) -> ackermann_out_gaa 14.27/4.50 U2_gga(M, ackermann_out_gga(Val1)) -> U3_gga(ackermann_in_gga(M, Val1)) 14.27/4.50 ackermann_in_gaa(0) -> ackermann_out_gaa 14.27/4.50 ackermann_in_gga(s(M), 0) -> U1_gga(ackermann_in_gga(M, s(0))) 14.27/4.50 U3_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) 14.27/4.50 U1_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) 14.27/4.50 14.27/4.50 The set Q consists of the following terms: 14.27/4.50 14.27/4.50 ackermann_in_gaa(x0) 14.27/4.50 U1_gaa(x0) 14.27/4.50 U2_gaa(x0, x1) 14.27/4.50 ackermann_in_gga(x0, x1) 14.27/4.50 U3_gaa(x0) 14.27/4.50 U2_gga(x0, x1) 14.27/4.50 U3_gga(x0) 14.27/4.50 U1_gga(x0) 14.27/4.50 14.27/4.50 We have to consider all (P,Q,R)-chains. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (19) QDPOrderProof (EQUIVALENT) 14.27/4.50 We use the reduction pair processor [LPAR04,JAR06]. 14.27/4.50 14.27/4.50 14.27/4.50 The following pairs can be oriented strictly and are deleted. 14.27/4.50 14.27/4.50 ACKERMANN_IN_GAA(s(M)) -> U2_GAA(M, ackermann_in_gaa(s(M))) 14.27/4.50 U2_GAA(M, ackermann_out_gaa) -> ACKERMANN_IN_GAA(M) 14.27/4.50 The remaining pairs can at least be oriented weakly. 14.27/4.50 Used ordering: Combined order from the following AFS and order. 14.27/4.50 ACKERMANN_IN_GAA(x1) = ACKERMANN_IN_GAA(x1) 14.27/4.50 14.27/4.50 s(x1) = s(x1) 14.27/4.50 14.27/4.50 U2_GAA(x1, x2) = U2_GAA(x1, x2) 14.27/4.50 14.27/4.50 ackermann_in_gaa(x1) = ackermann_in_gaa(x1) 14.27/4.50 14.27/4.50 ackermann_out_gaa = ackermann_out_gaa 14.27/4.50 14.27/4.50 U1_gaa(x1) = U1_gaa(x1) 14.27/4.50 14.27/4.50 ackermann_in_gga(x1, x2) = x2 14.27/4.50 14.27/4.50 0 = 0 14.27/4.50 14.27/4.50 U2_gaa(x1, x2) = U2_gaa(x1) 14.27/4.50 14.27/4.50 U3_gaa(x1) = x1 14.27/4.50 14.27/4.50 ackermann_out_gga(x1) = ackermann_out_gga 14.27/4.50 14.27/4.50 U2_gga(x1, x2) = U2_gga(x2) 14.27/4.50 14.27/4.50 U1_gga(x1) = U1_gga 14.27/4.50 14.27/4.50 U3_gga(x1) = U3_gga 14.27/4.50 14.27/4.50 14.27/4.50 Recursive path order with status [RPO]. 14.27/4.50 Quasi-Precedence: s_1 > [ACKERMANN_IN_GAA_1, U2_GAA_2] > [ackermann_in_gaa_1, ackermann_out_gaa, U1_gaa_1, U2_gaa_1] > [0, ackermann_out_gga, U1_gga, U3_gga] 14.27/4.50 s_1 > U2_gga_1 > [0, ackermann_out_gga, U1_gga, U3_gga] 14.27/4.50 14.27/4.50 Status: ACKERMANN_IN_GAA_1: [1] 14.27/4.50 s_1: [1] 14.27/4.50 U2_GAA_2: [1,2] 14.27/4.50 ackermann_in_gaa_1: multiset status 14.27/4.50 ackermann_out_gaa: multiset status 14.27/4.50 U1_gaa_1: multiset status 14.27/4.50 0: multiset status 14.27/4.50 U2_gaa_1: multiset status 14.27/4.50 ackermann_out_gga: multiset status 14.27/4.50 U2_gga_1: multiset status 14.27/4.50 U1_gga: multiset status 14.27/4.50 U3_gga: multiset status 14.27/4.50 14.27/4.50 14.27/4.50 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 14.27/4.50 14.27/4.50 ackermann_in_gaa(s(M)) -> U1_gaa(ackermann_in_gga(M, s(0))) 14.27/4.50 ackermann_in_gaa(s(M)) -> U2_gaa(M, ackermann_in_gaa(s(M))) 14.27/4.50 U2_gaa(M, ackermann_out_gaa) -> U3_gaa(ackermann_in_gaa(M)) 14.27/4.50 ackermann_in_gaa(0) -> ackermann_out_gaa 14.27/4.50 U3_gaa(ackermann_out_gaa) -> ackermann_out_gaa 14.27/4.50 ackermann_in_gga(0, N) -> ackermann_out_gga(s(N)) 14.27/4.50 ackermann_in_gga(s(M), s(N)) -> U2_gga(M, ackermann_in_gga(s(M), N)) 14.27/4.50 U1_gaa(ackermann_out_gga(Val)) -> ackermann_out_gaa 14.27/4.50 ackermann_in_gga(s(M), 0) -> U1_gga(ackermann_in_gga(M, s(0))) 14.27/4.50 U1_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) 14.27/4.50 U2_gga(M, ackermann_out_gga(Val1)) -> U3_gga(ackermann_in_gga(M, Val1)) 14.27/4.50 U3_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) 14.27/4.50 14.27/4.50 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (20) 14.27/4.50 Obligation: 14.27/4.50 Q DP problem: 14.27/4.50 The TRS P consists of the following rules: 14.27/4.50 14.27/4.50 ACKERMANN_IN_GAA(s(M)) -> ACKERMANN_IN_GAA(s(M)) 14.27/4.50 14.27/4.50 The TRS R consists of the following rules: 14.27/4.50 14.27/4.50 ackermann_in_gaa(s(M)) -> U1_gaa(ackermann_in_gga(M, s(0))) 14.27/4.50 ackermann_in_gaa(s(M)) -> U2_gaa(M, ackermann_in_gaa(s(M))) 14.27/4.50 U1_gaa(ackermann_out_gga(Val)) -> ackermann_out_gaa 14.27/4.50 U2_gaa(M, ackermann_out_gaa) -> U3_gaa(ackermann_in_gaa(M)) 14.27/4.50 ackermann_in_gga(0, N) -> ackermann_out_gga(s(N)) 14.27/4.50 ackermann_in_gga(s(M), s(N)) -> U2_gga(M, ackermann_in_gga(s(M), N)) 14.27/4.50 U3_gaa(ackermann_out_gaa) -> ackermann_out_gaa 14.27/4.50 U2_gga(M, ackermann_out_gga(Val1)) -> U3_gga(ackermann_in_gga(M, Val1)) 14.27/4.50 ackermann_in_gaa(0) -> ackermann_out_gaa 14.27/4.50 ackermann_in_gga(s(M), 0) -> U1_gga(ackermann_in_gga(M, s(0))) 14.27/4.50 U3_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) 14.27/4.50 U1_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) 14.27/4.50 14.27/4.50 The set Q consists of the following terms: 14.27/4.50 14.27/4.50 ackermann_in_gaa(x0) 14.27/4.50 U1_gaa(x0) 14.27/4.50 U2_gaa(x0, x1) 14.27/4.50 ackermann_in_gga(x0, x1) 14.27/4.50 U3_gaa(x0) 14.27/4.50 U2_gga(x0, x1) 14.27/4.50 U3_gga(x0) 14.27/4.50 U1_gga(x0) 14.27/4.50 14.27/4.50 We have to consider all (P,Q,R)-chains. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (21) UsableRulesProof (EQUIVALENT) 14.27/4.50 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. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (22) 14.27/4.50 Obligation: 14.27/4.50 Q DP problem: 14.27/4.50 The TRS P consists of the following rules: 14.27/4.50 14.27/4.50 ACKERMANN_IN_GAA(s(M)) -> ACKERMANN_IN_GAA(s(M)) 14.27/4.50 14.27/4.50 R is empty. 14.27/4.50 The set Q consists of the following terms: 14.27/4.50 14.27/4.50 ackermann_in_gaa(x0) 14.27/4.50 U1_gaa(x0) 14.27/4.50 U2_gaa(x0, x1) 14.27/4.50 ackermann_in_gga(x0, x1) 14.27/4.50 U3_gaa(x0) 14.27/4.50 U2_gga(x0, x1) 14.27/4.50 U3_gga(x0) 14.27/4.50 U1_gga(x0) 14.27/4.50 14.27/4.50 We have to consider all (P,Q,R)-chains. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (23) QReductionProof (EQUIVALENT) 14.27/4.50 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 14.27/4.50 14.27/4.50 ackermann_in_gaa(x0) 14.27/4.50 U1_gaa(x0) 14.27/4.50 U2_gaa(x0, x1) 14.27/4.50 ackermann_in_gga(x0, x1) 14.27/4.50 U3_gaa(x0) 14.27/4.50 U2_gga(x0, x1) 14.27/4.50 U3_gga(x0) 14.27/4.50 U1_gga(x0) 14.27/4.50 14.27/4.50 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (24) 14.27/4.50 Obligation: 14.27/4.50 Q DP problem: 14.27/4.50 The TRS P consists of the following rules: 14.27/4.50 14.27/4.50 ACKERMANN_IN_GAA(s(M)) -> ACKERMANN_IN_GAA(s(M)) 14.27/4.50 14.27/4.50 R is empty. 14.27/4.50 Q is empty. 14.27/4.50 We have to consider all (P,Q,R)-chains. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (25) NonTerminationLoopProof (COMPLETE) 14.27/4.50 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 14.27/4.50 Found a loop by semiunifying a rule from P directly. 14.27/4.50 14.27/4.50 s = ACKERMANN_IN_GAA(s(M)) evaluates to t =ACKERMANN_IN_GAA(s(M)) 14.27/4.50 14.27/4.50 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 14.27/4.50 * Matcher: [ ] 14.27/4.50 * Semiunifier: [ ] 14.27/4.50 14.27/4.50 -------------------------------------------------------------------------------- 14.27/4.50 Rewriting sequence 14.27/4.50 14.27/4.50 The DP semiunifies directly so there is only one rewrite step from ACKERMANN_IN_GAA(s(M)) to ACKERMANN_IN_GAA(s(M)). 14.27/4.50 14.27/4.50 14.27/4.50 14.27/4.50 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (26) 14.27/4.50 NO 14.27/4.50 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (27) 14.27/4.50 Obligation: 14.27/4.50 Pi DP problem: 14.27/4.50 The TRS P consists of the following rules: 14.27/4.50 14.27/4.50 ACKERMANN_IN_GGG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.50 ACKERMANN_IN_GGG(s(M), s(N), Val) -> U2_GGG(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGG(M, Val1, Val) 14.27/4.50 14.27/4.50 The TRS R consists of the following rules: 14.27/4.50 14.27/4.50 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.50 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.50 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.50 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.50 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.50 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.50 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.50 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.50 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.50 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.50 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.50 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.50 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.50 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.50 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.50 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.50 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.50 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.50 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.50 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.50 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.50 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.50 14.27/4.50 The argument filtering Pi contains the following mapping: 14.27/4.50 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.50 14.27/4.50 0 = 0 14.27/4.50 14.27/4.50 s(x1) = s(x1) 14.27/4.50 14.27/4.50 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag 14.27/4.50 14.27/4.50 U1_gag(x1, x2, x3) = U1_gag(x3) 14.27/4.50 14.27/4.50 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.50 14.27/4.50 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg 14.27/4.50 14.27/4.50 U1_ggg(x1, x2, x3) = U1_ggg(x3) 14.27/4.50 14.27/4.50 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x3, x4) 14.27/4.50 14.27/4.50 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.50 14.27/4.50 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) 14.27/4.50 14.27/4.50 U1_gga(x1, x2, x3) = U1_gga(x3) 14.27/4.50 14.27/4.50 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) 14.27/4.50 14.27/4.50 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 14.27/4.50 14.27/4.50 U3_ggg(x1, x2, x3, x4) = U3_ggg(x4) 14.27/4.50 14.27/4.50 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.50 14.27/4.50 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.50 14.27/4.50 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa 14.27/4.50 14.27/4.50 U1_gaa(x1, x2, x3) = U1_gaa(x3) 14.27/4.50 14.27/4.50 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.50 14.27/4.50 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 14.27/4.50 14.27/4.50 U3_gag(x1, x2, x3, x4) = U3_gag(x4) 14.27/4.50 14.27/4.50 ACKERMANN_IN_GGG(x1, x2, x3) = ACKERMANN_IN_GGG(x1, x2, x3) 14.27/4.50 14.27/4.50 U2_GGG(x1, x2, x3, x4) = U2_GGG(x1, x3, x4) 14.27/4.50 14.27/4.50 14.27/4.50 We have to consider all (P,R,Pi)-chains 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (28) UsableRulesProof (EQUIVALENT) 14.27/4.50 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (29) 14.27/4.50 Obligation: 14.27/4.50 Pi DP problem: 14.27/4.50 The TRS P consists of the following rules: 14.27/4.50 14.27/4.50 ACKERMANN_IN_GGG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.50 ACKERMANN_IN_GGG(s(M), s(N), Val) -> U2_GGG(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGG(M, Val1, Val) 14.27/4.50 14.27/4.50 The TRS R consists of the following rules: 14.27/4.50 14.27/4.50 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.50 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.50 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.50 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.50 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.50 14.27/4.50 The argument filtering Pi contains the following mapping: 14.27/4.50 0 = 0 14.27/4.50 14.27/4.50 s(x1) = s(x1) 14.27/4.50 14.27/4.50 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.50 14.27/4.50 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) 14.27/4.50 14.27/4.50 U1_gga(x1, x2, x3) = U1_gga(x3) 14.27/4.50 14.27/4.50 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) 14.27/4.50 14.27/4.50 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 14.27/4.50 14.27/4.50 ACKERMANN_IN_GGG(x1, x2, x3) = ACKERMANN_IN_GGG(x1, x2, x3) 14.27/4.50 14.27/4.50 U2_GGG(x1, x2, x3, x4) = U2_GGG(x1, x3, x4) 14.27/4.50 14.27/4.50 14.27/4.50 We have to consider all (P,R,Pi)-chains 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (30) PiDPToQDPProof (SOUND) 14.27/4.50 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (31) 14.27/4.50 Obligation: 14.27/4.50 Q DP problem: 14.27/4.50 The TRS P consists of the following rules: 14.27/4.50 14.27/4.50 ACKERMANN_IN_GGG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.50 ACKERMANN_IN_GGG(s(M), s(N), Val) -> U2_GGG(M, Val, ackermann_in_gga(s(M), N)) 14.27/4.50 U2_GGG(M, Val, ackermann_out_gga(Val1)) -> ACKERMANN_IN_GGG(M, Val1, Val) 14.27/4.50 14.27/4.50 The TRS R consists of the following rules: 14.27/4.50 14.27/4.50 ackermann_in_gga(s(M), 0) -> U1_gga(ackermann_in_gga(M, s(0))) 14.27/4.50 ackermann_in_gga(s(M), s(N)) -> U2_gga(M, ackermann_in_gga(s(M), N)) 14.27/4.50 U1_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) 14.27/4.50 U2_gga(M, ackermann_out_gga(Val1)) -> U3_gga(ackermann_in_gga(M, Val1)) 14.27/4.50 ackermann_in_gga(0, N) -> ackermann_out_gga(s(N)) 14.27/4.50 U3_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) 14.27/4.50 14.27/4.50 The set Q consists of the following terms: 14.27/4.50 14.27/4.50 ackermann_in_gga(x0, x1) 14.27/4.50 U1_gga(x0) 14.27/4.50 U2_gga(x0, x1) 14.27/4.50 U3_gga(x0) 14.27/4.50 14.27/4.50 We have to consider all (P,Q,R)-chains. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (32) QDPSizeChangeProof (EQUIVALENT) 14.27/4.50 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. 14.27/4.50 14.27/4.50 From the DPs we obtained the following set of size-change graphs: 14.27/4.50 *ACKERMANN_IN_GGG(s(M), s(N), Val) -> U2_GGG(M, Val, ackermann_in_gga(s(M), N)) 14.27/4.50 The graph contains the following edges 1 > 1, 3 >= 2 14.27/4.50 14.27/4.50 14.27/4.50 *U2_GGG(M, Val, ackermann_out_gga(Val1)) -> ACKERMANN_IN_GGG(M, Val1, Val) 14.27/4.50 The graph contains the following edges 1 >= 1, 3 > 2, 2 >= 3 14.27/4.50 14.27/4.50 14.27/4.50 *ACKERMANN_IN_GGG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.50 The graph contains the following edges 1 > 1, 3 >= 3 14.27/4.50 14.27/4.50 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (33) 14.27/4.50 YES 14.27/4.50 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (34) 14.27/4.50 Obligation: 14.27/4.50 Pi DP problem: 14.27/4.50 The TRS P consists of the following rules: 14.27/4.50 14.27/4.50 ACKERMANN_IN_GAG(s(M), s(N), Val) -> U2_GAG(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.50 U2_GAG(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAG(M, Val1, Val) 14.27/4.50 14.27/4.50 The TRS R consists of the following rules: 14.27/4.50 14.27/4.50 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.50 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.50 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.50 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.50 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.50 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.50 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.50 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.50 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.50 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.50 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.50 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.50 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.50 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.50 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.50 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.50 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.50 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.50 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.50 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.50 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.50 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.50 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.50 14.27/4.50 The argument filtering Pi contains the following mapping: 14.27/4.50 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.50 14.27/4.50 0 = 0 14.27/4.50 14.27/4.50 s(x1) = s(x1) 14.27/4.50 14.27/4.50 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag 14.27/4.50 14.27/4.50 U1_gag(x1, x2, x3) = U1_gag(x3) 14.27/4.50 14.27/4.50 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.50 14.27/4.50 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg 14.27/4.50 14.27/4.50 U1_ggg(x1, x2, x3) = U1_ggg(x3) 14.27/4.50 14.27/4.50 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x3, x4) 14.27/4.50 14.27/4.50 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.50 14.27/4.50 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) 14.27/4.50 14.27/4.50 U1_gga(x1, x2, x3) = U1_gga(x3) 14.27/4.50 14.27/4.50 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) 14.27/4.50 14.27/4.50 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 14.27/4.50 14.27/4.50 U3_ggg(x1, x2, x3, x4) = U3_ggg(x4) 14.27/4.50 14.27/4.50 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.50 14.27/4.50 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.50 14.27/4.50 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa 14.27/4.50 14.27/4.50 U1_gaa(x1, x2, x3) = U1_gaa(x3) 14.27/4.50 14.27/4.50 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.50 14.27/4.50 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 14.27/4.50 14.27/4.50 U3_gag(x1, x2, x3, x4) = U3_gag(x4) 14.27/4.50 14.27/4.50 ACKERMANN_IN_GAG(x1, x2, x3) = ACKERMANN_IN_GAG(x1, x3) 14.27/4.50 14.27/4.50 U2_GAG(x1, x2, x3, x4) = U2_GAG(x1, x3, x4) 14.27/4.50 14.27/4.50 14.27/4.50 We have to consider all (P,R,Pi)-chains 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (35) UsableRulesProof (EQUIVALENT) 14.27/4.50 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 14.27/4.50 ---------------------------------------- 14.27/4.50 14.27/4.50 (36) 14.27/4.50 Obligation: 14.27/4.50 Pi DP problem: 14.27/4.52 The TRS P consists of the following rules: 14.27/4.52 14.27/4.52 ACKERMANN_IN_GAG(s(M), s(N), Val) -> U2_GAG(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 U2_GAG(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAG(M, Val1, Val) 14.27/4.52 14.27/4.52 The TRS R consists of the following rules: 14.27/4.52 14.27/4.52 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.52 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.52 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.52 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.52 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.52 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.52 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.52 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.52 14.27/4.52 The argument filtering Pi contains the following mapping: 14.27/4.52 0 = 0 14.27/4.52 14.27/4.52 s(x1) = s(x1) 14.27/4.52 14.27/4.52 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.52 14.27/4.52 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) 14.27/4.52 14.27/4.52 U1_gga(x1, x2, x3) = U1_gga(x3) 14.27/4.52 14.27/4.52 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) 14.27/4.52 14.27/4.52 U3_gga(x1, x2, x3, x4) = U3_gga(x4) 14.27/4.52 14.27/4.52 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.52 14.27/4.52 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa 14.27/4.52 14.27/4.52 U1_gaa(x1, x2, x3) = U1_gaa(x3) 14.27/4.52 14.27/4.52 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.52 14.27/4.52 U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) 14.27/4.52 14.27/4.52 ACKERMANN_IN_GAG(x1, x2, x3) = ACKERMANN_IN_GAG(x1, x3) 14.27/4.52 14.27/4.52 U2_GAG(x1, x2, x3, x4) = U2_GAG(x1, x3, x4) 14.27/4.52 14.27/4.52 14.27/4.52 We have to consider all (P,R,Pi)-chains 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (37) PiDPToQDPProof (SOUND) 14.27/4.52 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (38) 14.27/4.52 Obligation: 14.27/4.52 Q DP problem: 14.27/4.52 The TRS P consists of the following rules: 14.27/4.52 14.27/4.52 ACKERMANN_IN_GAG(s(M), Val) -> U2_GAG(M, Val, ackermann_in_gaa(s(M))) 14.27/4.52 U2_GAG(M, Val, ackermann_out_gaa) -> ACKERMANN_IN_GAG(M, Val) 14.27/4.52 14.27/4.52 The TRS R consists of the following rules: 14.27/4.52 14.27/4.52 ackermann_in_gaa(s(M)) -> U1_gaa(ackermann_in_gga(M, s(0))) 14.27/4.52 ackermann_in_gaa(s(M)) -> U2_gaa(M, ackermann_in_gaa(s(M))) 14.27/4.52 U1_gaa(ackermann_out_gga(Val)) -> ackermann_out_gaa 14.27/4.52 U2_gaa(M, ackermann_out_gaa) -> U3_gaa(ackermann_in_gaa(M)) 14.27/4.52 ackermann_in_gga(0, N) -> ackermann_out_gga(s(N)) 14.27/4.52 ackermann_in_gga(s(M), s(N)) -> U2_gga(M, ackermann_in_gga(s(M), N)) 14.27/4.52 U3_gaa(ackermann_out_gaa) -> ackermann_out_gaa 14.27/4.52 U2_gga(M, ackermann_out_gga(Val1)) -> U3_gga(ackermann_in_gga(M, Val1)) 14.27/4.52 ackermann_in_gaa(0) -> ackermann_out_gaa 14.27/4.52 ackermann_in_gga(s(M), 0) -> U1_gga(ackermann_in_gga(M, s(0))) 14.27/4.52 U3_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) 14.27/4.52 U1_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) 14.27/4.52 14.27/4.52 The set Q consists of the following terms: 14.27/4.52 14.27/4.52 ackermann_in_gaa(x0) 14.27/4.52 U1_gaa(x0) 14.27/4.52 U2_gaa(x0, x1) 14.27/4.52 ackermann_in_gga(x0, x1) 14.27/4.52 U3_gaa(x0) 14.27/4.52 U2_gga(x0, x1) 14.27/4.52 U3_gga(x0) 14.27/4.52 U1_gga(x0) 14.27/4.52 14.27/4.52 We have to consider all (P,Q,R)-chains. 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (39) QDPSizeChangeProof (EQUIVALENT) 14.27/4.52 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. 14.27/4.52 14.27/4.52 From the DPs we obtained the following set of size-change graphs: 14.27/4.52 *U2_GAG(M, Val, ackermann_out_gaa) -> ACKERMANN_IN_GAG(M, Val) 14.27/4.52 The graph contains the following edges 1 >= 1, 2 >= 2 14.27/4.52 14.27/4.52 14.27/4.52 *ACKERMANN_IN_GAG(s(M), Val) -> U2_GAG(M, Val, ackermann_in_gaa(s(M))) 14.27/4.52 The graph contains the following edges 1 > 1, 2 >= 2 14.27/4.52 14.27/4.52 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (40) 14.27/4.52 YES 14.27/4.52 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (41) PrologToPiTRSProof (SOUND) 14.27/4.52 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 14.27/4.52 14.27/4.52 ackermann_in_3: (b,f,b) (b,b,b) (b,b,f) (b,f,f) 14.27/4.52 14.27/4.52 Transforming Prolog into the following Term Rewriting System: 14.27/4.52 14.27/4.52 Pi-finite rewrite system: 14.27/4.52 The TRS R consists of the following rules: 14.27/4.52 14.27/4.52 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.52 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.52 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.52 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.52 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.52 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.52 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.52 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.52 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.52 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.52 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.52 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.52 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.52 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.52 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.52 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.52 14.27/4.52 The argument filtering Pi contains the following mapping: 14.27/4.52 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.52 14.27/4.52 0 = 0 14.27/4.52 14.27/4.52 s(x1) = s(x1) 14.27/4.52 14.27/4.52 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag(x1, x3) 14.27/4.52 14.27/4.52 U1_gag(x1, x2, x3) = U1_gag(x1, x2, x3) 14.27/4.52 14.27/4.52 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 U1_ggg(x1, x2, x3) = U1_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.52 14.27/4.52 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x1, x2, x3) 14.27/4.52 14.27/4.52 U1_gga(x1, x2, x3) = U1_gga(x1, x3) 14.27/4.52 14.27/4.52 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.52 14.27/4.52 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.52 14.27/4.52 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa(x1) 14.27/4.52 14.27/4.52 U1_gaa(x1, x2, x3) = U1_gaa(x1, x3) 14.27/4.52 14.27/4.52 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.52 14.27/4.52 U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) 14.27/4.52 14.27/4.52 U3_gag(x1, x2, x3, x4) = U3_gag(x1, x3, x4) 14.27/4.52 14.27/4.52 14.27/4.52 14.27/4.52 14.27/4.52 14.27/4.52 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 14.27/4.52 14.27/4.52 14.27/4.52 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (42) 14.27/4.52 Obligation: 14.27/4.52 Pi-finite rewrite system: 14.27/4.52 The TRS R consists of the following rules: 14.27/4.52 14.27/4.52 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.52 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.52 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.52 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.52 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.52 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.52 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.52 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.52 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.52 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.52 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.52 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.52 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.52 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.52 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.52 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.52 14.27/4.52 The argument filtering Pi contains the following mapping: 14.27/4.52 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.52 14.27/4.52 0 = 0 14.27/4.52 14.27/4.52 s(x1) = s(x1) 14.27/4.52 14.27/4.52 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag(x1, x3) 14.27/4.52 14.27/4.52 U1_gag(x1, x2, x3) = U1_gag(x1, x2, x3) 14.27/4.52 14.27/4.52 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 U1_ggg(x1, x2, x3) = U1_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.52 14.27/4.52 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x1, x2, x3) 14.27/4.52 14.27/4.52 U1_gga(x1, x2, x3) = U1_gga(x1, x3) 14.27/4.52 14.27/4.52 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.52 14.27/4.52 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.52 14.27/4.52 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa(x1) 14.27/4.52 14.27/4.52 U1_gaa(x1, x2, x3) = U1_gaa(x1, x3) 14.27/4.52 14.27/4.52 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.52 14.27/4.52 U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) 14.27/4.52 14.27/4.52 U3_gag(x1, x2, x3, x4) = U3_gag(x1, x3, x4) 14.27/4.52 14.27/4.52 14.27/4.52 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (43) DependencyPairsProof (EQUIVALENT) 14.27/4.52 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 14.27/4.52 Pi DP problem: 14.27/4.52 The TRS P consists of the following rules: 14.27/4.52 14.27/4.52 ACKERMANN_IN_GAG(s(M), 0, Val) -> U1_GAG(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ACKERMANN_IN_GAG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.52 ACKERMANN_IN_GGG(s(M), 0, Val) -> U1_GGG(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ACKERMANN_IN_GGG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.52 ACKERMANN_IN_GGG(s(M), s(N), Val) -> U2_GGG(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 ACKERMANN_IN_GGG(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) 14.27/4.52 ACKERMANN_IN_GGA(s(M), 0, Val) -> U1_GGA(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 ACKERMANN_IN_GGA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) 14.27/4.52 ACKERMANN_IN_GGA(s(M), s(N), Val) -> U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 ACKERMANN_IN_GGA(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) 14.27/4.52 U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_GGA(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.52 U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGA(M, Val1, Val) 14.27/4.52 U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_GGG(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.52 U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGG(M, Val1, Val) 14.27/4.52 ACKERMANN_IN_GAG(s(M), s(N), Val) -> U2_GAG(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 ACKERMANN_IN_GAG(s(M), s(N), Val) -> ACKERMANN_IN_GAA(s(M), N, Val1) 14.27/4.52 ACKERMANN_IN_GAA(s(M), 0, Val) -> U1_GAA(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 ACKERMANN_IN_GAA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) 14.27/4.52 ACKERMANN_IN_GAA(s(M), s(N), Val) -> U2_GAA(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 ACKERMANN_IN_GAA(s(M), s(N), Val) -> ACKERMANN_IN_GAA(s(M), N, Val1) 14.27/4.52 U2_GAA(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_GAA(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.52 U2_GAA(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAA(M, Val1, Val) 14.27/4.52 U2_GAG(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_GAG(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.52 U2_GAG(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAG(M, Val1, Val) 14.27/4.52 14.27/4.52 The TRS R consists of the following rules: 14.27/4.52 14.27/4.52 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.52 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.52 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.52 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.52 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.52 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.52 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.52 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.52 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.52 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.52 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.52 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.52 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.52 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.52 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.52 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.52 14.27/4.52 The argument filtering Pi contains the following mapping: 14.27/4.52 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.52 14.27/4.52 0 = 0 14.27/4.52 14.27/4.52 s(x1) = s(x1) 14.27/4.52 14.27/4.52 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag(x1, x3) 14.27/4.52 14.27/4.52 U1_gag(x1, x2, x3) = U1_gag(x1, x2, x3) 14.27/4.52 14.27/4.52 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 U1_ggg(x1, x2, x3) = U1_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.52 14.27/4.52 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x1, x2, x3) 14.27/4.52 14.27/4.52 U1_gga(x1, x2, x3) = U1_gga(x1, x3) 14.27/4.52 14.27/4.52 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.52 14.27/4.52 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.52 14.27/4.52 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa(x1) 14.27/4.52 14.27/4.52 U1_gaa(x1, x2, x3) = U1_gaa(x1, x3) 14.27/4.52 14.27/4.52 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.52 14.27/4.52 U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) 14.27/4.52 14.27/4.52 U3_gag(x1, x2, x3, x4) = U3_gag(x1, x3, x4) 14.27/4.52 14.27/4.52 ACKERMANN_IN_GAG(x1, x2, x3) = ACKERMANN_IN_GAG(x1, x3) 14.27/4.52 14.27/4.52 U1_GAG(x1, x2, x3) = U1_GAG(x1, x2, x3) 14.27/4.52 14.27/4.52 ACKERMANN_IN_GGG(x1, x2, x3) = ACKERMANN_IN_GGG(x1, x2, x3) 14.27/4.52 14.27/4.52 U1_GGG(x1, x2, x3) = U1_GGG(x1, x2, x3) 14.27/4.52 14.27/4.52 U2_GGG(x1, x2, x3, x4) = U2_GGG(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2) 14.27/4.52 14.27/4.52 U1_GGA(x1, x2, x3) = U1_GGA(x1, x3) 14.27/4.52 14.27/4.52 U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_GGG(x1, x2, x3, x4) = U3_GGG(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 U2_GAG(x1, x2, x3, x4) = U2_GAG(x1, x3, x4) 14.27/4.52 14.27/4.52 ACKERMANN_IN_GAA(x1, x2, x3) = ACKERMANN_IN_GAA(x1) 14.27/4.52 14.27/4.52 U1_GAA(x1, x2, x3) = U1_GAA(x1, x3) 14.27/4.52 14.27/4.52 U2_GAA(x1, x2, x3, x4) = U2_GAA(x1, x4) 14.27/4.52 14.27/4.52 U3_GAA(x1, x2, x3, x4) = U3_GAA(x1, x4) 14.27/4.52 14.27/4.52 U3_GAG(x1, x2, x3, x4) = U3_GAG(x1, x3, x4) 14.27/4.52 14.27/4.52 14.27/4.52 We have to consider all (P,R,Pi)-chains 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (44) 14.27/4.52 Obligation: 14.27/4.52 Pi DP problem: 14.27/4.52 The TRS P consists of the following rules: 14.27/4.52 14.27/4.52 ACKERMANN_IN_GAG(s(M), 0, Val) -> U1_GAG(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ACKERMANN_IN_GAG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.52 ACKERMANN_IN_GGG(s(M), 0, Val) -> U1_GGG(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ACKERMANN_IN_GGG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.52 ACKERMANN_IN_GGG(s(M), s(N), Val) -> U2_GGG(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 ACKERMANN_IN_GGG(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) 14.27/4.52 ACKERMANN_IN_GGA(s(M), 0, Val) -> U1_GGA(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 ACKERMANN_IN_GGA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) 14.27/4.52 ACKERMANN_IN_GGA(s(M), s(N), Val) -> U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 ACKERMANN_IN_GGA(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) 14.27/4.52 U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_GGA(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.52 U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGA(M, Val1, Val) 14.27/4.52 U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_GGG(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.52 U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGG(M, Val1, Val) 14.27/4.52 ACKERMANN_IN_GAG(s(M), s(N), Val) -> U2_GAG(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 ACKERMANN_IN_GAG(s(M), s(N), Val) -> ACKERMANN_IN_GAA(s(M), N, Val1) 14.27/4.52 ACKERMANN_IN_GAA(s(M), 0, Val) -> U1_GAA(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 ACKERMANN_IN_GAA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) 14.27/4.52 ACKERMANN_IN_GAA(s(M), s(N), Val) -> U2_GAA(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 ACKERMANN_IN_GAA(s(M), s(N), Val) -> ACKERMANN_IN_GAA(s(M), N, Val1) 14.27/4.52 U2_GAA(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_GAA(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.52 U2_GAA(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAA(M, Val1, Val) 14.27/4.52 U2_GAG(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_GAG(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.52 U2_GAG(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAG(M, Val1, Val) 14.27/4.52 14.27/4.52 The TRS R consists of the following rules: 14.27/4.52 14.27/4.52 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.52 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.52 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.52 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.52 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.52 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.52 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.52 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.52 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.52 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.52 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.52 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.52 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.52 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.52 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.52 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.52 14.27/4.52 The argument filtering Pi contains the following mapping: 14.27/4.52 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.52 14.27/4.52 0 = 0 14.27/4.52 14.27/4.52 s(x1) = s(x1) 14.27/4.52 14.27/4.52 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag(x1, x3) 14.27/4.52 14.27/4.52 U1_gag(x1, x2, x3) = U1_gag(x1, x2, x3) 14.27/4.52 14.27/4.52 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 U1_ggg(x1, x2, x3) = U1_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.52 14.27/4.52 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x1, x2, x3) 14.27/4.52 14.27/4.52 U1_gga(x1, x2, x3) = U1_gga(x1, x3) 14.27/4.52 14.27/4.52 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.52 14.27/4.52 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.52 14.27/4.52 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa(x1) 14.27/4.52 14.27/4.52 U1_gaa(x1, x2, x3) = U1_gaa(x1, x3) 14.27/4.52 14.27/4.52 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.52 14.27/4.52 U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) 14.27/4.52 14.27/4.52 U3_gag(x1, x2, x3, x4) = U3_gag(x1, x3, x4) 14.27/4.52 14.27/4.52 ACKERMANN_IN_GAG(x1, x2, x3) = ACKERMANN_IN_GAG(x1, x3) 14.27/4.52 14.27/4.52 U1_GAG(x1, x2, x3) = U1_GAG(x1, x2, x3) 14.27/4.52 14.27/4.52 ACKERMANN_IN_GGG(x1, x2, x3) = ACKERMANN_IN_GGG(x1, x2, x3) 14.27/4.52 14.27/4.52 U1_GGG(x1, x2, x3) = U1_GGG(x1, x2, x3) 14.27/4.52 14.27/4.52 U2_GGG(x1, x2, x3, x4) = U2_GGG(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2) 14.27/4.52 14.27/4.52 U1_GGA(x1, x2, x3) = U1_GGA(x1, x3) 14.27/4.52 14.27/4.52 U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_GGG(x1, x2, x3, x4) = U3_GGG(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 U2_GAG(x1, x2, x3, x4) = U2_GAG(x1, x3, x4) 14.27/4.52 14.27/4.52 ACKERMANN_IN_GAA(x1, x2, x3) = ACKERMANN_IN_GAA(x1) 14.27/4.52 14.27/4.52 U1_GAA(x1, x2, x3) = U1_GAA(x1, x3) 14.27/4.52 14.27/4.52 U2_GAA(x1, x2, x3, x4) = U2_GAA(x1, x4) 14.27/4.52 14.27/4.52 U3_GAA(x1, x2, x3, x4) = U3_GAA(x1, x4) 14.27/4.52 14.27/4.52 U3_GAG(x1, x2, x3, x4) = U3_GAG(x1, x3, x4) 14.27/4.52 14.27/4.52 14.27/4.52 We have to consider all (P,R,Pi)-chains 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (45) DependencyGraphProof (EQUIVALENT) 14.27/4.52 The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 12 less nodes. 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (46) 14.27/4.52 Complex Obligation (AND) 14.27/4.52 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (47) 14.27/4.52 Obligation: 14.27/4.52 Pi DP problem: 14.27/4.52 The TRS P consists of the following rules: 14.27/4.52 14.27/4.52 ACKERMANN_IN_GGA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) 14.27/4.52 ACKERMANN_IN_GGA(s(M), s(N), Val) -> U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGA(M, Val1, Val) 14.27/4.52 ACKERMANN_IN_GGA(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) 14.27/4.52 14.27/4.52 The TRS R consists of the following rules: 14.27/4.52 14.27/4.52 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.52 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.52 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.52 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.52 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.52 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.52 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.52 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.52 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.52 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.52 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.52 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.52 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.52 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.52 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.52 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.52 14.27/4.52 The argument filtering Pi contains the following mapping: 14.27/4.52 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.52 14.27/4.52 0 = 0 14.27/4.52 14.27/4.52 s(x1) = s(x1) 14.27/4.52 14.27/4.52 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag(x1, x3) 14.27/4.52 14.27/4.52 U1_gag(x1, x2, x3) = U1_gag(x1, x2, x3) 14.27/4.52 14.27/4.52 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 U1_ggg(x1, x2, x3) = U1_ggg(x1, x2, x3) 14.27/4.52 14.27/4.52 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.52 14.27/4.52 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x1, x2, x3) 14.27/4.52 14.27/4.52 U1_gga(x1, x2, x3) = U1_gga(x1, x3) 14.27/4.52 14.27/4.52 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4) 14.27/4.52 14.27/4.52 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.52 14.27/4.52 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.52 14.27/4.52 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa(x1) 14.27/4.52 14.27/4.52 U1_gaa(x1, x2, x3) = U1_gaa(x1, x3) 14.27/4.52 14.27/4.52 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.52 14.27/4.52 U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) 14.27/4.52 14.27/4.52 U3_gag(x1, x2, x3, x4) = U3_gag(x1, x3, x4) 14.27/4.52 14.27/4.52 ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2) 14.27/4.52 14.27/4.52 U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) 14.27/4.52 14.27/4.52 14.27/4.52 We have to consider all (P,R,Pi)-chains 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (48) UsableRulesProof (EQUIVALENT) 14.27/4.52 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (49) 14.27/4.52 Obligation: 14.27/4.52 Pi DP problem: 14.27/4.52 The TRS P consists of the following rules: 14.27/4.52 14.27/4.52 ACKERMANN_IN_GGA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) 14.27/4.52 ACKERMANN_IN_GGA(s(M), s(N), Val) -> U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGA(M, Val1, Val) 14.27/4.52 ACKERMANN_IN_GGA(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) 14.27/4.52 14.27/4.52 The TRS R consists of the following rules: 14.27/4.52 14.27/4.52 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.52 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.52 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.52 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.52 14.27/4.52 The argument filtering Pi contains the following mapping: 14.27/4.52 0 = 0 14.27/4.52 14.27/4.52 s(x1) = s(x1) 14.27/4.52 14.27/4.52 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.52 14.27/4.52 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x1, x2, x3) 14.27/4.52 14.27/4.52 U1_gga(x1, x2, x3) = U1_gga(x1, x3) 14.27/4.52 14.27/4.52 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 14.27/4.52 14.27/4.52 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 14.27/4.52 14.27/4.52 ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2) 14.27/4.52 14.27/4.52 U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) 14.27/4.52 14.27/4.52 14.27/4.52 We have to consider all (P,R,Pi)-chains 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (50) PiDPToQDPProof (SOUND) 14.27/4.52 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (51) 14.27/4.52 Obligation: 14.27/4.52 Q DP problem: 14.27/4.52 The TRS P consists of the following rules: 14.27/4.52 14.27/4.52 ACKERMANN_IN_GGA(s(M), 0) -> ACKERMANN_IN_GGA(M, s(0)) 14.27/4.52 ACKERMANN_IN_GGA(s(M), s(N)) -> U2_GGA(M, N, ackermann_in_gga(s(M), N)) 14.27/4.52 U2_GGA(M, N, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGA(M, Val1) 14.27/4.52 ACKERMANN_IN_GGA(s(M), s(N)) -> ACKERMANN_IN_GGA(s(M), N) 14.27/4.52 14.27/4.52 The TRS R consists of the following rules: 14.27/4.52 14.27/4.52 ackermann_in_gga(s(M), 0) -> U1_gga(M, ackermann_in_gga(M, s(0))) 14.27/4.52 ackermann_in_gga(s(M), s(N)) -> U2_gga(M, N, ackermann_in_gga(s(M), N)) 14.27/4.52 U1_gga(M, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.52 U2_gga(M, N, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, ackermann_in_gga(M, Val1)) 14.27/4.52 ackermann_in_gga(0, N) -> ackermann_out_gga(0, N, s(N)) 14.27/4.52 U3_gga(M, N, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.52 14.27/4.52 The set Q consists of the following terms: 14.27/4.52 14.27/4.52 ackermann_in_gga(x0, x1) 14.27/4.52 U1_gga(x0, x1) 14.27/4.52 U2_gga(x0, x1, x2) 14.27/4.52 U3_gga(x0, x1, x2) 14.27/4.52 14.27/4.52 We have to consider all (P,Q,R)-chains. 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (52) QDPSizeChangeProof (EQUIVALENT) 14.27/4.52 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. 14.27/4.52 14.27/4.52 From the DPs we obtained the following set of size-change graphs: 14.27/4.52 *ACKERMANN_IN_GGA(s(M), s(N)) -> ACKERMANN_IN_GGA(s(M), N) 14.27/4.52 The graph contains the following edges 1 >= 1, 2 > 2 14.27/4.52 14.27/4.52 14.27/4.52 *ACKERMANN_IN_GGA(s(M), s(N)) -> U2_GGA(M, N, ackermann_in_gga(s(M), N)) 14.27/4.52 The graph contains the following edges 1 > 1, 2 > 2 14.27/4.52 14.27/4.52 14.27/4.52 *U2_GGA(M, N, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGA(M, Val1) 14.27/4.52 The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2 14.27/4.52 14.27/4.52 14.27/4.52 *ACKERMANN_IN_GGA(s(M), 0) -> ACKERMANN_IN_GGA(M, s(0)) 14.27/4.52 The graph contains the following edges 1 > 1 14.27/4.52 14.27/4.52 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (53) 14.27/4.52 YES 14.27/4.52 14.27/4.52 ---------------------------------------- 14.27/4.52 14.27/4.52 (54) 14.27/4.52 Obligation: 14.27/4.52 Pi DP problem: 14.27/4.52 The TRS P consists of the following rules: 14.27/4.52 14.27/4.52 ACKERMANN_IN_GAA(s(M), s(N), Val) -> U2_GAA(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.52 U2_GAA(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAA(M, Val1, Val) 14.27/4.52 ACKERMANN_IN_GAA(s(M), s(N), Val) -> ACKERMANN_IN_GAA(s(M), N, Val1) 14.27/4.52 14.27/4.52 The TRS R consists of the following rules: 14.27/4.52 14.27/4.52 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.52 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.52 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.52 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.52 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.52 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.52 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.52 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.53 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.53 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.53 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.53 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.53 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.53 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.53 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.53 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.53 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.53 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.53 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.53 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.53 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.53 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.53 14.27/4.53 The argument filtering Pi contains the following mapping: 14.27/4.53 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.53 14.27/4.53 0 = 0 14.27/4.53 14.27/4.53 s(x1) = s(x1) 14.27/4.53 14.27/4.53 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag(x1, x3) 14.27/4.53 14.27/4.53 U1_gag(x1, x2, x3) = U1_gag(x1, x2, x3) 14.27/4.53 14.27/4.53 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.53 14.27/4.53 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg(x1, x2, x3) 14.27/4.53 14.27/4.53 U1_ggg(x1, x2, x3) = U1_ggg(x1, x2, x3) 14.27/4.53 14.27/4.53 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x2, x3, x4) 14.27/4.53 14.27/4.53 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.53 14.27/4.53 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x1, x2, x3) 14.27/4.53 14.27/4.53 U1_gga(x1, x2, x3) = U1_gga(x1, x3) 14.27/4.53 14.27/4.53 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 14.27/4.53 14.27/4.53 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 14.27/4.53 14.27/4.53 U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4) 14.27/4.53 14.27/4.53 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.53 14.27/4.53 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.53 14.27/4.53 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa(x1) 14.27/4.53 14.27/4.53 U1_gaa(x1, x2, x3) = U1_gaa(x1, x3) 14.27/4.53 14.27/4.53 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.53 14.27/4.53 U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) 14.27/4.53 14.27/4.53 U3_gag(x1, x2, x3, x4) = U3_gag(x1, x3, x4) 14.27/4.53 14.27/4.53 ACKERMANN_IN_GAA(x1, x2, x3) = ACKERMANN_IN_GAA(x1) 14.27/4.53 14.27/4.53 U2_GAA(x1, x2, x3, x4) = U2_GAA(x1, x4) 14.27/4.53 14.27/4.53 14.27/4.53 We have to consider all (P,R,Pi)-chains 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (55) UsableRulesProof (EQUIVALENT) 14.27/4.53 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (56) 14.27/4.53 Obligation: 14.27/4.53 Pi DP problem: 14.27/4.53 The TRS P consists of the following rules: 14.27/4.53 14.27/4.53 ACKERMANN_IN_GAA(s(M), s(N), Val) -> U2_GAA(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.53 U2_GAA(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAA(M, Val1, Val) 14.27/4.53 ACKERMANN_IN_GAA(s(M), s(N), Val) -> ACKERMANN_IN_GAA(s(M), N, Val1) 14.27/4.53 14.27/4.53 The TRS R consists of the following rules: 14.27/4.53 14.27/4.53 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.53 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.53 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.53 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.53 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.53 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.53 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.53 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.53 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.53 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.53 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.53 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.53 14.27/4.53 The argument filtering Pi contains the following mapping: 14.27/4.53 0 = 0 14.27/4.53 14.27/4.53 s(x1) = s(x1) 14.27/4.53 14.27/4.53 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.53 14.27/4.53 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x1, x2, x3) 14.27/4.53 14.27/4.53 U1_gga(x1, x2, x3) = U1_gga(x1, x3) 14.27/4.53 14.27/4.53 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 14.27/4.53 14.27/4.53 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 14.27/4.53 14.27/4.53 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.53 14.27/4.53 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa(x1) 14.27/4.53 14.27/4.53 U1_gaa(x1, x2, x3) = U1_gaa(x1, x3) 14.27/4.53 14.27/4.53 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.53 14.27/4.53 U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) 14.27/4.53 14.27/4.53 ACKERMANN_IN_GAA(x1, x2, x3) = ACKERMANN_IN_GAA(x1) 14.27/4.53 14.27/4.53 U2_GAA(x1, x2, x3, x4) = U2_GAA(x1, x4) 14.27/4.53 14.27/4.53 14.27/4.53 We have to consider all (P,R,Pi)-chains 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (57) PiDPToQDPProof (SOUND) 14.27/4.53 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (58) 14.27/4.53 Obligation: 14.27/4.53 Q DP problem: 14.27/4.53 The TRS P consists of the following rules: 14.27/4.53 14.27/4.53 ACKERMANN_IN_GAA(s(M)) -> U2_GAA(M, ackermann_in_gaa(s(M))) 14.27/4.53 U2_GAA(M, ackermann_out_gaa(s(M))) -> ACKERMANN_IN_GAA(M) 14.27/4.53 ACKERMANN_IN_GAA(s(M)) -> ACKERMANN_IN_GAA(s(M)) 14.27/4.53 14.27/4.53 The TRS R consists of the following rules: 14.27/4.53 14.27/4.53 ackermann_in_gaa(s(M)) -> U1_gaa(M, ackermann_in_gga(M, s(0))) 14.27/4.53 ackermann_in_gaa(s(M)) -> U2_gaa(M, ackermann_in_gaa(s(M))) 14.27/4.53 U1_gaa(M, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M)) 14.27/4.53 U2_gaa(M, ackermann_out_gaa(s(M))) -> U3_gaa(M, ackermann_in_gaa(M)) 14.27/4.53 ackermann_in_gga(0, N) -> ackermann_out_gga(0, N, s(N)) 14.27/4.53 ackermann_in_gga(s(M), s(N)) -> U2_gga(M, N, ackermann_in_gga(s(M), N)) 14.27/4.53 U3_gaa(M, ackermann_out_gaa(M)) -> ackermann_out_gaa(s(M)) 14.27/4.53 U2_gga(M, N, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, ackermann_in_gga(M, Val1)) 14.27/4.53 ackermann_in_gaa(0) -> ackermann_out_gaa(0) 14.27/4.53 ackermann_in_gga(s(M), 0) -> U1_gga(M, ackermann_in_gga(M, s(0))) 14.27/4.53 U3_gga(M, N, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.53 U1_gga(M, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.53 14.27/4.53 The set Q consists of the following terms: 14.27/4.53 14.27/4.53 ackermann_in_gaa(x0) 14.27/4.53 U1_gaa(x0, x1) 14.27/4.53 U2_gaa(x0, x1) 14.27/4.53 ackermann_in_gga(x0, x1) 14.27/4.53 U3_gaa(x0, x1) 14.27/4.53 U2_gga(x0, x1, x2) 14.27/4.53 U3_gga(x0, x1, x2) 14.27/4.53 U1_gga(x0, x1) 14.27/4.53 14.27/4.53 We have to consider all (P,Q,R)-chains. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (59) QDPOrderProof (EQUIVALENT) 14.27/4.53 We use the reduction pair processor [LPAR04,JAR06]. 14.27/4.53 14.27/4.53 14.27/4.53 The following pairs can be oriented strictly and are deleted. 14.27/4.53 14.27/4.53 ACKERMANN_IN_GAA(s(M)) -> U2_GAA(M, ackermann_in_gaa(s(M))) 14.27/4.53 The remaining pairs can at least be oriented weakly. 14.27/4.53 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 14.27/4.53 14.27/4.53 POL( U2_GAA_2(x_1, x_2) ) = 2x_1 14.27/4.53 POL( ackermann_in_gaa_1(x_1) ) = 2 14.27/4.53 POL( s_1(x_1) ) = x_1 + 2 14.27/4.53 POL( U1_gaa_2(x_1, x_2) ) = max{0, -2} 14.27/4.53 POL( ackermann_in_gga_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} 14.27/4.53 POL( 0 ) = 2 14.27/4.53 POL( U2_gaa_2(x_1, x_2) ) = 2x_2 + 2 14.27/4.53 POL( ackermann_out_gaa_1(x_1) ) = x_1 + 2 14.27/4.53 POL( U3_gaa_2(x_1, x_2) ) = max{0, -2} 14.27/4.53 POL( ackermann_out_gga_3(x_1, ..., x_3) ) = max{0, x_3 - 2} 14.27/4.53 POL( U2_gga_3(x_1, ..., x_3) ) = 2x_1 + 2x_3 + 2 14.27/4.53 POL( U1_gga_2(x_1, x_2) ) = 2x_1 + 1 14.27/4.53 POL( U3_gga_3(x_1, ..., x_3) ) = x_2 + 2 14.27/4.53 POL( ACKERMANN_IN_GAA_1(x_1) ) = 2x_1 14.27/4.53 14.27/4.53 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 14.27/4.53 none 14.27/4.53 14.27/4.53 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (60) 14.27/4.53 Obligation: 14.27/4.53 Q DP problem: 14.27/4.53 The TRS P consists of the following rules: 14.27/4.53 14.27/4.53 U2_GAA(M, ackermann_out_gaa(s(M))) -> ACKERMANN_IN_GAA(M) 14.27/4.53 ACKERMANN_IN_GAA(s(M)) -> ACKERMANN_IN_GAA(s(M)) 14.27/4.53 14.27/4.53 The TRS R consists of the following rules: 14.27/4.53 14.27/4.53 ackermann_in_gaa(s(M)) -> U1_gaa(M, ackermann_in_gga(M, s(0))) 14.27/4.53 ackermann_in_gaa(s(M)) -> U2_gaa(M, ackermann_in_gaa(s(M))) 14.27/4.53 U1_gaa(M, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M)) 14.27/4.53 U2_gaa(M, ackermann_out_gaa(s(M))) -> U3_gaa(M, ackermann_in_gaa(M)) 14.27/4.53 ackermann_in_gga(0, N) -> ackermann_out_gga(0, N, s(N)) 14.27/4.53 ackermann_in_gga(s(M), s(N)) -> U2_gga(M, N, ackermann_in_gga(s(M), N)) 14.27/4.53 U3_gaa(M, ackermann_out_gaa(M)) -> ackermann_out_gaa(s(M)) 14.27/4.53 U2_gga(M, N, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, ackermann_in_gga(M, Val1)) 14.27/4.53 ackermann_in_gaa(0) -> ackermann_out_gaa(0) 14.27/4.53 ackermann_in_gga(s(M), 0) -> U1_gga(M, ackermann_in_gga(M, s(0))) 14.27/4.53 U3_gga(M, N, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.53 U1_gga(M, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.53 14.27/4.53 The set Q consists of the following terms: 14.27/4.53 14.27/4.53 ackermann_in_gaa(x0) 14.27/4.53 U1_gaa(x0, x1) 14.27/4.53 U2_gaa(x0, x1) 14.27/4.53 ackermann_in_gga(x0, x1) 14.27/4.53 U3_gaa(x0, x1) 14.27/4.53 U2_gga(x0, x1, x2) 14.27/4.53 U3_gga(x0, x1, x2) 14.27/4.53 U1_gga(x0, x1) 14.27/4.53 14.27/4.53 We have to consider all (P,Q,R)-chains. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (61) DependencyGraphProof (EQUIVALENT) 14.27/4.53 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (62) 14.27/4.53 Obligation: 14.27/4.53 Q DP problem: 14.27/4.53 The TRS P consists of the following rules: 14.27/4.53 14.27/4.53 ACKERMANN_IN_GAA(s(M)) -> ACKERMANN_IN_GAA(s(M)) 14.27/4.53 14.27/4.53 The TRS R consists of the following rules: 14.27/4.53 14.27/4.53 ackermann_in_gaa(s(M)) -> U1_gaa(M, ackermann_in_gga(M, s(0))) 14.27/4.53 ackermann_in_gaa(s(M)) -> U2_gaa(M, ackermann_in_gaa(s(M))) 14.27/4.53 U1_gaa(M, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M)) 14.27/4.53 U2_gaa(M, ackermann_out_gaa(s(M))) -> U3_gaa(M, ackermann_in_gaa(M)) 14.27/4.53 ackermann_in_gga(0, N) -> ackermann_out_gga(0, N, s(N)) 14.27/4.53 ackermann_in_gga(s(M), s(N)) -> U2_gga(M, N, ackermann_in_gga(s(M), N)) 14.27/4.53 U3_gaa(M, ackermann_out_gaa(M)) -> ackermann_out_gaa(s(M)) 14.27/4.53 U2_gga(M, N, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, ackermann_in_gga(M, Val1)) 14.27/4.53 ackermann_in_gaa(0) -> ackermann_out_gaa(0) 14.27/4.53 ackermann_in_gga(s(M), 0) -> U1_gga(M, ackermann_in_gga(M, s(0))) 14.27/4.53 U3_gga(M, N, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.53 U1_gga(M, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.53 14.27/4.53 The set Q consists of the following terms: 14.27/4.53 14.27/4.53 ackermann_in_gaa(x0) 14.27/4.53 U1_gaa(x0, x1) 14.27/4.53 U2_gaa(x0, x1) 14.27/4.53 ackermann_in_gga(x0, x1) 14.27/4.53 U3_gaa(x0, x1) 14.27/4.53 U2_gga(x0, x1, x2) 14.27/4.53 U3_gga(x0, x1, x2) 14.27/4.53 U1_gga(x0, x1) 14.27/4.53 14.27/4.53 We have to consider all (P,Q,R)-chains. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (63) UsableRulesProof (EQUIVALENT) 14.27/4.53 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. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (64) 14.27/4.53 Obligation: 14.27/4.53 Q DP problem: 14.27/4.53 The TRS P consists of the following rules: 14.27/4.53 14.27/4.53 ACKERMANN_IN_GAA(s(M)) -> ACKERMANN_IN_GAA(s(M)) 14.27/4.53 14.27/4.53 R is empty. 14.27/4.53 The set Q consists of the following terms: 14.27/4.53 14.27/4.53 ackermann_in_gaa(x0) 14.27/4.53 U1_gaa(x0, x1) 14.27/4.53 U2_gaa(x0, x1) 14.27/4.53 ackermann_in_gga(x0, x1) 14.27/4.53 U3_gaa(x0, x1) 14.27/4.53 U2_gga(x0, x1, x2) 14.27/4.53 U3_gga(x0, x1, x2) 14.27/4.53 U1_gga(x0, x1) 14.27/4.53 14.27/4.53 We have to consider all (P,Q,R)-chains. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (65) QReductionProof (EQUIVALENT) 14.27/4.53 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 14.27/4.53 14.27/4.53 ackermann_in_gaa(x0) 14.27/4.53 U1_gaa(x0, x1) 14.27/4.53 U2_gaa(x0, x1) 14.27/4.53 ackermann_in_gga(x0, x1) 14.27/4.53 U3_gaa(x0, x1) 14.27/4.53 U2_gga(x0, x1, x2) 14.27/4.53 U3_gga(x0, x1, x2) 14.27/4.53 U1_gga(x0, x1) 14.27/4.53 14.27/4.53 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (66) 14.27/4.53 Obligation: 14.27/4.53 Q DP problem: 14.27/4.53 The TRS P consists of the following rules: 14.27/4.53 14.27/4.53 ACKERMANN_IN_GAA(s(M)) -> ACKERMANN_IN_GAA(s(M)) 14.27/4.53 14.27/4.53 R is empty. 14.27/4.53 Q is empty. 14.27/4.53 We have to consider all (P,Q,R)-chains. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (67) NonTerminationLoopProof (COMPLETE) 14.27/4.53 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 14.27/4.53 Found a loop by semiunifying a rule from P directly. 14.27/4.53 14.27/4.53 s = ACKERMANN_IN_GAA(s(M)) evaluates to t =ACKERMANN_IN_GAA(s(M)) 14.27/4.53 14.27/4.53 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 14.27/4.53 * Matcher: [ ] 14.27/4.53 * Semiunifier: [ ] 14.27/4.53 14.27/4.53 -------------------------------------------------------------------------------- 14.27/4.53 Rewriting sequence 14.27/4.53 14.27/4.53 The DP semiunifies directly so there is only one rewrite step from ACKERMANN_IN_GAA(s(M)) to ACKERMANN_IN_GAA(s(M)). 14.27/4.53 14.27/4.53 14.27/4.53 14.27/4.53 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (68) 14.27/4.53 NO 14.27/4.53 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (69) 14.27/4.53 Obligation: 14.27/4.53 Pi DP problem: 14.27/4.53 The TRS P consists of the following rules: 14.27/4.53 14.27/4.53 ACKERMANN_IN_GGG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.53 ACKERMANN_IN_GGG(s(M), s(N), Val) -> U2_GGG(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.53 U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGG(M, Val1, Val) 14.27/4.53 14.27/4.53 The TRS R consists of the following rules: 14.27/4.53 14.27/4.53 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.53 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.53 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.53 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.53 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.53 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.53 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.53 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.53 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.53 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.53 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.53 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.53 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.53 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.53 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.53 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.53 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.53 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.53 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.53 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.53 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.53 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.53 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.53 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.53 14.27/4.53 The argument filtering Pi contains the following mapping: 14.27/4.53 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.53 14.27/4.53 0 = 0 14.27/4.53 14.27/4.53 s(x1) = s(x1) 14.27/4.53 14.27/4.53 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag(x1, x3) 14.27/4.53 14.27/4.53 U1_gag(x1, x2, x3) = U1_gag(x1, x2, x3) 14.27/4.53 14.27/4.53 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.53 14.27/4.53 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg(x1, x2, x3) 14.27/4.53 14.27/4.53 U1_ggg(x1, x2, x3) = U1_ggg(x1, x2, x3) 14.27/4.53 14.27/4.53 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x2, x3, x4) 14.27/4.53 14.27/4.53 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.53 14.27/4.53 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x1, x2, x3) 14.27/4.53 14.27/4.53 U1_gga(x1, x2, x3) = U1_gga(x1, x3) 14.27/4.53 14.27/4.53 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 14.27/4.53 14.27/4.53 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 14.27/4.53 14.27/4.53 U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4) 14.27/4.53 14.27/4.53 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.53 14.27/4.53 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.53 14.27/4.53 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa(x1) 14.27/4.53 14.27/4.53 U1_gaa(x1, x2, x3) = U1_gaa(x1, x3) 14.27/4.53 14.27/4.53 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.53 14.27/4.53 U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) 14.27/4.53 14.27/4.53 U3_gag(x1, x2, x3, x4) = U3_gag(x1, x3, x4) 14.27/4.53 14.27/4.53 ACKERMANN_IN_GGG(x1, x2, x3) = ACKERMANN_IN_GGG(x1, x2, x3) 14.27/4.53 14.27/4.53 U2_GGG(x1, x2, x3, x4) = U2_GGG(x1, x2, x3, x4) 14.27/4.53 14.27/4.53 14.27/4.53 We have to consider all (P,R,Pi)-chains 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (70) UsableRulesProof (EQUIVALENT) 14.27/4.53 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (71) 14.27/4.53 Obligation: 14.27/4.53 Pi DP problem: 14.27/4.53 The TRS P consists of the following rules: 14.27/4.53 14.27/4.53 ACKERMANN_IN_GGG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.53 ACKERMANN_IN_GGG(s(M), s(N), Val) -> U2_GGG(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.53 U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGG(M, Val1, Val) 14.27/4.53 14.27/4.53 The TRS R consists of the following rules: 14.27/4.53 14.27/4.53 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.53 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.53 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.53 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.53 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.53 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.53 14.27/4.53 The argument filtering Pi contains the following mapping: 14.27/4.53 0 = 0 14.27/4.53 14.27/4.53 s(x1) = s(x1) 14.27/4.53 14.27/4.53 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.53 14.27/4.53 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x1, x2, x3) 14.27/4.53 14.27/4.53 U1_gga(x1, x2, x3) = U1_gga(x1, x3) 14.27/4.53 14.27/4.53 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 14.27/4.53 14.27/4.53 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 14.27/4.53 14.27/4.53 ACKERMANN_IN_GGG(x1, x2, x3) = ACKERMANN_IN_GGG(x1, x2, x3) 14.27/4.53 14.27/4.53 U2_GGG(x1, x2, x3, x4) = U2_GGG(x1, x2, x3, x4) 14.27/4.53 14.27/4.53 14.27/4.53 We have to consider all (P,R,Pi)-chains 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (72) PiDPToQDPProof (SOUND) 14.27/4.53 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (73) 14.27/4.53 Obligation: 14.27/4.53 Q DP problem: 14.27/4.53 The TRS P consists of the following rules: 14.27/4.53 14.27/4.53 ACKERMANN_IN_GGG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.53 ACKERMANN_IN_GGG(s(M), s(N), Val) -> U2_GGG(M, N, Val, ackermann_in_gga(s(M), N)) 14.27/4.53 U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGG(M, Val1, Val) 14.27/4.53 14.27/4.53 The TRS R consists of the following rules: 14.27/4.53 14.27/4.53 ackermann_in_gga(s(M), 0) -> U1_gga(M, ackermann_in_gga(M, s(0))) 14.27/4.53 ackermann_in_gga(s(M), s(N)) -> U2_gga(M, N, ackermann_in_gga(s(M), N)) 14.27/4.53 U1_gga(M, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.53 U2_gga(M, N, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, ackermann_in_gga(M, Val1)) 14.27/4.53 ackermann_in_gga(0, N) -> ackermann_out_gga(0, N, s(N)) 14.27/4.53 U3_gga(M, N, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.53 14.27/4.53 The set Q consists of the following terms: 14.27/4.53 14.27/4.53 ackermann_in_gga(x0, x1) 14.27/4.53 U1_gga(x0, x1) 14.27/4.53 U2_gga(x0, x1, x2) 14.27/4.53 U3_gga(x0, x1, x2) 14.27/4.53 14.27/4.53 We have to consider all (P,Q,R)-chains. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (74) QDPSizeChangeProof (EQUIVALENT) 14.27/4.53 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. 14.27/4.53 14.27/4.53 From the DPs we obtained the following set of size-change graphs: 14.27/4.53 *ACKERMANN_IN_GGG(s(M), s(N), Val) -> U2_GGG(M, N, Val, ackermann_in_gga(s(M), N)) 14.27/4.53 The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 14.27/4.53 14.27/4.53 14.27/4.53 *U2_GGG(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGG(M, Val1, Val) 14.27/4.53 The graph contains the following edges 1 >= 1, 4 > 1, 4 > 2, 3 >= 3 14.27/4.53 14.27/4.53 14.27/4.53 *ACKERMANN_IN_GGG(s(M), 0, Val) -> ACKERMANN_IN_GGG(M, s(0), Val) 14.27/4.53 The graph contains the following edges 1 > 1, 3 >= 3 14.27/4.53 14.27/4.53 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (75) 14.27/4.53 YES 14.27/4.53 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (76) 14.27/4.53 Obligation: 14.27/4.53 Pi DP problem: 14.27/4.53 The TRS P consists of the following rules: 14.27/4.53 14.27/4.53 ACKERMANN_IN_GAG(s(M), s(N), Val) -> U2_GAG(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.53 U2_GAG(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAG(M, Val1, Val) 14.27/4.53 14.27/4.53 The TRS R consists of the following rules: 14.27/4.53 14.27/4.53 ackermann_in_gag(0, N, s(N)) -> ackermann_out_gag(0, N, s(N)) 14.27/4.53 ackermann_in_gag(s(M), 0, Val) -> U1_gag(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.53 ackermann_in_ggg(0, N, s(N)) -> ackermann_out_ggg(0, N, s(N)) 14.27/4.53 ackermann_in_ggg(s(M), 0, Val) -> U1_ggg(M, Val, ackermann_in_ggg(M, s(0), Val)) 14.27/4.53 ackermann_in_ggg(s(M), s(N), Val) -> U2_ggg(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.53 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.53 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.53 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.53 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.53 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.53 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.53 U2_ggg(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_ggg(M, N, Val, ackermann_in_ggg(M, Val1, Val)) 14.27/4.53 U3_ggg(M, N, Val, ackermann_out_ggg(M, Val1, Val)) -> ackermann_out_ggg(s(M), s(N), Val) 14.27/4.53 U1_ggg(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_ggg(s(M), 0, Val) 14.27/4.53 U1_gag(M, Val, ackermann_out_ggg(M, s(0), Val)) -> ackermann_out_gag(s(M), 0, Val) 14.27/4.53 ackermann_in_gag(s(M), s(N), Val) -> U2_gag(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.53 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.53 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.53 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.53 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.53 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.53 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.53 U2_gag(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gag(M, N, Val, ackermann_in_gag(M, Val1, Val)) 14.27/4.53 U3_gag(M, N, Val, ackermann_out_gag(M, Val1, Val)) -> ackermann_out_gag(s(M), s(N), Val) 14.27/4.53 14.27/4.53 The argument filtering Pi contains the following mapping: 14.27/4.53 ackermann_in_gag(x1, x2, x3) = ackermann_in_gag(x1, x3) 14.27/4.53 14.27/4.53 0 = 0 14.27/4.53 14.27/4.53 s(x1) = s(x1) 14.27/4.53 14.27/4.53 ackermann_out_gag(x1, x2, x3) = ackermann_out_gag(x1, x3) 14.27/4.53 14.27/4.53 U1_gag(x1, x2, x3) = U1_gag(x1, x2, x3) 14.27/4.53 14.27/4.53 ackermann_in_ggg(x1, x2, x3) = ackermann_in_ggg(x1, x2, x3) 14.27/4.53 14.27/4.53 ackermann_out_ggg(x1, x2, x3) = ackermann_out_ggg(x1, x2, x3) 14.27/4.53 14.27/4.53 U1_ggg(x1, x2, x3) = U1_ggg(x1, x2, x3) 14.27/4.53 14.27/4.53 U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x2, x3, x4) 14.27/4.53 14.27/4.53 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.53 14.27/4.53 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x1, x2, x3) 14.27/4.53 14.27/4.53 U1_gga(x1, x2, x3) = U1_gga(x1, x3) 14.27/4.53 14.27/4.53 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 14.27/4.53 14.27/4.53 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 14.27/4.53 14.27/4.53 U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4) 14.27/4.53 14.27/4.53 U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4) 14.27/4.53 14.27/4.53 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.53 14.27/4.53 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa(x1) 14.27/4.53 14.27/4.53 U1_gaa(x1, x2, x3) = U1_gaa(x1, x3) 14.27/4.53 14.27/4.53 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.53 14.27/4.53 U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) 14.27/4.53 14.27/4.53 U3_gag(x1, x2, x3, x4) = U3_gag(x1, x3, x4) 14.27/4.53 14.27/4.53 ACKERMANN_IN_GAG(x1, x2, x3) = ACKERMANN_IN_GAG(x1, x3) 14.27/4.53 14.27/4.53 U2_GAG(x1, x2, x3, x4) = U2_GAG(x1, x3, x4) 14.27/4.53 14.27/4.53 14.27/4.53 We have to consider all (P,R,Pi)-chains 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (77) UsableRulesProof (EQUIVALENT) 14.27/4.53 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (78) 14.27/4.53 Obligation: 14.27/4.53 Pi DP problem: 14.27/4.53 The TRS P consists of the following rules: 14.27/4.53 14.27/4.53 ACKERMANN_IN_GAG(s(M), s(N), Val) -> U2_GAG(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.53 U2_GAG(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> ACKERMANN_IN_GAG(M, Val1, Val) 14.27/4.53 14.27/4.53 The TRS R consists of the following rules: 14.27/4.53 14.27/4.53 ackermann_in_gaa(s(M), 0, Val) -> U1_gaa(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.53 ackermann_in_gaa(s(M), s(N), Val) -> U2_gaa(M, N, Val, ackermann_in_gaa(s(M), N, Val1)) 14.27/4.53 U1_gaa(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M), 0, Val) 14.27/4.53 U2_gaa(M, N, Val, ackermann_out_gaa(s(M), N, Val1)) -> U3_gaa(M, N, Val, ackermann_in_gaa(M, Val1, Val)) 14.27/4.53 ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) 14.27/4.53 ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) 14.27/4.53 U3_gaa(M, N, Val, ackermann_out_gaa(M, Val1, Val)) -> ackermann_out_gaa(s(M), s(N), Val) 14.27/4.53 U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) 14.27/4.53 ackermann_in_gaa(0, N, s(N)) -> ackermann_out_gaa(0, N, s(N)) 14.27/4.53 ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) 14.27/4.53 U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.53 U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.53 14.27/4.53 The argument filtering Pi contains the following mapping: 14.27/4.53 0 = 0 14.27/4.53 14.27/4.53 s(x1) = s(x1) 14.27/4.53 14.27/4.53 ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 14.27/4.53 14.27/4.53 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x1, x2, x3) 14.27/4.53 14.27/4.53 U1_gga(x1, x2, x3) = U1_gga(x1, x3) 14.27/4.53 14.27/4.53 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 14.27/4.53 14.27/4.53 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 14.27/4.53 14.27/4.53 ackermann_in_gaa(x1, x2, x3) = ackermann_in_gaa(x1) 14.27/4.53 14.27/4.53 ackermann_out_gaa(x1, x2, x3) = ackermann_out_gaa(x1) 14.27/4.53 14.27/4.53 U1_gaa(x1, x2, x3) = U1_gaa(x1, x3) 14.27/4.53 14.27/4.53 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) 14.27/4.53 14.27/4.53 U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) 14.27/4.53 14.27/4.53 ACKERMANN_IN_GAG(x1, x2, x3) = ACKERMANN_IN_GAG(x1, x3) 14.27/4.53 14.27/4.53 U2_GAG(x1, x2, x3, x4) = U2_GAG(x1, x3, x4) 14.27/4.53 14.27/4.53 14.27/4.53 We have to consider all (P,R,Pi)-chains 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (79) PiDPToQDPProof (SOUND) 14.27/4.53 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (80) 14.27/4.53 Obligation: 14.27/4.53 Q DP problem: 14.27/4.53 The TRS P consists of the following rules: 14.27/4.53 14.27/4.53 ACKERMANN_IN_GAG(s(M), Val) -> U2_GAG(M, Val, ackermann_in_gaa(s(M))) 14.27/4.53 U2_GAG(M, Val, ackermann_out_gaa(s(M))) -> ACKERMANN_IN_GAG(M, Val) 14.27/4.53 14.27/4.53 The TRS R consists of the following rules: 14.27/4.53 14.27/4.53 ackermann_in_gaa(s(M)) -> U1_gaa(M, ackermann_in_gga(M, s(0))) 14.27/4.53 ackermann_in_gaa(s(M)) -> U2_gaa(M, ackermann_in_gaa(s(M))) 14.27/4.53 U1_gaa(M, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gaa(s(M)) 14.27/4.53 U2_gaa(M, ackermann_out_gaa(s(M))) -> U3_gaa(M, ackermann_in_gaa(M)) 14.27/4.53 ackermann_in_gga(0, N) -> ackermann_out_gga(0, N, s(N)) 14.27/4.53 ackermann_in_gga(s(M), s(N)) -> U2_gga(M, N, ackermann_in_gga(s(M), N)) 14.27/4.53 U3_gaa(M, ackermann_out_gaa(M)) -> ackermann_out_gaa(s(M)) 14.27/4.53 U2_gga(M, N, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, ackermann_in_gga(M, Val1)) 14.27/4.53 ackermann_in_gaa(0) -> ackermann_out_gaa(0) 14.27/4.53 ackermann_in_gga(s(M), 0) -> U1_gga(M, ackermann_in_gga(M, s(0))) 14.27/4.53 U3_gga(M, N, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) 14.27/4.53 U1_gga(M, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) 14.27/4.53 14.27/4.53 The set Q consists of the following terms: 14.27/4.53 14.27/4.53 ackermann_in_gaa(x0) 14.27/4.53 U1_gaa(x0, x1) 14.27/4.53 U2_gaa(x0, x1) 14.27/4.53 ackermann_in_gga(x0, x1) 14.27/4.53 U3_gaa(x0, x1) 14.27/4.53 U2_gga(x0, x1, x2) 14.27/4.53 U3_gga(x0, x1, x2) 14.27/4.53 U1_gga(x0, x1) 14.27/4.53 14.27/4.53 We have to consider all (P,Q,R)-chains. 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (81) QDPSizeChangeProof (EQUIVALENT) 14.27/4.53 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. 14.27/4.53 14.27/4.53 From the DPs we obtained the following set of size-change graphs: 14.27/4.53 *U2_GAG(M, Val, ackermann_out_gaa(s(M))) -> ACKERMANN_IN_GAG(M, Val) 14.27/4.53 The graph contains the following edges 1 >= 1, 3 > 1, 2 >= 2 14.27/4.53 14.27/4.53 14.27/4.53 *ACKERMANN_IN_GAG(s(M), Val) -> U2_GAG(M, Val, ackermann_in_gaa(s(M))) 14.27/4.53 The graph contains the following edges 1 > 1, 2 >= 2 14.27/4.53 14.27/4.53 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (82) 14.27/4.53 YES 14.27/4.53 14.27/4.53 ---------------------------------------- 14.27/4.53 14.27/4.53 (83) PrologToDTProblemTransformerProof (SOUND) 14.27/4.53 Built DT problem from termination graph DT10. 14.27/4.53 14.27/4.53 { 14.27/4.53 "root": 8, 14.27/4.53 "program": { 14.27/4.53 "directives": [], 14.27/4.53 "clauses": [ 14.27/4.53 [ 14.27/4.53 "(ackermann (0) N (s N))", 14.27/4.53 null 14.27/4.53 ], 14.27/4.53 [ 14.27/4.53 "(ackermann (s M) (0) Val)", 14.27/4.53 "(ackermann M (s (0)) Val)" 14.27/4.53 ], 14.27/4.53 [ 14.27/4.53 "(ackermann (s M) (s N) Val)", 14.27/4.53 "(',' (ackermann (s M) N Val1) (ackermann M Val1 Val))" 14.27/4.53 ] 14.27/4.53 ] 14.27/4.53 }, 14.27/4.53 "graph": { 14.27/4.53 "nodes": { 14.27/4.53 "type": "Nodes", 14.27/4.53 "351": { 14.27/4.53 "goal": [], 14.27/4.53 "kb": { 14.27/4.53 "nonunifying": [], 14.27/4.53 "intvars": {}, 14.27/4.53 "arithmetic": { 14.27/4.53 "type": "PlainIntegerRelationState", 14.27/4.53 "relations": [] 14.27/4.53 }, 14.27/4.53 "ground": [], 14.27/4.53 "free": [], 14.27/4.53 "exprvars": [] 14.27/4.53 } 14.27/4.53 }, 14.27/4.53 "112": { 14.27/4.53 "goal": [ 14.27/4.53 { 14.27/4.53 "clause": 1, 14.27/4.53 "scope": 1, 14.27/4.53 "term": "(ackermann T1 T2 T3)" 14.27/4.53 }, 14.27/4.53 { 14.27/4.53 "clause": 2, 14.27/4.53 "scope": 1, 14.27/4.53 "term": "(ackermann T1 T2 T3)" 14.27/4.53 } 14.27/4.53 ], 14.27/4.53 "kb": { 14.27/4.53 "nonunifying": [[ 14.27/4.53 "(ackermann T1 T2 T3)", 14.27/4.53 "(ackermann (0) X2 (s X2))" 14.27/4.53 ]], 14.27/4.53 "intvars": {}, 14.27/4.53 "arithmetic": { 14.27/4.53 "type": "PlainIntegerRelationState", 14.27/4.53 "relations": [] 14.27/4.53 }, 14.27/4.53 "ground": [ 14.27/4.53 "T1", 14.27/4.53 "T3" 14.27/4.53 ], 14.27/4.53 "free": ["X2"], 14.27/4.53 "exprvars": [] 14.27/4.53 } 14.27/4.53 }, 14.27/4.53 "354": { 14.27/4.53 "goal": [{ 14.27/4.53 "clause": 2, 14.27/4.53 "scope": 1, 14.27/4.53 "term": "(ackermann (s T8) T2 T9)" 14.27/4.53 }], 14.27/4.53 "kb": { 14.27/4.53 "nonunifying": [], 14.27/4.53 "intvars": {}, 14.27/4.53 "arithmetic": { 14.27/4.53 "type": "PlainIntegerRelationState", 14.27/4.53 "relations": [] 14.27/4.53 }, 14.27/4.53 "ground": [ 14.27/4.53 "T8", 14.27/4.53 "T9" 14.27/4.53 ], 14.27/4.53 "free": [], 14.27/4.53 "exprvars": [] 14.27/4.53 } 14.27/4.53 }, 14.27/4.53 "475": { 14.27/4.53 "goal": [ 14.27/4.53 { 14.27/4.53 "clause": 0, 14.27/4.53 "scope": 7, 14.27/4.53 "term": "(ackermann T76 T79 X183)" 14.27/4.53 }, 14.27/4.53 { 14.27/4.53 "clause": 1, 14.27/4.53 "scope": 7, 14.27/4.53 "term": "(ackermann T76 T79 X183)" 14.27/4.53 }, 14.27/4.53 { 14.27/4.53 "clause": 2, 14.27/4.53 "scope": 7, 14.27/4.53 "term": "(ackermann T76 T79 X183)" 14.27/4.53 } 14.27/4.53 ], 14.27/4.53 "kb": { 14.27/4.53 "nonunifying": [], 14.27/4.53 "intvars": {}, 14.27/4.53 "arithmetic": { 14.27/4.53 "type": "PlainIntegerRelationState", 14.27/4.53 "relations": [] 14.27/4.53 }, 14.27/4.53 "ground": ["T76"], 14.27/4.53 "free": ["X183"], 14.27/4.53 "exprvars": [] 14.27/4.53 } 14.27/4.53 }, 14.27/4.53 "476": { 14.27/4.53 "goal": [{ 14.27/4.53 "clause": 0, 14.27/4.53 "scope": 7, 14.27/4.53 "term": "(ackermann T76 T79 X183)" 14.27/4.53 }], 14.27/4.53 "kb": { 14.27/4.53 "nonunifying": [], 14.27/4.53 "intvars": {}, 14.27/4.53 "arithmetic": { 14.27/4.53 "type": "PlainIntegerRelationState", 14.27/4.53 "relations": [] 14.27/4.53 }, 14.27/4.53 "ground": ["T76"], 14.27/4.53 "free": ["X183"], 14.27/4.53 "exprvars": [] 14.27/4.53 } 14.27/4.53 }, 14.27/4.53 "477": { 14.27/4.53 "goal": [ 14.27/4.53 { 14.27/4.53 "clause": 1, 14.27/4.53 "scope": 7, 14.27/4.53 "term": "(ackermann T76 T79 X183)" 14.27/4.53 }, 14.27/4.53 { 14.27/4.53 "clause": 2, 14.27/4.53 "scope": 7, 14.27/4.53 "term": "(ackermann T76 T79 X183)" 14.27/4.53 } 14.27/4.53 ], 14.27/4.53 "kb": { 14.27/4.53 "nonunifying": [], 14.27/4.53 "intvars": {}, 14.27/4.53 "arithmetic": { 14.27/4.53 "type": "PlainIntegerRelationState", 14.27/4.53 "relations": [] 14.27/4.53 }, 14.27/4.53 "ground": ["T76"], 14.27/4.53 "free": ["X183"], 14.27/4.53 "exprvars": [] 14.27/4.53 } 14.27/4.53 }, 14.27/4.53 "357": { 14.27/4.53 "goal": [{ 14.27/4.53 "clause": -1, 14.27/4.53 "scope": -1, 14.27/4.53 "term": "(',' (ackermann (s T62) T65 X147) (ackermann T62 X147 T64))" 14.27/4.53 }], 14.27/4.53 "kb": { 14.27/4.53 "nonunifying": [], 14.27/4.53 "intvars": {}, 14.27/4.53 "arithmetic": { 14.27/4.53 "type": "PlainIntegerRelationState", 14.27/4.53 "relations": [] 14.27/4.53 }, 14.27/4.53 "ground": [ 14.27/4.53 "T62", 14.27/4.53 "T64" 14.27/4.53 ], 14.27/4.53 "free": ["X147"], 14.27/4.53 "exprvars": [] 14.27/4.53 } 14.27/4.53 }, 14.27/4.53 "358": { 14.27/4.53 "goal": [], 14.27/4.53 "kb": { 14.27/4.53 "nonunifying": [], 14.27/4.53 "intvars": {}, 14.27/4.53 "arithmetic": { 14.27/4.53 "type": "PlainIntegerRelationState", 14.27/4.53 "relations": [] 14.27/4.53 }, 14.27/4.53 "ground": [], 14.27/4.53 "free": [], 14.27/4.53 "exprvars": [] 14.27/4.53 } 14.27/4.53 }, 14.27/4.53 "479": { 14.27/4.53 "goal": [{ 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14.27/4.54 { 14.27/4.54 "clause": 1, 14.27/4.54 "scope": 1, 14.27/4.54 "term": "(ackermann (0) T2 (s T5))" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "clause": 2, 14.27/4.54 "scope": 1, 14.27/4.54 "term": "(ackermann (0) T2 (s T5))" 14.27/4.54 } 14.27/4.54 ], 14.27/4.54 "kb": { 14.27/4.54 "nonunifying": [], 14.27/4.54 "intvars": {}, 14.27/4.54 "arithmetic": { 14.27/4.54 "type": "PlainIntegerRelationState", 14.27/4.54 "relations": [] 14.27/4.54 }, 14.27/4.54 "ground": ["T5"], 14.27/4.54 "free": [], 14.27/4.54 "exprvars": [] 14.27/4.54 } 14.27/4.54 }, 14.27/4.54 "506": { 14.27/4.54 "goal": [{ 14.27/4.54 "clause": 2, 14.27/4.54 "scope": 8, 14.27/4.54 "term": "(',' (ackermann (s T104) T107 X236) (ackermann T104 X236 T106))" 14.27/4.54 }], 14.27/4.54 "kb": { 14.27/4.54 "nonunifying": [[ 14.27/4.54 "(ackermann (s T104) T2 T106)", 14.27/4.54 "(ackermann (s X10) (0) X11)" 14.27/4.54 ]], 14.27/4.54 "intvars": {}, 14.27/4.54 "arithmetic": { 14.27/4.54 "type": "PlainIntegerRelationState", 14.27/4.54 "relations": [] 14.27/4.54 }, 14.27/4.54 "ground": [ 14.27/4.54 "T104", 14.27/4.54 "T106" 14.27/4.54 ], 14.27/4.54 "free": [ 14.27/4.54 "X10", 14.27/4.54 "X11", 14.27/4.54 "X236" 14.27/4.54 ], 14.27/4.54 "exprvars": [] 14.27/4.54 } 14.27/4.54 }, 14.27/4.54 "507": { 14.27/4.54 "goal": [{ 14.27/4.54 "clause": -1, 14.27/4.54 "scope": -1, 14.27/4.54 "term": "(',' (ackermann T112 (s (0)) X252) (ackermann T112 X252 T106))" 14.27/4.54 }], 14.27/4.54 "kb": { 14.27/4.54 "nonunifying": [[ 14.27/4.54 "(ackermann (s T112) T2 T106)", 14.27/4.54 "(ackermann (s X10) (0) X11)" 14.27/4.54 ]], 14.27/4.54 "intvars": {}, 14.27/4.54 "arithmetic": { 14.27/4.54 "type": "PlainIntegerRelationState", 14.27/4.54 "relations": [] 14.27/4.54 }, 14.27/4.54 "ground": [ 14.27/4.54 "T106", 14.27/4.54 "T112" 14.27/4.54 ], 14.27/4.54 "free": [ 14.27/4.54 "X10", 14.27/4.54 "X11", 14.27/4.54 "X252" 14.27/4.54 ], 14.27/4.54 "exprvars": [] 14.27/4.54 } 14.27/4.54 }, 14.27/4.54 "508": { 14.27/4.54 "goal": [], 14.27/4.54 "kb": { 14.27/4.54 "nonunifying": [], 14.27/4.54 "intvars": {}, 14.27/4.54 "arithmetic": { 14.27/4.54 "type": "PlainIntegerRelationState", 14.27/4.54 "relations": [] 14.27/4.54 }, 14.27/4.54 "ground": [], 14.27/4.54 "free": [], 14.27/4.54 "exprvars": [] 14.27/4.54 } 14.27/4.54 } 14.27/4.54 }, 14.27/4.54 "edges": [ 14.27/4.54 { 14.27/4.54 "from": 8, 14.27/4.54 "to": 9, 14.27/4.54 "label": "CASE" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 9, 14.27/4.54 "to": 108, 14.27/4.54 "label": "EVAL with clause\nackermann(0, X2, s(X2)).\nand substitutionT1 -> 0,\nT2 -> T5,\nX2 -> T5,\nT3 -> s(T5)" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 9, 14.27/4.54 "to": 112, 14.27/4.54 "label": "EVAL-BACKTRACK" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 108, 14.27/4.54 "to": 118, 14.27/4.54 "label": "SUCCESS" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 112, 14.27/4.54 "to": 131, 14.27/4.54 "label": "EVAL with clause\nackermann(s(X10), 0, X11) :- ackermann(X10, s(0), X11).\nand substitutionX10 -> T8,\nT1 -> s(T8),\nT2 -> 0,\nT3 -> T9,\nX11 -> T9" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 112, 14.27/4.54 "to": 137, 14.27/4.54 "label": "EVAL-BACKTRACK" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 118, 14.27/4.54 "to": 123, 14.27/4.54 "label": "BACKTRACK\nfor clause: ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val)because of non-unification" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 123, 14.27/4.54 "to": 124, 14.27/4.54 "label": "BACKTRACK\nfor clause: ackermann(s(M), s(N), Val) :- ','(ackermann(s(M), N, Val1), ackermann(M, Val1, Val))because of non-unification" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 131, 14.27/4.54 "to": 140, 14.27/4.54 "label": "CASE" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 137, 14.27/4.54 "to": 499, 14.27/4.54 "label": "EVAL with clause\nackermann(s(X233), s(X234), X235) :- ','(ackermann(s(X233), X234, X236), ackermann(X233, X236, X235)).\nand substitutionX233 -> T104,\nT1 -> s(T104),\nX234 -> T107,\nT2 -> s(T107),\nT3 -> T106,\nX235 -> T106,\nT105 -> T107" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 137, 14.27/4.54 "to": 500, 14.27/4.54 "label": "EVAL-BACKTRACK" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 140, 14.27/4.54 "to": 144, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 140, 14.27/4.54 "to": 146, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 144, 14.27/4.54 "to": 151, 14.27/4.54 "label": "EVAL with clause\nackermann(0, X16, s(X16)).\nand substitutionT8 -> 0,\nX16 -> s(0),\nT9 -> s(s(0))" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 144, 14.27/4.54 "to": 154, 14.27/4.54 "label": "EVAL-BACKTRACK" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 146, 14.27/4.54 "to": 159, 14.27/4.54 "label": "BACKTRACK\nfor clause: ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val)because of non-unification" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 151, 14.27/4.54 "to": 156, 14.27/4.54 "label": "SUCCESS" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 159, 14.27/4.54 "to": 165, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 159, 14.27/4.54 "to": 166, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 165, 14.27/4.54 "to": 262, 14.27/4.54 "label": "EVAL with clause\nackermann(s(X33), s(X34), X35) :- ','(ackermann(s(X33), X34, X36), ackermann(X33, X36, X35)).\nand substitutionX33 -> T19,\nT8 -> s(T19),\nX34 -> 0,\nT9 -> T20,\nX35 -> T20" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 165, 14.27/4.54 "to": 263, 14.27/4.54 "label": "EVAL-BACKTRACK" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 166, 14.27/4.54 "to": 354, 14.27/4.54 "label": "FAILURE" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 262, 14.27/4.54 "to": 264, 14.27/4.54 "label": "SPLIT 1" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 262, 14.27/4.54 "to": 265, 14.27/4.54 "label": "SPLIT 2\nnew knowledge:\nT19 is ground\nT21 is ground\nreplacements:X36 -> T21" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 264, 14.27/4.54 "to": 269, 14.27/4.54 "label": "CASE" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 265, 14.27/4.54 "to": 8, 14.27/4.54 "label": "INSTANCE with matching:\nT1 -> T19\nT2 -> T21\nT3 -> T20" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 269, 14.27/4.54 "to": 270, 14.27/4.54 "label": "BACKTRACK\nfor clause: ackermann(0, N, s(N))because of non-unification" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 270, 14.27/4.54 "to": 275, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 270, 14.27/4.54 "to": 276, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 275, 14.27/4.54 "to": 279, 14.27/4.54 "label": "ONLY EVAL with clause\nackermann(s(X58), 0, X59) :- ackermann(X58, s(0), X59).\nand substitutionT19 -> T26,\nX58 -> T26,\nX36 -> X60,\nX59 -> X60" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 276, 14.27/4.54 "to": 351, 14.27/4.54 "label": "BACKTRACK\nfor clause: ackermann(s(M), s(N), Val) :- ','(ackermann(s(M), N, Val1), ackermann(M, Val1, Val))because of non-unification" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 279, 14.27/4.54 "to": 284, 14.27/4.54 "label": "CASE" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 284, 14.27/4.54 "to": 285, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 284, 14.27/4.54 "to": 286, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 285, 14.27/4.54 "to": 307, 14.27/4.54 "label": "EVAL with clause\nackermann(0, X67, s(X67)).\nand substitutionT26 -> 0,\nX67 -> s(0),\nX60 -> s(s(0))" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 285, 14.27/4.54 "to": 308, 14.27/4.54 "label": "EVAL-BACKTRACK" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 286, 14.27/4.54 "to": 310, 14.27/4.54 "label": "BACKTRACK\nfor clause: ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val)because of non-unification" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 307, 14.27/4.54 "to": 309, 14.27/4.54 "label": "SUCCESS" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 310, 14.27/4.54 "to": 311, 14.27/4.54 "label": "EVAL with clause\nackermann(s(X79), s(X80), X81) :- ','(ackermann(s(X79), X80, X82), ackermann(X79, X82, X81)).\nand substitutionX79 -> T30,\nT26 -> s(T30),\nX80 -> 0,\nX60 -> X83,\nX81 -> X83" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 310, 14.27/4.54 "to": 312, 14.27/4.54 "label": "EVAL-BACKTRACK" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 311, 14.27/4.54 "to": 313, 14.27/4.54 "label": "SPLIT 1" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 311, 14.27/4.54 "to": 314, 14.27/4.54 "label": "SPLIT 2\nnew knowledge:\nT30 is ground\nT31 is ground\nreplacements:X82 -> T31" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 313, 14.27/4.54 "to": 264, 14.27/4.54 "label": "INSTANCE with matching:\nT19 -> T30\nX36 -> X82" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 314, 14.27/4.54 "to": 319, 14.27/4.54 "label": "CASE" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 319, 14.27/4.54 "to": 321, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 319, 14.27/4.54 "to": 322, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 321, 14.27/4.54 "to": 326, 14.27/4.54 "label": "EVAL with clause\nackermann(0, X94, s(X94)).\nand substitutionT30 -> 0,\nT31 -> T38,\nX94 -> T38,\nX83 -> s(T38)" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 321, 14.27/4.54 "to": 327, 14.27/4.54 "label": "EVAL-BACKTRACK" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 322, 14.27/4.54 "to": 331, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 322, 14.27/4.54 "to": 332, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 326, 14.27/4.54 "to": 328, 14.27/4.54 "label": "SUCCESS" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 331, 14.27/4.54 "to": 335, 14.27/4.54 "label": "EVAL with clause\nackermann(s(X107), 0, X108) :- ackermann(X107, s(0), X108).\nand substitutionX107 -> T43,\nT30 -> s(T43),\nT31 -> 0,\nX83 -> X109,\nX108 -> X109" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 331, 14.27/4.54 "to": 336, 14.27/4.54 "label": "EVAL-BACKTRACK" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 332, 14.27/4.54 "to": 341, 14.27/4.54 "label": "EVAL with clause\nackermann(s(X121), s(X122), X123) :- ','(ackermann(s(X121), X122, X124), ackermann(X121, X124, X123)).\nand substitutionX121 -> T48,\nT30 -> s(T48),\nX122 -> T49,\nT31 -> s(T49),\nX83 -> X125,\nX123 -> X125" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 332, 14.27/4.54 "to": 343, 14.27/4.54 "label": "EVAL-BACKTRACK" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 335, 14.27/4.54 "to": 279, 14.27/4.54 "label": "INSTANCE with matching:\nT26 -> T43\nX60 -> X109" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 341, 14.27/4.54 "to": 347, 14.27/4.54 "label": "SPLIT 1" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 341, 14.27/4.54 "to": 348, 14.27/4.54 "label": "SPLIT 2\nnew knowledge:\nT48 is ground\nT49 is ground\nT50 is ground\nreplacements:X124 -> T50" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 347, 14.27/4.54 "to": 314, 14.27/4.54 "label": "INSTANCE with matching:\nT30 -> s(T48)\nT31 -> T49\nX83 -> X124" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 348, 14.27/4.54 "to": 314, 14.27/4.54 "label": "INSTANCE with matching:\nT30 -> T48\nT31 -> T50\nX83 -> X125" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 354, 14.27/4.54 "to": 357, 14.27/4.54 "label": "EVAL with clause\nackermann(s(X144), s(X145), X146) :- ','(ackermann(s(X144), X145, X147), ackermann(X144, X147, X146)).\nand substitutionT8 -> T62,\nX144 -> T62,\nX145 -> T65,\nT2 -> s(T65),\nT9 -> T64,\nX146 -> T64,\nT63 -> T65" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 354, 14.27/4.54 "to": 358, 14.27/4.54 "label": "EVAL-BACKTRACK" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 357, 14.27/4.54 "to": 359, 14.27/4.54 "label": "SPLIT 1" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 357, 14.27/4.54 "to": 360, 14.27/4.54 "label": "SPLIT 2\nnew knowledge:\nT62 is ground\nreplacements:X147 -> T66" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 359, 14.27/4.54 "to": 385, 14.27/4.54 "label": "CASE" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 360, 14.27/4.54 "to": 8, 14.27/4.54 "label": "INSTANCE with matching:\nT1 -> T62\nT2 -> T66\nT3 -> T64" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 385, 14.27/4.54 "to": 386, 14.27/4.54 "label": "BACKTRACK\nfor clause: ackermann(0, N, s(N))because of non-unification" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 386, 14.27/4.54 "to": 387, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 386, 14.27/4.54 "to": 388, 14.27/4.54 "label": "PARALLEL" 14.27/4.54 }, 14.27/4.54 { 14.27/4.54 "from": 387, 14.27/4.54 "to": 413, 14.27/4.54 "label": "EVAL with clause\nackermann(s(X165), 0, X166) :- ackermann(X165, s(0), X166).\nand substitutionT62 -> T71,\nX165 -> T71,\nT65 -> 0,\nX147 -> X167,\nX166 -> X167" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 387, 14.27/4.55 "to": 414, 14.27/4.55 "label": "EVAL-BACKTRACK" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 388, 14.27/4.55 "to": 415, 14.27/4.55 "label": "EVAL with clause\nackermann(s(X179), s(X180), X181) :- ','(ackermann(s(X179), X180, X182), ackermann(X179, X182, X181)).\nand substitutionT62 -> T76,\nX179 -> T76,\nX180 -> T78,\nT65 -> s(T78),\nX147 -> X183,\nX181 -> X183,\nT77 -> T78" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 388, 14.27/4.55 "to": 416, 14.27/4.55 "label": "EVAL-BACKTRACK" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 413, 14.27/4.55 "to": 279, 14.27/4.55 "label": "INSTANCE with matching:\nT26 -> T71\nX60 -> X167" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 415, 14.27/4.55 "to": 417, 14.27/4.55 "label": "SPLIT 1" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 415, 14.27/4.55 "to": 418, 14.27/4.55 "label": "SPLIT 2\nnew knowledge:\nT76 is ground\nreplacements:X182 -> T79" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 417, 14.27/4.55 "to": 359, 14.27/4.55 "label": "INSTANCE with matching:\nT62 -> T76\nT65 -> T78\nX147 -> X182" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 418, 14.27/4.55 "to": 475, 14.27/4.55 "label": "CASE" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 475, 14.27/4.55 "to": 476, 14.27/4.55 "label": "PARALLEL" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 475, 14.27/4.55 "to": 477, 14.27/4.55 "label": "PARALLEL" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 476, 14.27/4.55 "to": 479, 14.27/4.55 "label": "EVAL with clause\nackermann(0, X194, s(X194)).\nand substitutionT76 -> 0,\nT79 -> T86,\nX194 -> T86,\nX183 -> s(T86)" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 476, 14.27/4.55 "to": 480, 14.27/4.55 "label": "EVAL-BACKTRACK" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 477, 14.27/4.55 "to": 484, 14.27/4.55 "label": "PARALLEL" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 477, 14.27/4.55 "to": 485, 14.27/4.55 "label": "PARALLEL" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 479, 14.27/4.55 "to": 481, 14.27/4.55 "label": "SUCCESS" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 484, 14.27/4.55 "to": 491, 14.27/4.55 "label": "EVAL with clause\nackermann(s(X207), 0, X208) :- ackermann(X207, s(0), X208).\nand substitutionX207 -> T91,\nT76 -> s(T91),\nT79 -> 0,\nX183 -> X209,\nX208 -> X209" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 484, 14.27/4.55 "to": 492, 14.27/4.55 "label": "EVAL-BACKTRACK" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 485, 14.27/4.55 "to": 495, 14.27/4.55 "label": "EVAL with clause\nackermann(s(X221), s(X222), X223) :- ','(ackermann(s(X221), X222, X224), ackermann(X221, X224, X223)).\nand substitutionX221 -> T96,\nT76 -> s(T96),\nX222 -> T98,\nT79 -> s(T98),\nX183 -> X225,\nX223 -> X225,\nT97 -> T98" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 485, 14.27/4.55 "to": 496, 14.27/4.55 "label": "EVAL-BACKTRACK" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 491, 14.27/4.55 "to": 279, 14.27/4.55 "label": "INSTANCE with matching:\nT26 -> T91\nX60 -> X209" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 495, 14.27/4.55 "to": 415, 14.27/4.55 "label": "INSTANCE with matching:\nT76 -> T96\nT78 -> T98\nX182 -> X224\nX183 -> X225" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 499, 14.27/4.55 "to": 501, 14.27/4.55 "label": "CASE" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 501, 14.27/4.55 "to": 502, 14.27/4.55 "label": "BACKTRACK\nfor clause: ackermann(0, N, s(N))because of non-unification" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 502, 14.27/4.55 "to": 503, 14.27/4.55 "label": "PARALLEL" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 502, 14.27/4.55 "to": 506, 14.27/4.55 "label": "PARALLEL" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 503, 14.27/4.55 "to": 507, 14.27/4.55 "label": "EVAL with clause\nackermann(s(X250), 0, X251) :- ackermann(X250, s(0), X251).\nand substitutionT104 -> T112,\nX250 -> T112,\nT107 -> 0,\nX236 -> X252,\nX251 -> X252" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 503, 14.27/4.55 "to": 508, 14.27/4.55 "label": "EVAL-BACKTRACK" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 506, 14.27/4.55 "to": 520, 14.27/4.55 "label": "EVAL with clause\nackermann(s(X268), s(X269), X270) :- ','(ackermann(s(X268), X269, X271), ackermann(X268, X271, X270)).\nand substitutionT104 -> T120,\nX268 -> T120,\nX269 -> T122,\nT107 -> s(T122),\nX236 -> X272,\nX270 -> X272,\nT121 -> T122" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 506, 14.27/4.55 "to": 521, 14.27/4.55 "label": "EVAL-BACKTRACK" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 507, 14.27/4.55 "to": 513, 14.27/4.55 "label": "SPLIT 1" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 507, 14.27/4.55 "to": 514, 14.27/4.55 "label": "SPLIT 2\nnew knowledge:\nT112 is ground\nT113 is ground\nreplacements:X252 -> T113" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 513, 14.27/4.55 "to": 279, 14.27/4.55 "label": "INSTANCE with matching:\nT26 -> T112\nX60 -> X252" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 514, 14.27/4.55 "to": 8, 14.27/4.55 "label": "INSTANCE with matching:\nT1 -> T112\nT2 -> T113\nT3 -> T106" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 520, 14.27/4.55 "to": 522, 14.27/4.55 "label": "SPLIT 1" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 520, 14.27/4.55 "to": 523, 14.27/4.55 "label": "SPLIT 2\nnew knowledge:\nT120 is ground\nreplacements:X271 -> T123,\nT2 -> T124" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 522, 14.27/4.55 "to": 359, 14.27/4.55 "label": "INSTANCE with matching:\nT62 -> T120\nT65 -> T122\nX147 -> X271" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 523, 14.27/4.55 "to": 524, 14.27/4.55 "label": "SPLIT 1" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 523, 14.27/4.55 "to": 525, 14.27/4.55 "label": "SPLIT 2\nnew knowledge:\nT120 is ground\nreplacements:X272 -> T127,\nT124 -> T128" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 524, 14.27/4.55 "to": 418, 14.27/4.55 "label": "INSTANCE with matching:\nT76 -> T120\nT79 -> T123\nX183 -> X272" 14.27/4.55 }, 14.27/4.55 { 14.27/4.55 "from": 525, 14.27/4.55 "to": 8, 14.27/4.55 "label": "INSTANCE with matching:\nT1 -> T120\nT2 -> T127\nT3 -> T106" 14.27/4.55 } 14.27/4.55 ], 14.27/4.55 "type": "Graph" 14.27/4.55 } 14.27/4.55 } 14.27/4.55 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (84) 14.27/4.55 Obligation: 14.27/4.55 Triples: 14.27/4.55 14.27/4.55 ackermannB(X1, X2) :- ackermannD(X1, X2). 14.27/4.55 ackermannD(s(X1), X2) :- ackermannB(X1, X3). 14.27/4.55 ackermannD(s(X1), X2) :- ','(ackermanncB(X1, X3), ackermannF(X1, X3, X2)). 14.27/4.55 ackermannF(s(X1), 0, X2) :- ackermannD(X1, X2). 14.27/4.55 ackermannF(s(X1), s(X2), X3) :- ackermannF(s(X1), X2, X4). 14.27/4.55 ackermannF(s(X1), s(X2), X3) :- ','(ackermanncF(s(X1), X2, X4), ackermannF(X1, X4, X3)). 14.27/4.55 ackermannC(X1, 0, X2) :- ackermannD(X1, X2). 14.27/4.55 ackermannC(X1, s(X2), X3) :- pG(X1, X2, X4, X3). 14.27/4.55 pG(X1, X2, X3, X4) :- ackermannC(X1, X2, X3). 14.27/4.55 pG(X1, X2, X3, X4) :- ','(ackermanncC(X1, X2, X3), ackermannE(X1, X3, X4)). 14.27/4.55 ackermannE(s(X1), 0, X2) :- ackermannD(X1, X2). 14.27/4.55 ackermannE(s(X1), s(X2), X3) :- pG(X1, X2, X4, X3). 14.27/4.55 ackermannA(s(s(X1)), 0, X2) :- ackermannB(X1, X3). 14.27/4.55 ackermannA(s(s(X1)), 0, X2) :- ','(ackermanncB(X1, X3), ackermannA(X1, X3, X2)). 14.27/4.55 ackermannA(s(X1), s(X2), X3) :- ackermannC(X1, X2, X4). 14.27/4.55 ackermannA(s(X1), s(X2), X3) :- ','(ackermanncC(X1, X2, X4), ackermannA(X1, X4, X3)). 14.27/4.55 ackermannA(s(X1), s(0), X2) :- ackermannD(X1, X3). 14.27/4.55 ackermannA(s(X1), s(0), X2) :- ','(ackermanncD(X1, X3), ackermannA(X1, X3, X2)). 14.27/4.55 ackermannA(s(X1), s(s(X2)), X3) :- ackermannC(X1, X2, X4). 14.27/4.55 ackermannA(s(X1), s(s(X2)), X3) :- ','(ackermanncC(X1, X2, X4), ackermannE(X1, X4, X5)). 14.27/4.55 ackermannA(s(X1), s(s(X2)), X3) :- ','(ackermanncC(X1, X2, X4), ','(ackermanncE(X1, X4, X5), ackermannA(X1, X5, X3))). 14.27/4.55 14.27/4.55 Clauses: 14.27/4.55 14.27/4.55 ackermanncA(0, X1, s(X1)). 14.27/4.55 ackermanncA(s(0), 0, s(s(0))). 14.27/4.55 ackermanncA(s(s(X1)), 0, X2) :- ','(ackermanncB(X1, X3), ackermanncA(X1, X3, X2)). 14.27/4.55 ackermanncA(s(X1), s(X2), X3) :- ','(ackermanncC(X1, X2, X4), ackermanncA(X1, X4, X3)). 14.27/4.55 ackermanncA(s(X1), s(0), X2) :- ','(ackermanncD(X1, X3), ackermanncA(X1, X3, X2)). 14.27/4.55 ackermanncA(s(X1), s(s(X2)), X3) :- ','(ackermanncC(X1, X2, X4), ','(ackermanncE(X1, X4, X5), ackermanncA(X1, X5, X3))). 14.27/4.55 ackermanncB(X1, X2) :- ackermanncD(X1, X2). 14.27/4.55 ackermanncD(0, s(s(0))). 14.27/4.55 ackermanncD(s(X1), X2) :- ','(ackermanncB(X1, X3), ackermanncF(X1, X3, X2)). 14.27/4.55 ackermanncF(0, X1, s(X1)). 14.27/4.55 ackermanncF(s(X1), 0, X2) :- ackermanncD(X1, X2). 14.27/4.55 ackermanncF(s(X1), s(X2), X3) :- ','(ackermanncF(s(X1), X2, X4), ackermanncF(X1, X4, X3)). 14.27/4.55 ackermanncC(X1, 0, X2) :- ackermanncD(X1, X2). 14.27/4.55 ackermanncC(X1, s(X2), X3) :- qcG(X1, X2, X4, X3). 14.27/4.55 qcG(X1, X2, X3, X4) :- ','(ackermanncC(X1, X2, X3), ackermanncE(X1, X3, X4)). 14.27/4.55 ackermanncE(0, X1, s(X1)). 14.27/4.55 ackermanncE(s(X1), 0, X2) :- ackermanncD(X1, X2). 14.27/4.55 ackermanncE(s(X1), s(X2), X3) :- qcG(X1, X2, X4, X3). 14.27/4.55 14.27/4.55 Afs: 14.27/4.55 14.27/4.55 ackermannA(x1, x2, x3) = ackermannA(x1, x3) 14.27/4.55 14.27/4.55 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (85) TriplesToPiDPProof (SOUND) 14.27/4.55 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 14.27/4.55 14.27/4.55 ackermannA_in_3: (b,f,b) (b,b,b) 14.27/4.55 14.27/4.55 ackermannB_in_2: (b,f) 14.27/4.55 14.27/4.55 ackermannD_in_2: (b,f) 14.27/4.55 14.27/4.55 ackermanncB_in_2: (b,f) 14.27/4.55 14.27/4.55 ackermanncD_in_2: (b,f) 14.27/4.55 14.27/4.55 ackermanncF_in_3: (b,b,f) 14.27/4.55 14.27/4.55 ackermannF_in_3: (b,b,f) 14.27/4.55 14.27/4.55 ackermannC_in_3: (b,b,f) (b,f,f) 14.27/4.55 14.27/4.55 pG_in_4: (b,b,f,f) (b,f,f,f) 14.27/4.55 14.27/4.55 ackermanncC_in_3: (b,b,f) (b,f,f) 14.27/4.55 14.27/4.55 qcG_in_4: (b,b,f,f) (b,f,f,f) 14.27/4.55 14.27/4.55 ackermanncE_in_3: (b,b,f) 14.27/4.55 14.27/4.55 ackermannE_in_3: (b,b,f) 14.27/4.55 14.27/4.55 Transforming TRIPLES into the following Term Rewriting System: 14.27/4.55 14.27/4.55 Pi DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 ACKERMANNA_IN_GAG(s(s(X1)), 0, X2) -> U16_GAG(X1, X2, ackermannB_in_ga(X1, X3)) 14.27/4.55 ACKERMANNA_IN_GAG(s(s(X1)), 0, X2) -> ACKERMANNB_IN_GA(X1, X3) 14.27/4.55 ACKERMANNB_IN_GA(X1, X2) -> U1_GA(X1, X2, ackermannD_in_ga(X1, X2)) 14.27/4.55 ACKERMANNB_IN_GA(X1, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANND_IN_GA(s(X1), X2) -> U2_GA(X1, X2, ackermannB_in_ga(X1, X3)) 14.27/4.55 ACKERMANND_IN_GA(s(X1), X2) -> ACKERMANNB_IN_GA(X1, X3) 14.27/4.55 ACKERMANND_IN_GA(s(X1), X2) -> U3_GA(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U3_GA(X1, X2, ackermanncB_out_ga(X1, X3)) -> U4_GA(X1, X2, ackermannF_in_gga(X1, X3, X2)) 14.27/4.55 U3_GA(X1, X2, ackermanncB_out_ga(X1, X3)) -> ACKERMANNF_IN_GGA(X1, X3, X2) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), 0, X2) -> U5_GGA(X1, X2, ackermannD_in_ga(X1, X2)) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), 0, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), s(X2), X3) -> U6_GGA(X1, X2, X3, ackermannF_in_gga(s(X1), X2, X4)) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), s(X2), X3) -> ACKERMANNF_IN_GGA(s(X1), X2, X4) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), s(X2), X3) -> U7_GGA(X1, X2, X3, ackermanncF_in_gga(s(X1), X2, X4)) 14.27/4.55 U7_GGA(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> U8_GGA(X1, X2, X3, ackermannF_in_gga(X1, X4, X3)) 14.27/4.55 U7_GGA(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> ACKERMANNF_IN_GGA(X1, X4, X3) 14.27/4.55 ACKERMANNA_IN_GAG(s(s(X1)), 0, X2) -> U17_GAG(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U17_GAG(X1, X2, ackermanncB_out_ga(X1, X3)) -> U18_GAG(X1, X2, ackermannA_in_ggg(X1, X3, X2)) 14.27/4.55 U17_GAG(X1, X2, ackermanncB_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.55 ACKERMANNA_IN_GGG(s(s(X1)), 0, X2) -> U16_GGG(X1, X2, ackermannB_in_ga(X1, X3)) 14.27/4.55 ACKERMANNA_IN_GGG(s(s(X1)), 0, X2) -> ACKERMANNB_IN_GA(X1, X3) 14.27/4.55 ACKERMANNA_IN_GGG(s(s(X1)), 0, X2) -> U17_GGG(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U17_GGG(X1, X2, ackermanncB_out_ga(X1, X3)) -> U18_GGG(X1, X2, ackermannA_in_ggg(X1, X3, X2)) 14.27/4.55 U17_GGG(X1, X2, ackermanncB_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(X2), X3) -> U19_GGG(X1, X2, X3, ackermannC_in_gga(X1, X2, X4)) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(X2), X3) -> ACKERMANNC_IN_GGA(X1, X2, X4) 14.27/4.55 ACKERMANNC_IN_GGA(X1, 0, X2) -> U9_GGA(X1, X2, ackermannD_in_ga(X1, X2)) 14.27/4.55 ACKERMANNC_IN_GGA(X1, 0, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANNC_IN_GGA(X1, s(X2), X3) -> U10_GGA(X1, X2, X3, pG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 ACKERMANNC_IN_GGA(X1, s(X2), X3) -> PG_IN_GGAA(X1, X2, X4, X3) 14.27/4.55 PG_IN_GGAA(X1, X2, X3, X4) -> U11_GGAA(X1, X2, X3, X4, ackermannC_in_gga(X1, X2, X3)) 14.27/4.55 PG_IN_GGAA(X1, X2, X3, X4) -> ACKERMANNC_IN_GGA(X1, X2, X3) 14.27/4.55 PG_IN_GGAA(X1, X2, X3, X4) -> U12_GGAA(X1, X2, X3, X4, ackermanncC_in_gga(X1, X2, X3)) 14.27/4.55 U12_GGAA(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> U13_GGAA(X1, X2, X3, X4, ackermannE_in_gga(X1, X3, X4)) 14.27/4.55 U12_GGAA(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> ACKERMANNE_IN_GGA(X1, X3, X4) 14.27/4.55 ACKERMANNE_IN_GGA(s(X1), 0, X2) -> U14_GGA(X1, X2, ackermannD_in_ga(X1, X2)) 14.27/4.55 ACKERMANNE_IN_GGA(s(X1), 0, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANNE_IN_GGA(s(X1), s(X2), X3) -> U15_GGA(X1, X2, X3, pG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 ACKERMANNE_IN_GGA(s(X1), s(X2), X3) -> PG_IN_GGAA(X1, X2, X4, X3) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(X2), X3) -> U20_GGG(X1, X2, X3, ackermanncC_in_gga(X1, X2, X4)) 14.27/4.55 U20_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> U21_GGG(X1, X2, X3, ackermannA_in_ggg(X1, X4, X3)) 14.27/4.55 U20_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> ACKERMANNA_IN_GGG(X1, X4, X3) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(0), X2) -> U22_GGG(X1, X2, ackermannD_in_ga(X1, X3)) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(0), X2) -> ACKERMANND_IN_GA(X1, X3) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(0), X2) -> U23_GGG(X1, X2, ackermanncD_in_ga(X1, X3)) 14.27/4.55 U23_GGG(X1, X2, ackermanncD_out_ga(X1, X3)) -> U24_GGG(X1, X2, ackermannA_in_ggg(X1, X3, X2)) 14.27/4.55 U23_GGG(X1, X2, ackermanncD_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(s(X2)), X3) -> U25_GGG(X1, X2, X3, ackermannC_in_gga(X1, X2, X4)) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(s(X2)), X3) -> ACKERMANNC_IN_GGA(X1, X2, X4) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(s(X2)), X3) -> U26_GGG(X1, X2, X3, ackermanncC_in_gga(X1, X2, X4)) 14.27/4.55 U26_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> U27_GGG(X1, X2, X3, ackermannE_in_gga(X1, X4, X5)) 14.27/4.55 U26_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> ACKERMANNE_IN_GGA(X1, X4, X5) 14.27/4.55 U26_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> U28_GGG(X1, X2, X3, ackermanncE_in_gga(X1, X4, X5)) 14.27/4.55 U28_GGG(X1, X2, X3, ackermanncE_out_gga(X1, X4, X5)) -> U29_GGG(X1, X2, X3, ackermannA_in_ggg(X1, X5, X3)) 14.27/4.55 U28_GGG(X1, X2, X3, ackermanncE_out_gga(X1, X4, X5)) -> ACKERMANNA_IN_GGG(X1, X5, X3) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(X2), X3) -> U19_GAG(X1, X2, X3, ackermannC_in_gaa(X1, X2, X4)) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(X2), X3) -> ACKERMANNC_IN_GAA(X1, X2, X4) 14.27/4.55 ACKERMANNC_IN_GAA(X1, 0, X2) -> U9_GAA(X1, X2, ackermannD_in_ga(X1, X2)) 14.27/4.55 ACKERMANNC_IN_GAA(X1, 0, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANNC_IN_GAA(X1, s(X2), X3) -> U10_GAA(X1, X2, X3, pG_in_gaaa(X1, X2, X4, X3)) 14.27/4.55 ACKERMANNC_IN_GAA(X1, s(X2), X3) -> PG_IN_GAAA(X1, X2, X4, X3) 14.27/4.55 PG_IN_GAAA(X1, X2, X3, X4) -> U11_GAAA(X1, X2, X3, X4, ackermannC_in_gaa(X1, X2, X3)) 14.27/4.55 PG_IN_GAAA(X1, X2, X3, X4) -> ACKERMANNC_IN_GAA(X1, X2, X3) 14.27/4.55 PG_IN_GAAA(X1, X2, X3, X4) -> U12_GAAA(X1, X2, X3, X4, ackermanncC_in_gaa(X1, X2, X3)) 14.27/4.55 U12_GAAA(X1, X2, X3, X4, ackermanncC_out_gaa(X1, X2, X3)) -> U13_GAAA(X1, X2, X3, X4, ackermannE_in_gga(X1, X3, X4)) 14.27/4.55 U12_GAAA(X1, X2, X3, X4, ackermanncC_out_gaa(X1, X2, X3)) -> ACKERMANNE_IN_GGA(X1, X3, X4) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(X2), X3) -> U20_GAG(X1, X2, X3, ackermanncC_in_gaa(X1, X2, X4)) 14.27/4.55 U20_GAG(X1, X2, X3, ackermanncC_out_gaa(X1, X2, X4)) -> U21_GAG(X1, X2, X3, ackermannA_in_ggg(X1, X4, X3)) 14.27/4.55 U20_GAG(X1, X2, X3, ackermanncC_out_gaa(X1, X2, X4)) -> ACKERMANNA_IN_GGG(X1, X4, X3) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(0), X2) -> U22_GAG(X1, X2, ackermannD_in_ga(X1, X3)) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(0), X2) -> ACKERMANND_IN_GA(X1, X3) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(0), X2) -> U23_GAG(X1, X2, ackermanncD_in_ga(X1, X3)) 14.27/4.55 U23_GAG(X1, X2, ackermanncD_out_ga(X1, X3)) -> U24_GAG(X1, X2, ackermannA_in_ggg(X1, X3, X2)) 14.27/4.55 U23_GAG(X1, X2, ackermanncD_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(s(X2)), X3) -> U25_GAG(X1, X2, X3, ackermannC_in_gaa(X1, X2, X4)) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(s(X2)), X3) -> ACKERMANNC_IN_GAA(X1, X2, X4) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(s(X2)), X3) -> U26_GAG(X1, X2, X3, ackermanncC_in_gaa(X1, X2, X4)) 14.27/4.55 U26_GAG(X1, X2, X3, ackermanncC_out_gaa(X1, X2, X4)) -> U27_GAG(X1, X2, X3, ackermannE_in_gga(X1, X4, X5)) 14.27/4.55 U26_GAG(X1, X2, X3, ackermanncC_out_gaa(X1, X2, X4)) -> ACKERMANNE_IN_GGA(X1, X4, X5) 14.27/4.55 U26_GAG(X1, X2, X3, ackermanncC_out_gaa(X1, X2, X4)) -> U28_GAG(X1, X2, X3, ackermanncE_in_gga(X1, X4, X5)) 14.27/4.55 U28_GAG(X1, X2, X3, ackermanncE_out_gga(X1, X4, X5)) -> U29_GAG(X1, X2, X3, ackermannA_in_ggg(X1, X5, X3)) 14.27/4.55 U28_GAG(X1, X2, X3, ackermanncE_out_gga(X1, X4, X5)) -> ACKERMANNA_IN_GGG(X1, X5, X3) 14.27/4.55 14.27/4.55 The TRS R consists of the following rules: 14.27/4.55 14.27/4.55 ackermanncB_in_ga(X1, X2) -> U40_ga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncD_in_ga(0, s(s(0))) -> ackermanncD_out_ga(0, s(s(0))) 14.27/4.55 ackermanncD_in_ga(s(X1), X2) -> U41_ga(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U41_ga(X1, X2, ackermanncB_out_ga(X1, X3)) -> U42_ga(X1, X2, ackermanncF_in_gga(X1, X3, X2)) 14.27/4.55 ackermanncF_in_gga(0, X1, s(X1)) -> ackermanncF_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncF_in_gga(s(X1), 0, X2) -> U43_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U43_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncF_out_gga(s(X1), 0, X2) 14.27/4.55 ackermanncF_in_gga(s(X1), s(X2), X3) -> U44_gga(X1, X2, X3, ackermanncF_in_gga(s(X1), X2, X4)) 14.27/4.55 U44_gga(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> U45_gga(X1, X2, X3, ackermanncF_in_gga(X1, X4, X3)) 14.27/4.55 U45_gga(X1, X2, X3, ackermanncF_out_gga(X1, X4, X3)) -> ackermanncF_out_gga(s(X1), s(X2), X3) 14.27/4.55 U42_ga(X1, X2, ackermanncF_out_gga(X1, X3, X2)) -> ackermanncD_out_ga(s(X1), X2) 14.27/4.55 U40_ga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncB_out_ga(X1, X2) 14.27/4.55 ackermanncC_in_gga(X1, 0, X2) -> U46_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U46_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gga(X1, 0, X2) 14.27/4.55 ackermanncC_in_gga(X1, s(X2), X3) -> U47_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 qcG_in_ggaa(X1, X2, X3, X4) -> U48_ggaa(X1, X2, X3, X4, ackermanncC_in_gga(X1, X2, X3)) 14.27/4.55 U48_ggaa(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> U49_ggaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 ackermanncE_in_gga(0, X1, s(X1)) -> ackermanncE_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncE_in_gga(s(X1), 0, X2) -> U50_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U50_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncE_out_gga(s(X1), 0, X2) 14.27/4.55 ackermanncE_in_gga(s(X1), s(X2), X3) -> U51_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 U51_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncE_out_gga(s(X1), s(X2), X3) 14.27/4.55 U49_ggaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_ggaa(X1, X2, X3, X4) 14.27/4.55 U47_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncC_out_gga(X1, s(X2), X3) 14.27/4.55 ackermanncC_in_gaa(X1, 0, X2) -> U46_gaa(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U46_gaa(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gaa(X1, 0, X2) 14.27/4.55 ackermanncC_in_gaa(X1, s(X2), X3) -> U47_gaa(X1, X2, X3, qcG_in_gaaa(X1, X2, X4, X3)) 14.27/4.55 qcG_in_gaaa(X1, X2, X3, X4) -> U48_gaaa(X1, X2, X3, X4, ackermanncC_in_gaa(X1, X2, X3)) 14.27/4.55 U48_gaaa(X1, X2, X3, X4, ackermanncC_out_gaa(X1, X2, X3)) -> U49_gaaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 U49_gaaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_gaaa(X1, X2, X3, X4) 14.27/4.55 U47_gaa(X1, X2, X3, qcG_out_gaaa(X1, X2, X4, X3)) -> ackermanncC_out_gaa(X1, s(X2), X3) 14.27/4.55 14.27/4.55 The argument filtering Pi contains the following mapping: 14.27/4.55 s(x1) = s(x1) 14.27/4.55 14.27/4.55 ackermannB_in_ga(x1, x2) = ackermannB_in_ga(x1) 14.27/4.55 14.27/4.55 ackermannD_in_ga(x1, x2) = ackermannD_in_ga(x1) 14.27/4.55 14.27/4.55 ackermanncB_in_ga(x1, x2) = ackermanncB_in_ga(x1) 14.27/4.55 14.27/4.55 U40_ga(x1, x2, x3) = U40_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncD_in_ga(x1, x2) = ackermanncD_in_ga(x1) 14.27/4.55 14.27/4.55 0 = 0 14.27/4.55 14.27/4.55 ackermanncD_out_ga(x1, x2) = ackermanncD_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U41_ga(x1, x2, x3) = U41_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncB_out_ga(x1, x2) = ackermanncB_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U42_ga(x1, x2, x3) = U42_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncF_in_gga(x1, x2, x3) = ackermanncF_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncF_out_gga(x1, x2, x3) = ackermanncF_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U43_gga(x1, x2, x3) = U43_gga(x1, x3) 14.27/4.55 14.27/4.55 U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 ackermannF_in_gga(x1, x2, x3) = ackermannF_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermannA_in_ggg(x1, x2, x3) = ackermannA_in_ggg(x1, x2, x3) 14.27/4.55 14.27/4.55 ackermannC_in_gga(x1, x2, x3) = ackermannC_in_gga(x1, x2) 14.27/4.55 14.27/4.55 pG_in_ggaa(x1, x2, x3, x4) = pG_in_ggaa(x1, x2) 14.27/4.55 14.27/4.55 ackermanncC_in_gga(x1, x2, x3) = ackermanncC_in_gga(x1, x2) 14.27/4.55 14.27/4.55 U46_gga(x1, x2, x3) = U46_gga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gga(x1, x2, x3) = ackermanncC_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_in_ggaa(x1, x2, x3, x4) = qcG_in_ggaa(x1, x2) 14.27/4.55 14.27/4.55 U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5) 14.27/4.55 14.27/4.55 U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 ackermanncE_in_gga(x1, x2, x3) = ackermanncE_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncE_out_gga(x1, x2, x3) = ackermanncE_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U50_gga(x1, x2, x3) = U50_gga(x1, x3) 14.27/4.55 14.27/4.55 U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_out_ggaa(x1, x2, x3, x4) = qcG_out_ggaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ackermannE_in_gga(x1, x2, x3) = ackermannE_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermannC_in_gaa(x1, x2, x3) = ackermannC_in_gaa(x1) 14.27/4.55 14.27/4.55 pG_in_gaaa(x1, x2, x3, x4) = pG_in_gaaa(x1) 14.27/4.55 14.27/4.55 ackermanncC_in_gaa(x1, x2, x3) = ackermanncC_in_gaa(x1) 14.27/4.55 14.27/4.55 U46_gaa(x1, x2, x3) = U46_gaa(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gaa(x1, x2, x3) = ackermanncC_out_gaa(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gaa(x1, x2, x3, x4) = U47_gaa(x1, x4) 14.27/4.55 14.27/4.55 qcG_in_gaaa(x1, x2, x3, x4) = qcG_in_gaaa(x1) 14.27/4.55 14.27/4.55 U48_gaaa(x1, x2, x3, x4, x5) = U48_gaaa(x1, x5) 14.27/4.55 14.27/4.55 U49_gaaa(x1, x2, x3, x4, x5) = U49_gaaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 qcG_out_gaaa(x1, x2, x3, x4) = qcG_out_gaaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ACKERMANNA_IN_GAG(x1, x2, x3) = ACKERMANNA_IN_GAG(x1, x3) 14.27/4.55 14.27/4.55 U16_GAG(x1, x2, x3) = U16_GAG(x1, x2, x3) 14.27/4.55 14.27/4.55 ACKERMANNB_IN_GA(x1, x2) = ACKERMANNB_IN_GA(x1) 14.27/4.55 14.27/4.55 U1_GA(x1, x2, x3) = U1_GA(x1, x3) 14.27/4.55 14.27/4.55 ACKERMANND_IN_GA(x1, x2) = ACKERMANND_IN_GA(x1) 14.27/4.55 14.27/4.55 U2_GA(x1, x2, x3) = U2_GA(x1, x3) 14.27/4.55 14.27/4.55 U3_GA(x1, x2, x3) = U3_GA(x1, x3) 14.27/4.55 14.27/4.55 U4_GA(x1, x2, x3) = U4_GA(x1, x3) 14.27/4.55 14.27/4.55 ACKERMANNF_IN_GGA(x1, x2, x3) = ACKERMANNF_IN_GGA(x1, x2) 14.27/4.55 14.27/4.55 U5_GGA(x1, x2, x3) = U5_GGA(x1, x3) 14.27/4.55 14.27/4.55 U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4) 14.27/4.55 14.27/4.55 U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4) 14.27/4.55 14.27/4.55 U8_GGA(x1, x2, x3, x4) = U8_GGA(x1, x2, x4) 14.27/4.55 14.27/4.55 U17_GAG(x1, x2, x3) = U17_GAG(x1, x2, x3) 14.27/4.55 14.27/4.55 U18_GAG(x1, x2, x3) = U18_GAG(x1, x2, x3) 14.27/4.55 14.27/4.55 ACKERMANNA_IN_GGG(x1, x2, x3) = ACKERMANNA_IN_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U16_GGG(x1, x2, x3) = U16_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U17_GGG(x1, x2, x3) = U17_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U18_GGG(x1, x2, x3) = U18_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U19_GGG(x1, x2, x3, x4) = U19_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ACKERMANNC_IN_GGA(x1, x2, x3) = ACKERMANNC_IN_GGA(x1, x2) 14.27/4.55 14.27/4.55 U9_GGA(x1, x2, x3) = U9_GGA(x1, x3) 14.27/4.55 14.27/4.55 U10_GGA(x1, x2, x3, x4) = U10_GGA(x1, x2, x4) 14.27/4.55 14.27/4.55 PG_IN_GGAA(x1, x2, x3, x4) = PG_IN_GGAA(x1, x2) 14.27/4.55 14.27/4.55 U11_GGAA(x1, x2, x3, x4, x5) = U11_GGAA(x1, x2, x5) 14.27/4.55 14.27/4.55 U12_GGAA(x1, x2, x3, x4, x5) = U12_GGAA(x1, x2, x5) 14.27/4.55 14.27/4.55 U13_GGAA(x1, x2, x3, x4, x5) = U13_GGAA(x1, x2, x5) 14.27/4.55 14.27/4.55 ACKERMANNE_IN_GGA(x1, x2, x3) = ACKERMANNE_IN_GGA(x1, x2) 14.27/4.55 14.27/4.55 U14_GGA(x1, x2, x3) = U14_GGA(x1, x3) 14.27/4.55 14.27/4.55 U15_GGA(x1, x2, x3, x4) = U15_GGA(x1, x2, x4) 14.27/4.55 14.27/4.55 U20_GGG(x1, x2, x3, x4) = U20_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U21_GGG(x1, x2, x3, x4) = U21_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U22_GGG(x1, x2, x3) = U22_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U23_GGG(x1, x2, x3) = U23_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U24_GGG(x1, x2, x3) = U24_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U25_GGG(x1, x2, x3, x4) = U25_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U26_GGG(x1, x2, x3, x4) = U26_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U27_GGG(x1, x2, x3, x4) = U27_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U28_GGG(x1, x2, x3, x4) = U28_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U29_GGG(x1, x2, x3, x4) = U29_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U19_GAG(x1, x2, x3, x4) = U19_GAG(x1, x3, x4) 14.27/4.55 14.27/4.55 ACKERMANNC_IN_GAA(x1, x2, x3) = ACKERMANNC_IN_GAA(x1) 14.27/4.55 14.27/4.55 U9_GAA(x1, x2, x3) = U9_GAA(x1, x3) 14.27/4.55 14.27/4.55 U10_GAA(x1, x2, x3, x4) = U10_GAA(x1, x4) 14.27/4.55 14.27/4.55 PG_IN_GAAA(x1, x2, x3, x4) = PG_IN_GAAA(x1) 14.27/4.55 14.27/4.55 U11_GAAA(x1, x2, x3, x4, x5) = U11_GAAA(x1, x5) 14.27/4.55 14.27/4.55 U12_GAAA(x1, x2, x3, x4, x5) = U12_GAAA(x1, x5) 14.27/4.55 14.27/4.55 U13_GAAA(x1, x2, x3, x4, x5) = U13_GAAA(x1, x2, x5) 14.27/4.55 14.27/4.55 U20_GAG(x1, x2, x3, x4) = U20_GAG(x1, x3, x4) 14.27/4.55 14.27/4.55 U21_GAG(x1, x2, x3, x4) = U21_GAG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U22_GAG(x1, x2, x3) = U22_GAG(x1, x2, x3) 14.27/4.55 14.27/4.55 U23_GAG(x1, x2, x3) = U23_GAG(x1, x2, x3) 14.27/4.55 14.27/4.55 U24_GAG(x1, x2, x3) = U24_GAG(x1, x2, x3) 14.27/4.55 14.27/4.55 U25_GAG(x1, x2, x3, x4) = U25_GAG(x1, x3, x4) 14.27/4.55 14.27/4.55 U26_GAG(x1, x2, x3, x4) = U26_GAG(x1, x3, x4) 14.27/4.55 14.27/4.55 U27_GAG(x1, x2, x3, x4) = U27_GAG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U28_GAG(x1, x2, x3, x4) = U28_GAG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U29_GAG(x1, x2, x3, x4) = U29_GAG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 14.27/4.55 We have to consider all (P,R,Pi)-chains 14.27/4.55 14.27/4.55 14.27/4.55 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 14.27/4.55 14.27/4.55 14.27/4.55 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (86) 14.27/4.55 Obligation: 14.27/4.55 Pi DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 ACKERMANNA_IN_GAG(s(s(X1)), 0, X2) -> U16_GAG(X1, X2, ackermannB_in_ga(X1, X3)) 14.27/4.55 ACKERMANNA_IN_GAG(s(s(X1)), 0, X2) -> ACKERMANNB_IN_GA(X1, X3) 14.27/4.55 ACKERMANNB_IN_GA(X1, X2) -> U1_GA(X1, X2, ackermannD_in_ga(X1, X2)) 14.27/4.55 ACKERMANNB_IN_GA(X1, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANND_IN_GA(s(X1), X2) -> U2_GA(X1, X2, ackermannB_in_ga(X1, X3)) 14.27/4.55 ACKERMANND_IN_GA(s(X1), X2) -> ACKERMANNB_IN_GA(X1, X3) 14.27/4.55 ACKERMANND_IN_GA(s(X1), X2) -> U3_GA(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U3_GA(X1, X2, ackermanncB_out_ga(X1, X3)) -> U4_GA(X1, X2, ackermannF_in_gga(X1, X3, X2)) 14.27/4.55 U3_GA(X1, X2, ackermanncB_out_ga(X1, X3)) -> ACKERMANNF_IN_GGA(X1, X3, X2) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), 0, X2) -> U5_GGA(X1, X2, ackermannD_in_ga(X1, X2)) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), 0, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), s(X2), X3) -> U6_GGA(X1, X2, X3, ackermannF_in_gga(s(X1), X2, X4)) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), s(X2), X3) -> ACKERMANNF_IN_GGA(s(X1), X2, X4) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), s(X2), X3) -> U7_GGA(X1, X2, X3, ackermanncF_in_gga(s(X1), X2, X4)) 14.27/4.55 U7_GGA(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> U8_GGA(X1, X2, X3, ackermannF_in_gga(X1, X4, X3)) 14.27/4.55 U7_GGA(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> ACKERMANNF_IN_GGA(X1, X4, X3) 14.27/4.55 ACKERMANNA_IN_GAG(s(s(X1)), 0, X2) -> U17_GAG(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U17_GAG(X1, X2, ackermanncB_out_ga(X1, X3)) -> U18_GAG(X1, X2, ackermannA_in_ggg(X1, X3, X2)) 14.27/4.55 U17_GAG(X1, X2, ackermanncB_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.55 ACKERMANNA_IN_GGG(s(s(X1)), 0, X2) -> U16_GGG(X1, X2, ackermannB_in_ga(X1, X3)) 14.27/4.55 ACKERMANNA_IN_GGG(s(s(X1)), 0, X2) -> ACKERMANNB_IN_GA(X1, X3) 14.27/4.55 ACKERMANNA_IN_GGG(s(s(X1)), 0, X2) -> U17_GGG(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U17_GGG(X1, X2, ackermanncB_out_ga(X1, X3)) -> U18_GGG(X1, X2, ackermannA_in_ggg(X1, X3, X2)) 14.27/4.55 U17_GGG(X1, X2, ackermanncB_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(X2), X3) -> U19_GGG(X1, X2, X3, ackermannC_in_gga(X1, X2, X4)) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(X2), X3) -> ACKERMANNC_IN_GGA(X1, X2, X4) 14.27/4.55 ACKERMANNC_IN_GGA(X1, 0, X2) -> U9_GGA(X1, X2, ackermannD_in_ga(X1, X2)) 14.27/4.55 ACKERMANNC_IN_GGA(X1, 0, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANNC_IN_GGA(X1, s(X2), X3) -> U10_GGA(X1, X2, X3, pG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 ACKERMANNC_IN_GGA(X1, s(X2), X3) -> PG_IN_GGAA(X1, X2, X4, X3) 14.27/4.55 PG_IN_GGAA(X1, X2, X3, X4) -> U11_GGAA(X1, X2, X3, X4, ackermannC_in_gga(X1, X2, X3)) 14.27/4.55 PG_IN_GGAA(X1, X2, X3, X4) -> ACKERMANNC_IN_GGA(X1, X2, X3) 14.27/4.55 PG_IN_GGAA(X1, X2, X3, X4) -> U12_GGAA(X1, X2, X3, X4, ackermanncC_in_gga(X1, X2, X3)) 14.27/4.55 U12_GGAA(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> U13_GGAA(X1, X2, X3, X4, ackermannE_in_gga(X1, X3, X4)) 14.27/4.55 U12_GGAA(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> ACKERMANNE_IN_GGA(X1, X3, X4) 14.27/4.55 ACKERMANNE_IN_GGA(s(X1), 0, X2) -> U14_GGA(X1, X2, ackermannD_in_ga(X1, X2)) 14.27/4.55 ACKERMANNE_IN_GGA(s(X1), 0, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANNE_IN_GGA(s(X1), s(X2), X3) -> U15_GGA(X1, X2, X3, pG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 ACKERMANNE_IN_GGA(s(X1), s(X2), X3) -> PG_IN_GGAA(X1, X2, X4, X3) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(X2), X3) -> U20_GGG(X1, X2, X3, ackermanncC_in_gga(X1, X2, X4)) 14.27/4.55 U20_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> U21_GGG(X1, X2, X3, ackermannA_in_ggg(X1, X4, X3)) 14.27/4.55 U20_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> ACKERMANNA_IN_GGG(X1, X4, X3) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(0), X2) -> U22_GGG(X1, X2, ackermannD_in_ga(X1, X3)) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(0), X2) -> ACKERMANND_IN_GA(X1, X3) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(0), X2) -> U23_GGG(X1, X2, ackermanncD_in_ga(X1, X3)) 14.27/4.55 U23_GGG(X1, X2, ackermanncD_out_ga(X1, X3)) -> U24_GGG(X1, X2, ackermannA_in_ggg(X1, X3, X2)) 14.27/4.55 U23_GGG(X1, X2, ackermanncD_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(s(X2)), X3) -> U25_GGG(X1, X2, X3, ackermannC_in_gga(X1, X2, X4)) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(s(X2)), X3) -> ACKERMANNC_IN_GGA(X1, X2, X4) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(s(X2)), X3) -> U26_GGG(X1, X2, X3, ackermanncC_in_gga(X1, X2, X4)) 14.27/4.55 U26_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> U27_GGG(X1, X2, X3, ackermannE_in_gga(X1, X4, X5)) 14.27/4.55 U26_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> ACKERMANNE_IN_GGA(X1, X4, X5) 14.27/4.55 U26_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> U28_GGG(X1, X2, X3, ackermanncE_in_gga(X1, X4, X5)) 14.27/4.55 U28_GGG(X1, X2, X3, ackermanncE_out_gga(X1, X4, X5)) -> U29_GGG(X1, X2, X3, ackermannA_in_ggg(X1, X5, X3)) 14.27/4.55 U28_GGG(X1, X2, X3, ackermanncE_out_gga(X1, X4, X5)) -> ACKERMANNA_IN_GGG(X1, X5, X3) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(X2), X3) -> U19_GAG(X1, X2, X3, ackermannC_in_gaa(X1, X2, X4)) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(X2), X3) -> ACKERMANNC_IN_GAA(X1, X2, X4) 14.27/4.55 ACKERMANNC_IN_GAA(X1, 0, X2) -> U9_GAA(X1, X2, ackermannD_in_ga(X1, X2)) 14.27/4.55 ACKERMANNC_IN_GAA(X1, 0, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANNC_IN_GAA(X1, s(X2), X3) -> U10_GAA(X1, X2, X3, pG_in_gaaa(X1, X2, X4, X3)) 14.27/4.55 ACKERMANNC_IN_GAA(X1, s(X2), X3) -> PG_IN_GAAA(X1, X2, X4, X3) 14.27/4.55 PG_IN_GAAA(X1, X2, X3, X4) -> U11_GAAA(X1, X2, X3, X4, ackermannC_in_gaa(X1, X2, X3)) 14.27/4.55 PG_IN_GAAA(X1, X2, X3, X4) -> ACKERMANNC_IN_GAA(X1, X2, X3) 14.27/4.55 PG_IN_GAAA(X1, X2, X3, X4) -> U12_GAAA(X1, X2, X3, X4, ackermanncC_in_gaa(X1, X2, X3)) 14.27/4.55 U12_GAAA(X1, X2, X3, X4, ackermanncC_out_gaa(X1, X2, X3)) -> U13_GAAA(X1, X2, X3, X4, ackermannE_in_gga(X1, X3, X4)) 14.27/4.55 U12_GAAA(X1, X2, X3, X4, ackermanncC_out_gaa(X1, X2, X3)) -> ACKERMANNE_IN_GGA(X1, X3, X4) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(X2), X3) -> U20_GAG(X1, X2, X3, ackermanncC_in_gaa(X1, X2, X4)) 14.27/4.55 U20_GAG(X1, X2, X3, ackermanncC_out_gaa(X1, X2, X4)) -> U21_GAG(X1, X2, X3, ackermannA_in_ggg(X1, X4, X3)) 14.27/4.55 U20_GAG(X1, X2, X3, ackermanncC_out_gaa(X1, X2, X4)) -> ACKERMANNA_IN_GGG(X1, X4, X3) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(0), X2) -> U22_GAG(X1, X2, ackermannD_in_ga(X1, X3)) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(0), X2) -> ACKERMANND_IN_GA(X1, X3) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(0), X2) -> U23_GAG(X1, X2, ackermanncD_in_ga(X1, X3)) 14.27/4.55 U23_GAG(X1, X2, ackermanncD_out_ga(X1, X3)) -> U24_GAG(X1, X2, ackermannA_in_ggg(X1, X3, X2)) 14.27/4.55 U23_GAG(X1, X2, ackermanncD_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(s(X2)), X3) -> U25_GAG(X1, X2, X3, ackermannC_in_gaa(X1, X2, X4)) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(s(X2)), X3) -> ACKERMANNC_IN_GAA(X1, X2, X4) 14.27/4.55 ACKERMANNA_IN_GAG(s(X1), s(s(X2)), X3) -> U26_GAG(X1, X2, X3, ackermanncC_in_gaa(X1, X2, X4)) 14.27/4.55 U26_GAG(X1, X2, X3, ackermanncC_out_gaa(X1, X2, X4)) -> U27_GAG(X1, X2, X3, ackermannE_in_gga(X1, X4, X5)) 14.27/4.55 U26_GAG(X1, X2, X3, ackermanncC_out_gaa(X1, X2, X4)) -> ACKERMANNE_IN_GGA(X1, X4, X5) 14.27/4.55 U26_GAG(X1, X2, X3, ackermanncC_out_gaa(X1, X2, X4)) -> U28_GAG(X1, X2, X3, ackermanncE_in_gga(X1, X4, X5)) 14.27/4.55 U28_GAG(X1, X2, X3, ackermanncE_out_gga(X1, X4, X5)) -> U29_GAG(X1, X2, X3, ackermannA_in_ggg(X1, X5, X3)) 14.27/4.55 U28_GAG(X1, X2, X3, ackermanncE_out_gga(X1, X4, X5)) -> ACKERMANNA_IN_GGG(X1, X5, X3) 14.27/4.55 14.27/4.55 The TRS R consists of the following rules: 14.27/4.55 14.27/4.55 ackermanncB_in_ga(X1, X2) -> U40_ga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncD_in_ga(0, s(s(0))) -> ackermanncD_out_ga(0, s(s(0))) 14.27/4.55 ackermanncD_in_ga(s(X1), X2) -> U41_ga(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U41_ga(X1, X2, ackermanncB_out_ga(X1, X3)) -> U42_ga(X1, X2, ackermanncF_in_gga(X1, X3, X2)) 14.27/4.55 ackermanncF_in_gga(0, X1, s(X1)) -> ackermanncF_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncF_in_gga(s(X1), 0, X2) -> U43_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U43_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncF_out_gga(s(X1), 0, X2) 14.27/4.55 ackermanncF_in_gga(s(X1), s(X2), X3) -> U44_gga(X1, X2, X3, ackermanncF_in_gga(s(X1), X2, X4)) 14.27/4.55 U44_gga(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> U45_gga(X1, X2, X3, ackermanncF_in_gga(X1, X4, X3)) 14.27/4.55 U45_gga(X1, X2, X3, ackermanncF_out_gga(X1, X4, X3)) -> ackermanncF_out_gga(s(X1), s(X2), X3) 14.27/4.55 U42_ga(X1, X2, ackermanncF_out_gga(X1, X3, X2)) -> ackermanncD_out_ga(s(X1), X2) 14.27/4.55 U40_ga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncB_out_ga(X1, X2) 14.27/4.55 ackermanncC_in_gga(X1, 0, X2) -> U46_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U46_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gga(X1, 0, X2) 14.27/4.55 ackermanncC_in_gga(X1, s(X2), X3) -> U47_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 qcG_in_ggaa(X1, X2, X3, X4) -> U48_ggaa(X1, X2, X3, X4, ackermanncC_in_gga(X1, X2, X3)) 14.27/4.55 U48_ggaa(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> U49_ggaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 ackermanncE_in_gga(0, X1, s(X1)) -> ackermanncE_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncE_in_gga(s(X1), 0, X2) -> U50_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U50_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncE_out_gga(s(X1), 0, X2) 14.27/4.55 ackermanncE_in_gga(s(X1), s(X2), X3) -> U51_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 U51_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncE_out_gga(s(X1), s(X2), X3) 14.27/4.55 U49_ggaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_ggaa(X1, X2, X3, X4) 14.27/4.55 U47_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncC_out_gga(X1, s(X2), X3) 14.27/4.55 ackermanncC_in_gaa(X1, 0, X2) -> U46_gaa(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U46_gaa(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gaa(X1, 0, X2) 14.27/4.55 ackermanncC_in_gaa(X1, s(X2), X3) -> U47_gaa(X1, X2, X3, qcG_in_gaaa(X1, X2, X4, X3)) 14.27/4.55 qcG_in_gaaa(X1, X2, X3, X4) -> U48_gaaa(X1, X2, X3, X4, ackermanncC_in_gaa(X1, X2, X3)) 14.27/4.55 U48_gaaa(X1, X2, X3, X4, ackermanncC_out_gaa(X1, X2, X3)) -> U49_gaaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 U49_gaaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_gaaa(X1, X2, X3, X4) 14.27/4.55 U47_gaa(X1, X2, X3, qcG_out_gaaa(X1, X2, X4, X3)) -> ackermanncC_out_gaa(X1, s(X2), X3) 14.27/4.55 14.27/4.55 The argument filtering Pi contains the following mapping: 14.27/4.55 s(x1) = s(x1) 14.27/4.55 14.27/4.55 ackermannB_in_ga(x1, x2) = ackermannB_in_ga(x1) 14.27/4.55 14.27/4.55 ackermannD_in_ga(x1, x2) = ackermannD_in_ga(x1) 14.27/4.55 14.27/4.55 ackermanncB_in_ga(x1, x2) = ackermanncB_in_ga(x1) 14.27/4.55 14.27/4.55 U40_ga(x1, x2, x3) = U40_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncD_in_ga(x1, x2) = ackermanncD_in_ga(x1) 14.27/4.55 14.27/4.55 0 = 0 14.27/4.55 14.27/4.55 ackermanncD_out_ga(x1, x2) = ackermanncD_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U41_ga(x1, x2, x3) = U41_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncB_out_ga(x1, x2) = ackermanncB_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U42_ga(x1, x2, x3) = U42_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncF_in_gga(x1, x2, x3) = ackermanncF_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncF_out_gga(x1, x2, x3) = ackermanncF_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U43_gga(x1, x2, x3) = U43_gga(x1, x3) 14.27/4.55 14.27/4.55 U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 ackermannF_in_gga(x1, x2, x3) = ackermannF_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermannA_in_ggg(x1, x2, x3) = ackermannA_in_ggg(x1, x2, x3) 14.27/4.55 14.27/4.55 ackermannC_in_gga(x1, x2, x3) = ackermannC_in_gga(x1, x2) 14.27/4.55 14.27/4.55 pG_in_ggaa(x1, x2, x3, x4) = pG_in_ggaa(x1, x2) 14.27/4.55 14.27/4.55 ackermanncC_in_gga(x1, x2, x3) = ackermanncC_in_gga(x1, x2) 14.27/4.55 14.27/4.55 U46_gga(x1, x2, x3) = U46_gga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gga(x1, x2, x3) = ackermanncC_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_in_ggaa(x1, x2, x3, x4) = qcG_in_ggaa(x1, x2) 14.27/4.55 14.27/4.55 U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5) 14.27/4.55 14.27/4.55 U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 ackermanncE_in_gga(x1, x2, x3) = ackermanncE_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncE_out_gga(x1, x2, x3) = ackermanncE_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U50_gga(x1, x2, x3) = U50_gga(x1, x3) 14.27/4.55 14.27/4.55 U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_out_ggaa(x1, x2, x3, x4) = qcG_out_ggaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ackermannE_in_gga(x1, x2, x3) = ackermannE_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermannC_in_gaa(x1, x2, x3) = ackermannC_in_gaa(x1) 14.27/4.55 14.27/4.55 pG_in_gaaa(x1, x2, x3, x4) = pG_in_gaaa(x1) 14.27/4.55 14.27/4.55 ackermanncC_in_gaa(x1, x2, x3) = ackermanncC_in_gaa(x1) 14.27/4.55 14.27/4.55 U46_gaa(x1, x2, x3) = U46_gaa(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gaa(x1, x2, x3) = ackermanncC_out_gaa(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gaa(x1, x2, x3, x4) = U47_gaa(x1, x4) 14.27/4.55 14.27/4.55 qcG_in_gaaa(x1, x2, x3, x4) = qcG_in_gaaa(x1) 14.27/4.55 14.27/4.55 U48_gaaa(x1, x2, x3, x4, x5) = U48_gaaa(x1, x5) 14.27/4.55 14.27/4.55 U49_gaaa(x1, x2, x3, x4, x5) = U49_gaaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 qcG_out_gaaa(x1, x2, x3, x4) = qcG_out_gaaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ACKERMANNA_IN_GAG(x1, x2, x3) = ACKERMANNA_IN_GAG(x1, x3) 14.27/4.55 14.27/4.55 U16_GAG(x1, x2, x3) = U16_GAG(x1, x2, x3) 14.27/4.55 14.27/4.55 ACKERMANNB_IN_GA(x1, x2) = ACKERMANNB_IN_GA(x1) 14.27/4.55 14.27/4.55 U1_GA(x1, x2, x3) = U1_GA(x1, x3) 14.27/4.55 14.27/4.55 ACKERMANND_IN_GA(x1, x2) = ACKERMANND_IN_GA(x1) 14.27/4.55 14.27/4.55 U2_GA(x1, x2, x3) = U2_GA(x1, x3) 14.27/4.55 14.27/4.55 U3_GA(x1, x2, x3) = U3_GA(x1, x3) 14.27/4.55 14.27/4.55 U4_GA(x1, x2, x3) = U4_GA(x1, x3) 14.27/4.55 14.27/4.55 ACKERMANNF_IN_GGA(x1, x2, x3) = ACKERMANNF_IN_GGA(x1, x2) 14.27/4.55 14.27/4.55 U5_GGA(x1, x2, x3) = U5_GGA(x1, x3) 14.27/4.55 14.27/4.55 U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4) 14.27/4.55 14.27/4.55 U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4) 14.27/4.55 14.27/4.55 U8_GGA(x1, x2, x3, x4) = U8_GGA(x1, x2, x4) 14.27/4.55 14.27/4.55 U17_GAG(x1, x2, x3) = U17_GAG(x1, x2, x3) 14.27/4.55 14.27/4.55 U18_GAG(x1, x2, x3) = U18_GAG(x1, x2, x3) 14.27/4.55 14.27/4.55 ACKERMANNA_IN_GGG(x1, x2, x3) = ACKERMANNA_IN_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U16_GGG(x1, x2, x3) = U16_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U17_GGG(x1, x2, x3) = U17_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U18_GGG(x1, x2, x3) = U18_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U19_GGG(x1, x2, x3, x4) = U19_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ACKERMANNC_IN_GGA(x1, x2, x3) = ACKERMANNC_IN_GGA(x1, x2) 14.27/4.55 14.27/4.55 U9_GGA(x1, x2, x3) = U9_GGA(x1, x3) 14.27/4.55 14.27/4.55 U10_GGA(x1, x2, x3, x4) = U10_GGA(x1, x2, x4) 14.27/4.55 14.27/4.55 PG_IN_GGAA(x1, x2, x3, x4) = PG_IN_GGAA(x1, x2) 14.27/4.55 14.27/4.55 U11_GGAA(x1, x2, x3, x4, x5) = U11_GGAA(x1, x2, x5) 14.27/4.55 14.27/4.55 U12_GGAA(x1, x2, x3, x4, x5) = U12_GGAA(x1, x2, x5) 14.27/4.55 14.27/4.55 U13_GGAA(x1, x2, x3, x4, x5) = U13_GGAA(x1, x2, x5) 14.27/4.55 14.27/4.55 ACKERMANNE_IN_GGA(x1, x2, x3) = ACKERMANNE_IN_GGA(x1, x2) 14.27/4.55 14.27/4.55 U14_GGA(x1, x2, x3) = U14_GGA(x1, x3) 14.27/4.55 14.27/4.55 U15_GGA(x1, x2, x3, x4) = U15_GGA(x1, x2, x4) 14.27/4.55 14.27/4.55 U20_GGG(x1, x2, x3, x4) = U20_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U21_GGG(x1, x2, x3, x4) = U21_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U22_GGG(x1, x2, x3) = U22_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U23_GGG(x1, x2, x3) = U23_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U24_GGG(x1, x2, x3) = U24_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U25_GGG(x1, x2, x3, x4) = U25_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U26_GGG(x1, x2, x3, x4) = U26_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U27_GGG(x1, x2, x3, x4) = U27_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U28_GGG(x1, x2, x3, x4) = U28_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U29_GGG(x1, x2, x3, x4) = U29_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U19_GAG(x1, x2, x3, x4) = U19_GAG(x1, x3, x4) 14.27/4.55 14.27/4.55 ACKERMANNC_IN_GAA(x1, x2, x3) = ACKERMANNC_IN_GAA(x1) 14.27/4.55 14.27/4.55 U9_GAA(x1, x2, x3) = U9_GAA(x1, x3) 14.27/4.55 14.27/4.55 U10_GAA(x1, x2, x3, x4) = U10_GAA(x1, x4) 14.27/4.55 14.27/4.55 PG_IN_GAAA(x1, x2, x3, x4) = PG_IN_GAAA(x1) 14.27/4.55 14.27/4.55 U11_GAAA(x1, x2, x3, x4, x5) = U11_GAAA(x1, x5) 14.27/4.55 14.27/4.55 U12_GAAA(x1, x2, x3, x4, x5) = U12_GAAA(x1, x5) 14.27/4.55 14.27/4.55 U13_GAAA(x1, x2, x3, x4, x5) = U13_GAAA(x1, x2, x5) 14.27/4.55 14.27/4.55 U20_GAG(x1, x2, x3, x4) = U20_GAG(x1, x3, x4) 14.27/4.55 14.27/4.55 U21_GAG(x1, x2, x3, x4) = U21_GAG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U22_GAG(x1, x2, x3) = U22_GAG(x1, x2, x3) 14.27/4.55 14.27/4.55 U23_GAG(x1, x2, x3) = U23_GAG(x1, x2, x3) 14.27/4.55 14.27/4.55 U24_GAG(x1, x2, x3) = U24_GAG(x1, x2, x3) 14.27/4.55 14.27/4.55 U25_GAG(x1, x2, x3, x4) = U25_GAG(x1, x3, x4) 14.27/4.55 14.27/4.55 U26_GAG(x1, x2, x3, x4) = U26_GAG(x1, x3, x4) 14.27/4.55 14.27/4.55 U27_GAG(x1, x2, x3, x4) = U27_GAG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U28_GAG(x1, x2, x3, x4) = U28_GAG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U29_GAG(x1, x2, x3, x4) = U29_GAG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 14.27/4.55 We have to consider all (P,R,Pi)-chains 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (87) DependencyGraphProof (EQUIVALENT) 14.27/4.55 The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 58 less nodes. 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (88) 14.27/4.55 Complex Obligation (AND) 14.27/4.55 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (89) 14.27/4.55 Obligation: 14.27/4.55 Pi DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 ACKERMANNB_IN_GA(X1, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANND_IN_GA(s(X1), X2) -> ACKERMANNB_IN_GA(X1, X3) 14.27/4.55 ACKERMANND_IN_GA(s(X1), X2) -> U3_GA(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U3_GA(X1, X2, ackermanncB_out_ga(X1, X3)) -> ACKERMANNF_IN_GGA(X1, X3, X2) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), 0, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), s(X2), X3) -> ACKERMANNF_IN_GGA(s(X1), X2, X4) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), s(X2), X3) -> U7_GGA(X1, X2, X3, ackermanncF_in_gga(s(X1), X2, X4)) 14.27/4.55 U7_GGA(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> ACKERMANNF_IN_GGA(X1, X4, X3) 14.27/4.55 14.27/4.55 The TRS R consists of the following rules: 14.27/4.55 14.27/4.55 ackermanncB_in_ga(X1, X2) -> U40_ga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncD_in_ga(0, s(s(0))) -> ackermanncD_out_ga(0, s(s(0))) 14.27/4.55 ackermanncD_in_ga(s(X1), X2) -> U41_ga(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U41_ga(X1, X2, ackermanncB_out_ga(X1, X3)) -> U42_ga(X1, X2, ackermanncF_in_gga(X1, X3, X2)) 14.27/4.55 ackermanncF_in_gga(0, X1, s(X1)) -> ackermanncF_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncF_in_gga(s(X1), 0, X2) -> U43_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U43_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncF_out_gga(s(X1), 0, X2) 14.27/4.55 ackermanncF_in_gga(s(X1), s(X2), X3) -> U44_gga(X1, X2, X3, ackermanncF_in_gga(s(X1), X2, X4)) 14.27/4.55 U44_gga(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> U45_gga(X1, X2, X3, ackermanncF_in_gga(X1, X4, X3)) 14.27/4.55 U45_gga(X1, X2, X3, ackermanncF_out_gga(X1, X4, X3)) -> ackermanncF_out_gga(s(X1), s(X2), X3) 14.27/4.55 U42_ga(X1, X2, ackermanncF_out_gga(X1, X3, X2)) -> ackermanncD_out_ga(s(X1), X2) 14.27/4.55 U40_ga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncB_out_ga(X1, X2) 14.27/4.55 ackermanncC_in_gga(X1, 0, X2) -> U46_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U46_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gga(X1, 0, X2) 14.27/4.55 ackermanncC_in_gga(X1, s(X2), X3) -> U47_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 qcG_in_ggaa(X1, X2, X3, X4) -> U48_ggaa(X1, X2, X3, X4, ackermanncC_in_gga(X1, X2, X3)) 14.27/4.55 U48_ggaa(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> U49_ggaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 ackermanncE_in_gga(0, X1, s(X1)) -> ackermanncE_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncE_in_gga(s(X1), 0, X2) -> U50_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U50_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncE_out_gga(s(X1), 0, X2) 14.27/4.55 ackermanncE_in_gga(s(X1), s(X2), X3) -> U51_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 U51_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncE_out_gga(s(X1), s(X2), X3) 14.27/4.55 U49_ggaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_ggaa(X1, X2, X3, X4) 14.27/4.55 U47_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncC_out_gga(X1, s(X2), X3) 14.27/4.55 ackermanncC_in_gaa(X1, 0, X2) -> U46_gaa(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U46_gaa(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gaa(X1, 0, X2) 14.27/4.55 ackermanncC_in_gaa(X1, s(X2), X3) -> U47_gaa(X1, X2, X3, qcG_in_gaaa(X1, X2, X4, X3)) 14.27/4.55 qcG_in_gaaa(X1, X2, X3, X4) -> U48_gaaa(X1, X2, X3, X4, ackermanncC_in_gaa(X1, X2, X3)) 14.27/4.55 U48_gaaa(X1, X2, X3, X4, ackermanncC_out_gaa(X1, X2, X3)) -> U49_gaaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 U49_gaaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_gaaa(X1, X2, X3, X4) 14.27/4.55 U47_gaa(X1, X2, X3, qcG_out_gaaa(X1, X2, X4, X3)) -> ackermanncC_out_gaa(X1, s(X2), X3) 14.27/4.55 14.27/4.55 The argument filtering Pi contains the following mapping: 14.27/4.55 s(x1) = s(x1) 14.27/4.55 14.27/4.55 ackermanncB_in_ga(x1, x2) = ackermanncB_in_ga(x1) 14.27/4.55 14.27/4.55 U40_ga(x1, x2, x3) = U40_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncD_in_ga(x1, x2) = ackermanncD_in_ga(x1) 14.27/4.55 14.27/4.55 0 = 0 14.27/4.55 14.27/4.55 ackermanncD_out_ga(x1, x2) = ackermanncD_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U41_ga(x1, x2, x3) = U41_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncB_out_ga(x1, x2) = ackermanncB_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U42_ga(x1, x2, x3) = U42_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncF_in_gga(x1, x2, x3) = ackermanncF_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncF_out_gga(x1, x2, x3) = ackermanncF_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U43_gga(x1, x2, x3) = U43_gga(x1, x3) 14.27/4.55 14.27/4.55 U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 ackermanncC_in_gga(x1, x2, x3) = ackermanncC_in_gga(x1, x2) 14.27/4.55 14.27/4.55 U46_gga(x1, x2, x3) = U46_gga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gga(x1, x2, x3) = ackermanncC_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_in_ggaa(x1, x2, x3, x4) = qcG_in_ggaa(x1, x2) 14.27/4.55 14.27/4.55 U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5) 14.27/4.55 14.27/4.55 U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 ackermanncE_in_gga(x1, x2, x3) = ackermanncE_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncE_out_gga(x1, x2, x3) = ackermanncE_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U50_gga(x1, x2, x3) = U50_gga(x1, x3) 14.27/4.55 14.27/4.55 U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_out_ggaa(x1, x2, x3, x4) = qcG_out_ggaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ackermanncC_in_gaa(x1, x2, x3) = ackermanncC_in_gaa(x1) 14.27/4.55 14.27/4.55 U46_gaa(x1, x2, x3) = U46_gaa(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gaa(x1, x2, x3) = ackermanncC_out_gaa(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gaa(x1, x2, x3, x4) = U47_gaa(x1, x4) 14.27/4.55 14.27/4.55 qcG_in_gaaa(x1, x2, x3, x4) = qcG_in_gaaa(x1) 14.27/4.55 14.27/4.55 U48_gaaa(x1, x2, x3, x4, x5) = U48_gaaa(x1, x5) 14.27/4.55 14.27/4.55 U49_gaaa(x1, x2, x3, x4, x5) = U49_gaaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 qcG_out_gaaa(x1, x2, x3, x4) = qcG_out_gaaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ACKERMANNB_IN_GA(x1, x2) = ACKERMANNB_IN_GA(x1) 14.27/4.55 14.27/4.55 ACKERMANND_IN_GA(x1, x2) = ACKERMANND_IN_GA(x1) 14.27/4.55 14.27/4.55 U3_GA(x1, x2, x3) = U3_GA(x1, x3) 14.27/4.55 14.27/4.55 ACKERMANNF_IN_GGA(x1, x2, x3) = ACKERMANNF_IN_GGA(x1, x2) 14.27/4.55 14.27/4.55 U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4) 14.27/4.55 14.27/4.55 14.27/4.55 We have to consider all (P,R,Pi)-chains 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (90) UsableRulesProof (EQUIVALENT) 14.27/4.55 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (91) 14.27/4.55 Obligation: 14.27/4.55 Pi DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 ACKERMANNB_IN_GA(X1, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANND_IN_GA(s(X1), X2) -> ACKERMANNB_IN_GA(X1, X3) 14.27/4.55 ACKERMANND_IN_GA(s(X1), X2) -> U3_GA(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U3_GA(X1, X2, ackermanncB_out_ga(X1, X3)) -> ACKERMANNF_IN_GGA(X1, X3, X2) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), 0, X2) -> ACKERMANND_IN_GA(X1, X2) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), s(X2), X3) -> ACKERMANNF_IN_GGA(s(X1), X2, X4) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), s(X2), X3) -> U7_GGA(X1, X2, X3, ackermanncF_in_gga(s(X1), X2, X4)) 14.27/4.55 U7_GGA(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> ACKERMANNF_IN_GGA(X1, X4, X3) 14.27/4.55 14.27/4.55 The TRS R consists of the following rules: 14.27/4.55 14.27/4.55 ackermanncB_in_ga(X1, X2) -> U40_ga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncF_in_gga(s(X1), 0, X2) -> U43_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncF_in_gga(s(X1), s(X2), X3) -> U44_gga(X1, X2, X3, ackermanncF_in_gga(s(X1), X2, X4)) 14.27/4.55 U40_ga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncB_out_ga(X1, X2) 14.27/4.55 U43_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncF_out_gga(s(X1), 0, X2) 14.27/4.55 U44_gga(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> U45_gga(X1, X2, X3, ackermanncF_in_gga(X1, X4, X3)) 14.27/4.55 ackermanncD_in_ga(0, s(s(0))) -> ackermanncD_out_ga(0, s(s(0))) 14.27/4.55 ackermanncD_in_ga(s(X1), X2) -> U41_ga(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U45_gga(X1, X2, X3, ackermanncF_out_gga(X1, X4, X3)) -> ackermanncF_out_gga(s(X1), s(X2), X3) 14.27/4.55 U41_ga(X1, X2, ackermanncB_out_ga(X1, X3)) -> U42_ga(X1, X2, ackermanncF_in_gga(X1, X3, X2)) 14.27/4.55 ackermanncF_in_gga(0, X1, s(X1)) -> ackermanncF_out_gga(0, X1, s(X1)) 14.27/4.55 U42_ga(X1, X2, ackermanncF_out_gga(X1, X3, X2)) -> ackermanncD_out_ga(s(X1), X2) 14.27/4.55 14.27/4.55 The argument filtering Pi contains the following mapping: 14.27/4.55 s(x1) = s(x1) 14.27/4.55 14.27/4.55 ackermanncB_in_ga(x1, x2) = ackermanncB_in_ga(x1) 14.27/4.55 14.27/4.55 U40_ga(x1, x2, x3) = U40_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncD_in_ga(x1, x2) = ackermanncD_in_ga(x1) 14.27/4.55 14.27/4.55 0 = 0 14.27/4.55 14.27/4.55 ackermanncD_out_ga(x1, x2) = ackermanncD_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U41_ga(x1, x2, x3) = U41_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncB_out_ga(x1, x2) = ackermanncB_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U42_ga(x1, x2, x3) = U42_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncF_in_gga(x1, x2, x3) = ackermanncF_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncF_out_gga(x1, x2, x3) = ackermanncF_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U43_gga(x1, x2, x3) = U43_gga(x1, x3) 14.27/4.55 14.27/4.55 U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 ACKERMANNB_IN_GA(x1, x2) = ACKERMANNB_IN_GA(x1) 14.27/4.55 14.27/4.55 ACKERMANND_IN_GA(x1, x2) = ACKERMANND_IN_GA(x1) 14.27/4.55 14.27/4.55 U3_GA(x1, x2, x3) = U3_GA(x1, x3) 14.27/4.55 14.27/4.55 ACKERMANNF_IN_GGA(x1, x2, x3) = ACKERMANNF_IN_GGA(x1, x2) 14.27/4.55 14.27/4.55 U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4) 14.27/4.55 14.27/4.55 14.27/4.55 We have to consider all (P,R,Pi)-chains 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (92) PiDPToQDPProof (SOUND) 14.27/4.55 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (93) 14.27/4.55 Obligation: 14.27/4.55 Q DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 ACKERMANNB_IN_GA(X1) -> ACKERMANND_IN_GA(X1) 14.27/4.55 ACKERMANND_IN_GA(s(X1)) -> ACKERMANNB_IN_GA(X1) 14.27/4.55 ACKERMANND_IN_GA(s(X1)) -> U3_GA(X1, ackermanncB_in_ga(X1)) 14.27/4.55 U3_GA(X1, ackermanncB_out_ga(X1, X3)) -> ACKERMANNF_IN_GGA(X1, X3) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), 0) -> ACKERMANND_IN_GA(X1) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), s(X2)) -> ACKERMANNF_IN_GGA(s(X1), X2) 14.27/4.55 ACKERMANNF_IN_GGA(s(X1), s(X2)) -> U7_GGA(X1, X2, ackermanncF_in_gga(s(X1), X2)) 14.27/4.55 U7_GGA(X1, X2, ackermanncF_out_gga(s(X1), X2, X4)) -> ACKERMANNF_IN_GGA(X1, X4) 14.27/4.55 14.27/4.55 The TRS R consists of the following rules: 14.27/4.55 14.27/4.55 ackermanncB_in_ga(X1) -> U40_ga(X1, ackermanncD_in_ga(X1)) 14.27/4.55 ackermanncF_in_gga(s(X1), 0) -> U43_gga(X1, ackermanncD_in_ga(X1)) 14.27/4.55 ackermanncF_in_gga(s(X1), s(X2)) -> U44_gga(X1, X2, ackermanncF_in_gga(s(X1), X2)) 14.27/4.55 U40_ga(X1, ackermanncD_out_ga(X1, X2)) -> ackermanncB_out_ga(X1, X2) 14.27/4.55 U43_gga(X1, ackermanncD_out_ga(X1, X2)) -> ackermanncF_out_gga(s(X1), 0, X2) 14.27/4.55 U44_gga(X1, X2, ackermanncF_out_gga(s(X1), X2, X4)) -> U45_gga(X1, X2, ackermanncF_in_gga(X1, X4)) 14.27/4.55 ackermanncD_in_ga(0) -> ackermanncD_out_ga(0, s(s(0))) 14.27/4.55 ackermanncD_in_ga(s(X1)) -> U41_ga(X1, ackermanncB_in_ga(X1)) 14.27/4.55 U45_gga(X1, X2, ackermanncF_out_gga(X1, X4, X3)) -> ackermanncF_out_gga(s(X1), s(X2), X3) 14.27/4.55 U41_ga(X1, ackermanncB_out_ga(X1, X3)) -> U42_ga(X1, ackermanncF_in_gga(X1, X3)) 14.27/4.55 ackermanncF_in_gga(0, X1) -> ackermanncF_out_gga(0, X1, s(X1)) 14.27/4.55 U42_ga(X1, ackermanncF_out_gga(X1, X3, X2)) -> ackermanncD_out_ga(s(X1), X2) 14.27/4.55 14.27/4.55 The set Q consists of the following terms: 14.27/4.55 14.27/4.55 ackermanncB_in_ga(x0) 14.27/4.55 ackermanncF_in_gga(x0, x1) 14.27/4.55 U40_ga(x0, x1) 14.27/4.55 U43_gga(x0, x1) 14.27/4.55 U44_gga(x0, x1, x2) 14.27/4.55 ackermanncD_in_ga(x0) 14.27/4.55 U45_gga(x0, x1, x2) 14.27/4.55 U41_ga(x0, x1) 14.27/4.55 U42_ga(x0, x1) 14.27/4.55 14.27/4.55 We have to consider all (P,Q,R)-chains. 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (94) QDPSizeChangeProof (EQUIVALENT) 14.27/4.55 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. 14.27/4.55 14.27/4.55 From the DPs we obtained the following set of size-change graphs: 14.27/4.55 *ACKERMANND_IN_GA(s(X1)) -> ACKERMANNB_IN_GA(X1) 14.27/4.55 The graph contains the following edges 1 > 1 14.27/4.55 14.27/4.55 14.27/4.55 *ACKERMANND_IN_GA(s(X1)) -> U3_GA(X1, ackermanncB_in_ga(X1)) 14.27/4.55 The graph contains the following edges 1 > 1 14.27/4.55 14.27/4.55 14.27/4.55 *ACKERMANNB_IN_GA(X1) -> ACKERMANND_IN_GA(X1) 14.27/4.55 The graph contains the following edges 1 >= 1 14.27/4.55 14.27/4.55 14.27/4.55 *ACKERMANNF_IN_GGA(s(X1), 0) -> ACKERMANND_IN_GA(X1) 14.27/4.55 The graph contains the following edges 1 > 1 14.27/4.55 14.27/4.55 14.27/4.55 *U3_GA(X1, ackermanncB_out_ga(X1, X3)) -> ACKERMANNF_IN_GGA(X1, X3) 14.27/4.55 The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2 14.27/4.55 14.27/4.55 14.27/4.55 *U7_GGA(X1, X2, ackermanncF_out_gga(s(X1), X2, X4)) -> ACKERMANNF_IN_GGA(X1, X4) 14.27/4.55 The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2 14.27/4.55 14.27/4.55 14.27/4.55 *ACKERMANNF_IN_GGA(s(X1), s(X2)) -> ACKERMANNF_IN_GGA(s(X1), X2) 14.27/4.55 The graph contains the following edges 1 >= 1, 2 > 2 14.27/4.55 14.27/4.55 14.27/4.55 *ACKERMANNF_IN_GGA(s(X1), s(X2)) -> U7_GGA(X1, X2, ackermanncF_in_gga(s(X1), X2)) 14.27/4.55 The graph contains the following edges 1 > 1, 2 > 2 14.27/4.55 14.27/4.55 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (95) 14.27/4.55 YES 14.27/4.55 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (96) 14.27/4.55 Obligation: 14.27/4.55 Pi DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 ACKERMANNC_IN_GGA(X1, s(X2), X3) -> PG_IN_GGAA(X1, X2, X4, X3) 14.27/4.55 PG_IN_GGAA(X1, X2, X3, X4) -> ACKERMANNC_IN_GGA(X1, X2, X3) 14.27/4.55 PG_IN_GGAA(X1, X2, X3, X4) -> U12_GGAA(X1, X2, X3, X4, ackermanncC_in_gga(X1, X2, X3)) 14.27/4.55 U12_GGAA(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> ACKERMANNE_IN_GGA(X1, X3, X4) 14.27/4.55 ACKERMANNE_IN_GGA(s(X1), s(X2), X3) -> PG_IN_GGAA(X1, X2, X4, X3) 14.27/4.55 14.27/4.55 The TRS R consists of the following rules: 14.27/4.55 14.27/4.55 ackermanncB_in_ga(X1, X2) -> U40_ga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncD_in_ga(0, s(s(0))) -> ackermanncD_out_ga(0, s(s(0))) 14.27/4.55 ackermanncD_in_ga(s(X1), X2) -> U41_ga(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U41_ga(X1, X2, ackermanncB_out_ga(X1, X3)) -> U42_ga(X1, X2, ackermanncF_in_gga(X1, X3, X2)) 14.27/4.55 ackermanncF_in_gga(0, X1, s(X1)) -> ackermanncF_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncF_in_gga(s(X1), 0, X2) -> U43_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U43_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncF_out_gga(s(X1), 0, X2) 14.27/4.55 ackermanncF_in_gga(s(X1), s(X2), X3) -> U44_gga(X1, X2, X3, ackermanncF_in_gga(s(X1), X2, X4)) 14.27/4.55 U44_gga(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> U45_gga(X1, X2, X3, ackermanncF_in_gga(X1, X4, X3)) 14.27/4.55 U45_gga(X1, X2, X3, ackermanncF_out_gga(X1, X4, X3)) -> ackermanncF_out_gga(s(X1), s(X2), X3) 14.27/4.55 U42_ga(X1, X2, ackermanncF_out_gga(X1, X3, X2)) -> ackermanncD_out_ga(s(X1), X2) 14.27/4.55 U40_ga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncB_out_ga(X1, X2) 14.27/4.55 ackermanncC_in_gga(X1, 0, X2) -> U46_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U46_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gga(X1, 0, X2) 14.27/4.55 ackermanncC_in_gga(X1, s(X2), X3) -> U47_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 qcG_in_ggaa(X1, X2, X3, X4) -> U48_ggaa(X1, X2, X3, X4, ackermanncC_in_gga(X1, X2, X3)) 14.27/4.55 U48_ggaa(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> U49_ggaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 ackermanncE_in_gga(0, X1, s(X1)) -> ackermanncE_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncE_in_gga(s(X1), 0, X2) -> U50_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U50_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncE_out_gga(s(X1), 0, X2) 14.27/4.55 ackermanncE_in_gga(s(X1), s(X2), X3) -> U51_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 U51_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncE_out_gga(s(X1), s(X2), X3) 14.27/4.55 U49_ggaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_ggaa(X1, X2, X3, X4) 14.27/4.55 U47_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncC_out_gga(X1, s(X2), X3) 14.27/4.55 ackermanncC_in_gaa(X1, 0, X2) -> U46_gaa(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U46_gaa(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gaa(X1, 0, X2) 14.27/4.55 ackermanncC_in_gaa(X1, s(X2), X3) -> U47_gaa(X1, X2, X3, qcG_in_gaaa(X1, X2, X4, X3)) 14.27/4.55 qcG_in_gaaa(X1, X2, X3, X4) -> U48_gaaa(X1, X2, X3, X4, ackermanncC_in_gaa(X1, X2, X3)) 14.27/4.55 U48_gaaa(X1, X2, X3, X4, ackermanncC_out_gaa(X1, X2, X3)) -> U49_gaaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 U49_gaaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_gaaa(X1, X2, X3, X4) 14.27/4.55 U47_gaa(X1, X2, X3, qcG_out_gaaa(X1, X2, X4, X3)) -> ackermanncC_out_gaa(X1, s(X2), X3) 14.27/4.55 14.27/4.55 The argument filtering Pi contains the following mapping: 14.27/4.55 s(x1) = s(x1) 14.27/4.55 14.27/4.55 ackermanncB_in_ga(x1, x2) = ackermanncB_in_ga(x1) 14.27/4.55 14.27/4.55 U40_ga(x1, x2, x3) = U40_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncD_in_ga(x1, x2) = ackermanncD_in_ga(x1) 14.27/4.55 14.27/4.55 0 = 0 14.27/4.55 14.27/4.55 ackermanncD_out_ga(x1, x2) = ackermanncD_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U41_ga(x1, x2, x3) = U41_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncB_out_ga(x1, x2) = ackermanncB_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U42_ga(x1, x2, x3) = U42_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncF_in_gga(x1, x2, x3) = ackermanncF_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncF_out_gga(x1, x2, x3) = ackermanncF_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U43_gga(x1, x2, x3) = U43_gga(x1, x3) 14.27/4.55 14.27/4.55 U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 ackermanncC_in_gga(x1, x2, x3) = ackermanncC_in_gga(x1, x2) 14.27/4.55 14.27/4.55 U46_gga(x1, x2, x3) = U46_gga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gga(x1, x2, x3) = ackermanncC_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_in_ggaa(x1, x2, x3, x4) = qcG_in_ggaa(x1, x2) 14.27/4.55 14.27/4.55 U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5) 14.27/4.55 14.27/4.55 U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 ackermanncE_in_gga(x1, x2, x3) = ackermanncE_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncE_out_gga(x1, x2, x3) = ackermanncE_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U50_gga(x1, x2, x3) = U50_gga(x1, x3) 14.27/4.55 14.27/4.55 U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_out_ggaa(x1, x2, x3, x4) = qcG_out_ggaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ackermanncC_in_gaa(x1, x2, x3) = ackermanncC_in_gaa(x1) 14.27/4.55 14.27/4.55 U46_gaa(x1, x2, x3) = U46_gaa(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gaa(x1, x2, x3) = ackermanncC_out_gaa(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gaa(x1, x2, x3, x4) = U47_gaa(x1, x4) 14.27/4.55 14.27/4.55 qcG_in_gaaa(x1, x2, x3, x4) = qcG_in_gaaa(x1) 14.27/4.55 14.27/4.55 U48_gaaa(x1, x2, x3, x4, x5) = U48_gaaa(x1, x5) 14.27/4.55 14.27/4.55 U49_gaaa(x1, x2, x3, x4, x5) = U49_gaaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 qcG_out_gaaa(x1, x2, x3, x4) = qcG_out_gaaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ACKERMANNC_IN_GGA(x1, x2, x3) = ACKERMANNC_IN_GGA(x1, x2) 14.27/4.55 14.27/4.55 PG_IN_GGAA(x1, x2, x3, x4) = PG_IN_GGAA(x1, x2) 14.27/4.55 14.27/4.55 U12_GGAA(x1, x2, x3, x4, x5) = U12_GGAA(x1, x2, x5) 14.27/4.55 14.27/4.55 ACKERMANNE_IN_GGA(x1, x2, x3) = ACKERMANNE_IN_GGA(x1, x2) 14.27/4.55 14.27/4.55 14.27/4.55 We have to consider all (P,R,Pi)-chains 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (97) UsableRulesProof (EQUIVALENT) 14.27/4.55 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (98) 14.27/4.55 Obligation: 14.27/4.55 Pi DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 ACKERMANNC_IN_GGA(X1, s(X2), X3) -> PG_IN_GGAA(X1, X2, X4, X3) 14.27/4.55 PG_IN_GGAA(X1, X2, X3, X4) -> ACKERMANNC_IN_GGA(X1, X2, X3) 14.27/4.55 PG_IN_GGAA(X1, X2, X3, X4) -> U12_GGAA(X1, X2, X3, X4, ackermanncC_in_gga(X1, X2, X3)) 14.27/4.55 U12_GGAA(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> ACKERMANNE_IN_GGA(X1, X3, X4) 14.27/4.55 ACKERMANNE_IN_GGA(s(X1), s(X2), X3) -> PG_IN_GGAA(X1, X2, X4, X3) 14.27/4.55 14.27/4.55 The TRS R consists of the following rules: 14.27/4.55 14.27/4.55 ackermanncC_in_gga(X1, 0, X2) -> U46_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncC_in_gga(X1, s(X2), X3) -> U47_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 U46_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gga(X1, 0, X2) 14.27/4.55 U47_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncC_out_gga(X1, s(X2), X3) 14.27/4.55 ackermanncD_in_ga(0, s(s(0))) -> ackermanncD_out_ga(0, s(s(0))) 14.27/4.55 ackermanncD_in_ga(s(X1), X2) -> U41_ga(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 qcG_in_ggaa(X1, X2, X3, X4) -> U48_ggaa(X1, X2, X3, X4, ackermanncC_in_gga(X1, X2, X3)) 14.27/4.55 U41_ga(X1, X2, ackermanncB_out_ga(X1, X3)) -> U42_ga(X1, X2, ackermanncF_in_gga(X1, X3, X2)) 14.27/4.55 U48_ggaa(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> U49_ggaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 ackermanncB_in_ga(X1, X2) -> U40_ga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U42_ga(X1, X2, ackermanncF_out_gga(X1, X3, X2)) -> ackermanncD_out_ga(s(X1), X2) 14.27/4.55 U49_ggaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_ggaa(X1, X2, X3, X4) 14.27/4.55 U40_ga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncB_out_ga(X1, X2) 14.27/4.55 ackermanncF_in_gga(0, X1, s(X1)) -> ackermanncF_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncF_in_gga(s(X1), 0, X2) -> U43_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncF_in_gga(s(X1), s(X2), X3) -> U44_gga(X1, X2, X3, ackermanncF_in_gga(s(X1), X2, X4)) 14.27/4.55 ackermanncE_in_gga(0, X1, s(X1)) -> ackermanncE_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncE_in_gga(s(X1), 0, X2) -> U50_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncE_in_gga(s(X1), s(X2), X3) -> U51_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 U43_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncF_out_gga(s(X1), 0, X2) 14.27/4.55 U44_gga(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> U45_gga(X1, X2, X3, ackermanncF_in_gga(X1, X4, X3)) 14.27/4.55 U50_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncE_out_gga(s(X1), 0, X2) 14.27/4.55 U51_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncE_out_gga(s(X1), s(X2), X3) 14.27/4.55 U45_gga(X1, X2, X3, ackermanncF_out_gga(X1, X4, X3)) -> ackermanncF_out_gga(s(X1), s(X2), X3) 14.27/4.55 14.27/4.55 The argument filtering Pi contains the following mapping: 14.27/4.55 s(x1) = s(x1) 14.27/4.55 14.27/4.55 ackermanncB_in_ga(x1, x2) = ackermanncB_in_ga(x1) 14.27/4.55 14.27/4.55 U40_ga(x1, x2, x3) = U40_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncD_in_ga(x1, x2) = ackermanncD_in_ga(x1) 14.27/4.55 14.27/4.55 0 = 0 14.27/4.55 14.27/4.55 ackermanncD_out_ga(x1, x2) = ackermanncD_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U41_ga(x1, x2, x3) = U41_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncB_out_ga(x1, x2) = ackermanncB_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U42_ga(x1, x2, x3) = U42_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncF_in_gga(x1, x2, x3) = ackermanncF_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncF_out_gga(x1, x2, x3) = ackermanncF_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U43_gga(x1, x2, x3) = U43_gga(x1, x3) 14.27/4.55 14.27/4.55 U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 ackermanncC_in_gga(x1, x2, x3) = ackermanncC_in_gga(x1, x2) 14.27/4.55 14.27/4.55 U46_gga(x1, x2, x3) = U46_gga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gga(x1, x2, x3) = ackermanncC_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_in_ggaa(x1, x2, x3, x4) = qcG_in_ggaa(x1, x2) 14.27/4.55 14.27/4.55 U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5) 14.27/4.55 14.27/4.55 U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 ackermanncE_in_gga(x1, x2, x3) = ackermanncE_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncE_out_gga(x1, x2, x3) = ackermanncE_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U50_gga(x1, x2, x3) = U50_gga(x1, x3) 14.27/4.55 14.27/4.55 U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_out_ggaa(x1, x2, x3, x4) = qcG_out_ggaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ACKERMANNC_IN_GGA(x1, x2, x3) = ACKERMANNC_IN_GGA(x1, x2) 14.27/4.55 14.27/4.55 PG_IN_GGAA(x1, x2, x3, x4) = PG_IN_GGAA(x1, x2) 14.27/4.55 14.27/4.55 U12_GGAA(x1, x2, x3, x4, x5) = U12_GGAA(x1, x2, x5) 14.27/4.55 14.27/4.55 ACKERMANNE_IN_GGA(x1, x2, x3) = ACKERMANNE_IN_GGA(x1, x2) 14.27/4.55 14.27/4.55 14.27/4.55 We have to consider all (P,R,Pi)-chains 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (99) PiDPToQDPProof (SOUND) 14.27/4.55 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (100) 14.27/4.55 Obligation: 14.27/4.55 Q DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 ACKERMANNC_IN_GGA(X1, s(X2)) -> PG_IN_GGAA(X1, X2) 14.27/4.55 PG_IN_GGAA(X1, X2) -> ACKERMANNC_IN_GGA(X1, X2) 14.27/4.55 PG_IN_GGAA(X1, X2) -> U12_GGAA(X1, X2, ackermanncC_in_gga(X1, X2)) 14.27/4.55 U12_GGAA(X1, X2, ackermanncC_out_gga(X1, X2, X3)) -> ACKERMANNE_IN_GGA(X1, X3) 14.27/4.55 ACKERMANNE_IN_GGA(s(X1), s(X2)) -> PG_IN_GGAA(X1, X2) 14.27/4.55 14.27/4.55 The TRS R consists of the following rules: 14.27/4.55 14.27/4.55 ackermanncC_in_gga(X1, 0) -> U46_gga(X1, ackermanncD_in_ga(X1)) 14.27/4.55 ackermanncC_in_gga(X1, s(X2)) -> U47_gga(X1, X2, qcG_in_ggaa(X1, X2)) 14.27/4.55 U46_gga(X1, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gga(X1, 0, X2) 14.27/4.55 U47_gga(X1, X2, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncC_out_gga(X1, s(X2), X3) 14.27/4.55 ackermanncD_in_ga(0) -> ackermanncD_out_ga(0, s(s(0))) 14.27/4.55 ackermanncD_in_ga(s(X1)) -> U41_ga(X1, ackermanncB_in_ga(X1)) 14.27/4.55 qcG_in_ggaa(X1, X2) -> U48_ggaa(X1, X2, ackermanncC_in_gga(X1, X2)) 14.27/4.55 U41_ga(X1, ackermanncB_out_ga(X1, X3)) -> U42_ga(X1, ackermanncF_in_gga(X1, X3)) 14.27/4.55 U48_ggaa(X1, X2, ackermanncC_out_gga(X1, X2, X3)) -> U49_ggaa(X1, X2, X3, ackermanncE_in_gga(X1, X3)) 14.27/4.55 ackermanncB_in_ga(X1) -> U40_ga(X1, ackermanncD_in_ga(X1)) 14.27/4.55 U42_ga(X1, ackermanncF_out_gga(X1, X3, X2)) -> ackermanncD_out_ga(s(X1), X2) 14.27/4.55 U49_ggaa(X1, X2, X3, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_ggaa(X1, X2, X3, X4) 14.27/4.55 U40_ga(X1, ackermanncD_out_ga(X1, X2)) -> ackermanncB_out_ga(X1, X2) 14.27/4.55 ackermanncF_in_gga(0, X1) -> ackermanncF_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncF_in_gga(s(X1), 0) -> U43_gga(X1, ackermanncD_in_ga(X1)) 14.27/4.55 ackermanncF_in_gga(s(X1), s(X2)) -> U44_gga(X1, X2, ackermanncF_in_gga(s(X1), X2)) 14.27/4.55 ackermanncE_in_gga(0, X1) -> ackermanncE_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncE_in_gga(s(X1), 0) -> U50_gga(X1, ackermanncD_in_ga(X1)) 14.27/4.55 ackermanncE_in_gga(s(X1), s(X2)) -> U51_gga(X1, X2, qcG_in_ggaa(X1, X2)) 14.27/4.55 U43_gga(X1, ackermanncD_out_ga(X1, X2)) -> ackermanncF_out_gga(s(X1), 0, X2) 14.27/4.55 U44_gga(X1, X2, ackermanncF_out_gga(s(X1), X2, X4)) -> U45_gga(X1, X2, ackermanncF_in_gga(X1, X4)) 14.27/4.55 U50_gga(X1, ackermanncD_out_ga(X1, X2)) -> ackermanncE_out_gga(s(X1), 0, X2) 14.27/4.55 U51_gga(X1, X2, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncE_out_gga(s(X1), s(X2), X3) 14.27/4.55 U45_gga(X1, X2, ackermanncF_out_gga(X1, X4, X3)) -> ackermanncF_out_gga(s(X1), s(X2), X3) 14.27/4.55 14.27/4.55 The set Q consists of the following terms: 14.27/4.55 14.27/4.55 ackermanncC_in_gga(x0, x1) 14.27/4.55 U46_gga(x0, x1) 14.27/4.55 U47_gga(x0, x1, x2) 14.27/4.55 ackermanncD_in_ga(x0) 14.27/4.55 qcG_in_ggaa(x0, x1) 14.27/4.55 U41_ga(x0, x1) 14.27/4.55 U48_ggaa(x0, x1, x2) 14.27/4.55 ackermanncB_in_ga(x0) 14.27/4.55 U42_ga(x0, x1) 14.27/4.55 U49_ggaa(x0, x1, x2, x3) 14.27/4.55 U40_ga(x0, x1) 14.27/4.55 ackermanncF_in_gga(x0, x1) 14.27/4.55 ackermanncE_in_gga(x0, x1) 14.27/4.55 U43_gga(x0, x1) 14.27/4.55 U44_gga(x0, x1, x2) 14.27/4.55 U50_gga(x0, x1) 14.27/4.55 U51_gga(x0, x1, x2) 14.27/4.55 U45_gga(x0, x1, x2) 14.27/4.55 14.27/4.55 We have to consider all (P,Q,R)-chains. 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (101) QDPSizeChangeProof (EQUIVALENT) 14.27/4.55 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. 14.27/4.55 14.27/4.55 From the DPs we obtained the following set of size-change graphs: 14.27/4.55 *PG_IN_GGAA(X1, X2) -> ACKERMANNC_IN_GGA(X1, X2) 14.27/4.55 The graph contains the following edges 1 >= 1, 2 >= 2 14.27/4.55 14.27/4.55 14.27/4.55 *PG_IN_GGAA(X1, X2) -> U12_GGAA(X1, X2, ackermanncC_in_gga(X1, X2)) 14.27/4.55 The graph contains the following edges 1 >= 1, 2 >= 2 14.27/4.55 14.27/4.55 14.27/4.55 *ACKERMANNC_IN_GGA(X1, s(X2)) -> PG_IN_GGAA(X1, X2) 14.27/4.55 The graph contains the following edges 1 >= 1, 2 > 2 14.27/4.55 14.27/4.55 14.27/4.55 *ACKERMANNE_IN_GGA(s(X1), s(X2)) -> PG_IN_GGAA(X1, X2) 14.27/4.55 The graph contains the following edges 1 > 1, 2 > 2 14.27/4.55 14.27/4.55 14.27/4.55 *U12_GGAA(X1, X2, ackermanncC_out_gga(X1, X2, X3)) -> ACKERMANNE_IN_GGA(X1, X3) 14.27/4.55 The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2 14.27/4.55 14.27/4.55 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (102) 14.27/4.55 YES 14.27/4.55 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (103) 14.27/4.55 Obligation: 14.27/4.55 Pi DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 PG_IN_GAAA(X1, X2, X3, X4) -> ACKERMANNC_IN_GAA(X1, X2, X3) 14.27/4.55 ACKERMANNC_IN_GAA(X1, s(X2), X3) -> PG_IN_GAAA(X1, X2, X4, X3) 14.27/4.55 14.27/4.55 The TRS R consists of the following rules: 14.27/4.55 14.27/4.55 ackermanncB_in_ga(X1, X2) -> U40_ga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncD_in_ga(0, s(s(0))) -> ackermanncD_out_ga(0, s(s(0))) 14.27/4.55 ackermanncD_in_ga(s(X1), X2) -> U41_ga(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U41_ga(X1, X2, ackermanncB_out_ga(X1, X3)) -> U42_ga(X1, X2, ackermanncF_in_gga(X1, X3, X2)) 14.27/4.55 ackermanncF_in_gga(0, X1, s(X1)) -> ackermanncF_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncF_in_gga(s(X1), 0, X2) -> U43_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U43_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncF_out_gga(s(X1), 0, X2) 14.27/4.55 ackermanncF_in_gga(s(X1), s(X2), X3) -> U44_gga(X1, X2, X3, ackermanncF_in_gga(s(X1), X2, X4)) 14.27/4.55 U44_gga(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> U45_gga(X1, X2, X3, ackermanncF_in_gga(X1, X4, X3)) 14.27/4.55 U45_gga(X1, X2, X3, ackermanncF_out_gga(X1, X4, X3)) -> ackermanncF_out_gga(s(X1), s(X2), X3) 14.27/4.55 U42_ga(X1, X2, ackermanncF_out_gga(X1, X3, X2)) -> ackermanncD_out_ga(s(X1), X2) 14.27/4.55 U40_ga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncB_out_ga(X1, X2) 14.27/4.55 ackermanncC_in_gga(X1, 0, X2) -> U46_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U46_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gga(X1, 0, X2) 14.27/4.55 ackermanncC_in_gga(X1, s(X2), X3) -> U47_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 qcG_in_ggaa(X1, X2, X3, X4) -> U48_ggaa(X1, X2, X3, X4, ackermanncC_in_gga(X1, X2, X3)) 14.27/4.55 U48_ggaa(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> U49_ggaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 ackermanncE_in_gga(0, X1, s(X1)) -> ackermanncE_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncE_in_gga(s(X1), 0, X2) -> U50_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U50_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncE_out_gga(s(X1), 0, X2) 14.27/4.55 ackermanncE_in_gga(s(X1), s(X2), X3) -> U51_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 U51_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncE_out_gga(s(X1), s(X2), X3) 14.27/4.55 U49_ggaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_ggaa(X1, X2, X3, X4) 14.27/4.55 U47_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncC_out_gga(X1, s(X2), X3) 14.27/4.55 ackermanncC_in_gaa(X1, 0, X2) -> U46_gaa(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U46_gaa(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gaa(X1, 0, X2) 14.27/4.55 ackermanncC_in_gaa(X1, s(X2), X3) -> U47_gaa(X1, X2, X3, qcG_in_gaaa(X1, X2, X4, X3)) 14.27/4.55 qcG_in_gaaa(X1, X2, X3, X4) -> U48_gaaa(X1, X2, X3, X4, ackermanncC_in_gaa(X1, X2, X3)) 14.27/4.55 U48_gaaa(X1, X2, X3, X4, ackermanncC_out_gaa(X1, X2, X3)) -> U49_gaaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 U49_gaaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_gaaa(X1, X2, X3, X4) 14.27/4.55 U47_gaa(X1, X2, X3, qcG_out_gaaa(X1, X2, X4, X3)) -> ackermanncC_out_gaa(X1, s(X2), X3) 14.27/4.55 14.27/4.55 The argument filtering Pi contains the following mapping: 14.27/4.55 s(x1) = s(x1) 14.27/4.55 14.27/4.55 ackermanncB_in_ga(x1, x2) = ackermanncB_in_ga(x1) 14.27/4.55 14.27/4.55 U40_ga(x1, x2, x3) = U40_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncD_in_ga(x1, x2) = ackermanncD_in_ga(x1) 14.27/4.55 14.27/4.55 0 = 0 14.27/4.55 14.27/4.55 ackermanncD_out_ga(x1, x2) = ackermanncD_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U41_ga(x1, x2, x3) = U41_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncB_out_ga(x1, x2) = ackermanncB_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U42_ga(x1, x2, x3) = U42_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncF_in_gga(x1, x2, x3) = ackermanncF_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncF_out_gga(x1, x2, x3) = ackermanncF_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U43_gga(x1, x2, x3) = U43_gga(x1, x3) 14.27/4.55 14.27/4.55 U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 ackermanncC_in_gga(x1, x2, x3) = ackermanncC_in_gga(x1, x2) 14.27/4.55 14.27/4.55 U46_gga(x1, x2, x3) = U46_gga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gga(x1, x2, x3) = ackermanncC_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_in_ggaa(x1, x2, x3, x4) = qcG_in_ggaa(x1, x2) 14.27/4.55 14.27/4.55 U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5) 14.27/4.55 14.27/4.55 U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 ackermanncE_in_gga(x1, x2, x3) = ackermanncE_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncE_out_gga(x1, x2, x3) = ackermanncE_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U50_gga(x1, x2, x3) = U50_gga(x1, x3) 14.27/4.55 14.27/4.55 U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_out_ggaa(x1, x2, x3, x4) = qcG_out_ggaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ackermanncC_in_gaa(x1, x2, x3) = ackermanncC_in_gaa(x1) 14.27/4.55 14.27/4.55 U46_gaa(x1, x2, x3) = U46_gaa(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gaa(x1, x2, x3) = ackermanncC_out_gaa(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gaa(x1, x2, x3, x4) = U47_gaa(x1, x4) 14.27/4.55 14.27/4.55 qcG_in_gaaa(x1, x2, x3, x4) = qcG_in_gaaa(x1) 14.27/4.55 14.27/4.55 U48_gaaa(x1, x2, x3, x4, x5) = U48_gaaa(x1, x5) 14.27/4.55 14.27/4.55 U49_gaaa(x1, x2, x3, x4, x5) = U49_gaaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 qcG_out_gaaa(x1, x2, x3, x4) = qcG_out_gaaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ACKERMANNC_IN_GAA(x1, x2, x3) = ACKERMANNC_IN_GAA(x1) 14.27/4.55 14.27/4.55 PG_IN_GAAA(x1, x2, x3, x4) = PG_IN_GAAA(x1) 14.27/4.55 14.27/4.55 14.27/4.55 We have to consider all (P,R,Pi)-chains 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (104) UsableRulesProof (EQUIVALENT) 14.27/4.55 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (105) 14.27/4.55 Obligation: 14.27/4.55 Pi DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 PG_IN_GAAA(X1, X2, X3, X4) -> ACKERMANNC_IN_GAA(X1, X2, X3) 14.27/4.55 ACKERMANNC_IN_GAA(X1, s(X2), X3) -> PG_IN_GAAA(X1, X2, X4, X3) 14.27/4.55 14.27/4.55 R is empty. 14.27/4.55 The argument filtering Pi contains the following mapping: 14.27/4.55 s(x1) = s(x1) 14.27/4.55 14.27/4.55 ACKERMANNC_IN_GAA(x1, x2, x3) = ACKERMANNC_IN_GAA(x1) 14.27/4.55 14.27/4.55 PG_IN_GAAA(x1, x2, x3, x4) = PG_IN_GAAA(x1) 14.27/4.55 14.27/4.55 14.27/4.55 We have to consider all (P,R,Pi)-chains 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (106) PiDPToQDPProof (SOUND) 14.27/4.55 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (107) 14.27/4.55 Obligation: 14.27/4.55 Q DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 PG_IN_GAAA(X1) -> ACKERMANNC_IN_GAA(X1) 14.27/4.55 ACKERMANNC_IN_GAA(X1) -> PG_IN_GAAA(X1) 14.27/4.55 14.27/4.55 R is empty. 14.27/4.55 Q is empty. 14.27/4.55 We have to consider all (P,Q,R)-chains. 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (108) NonTerminationLoopProof (COMPLETE) 14.27/4.55 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 14.27/4.55 Found a loop by narrowing to the left: 14.27/4.55 14.27/4.55 s = ACKERMANNC_IN_GAA(X1') evaluates to t =ACKERMANNC_IN_GAA(X1') 14.27/4.55 14.27/4.55 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 14.27/4.55 * Matcher: [ ] 14.27/4.55 * Semiunifier: [ ] 14.27/4.55 14.27/4.55 -------------------------------------------------------------------------------- 14.27/4.55 Rewriting sequence 14.27/4.55 14.27/4.55 ACKERMANNC_IN_GAA(X1') -> PG_IN_GAAA(X1') 14.27/4.55 with rule ACKERMANNC_IN_GAA(X1'') -> PG_IN_GAAA(X1'') at position [] and matcher [X1'' / X1'] 14.27/4.55 14.27/4.55 PG_IN_GAAA(X1') -> ACKERMANNC_IN_GAA(X1') 14.27/4.55 with rule PG_IN_GAAA(X1) -> ACKERMANNC_IN_GAA(X1) 14.27/4.55 14.27/4.55 Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence 14.27/4.55 14.27/4.55 14.27/4.55 All these steps are and every following step will be a correct step w.r.t to Q. 14.27/4.55 14.27/4.55 14.27/4.55 14.27/4.55 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (109) 14.27/4.55 NO 14.27/4.55 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (110) 14.27/4.55 Obligation: 14.27/4.55 Pi DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 ACKERMANNA_IN_GGG(s(s(X1)), 0, X2) -> U17_GGG(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U17_GGG(X1, X2, ackermanncB_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(X2), X3) -> U20_GGG(X1, X2, X3, ackermanncC_in_gga(X1, X2, X4)) 14.27/4.55 U20_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> ACKERMANNA_IN_GGG(X1, X4, X3) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(0), X2) -> U23_GGG(X1, X2, ackermanncD_in_ga(X1, X3)) 14.27/4.55 U23_GGG(X1, X2, ackermanncD_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(s(X2)), X3) -> U26_GGG(X1, X2, X3, ackermanncC_in_gga(X1, X2, X4)) 14.27/4.55 U26_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> U28_GGG(X1, X2, X3, ackermanncE_in_gga(X1, X4, X5)) 14.27/4.55 U28_GGG(X1, X2, X3, ackermanncE_out_gga(X1, X4, X5)) -> ACKERMANNA_IN_GGG(X1, X5, X3) 14.27/4.55 14.27/4.55 The TRS R consists of the following rules: 14.27/4.55 14.27/4.55 ackermanncB_in_ga(X1, X2) -> U40_ga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncD_in_ga(0, s(s(0))) -> ackermanncD_out_ga(0, s(s(0))) 14.27/4.55 ackermanncD_in_ga(s(X1), X2) -> U41_ga(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U41_ga(X1, X2, ackermanncB_out_ga(X1, X3)) -> U42_ga(X1, X2, ackermanncF_in_gga(X1, X3, X2)) 14.27/4.55 ackermanncF_in_gga(0, X1, s(X1)) -> ackermanncF_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncF_in_gga(s(X1), 0, X2) -> U43_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U43_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncF_out_gga(s(X1), 0, X2) 14.27/4.55 ackermanncF_in_gga(s(X1), s(X2), X3) -> U44_gga(X1, X2, X3, ackermanncF_in_gga(s(X1), X2, X4)) 14.27/4.55 U44_gga(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> U45_gga(X1, X2, X3, ackermanncF_in_gga(X1, X4, X3)) 14.27/4.55 U45_gga(X1, X2, X3, ackermanncF_out_gga(X1, X4, X3)) -> ackermanncF_out_gga(s(X1), s(X2), X3) 14.27/4.55 U42_ga(X1, X2, ackermanncF_out_gga(X1, X3, X2)) -> ackermanncD_out_ga(s(X1), X2) 14.27/4.55 U40_ga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncB_out_ga(X1, X2) 14.27/4.55 ackermanncC_in_gga(X1, 0, X2) -> U46_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U46_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gga(X1, 0, X2) 14.27/4.55 ackermanncC_in_gga(X1, s(X2), X3) -> U47_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 qcG_in_ggaa(X1, X2, X3, X4) -> U48_ggaa(X1, X2, X3, X4, ackermanncC_in_gga(X1, X2, X3)) 14.27/4.55 U48_ggaa(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> U49_ggaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 ackermanncE_in_gga(0, X1, s(X1)) -> ackermanncE_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncE_in_gga(s(X1), 0, X2) -> U50_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U50_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncE_out_gga(s(X1), 0, X2) 14.27/4.55 ackermanncE_in_gga(s(X1), s(X2), X3) -> U51_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 U51_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncE_out_gga(s(X1), s(X2), X3) 14.27/4.55 U49_ggaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_ggaa(X1, X2, X3, X4) 14.27/4.55 U47_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncC_out_gga(X1, s(X2), X3) 14.27/4.55 ackermanncC_in_gaa(X1, 0, X2) -> U46_gaa(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 U46_gaa(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gaa(X1, 0, X2) 14.27/4.55 ackermanncC_in_gaa(X1, s(X2), X3) -> U47_gaa(X1, X2, X3, qcG_in_gaaa(X1, X2, X4, X3)) 14.27/4.55 qcG_in_gaaa(X1, X2, X3, X4) -> U48_gaaa(X1, X2, X3, X4, ackermanncC_in_gaa(X1, X2, X3)) 14.27/4.55 U48_gaaa(X1, X2, X3, X4, ackermanncC_out_gaa(X1, X2, X3)) -> U49_gaaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 U49_gaaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_gaaa(X1, X2, X3, X4) 14.27/4.55 U47_gaa(X1, X2, X3, qcG_out_gaaa(X1, X2, X4, X3)) -> ackermanncC_out_gaa(X1, s(X2), X3) 14.27/4.55 14.27/4.55 The argument filtering Pi contains the following mapping: 14.27/4.55 s(x1) = s(x1) 14.27/4.55 14.27/4.55 ackermanncB_in_ga(x1, x2) = ackermanncB_in_ga(x1) 14.27/4.55 14.27/4.55 U40_ga(x1, x2, x3) = U40_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncD_in_ga(x1, x2) = ackermanncD_in_ga(x1) 14.27/4.55 14.27/4.55 0 = 0 14.27/4.55 14.27/4.55 ackermanncD_out_ga(x1, x2) = ackermanncD_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U41_ga(x1, x2, x3) = U41_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncB_out_ga(x1, x2) = ackermanncB_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U42_ga(x1, x2, x3) = U42_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncF_in_gga(x1, x2, x3) = ackermanncF_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncF_out_gga(x1, x2, x3) = ackermanncF_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U43_gga(x1, x2, x3) = U43_gga(x1, x3) 14.27/4.55 14.27/4.55 U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 ackermanncC_in_gga(x1, x2, x3) = ackermanncC_in_gga(x1, x2) 14.27/4.55 14.27/4.55 U46_gga(x1, x2, x3) = U46_gga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gga(x1, x2, x3) = ackermanncC_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_in_ggaa(x1, x2, x3, x4) = qcG_in_ggaa(x1, x2) 14.27/4.55 14.27/4.55 U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5) 14.27/4.55 14.27/4.55 U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 ackermanncE_in_gga(x1, x2, x3) = ackermanncE_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncE_out_gga(x1, x2, x3) = ackermanncE_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U50_gga(x1, x2, x3) = U50_gga(x1, x3) 14.27/4.55 14.27/4.55 U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_out_ggaa(x1, x2, x3, x4) = qcG_out_ggaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ackermanncC_in_gaa(x1, x2, x3) = ackermanncC_in_gaa(x1) 14.27/4.55 14.27/4.55 U46_gaa(x1, x2, x3) = U46_gaa(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gaa(x1, x2, x3) = ackermanncC_out_gaa(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gaa(x1, x2, x3, x4) = U47_gaa(x1, x4) 14.27/4.55 14.27/4.55 qcG_in_gaaa(x1, x2, x3, x4) = qcG_in_gaaa(x1) 14.27/4.55 14.27/4.55 U48_gaaa(x1, x2, x3, x4, x5) = U48_gaaa(x1, x5) 14.27/4.55 14.27/4.55 U49_gaaa(x1, x2, x3, x4, x5) = U49_gaaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 qcG_out_gaaa(x1, x2, x3, x4) = qcG_out_gaaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ACKERMANNA_IN_GGG(x1, x2, x3) = ACKERMANNA_IN_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U17_GGG(x1, x2, x3) = U17_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U20_GGG(x1, x2, x3, x4) = U20_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U23_GGG(x1, x2, x3) = U23_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U26_GGG(x1, x2, x3, x4) = U26_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U28_GGG(x1, x2, x3, x4) = U28_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 14.27/4.55 We have to consider all (P,R,Pi)-chains 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (111) UsableRulesProof (EQUIVALENT) 14.27/4.55 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (112) 14.27/4.55 Obligation: 14.27/4.55 Pi DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 ACKERMANNA_IN_GGG(s(s(X1)), 0, X2) -> U17_GGG(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 U17_GGG(X1, X2, ackermanncB_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(X2), X3) -> U20_GGG(X1, X2, X3, ackermanncC_in_gga(X1, X2, X4)) 14.27/4.55 U20_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> ACKERMANNA_IN_GGG(X1, X4, X3) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(0), X2) -> U23_GGG(X1, X2, ackermanncD_in_ga(X1, X3)) 14.27/4.55 U23_GGG(X1, X2, ackermanncD_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(s(X2)), X3) -> U26_GGG(X1, X2, X3, ackermanncC_in_gga(X1, X2, X4)) 14.27/4.55 U26_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> U28_GGG(X1, X2, X3, ackermanncE_in_gga(X1, X4, X5)) 14.27/4.55 U28_GGG(X1, X2, X3, ackermanncE_out_gga(X1, X4, X5)) -> ACKERMANNA_IN_GGG(X1, X5, X3) 14.27/4.55 14.27/4.55 The TRS R consists of the following rules: 14.27/4.55 14.27/4.55 ackermanncB_in_ga(X1, X2) -> U40_ga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncC_in_gga(X1, 0, X2) -> U46_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncC_in_gga(X1, s(X2), X3) -> U47_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 ackermanncD_in_ga(0, s(s(0))) -> ackermanncD_out_ga(0, s(s(0))) 14.27/4.55 ackermanncD_in_ga(s(X1), X2) -> U41_ga(X1, X2, ackermanncB_in_ga(X1, X3)) 14.27/4.55 ackermanncE_in_gga(0, X1, s(X1)) -> ackermanncE_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncE_in_gga(s(X1), 0, X2) -> U50_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncE_in_gga(s(X1), s(X2), X3) -> U51_gga(X1, X2, X3, qcG_in_ggaa(X1, X2, X4, X3)) 14.27/4.55 U40_ga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncB_out_ga(X1, X2) 14.27/4.55 U46_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gga(X1, 0, X2) 14.27/4.55 U47_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncC_out_gga(X1, s(X2), X3) 14.27/4.55 U41_ga(X1, X2, ackermanncB_out_ga(X1, X3)) -> U42_ga(X1, X2, ackermanncF_in_gga(X1, X3, X2)) 14.27/4.55 U50_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncE_out_gga(s(X1), 0, X2) 14.27/4.55 U51_gga(X1, X2, X3, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncE_out_gga(s(X1), s(X2), X3) 14.27/4.55 qcG_in_ggaa(X1, X2, X3, X4) -> U48_ggaa(X1, X2, X3, X4, ackermanncC_in_gga(X1, X2, X3)) 14.27/4.55 U42_ga(X1, X2, ackermanncF_out_gga(X1, X3, X2)) -> ackermanncD_out_ga(s(X1), X2) 14.27/4.55 U48_ggaa(X1, X2, X3, X4, ackermanncC_out_gga(X1, X2, X3)) -> U49_ggaa(X1, X2, X3, X4, ackermanncE_in_gga(X1, X3, X4)) 14.27/4.55 ackermanncF_in_gga(0, X1, s(X1)) -> ackermanncF_out_gga(0, X1, s(X1)) 14.27/4.55 ackermanncF_in_gga(s(X1), 0, X2) -> U43_gga(X1, X2, ackermanncD_in_ga(X1, X2)) 14.27/4.55 ackermanncF_in_gga(s(X1), s(X2), X3) -> U44_gga(X1, X2, X3, ackermanncF_in_gga(s(X1), X2, X4)) 14.27/4.55 U49_ggaa(X1, X2, X3, X4, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_ggaa(X1, X2, X3, X4) 14.27/4.55 U43_gga(X1, X2, ackermanncD_out_ga(X1, X2)) -> ackermanncF_out_gga(s(X1), 0, X2) 14.27/4.55 U44_gga(X1, X2, X3, ackermanncF_out_gga(s(X1), X2, X4)) -> U45_gga(X1, X2, X3, ackermanncF_in_gga(X1, X4, X3)) 14.27/4.55 U45_gga(X1, X2, X3, ackermanncF_out_gga(X1, X4, X3)) -> ackermanncF_out_gga(s(X1), s(X2), X3) 14.27/4.55 14.27/4.55 The argument filtering Pi contains the following mapping: 14.27/4.55 s(x1) = s(x1) 14.27/4.55 14.27/4.55 ackermanncB_in_ga(x1, x2) = ackermanncB_in_ga(x1) 14.27/4.55 14.27/4.55 U40_ga(x1, x2, x3) = U40_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncD_in_ga(x1, x2) = ackermanncD_in_ga(x1) 14.27/4.55 14.27/4.55 0 = 0 14.27/4.55 14.27/4.55 ackermanncD_out_ga(x1, x2) = ackermanncD_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U41_ga(x1, x2, x3) = U41_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncB_out_ga(x1, x2) = ackermanncB_out_ga(x1, x2) 14.27/4.55 14.27/4.55 U42_ga(x1, x2, x3) = U42_ga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncF_in_gga(x1, x2, x3) = ackermanncF_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncF_out_gga(x1, x2, x3) = ackermanncF_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U43_gga(x1, x2, x3) = U43_gga(x1, x3) 14.27/4.55 14.27/4.55 U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 ackermanncC_in_gga(x1, x2, x3) = ackermanncC_in_gga(x1, x2) 14.27/4.55 14.27/4.55 U46_gga(x1, x2, x3) = U46_gga(x1, x3) 14.27/4.55 14.27/4.55 ackermanncC_out_gga(x1, x2, x3) = ackermanncC_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_in_ggaa(x1, x2, x3, x4) = qcG_in_ggaa(x1, x2) 14.27/4.55 14.27/4.55 U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5) 14.27/4.55 14.27/4.55 U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5) 14.27/4.55 14.27/4.55 ackermanncE_in_gga(x1, x2, x3) = ackermanncE_in_gga(x1, x2) 14.27/4.55 14.27/4.55 ackermanncE_out_gga(x1, x2, x3) = ackermanncE_out_gga(x1, x2, x3) 14.27/4.55 14.27/4.55 U50_gga(x1, x2, x3) = U50_gga(x1, x3) 14.27/4.55 14.27/4.55 U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4) 14.27/4.55 14.27/4.55 qcG_out_ggaa(x1, x2, x3, x4) = qcG_out_ggaa(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 ACKERMANNA_IN_GGG(x1, x2, x3) = ACKERMANNA_IN_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U17_GGG(x1, x2, x3) = U17_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U20_GGG(x1, x2, x3, x4) = U20_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U23_GGG(x1, x2, x3) = U23_GGG(x1, x2, x3) 14.27/4.55 14.27/4.55 U26_GGG(x1, x2, x3, x4) = U26_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 U28_GGG(x1, x2, x3, x4) = U28_GGG(x1, x2, x3, x4) 14.27/4.55 14.27/4.55 14.27/4.55 We have to consider all (P,R,Pi)-chains 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (113) PiDPToQDPProof (SOUND) 14.27/4.55 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 14.27/4.55 ---------------------------------------- 14.27/4.55 14.27/4.55 (114) 14.27/4.55 Obligation: 14.27/4.55 Q DP problem: 14.27/4.55 The TRS P consists of the following rules: 14.27/4.55 14.27/4.55 ACKERMANNA_IN_GGG(s(s(X1)), 0, X2) -> U17_GGG(X1, X2, ackermanncB_in_ga(X1)) 14.27/4.55 U17_GGG(X1, X2, ackermanncB_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(X2), X3) -> U20_GGG(X1, X2, X3, ackermanncC_in_gga(X1, X2)) 14.27/4.55 U20_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> ACKERMANNA_IN_GGG(X1, X4, X3) 14.27/4.55 ACKERMANNA_IN_GGG(s(X1), s(0), X2) -> U23_GGG(X1, X2, ackermanncD_in_ga(X1)) 14.27/4.56 U23_GGG(X1, X2, ackermanncD_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.56 ACKERMANNA_IN_GGG(s(X1), s(s(X2)), X3) -> U26_GGG(X1, X2, X3, ackermanncC_in_gga(X1, X2)) 14.27/4.56 U26_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> U28_GGG(X1, X2, X3, ackermanncE_in_gga(X1, X4)) 14.27/4.56 U28_GGG(X1, X2, X3, ackermanncE_out_gga(X1, X4, X5)) -> ACKERMANNA_IN_GGG(X1, X5, X3) 14.27/4.56 14.27/4.56 The TRS R consists of the following rules: 14.27/4.56 14.27/4.56 ackermanncB_in_ga(X1) -> U40_ga(X1, ackermanncD_in_ga(X1)) 14.27/4.56 ackermanncC_in_gga(X1, 0) -> U46_gga(X1, ackermanncD_in_ga(X1)) 14.27/4.56 ackermanncC_in_gga(X1, s(X2)) -> U47_gga(X1, X2, qcG_in_ggaa(X1, X2)) 14.27/4.56 ackermanncD_in_ga(0) -> ackermanncD_out_ga(0, s(s(0))) 14.27/4.56 ackermanncD_in_ga(s(X1)) -> U41_ga(X1, ackermanncB_in_ga(X1)) 14.27/4.56 ackermanncE_in_gga(0, X1) -> ackermanncE_out_gga(0, X1, s(X1)) 14.27/4.56 ackermanncE_in_gga(s(X1), 0) -> U50_gga(X1, ackermanncD_in_ga(X1)) 14.27/4.56 ackermanncE_in_gga(s(X1), s(X2)) -> U51_gga(X1, X2, qcG_in_ggaa(X1, X2)) 14.27/4.56 U40_ga(X1, ackermanncD_out_ga(X1, X2)) -> ackermanncB_out_ga(X1, X2) 14.27/4.56 U46_gga(X1, ackermanncD_out_ga(X1, X2)) -> ackermanncC_out_gga(X1, 0, X2) 14.27/4.56 U47_gga(X1, X2, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncC_out_gga(X1, s(X2), X3) 14.27/4.56 U41_ga(X1, ackermanncB_out_ga(X1, X3)) -> U42_ga(X1, ackermanncF_in_gga(X1, X3)) 14.27/4.56 U50_gga(X1, ackermanncD_out_ga(X1, X2)) -> ackermanncE_out_gga(s(X1), 0, X2) 14.27/4.56 U51_gga(X1, X2, qcG_out_ggaa(X1, X2, X4, X3)) -> ackermanncE_out_gga(s(X1), s(X2), X3) 14.27/4.56 qcG_in_ggaa(X1, X2) -> U48_ggaa(X1, X2, ackermanncC_in_gga(X1, X2)) 14.27/4.56 U42_ga(X1, ackermanncF_out_gga(X1, X3, X2)) -> ackermanncD_out_ga(s(X1), X2) 14.27/4.56 U48_ggaa(X1, X2, ackermanncC_out_gga(X1, X2, X3)) -> U49_ggaa(X1, X2, X3, ackermanncE_in_gga(X1, X3)) 14.27/4.56 ackermanncF_in_gga(0, X1) -> ackermanncF_out_gga(0, X1, s(X1)) 14.27/4.56 ackermanncF_in_gga(s(X1), 0) -> U43_gga(X1, ackermanncD_in_ga(X1)) 14.27/4.56 ackermanncF_in_gga(s(X1), s(X2)) -> U44_gga(X1, X2, ackermanncF_in_gga(s(X1), X2)) 14.27/4.56 U49_ggaa(X1, X2, X3, ackermanncE_out_gga(X1, X3, X4)) -> qcG_out_ggaa(X1, X2, X3, X4) 14.27/4.56 U43_gga(X1, ackermanncD_out_ga(X1, X2)) -> ackermanncF_out_gga(s(X1), 0, X2) 14.27/4.56 U44_gga(X1, X2, ackermanncF_out_gga(s(X1), X2, X4)) -> U45_gga(X1, X2, ackermanncF_in_gga(X1, X4)) 14.27/4.56 U45_gga(X1, X2, ackermanncF_out_gga(X1, X4, X3)) -> ackermanncF_out_gga(s(X1), s(X2), X3) 14.27/4.56 14.27/4.56 The set Q consists of the following terms: 14.27/4.56 14.27/4.56 ackermanncB_in_ga(x0) 14.27/4.56 ackermanncC_in_gga(x0, x1) 14.27/4.56 ackermanncD_in_ga(x0) 14.27/4.56 ackermanncE_in_gga(x0, x1) 14.27/4.56 U40_ga(x0, x1) 14.27/4.56 U46_gga(x0, x1) 14.27/4.56 U47_gga(x0, x1, x2) 14.27/4.56 U41_ga(x0, x1) 14.27/4.56 U50_gga(x0, x1) 14.27/4.56 U51_gga(x0, x1, x2) 14.27/4.56 qcG_in_ggaa(x0, x1) 14.27/4.56 U42_ga(x0, x1) 14.27/4.56 U48_ggaa(x0, x1, x2) 14.27/4.56 ackermanncF_in_gga(x0, x1) 14.27/4.56 U49_ggaa(x0, x1, x2, x3) 14.27/4.56 U43_gga(x0, x1) 14.27/4.56 U44_gga(x0, x1, x2) 14.27/4.56 U45_gga(x0, x1, x2) 14.27/4.56 14.27/4.56 We have to consider all (P,Q,R)-chains. 14.27/4.56 ---------------------------------------- 14.27/4.56 14.27/4.56 (115) QDPSizeChangeProof (EQUIVALENT) 14.27/4.56 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. 14.27/4.56 14.27/4.56 From the DPs we obtained the following set of size-change graphs: 14.27/4.56 *U17_GGG(X1, X2, ackermanncB_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.56 The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2, 2 >= 3 14.27/4.56 14.27/4.56 14.27/4.56 *ACKERMANNA_IN_GGG(s(s(X1)), 0, X2) -> U17_GGG(X1, X2, ackermanncB_in_ga(X1)) 14.27/4.56 The graph contains the following edges 1 > 1, 3 >= 2 14.27/4.56 14.27/4.56 14.27/4.56 *U26_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> U28_GGG(X1, X2, X3, ackermanncE_in_gga(X1, X4)) 14.27/4.56 The graph contains the following edges 1 >= 1, 4 > 1, 2 >= 2, 4 > 2, 3 >= 3 14.27/4.56 14.27/4.56 14.27/4.56 *U20_GGG(X1, X2, X3, ackermanncC_out_gga(X1, X2, X4)) -> ACKERMANNA_IN_GGG(X1, X4, X3) 14.27/4.56 The graph contains the following edges 1 >= 1, 4 > 1, 4 > 2, 3 >= 3 14.27/4.56 14.27/4.56 14.27/4.56 *ACKERMANNA_IN_GGG(s(X1), s(X2), X3) -> U20_GGG(X1, X2, X3, ackermanncC_in_gga(X1, X2)) 14.27/4.56 The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 14.27/4.56 14.27/4.56 14.27/4.56 *U23_GGG(X1, X2, ackermanncD_out_ga(X1, X3)) -> ACKERMANNA_IN_GGG(X1, X3, X2) 14.27/4.56 The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2, 2 >= 3 14.27/4.56 14.27/4.56 14.27/4.56 *U28_GGG(X1, X2, X3, ackermanncE_out_gga(X1, X4, X5)) -> ACKERMANNA_IN_GGG(X1, X5, X3) 14.27/4.56 The graph contains the following edges 1 >= 1, 4 > 1, 4 > 2, 3 >= 3 14.27/4.56 14.27/4.56 14.27/4.56 *ACKERMANNA_IN_GGG(s(X1), s(0), X2) -> U23_GGG(X1, X2, ackermanncD_in_ga(X1)) 14.27/4.56 The graph contains the following edges 1 > 1, 3 >= 2 14.27/4.56 14.27/4.56 14.27/4.56 *ACKERMANNA_IN_GGG(s(X1), s(s(X2)), X3) -> U26_GGG(X1, X2, X3, ackermanncC_in_gga(X1, X2)) 14.27/4.56 The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 14.27/4.56 14.27/4.56 14.27/4.56 ---------------------------------------- 14.27/4.56 14.27/4.56 (116) 14.27/4.56 YES 14.27/4.56 14.27/4.56 ---------------------------------------- 14.27/4.56 14.27/4.56 (117) PrologToTRSTransformerProof (SOUND) 14.27/4.56 Transformed Prolog program to TRS. 14.27/4.56 14.27/4.56 { 14.27/4.56 "root": 10, 14.27/4.56 "program": { 14.27/4.56 "directives": [], 14.27/4.56 "clauses": [ 14.27/4.56 [ 14.27/4.56 "(ackermann (0) N (s N))", 14.27/4.56 null 14.27/4.56 ], 14.27/4.56 [ 14.27/4.56 "(ackermann (s M) (0) Val)", 14.27/4.56 "(ackermann M (s (0)) Val)" 14.27/4.56 ], 14.27/4.56 [ 14.27/4.56 "(ackermann (s M) (s N) Val)", 14.27/4.56 "(',' (ackermann (s M) N Val1) (ackermann M Val1 Val))" 14.27/4.56 ] 14.27/4.56 ] 14.27/4.56 }, 14.27/4.56 "graph": { 14.27/4.56 "nodes": { 14.27/4.56 "type": "Nodes", 14.27/4.56 "271": { 14.27/4.56 "goal": [{ 14.27/4.56 "clause": -1, 14.27/4.56 "scope": -1, 14.27/4.56 "term": "(true)" 14.27/4.56 }], 14.27/4.56 "kb": { 14.27/4.56 "nonunifying": [], 14.27/4.56 "intvars": {}, 14.27/4.56 "arithmetic": { 14.27/4.56 "type": "PlainIntegerRelationState", 14.27/4.56 "relations": [] 14.27/4.56 }, 14.27/4.56 "ground": [], 14.27/4.56 "free": [], 14.27/4.56 "exprvars": [] 14.27/4.56 } 14.27/4.56 }, 14.27/4.56 "272": { 14.27/4.56 "goal": [], 14.27/4.56 "kb": { 14.27/4.56 "nonunifying": [], 14.27/4.56 "intvars": {}, 14.27/4.56 "arithmetic": { 14.27/4.56 "type": "PlainIntegerRelationState", 14.27/4.56 "relations": [] 14.27/4.56 }, 14.27/4.56 "ground": [], 14.27/4.56 "free": [], 14.27/4.56 "exprvars": [] 14.27/4.56 } 14.27/4.56 }, 14.27/4.56 "470": { 14.27/4.56 "goal": [], 14.27/4.56 "kb": { 14.27/4.56 "nonunifying": [], 14.27/4.56 "intvars": {}, 14.27/4.56 "arithmetic": { 14.27/4.56 "type": "PlainIntegerRelationState", 14.27/4.56 "relations": [] 14.27/4.56 }, 14.27/4.56 "ground": [], 14.27/4.56 "free": [], 14.27/4.56 "exprvars": [] 14.27/4.56 } 14.27/4.56 }, 14.27/4.56 "196": { 14.27/4.56 "goal": [{ 14.27/4.56 "clause": -1, 14.27/4.56 "scope": -1, 14.27/4.56 "term": "(ackermann T17 (s (0)) T18)" 14.27/4.56 }], 14.27/4.56 "kb": { 14.27/4.56 "nonunifying": [], 14.27/4.56 "intvars": {}, 14.27/4.56 "arithmetic": { 14.27/4.56 "type": "PlainIntegerRelationState", 14.27/4.56 "relations": [] 14.27/4.56 }, 14.27/4.56 "ground": [ 14.27/4.56 "T17", 14.27/4.56 "T18" 14.27/4.56 ], 14.27/4.56 "free": [], 14.27/4.56 "exprvars": [] 14.27/4.56 } 14.27/4.56 }, 14.27/4.56 "273": { 14.27/4.56 "goal": [], 14.27/4.56 "kb": { 14.27/4.56 "nonunifying": [], 14.27/4.56 "intvars": {}, 14.27/4.56 "arithmetic": { 14.27/4.56 "type": "PlainIntegerRelationState", 14.27/4.56 "relations": [] 14.27/4.56 }, 14.27/4.56 "ground": [], 14.27/4.56 "free": [], 14.27/4.56 "exprvars": [] 14.27/4.56 } 14.27/4.56 }, 14.27/4.56 "471": { 14.27/4.56 "goal": [{ 14.27/4.56 "clause": -1, 14.27/4.56 "scope": -1, 14.27/4.56 "term": "(',' (ackermann (s T108) T110 X215) (ackermann T108 X215 X216))" 14.27/4.56 }], 14.27/4.56 "kb": { 14.27/4.56 "nonunifying": [], 14.27/4.56 "intvars": {}, 14.27/4.56 "arithmetic": { 14.27/4.56 "type": "PlainIntegerRelationState", 14.52/4.56 "relations": [] 14.52/4.56 }, 14.52/4.56 "ground": ["T108"], 14.52/4.56 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14.52/4.56 ], 14.52/4.56 "free": [], 14.52/4.56 "exprvars": [] 14.52/4.56 } 14.52/4.56 }, 14.52/4.56 "466": { 14.52/4.56 "goal": [ 14.52/4.56 { 14.52/4.56 "clause": 1, 14.52/4.56 "scope": 7, 14.52/4.56 "term": "(ackermann (s T94) T97 X180)" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "clause": 2, 14.52/4.56 "scope": 7, 14.52/4.56 "term": "(ackermann (s T94) T97 X180)" 14.52/4.56 } 14.52/4.56 ], 14.52/4.56 "kb": { 14.52/4.56 "nonunifying": [], 14.52/4.56 "intvars": {}, 14.52/4.56 "arithmetic": { 14.52/4.56 "type": "PlainIntegerRelationState", 14.52/4.56 "relations": [] 14.52/4.56 }, 14.52/4.56 "ground": ["T94"], 14.52/4.56 "free": ["X180"], 14.52/4.56 "exprvars": [] 14.52/4.56 } 14.52/4.56 }, 14.52/4.56 "467": { 14.52/4.56 "goal": [{ 14.52/4.56 "clause": 1, 14.52/4.56 "scope": 7, 14.52/4.56 "term": "(ackermann (s T94) T97 X180)" 14.52/4.56 }], 14.52/4.56 "kb": { 14.52/4.56 "nonunifying": [], 14.52/4.56 "intvars": {}, 14.52/4.56 "arithmetic": { 14.52/4.56 "type": "PlainIntegerRelationState", 14.52/4.56 "relations": [] 14.52/4.56 }, 14.52/4.56 "ground": ["T94"], 14.52/4.56 "free": ["X180"], 14.52/4.56 "exprvars": [] 14.52/4.56 } 14.52/4.56 }, 14.52/4.56 "468": { 14.52/4.56 "goal": [{ 14.52/4.56 "clause": 2, 14.52/4.56 "scope": 7, 14.52/4.56 "term": "(ackermann (s T94) T97 X180)" 14.52/4.56 }], 14.52/4.56 "kb": { 14.52/4.56 "nonunifying": [], 14.52/4.56 "intvars": {}, 14.52/4.56 "arithmetic": { 14.52/4.56 "type": "PlainIntegerRelationState", 14.52/4.56 "relations": [] 14.52/4.56 }, 14.52/4.56 "ground": ["T94"], 14.52/4.56 "free": ["X180"], 14.52/4.56 "exprvars": [] 14.52/4.56 } 14.52/4.56 }, 14.52/4.56 "469": { 14.52/4.56 "goal": [{ 14.52/4.56 "clause": -1, 14.52/4.56 "scope": -1, 14.52/4.56 "term": "(ackermann T103 (s (0)) X200)" 14.52/4.56 }], 14.52/4.56 "kb": { 14.52/4.56 "nonunifying": [], 14.52/4.56 "intvars": {}, 14.52/4.56 "arithmetic": { 14.52/4.56 "type": "PlainIntegerRelationState", 14.52/4.56 "relations": [] 14.52/4.56 }, 14.52/4.56 "ground": ["T103"], 14.52/4.56 "free": ["X200"], 14.52/4.56 "exprvars": [] 14.52/4.56 } 14.52/4.56 } 14.52/4.56 }, 14.52/4.56 "edges": [ 14.52/4.56 { 14.52/4.56 "from": 10, 14.52/4.56 "to": 11, 14.52/4.56 "label": "CASE" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 11, 14.52/4.56 "to": 29, 14.52/4.56 "label": "PARALLEL" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 11, 14.52/4.56 "to": 32, 14.52/4.56 "label": "PARALLEL" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 29, 14.52/4.56 "to": 169, 14.52/4.56 "label": "EVAL with clause\nackermann(0, X5, s(X5)).\nand substitutionT1 -> 0,\nT2 -> T8,\nX5 -> T8,\nT3 -> s(T8)" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 29, 14.52/4.56 "to": 174, 14.52/4.56 "label": "EVAL-BACKTRACK" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 32, 14.52/4.56 "to": 183, 14.52/4.56 "label": "PARALLEL" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 32, 14.52/4.56 "to": 185, 14.52/4.56 "label": "PARALLEL" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 169, 14.52/4.56 "to": 176, 14.52/4.56 "label": "SUCCESS" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 183, 14.52/4.56 "to": 196, 14.52/4.56 "label": "EVAL with clause\nackermann(s(X14), 0, X15) :- ackermann(X14, s(0), X15).\nand substitutionX14 -> T17,\nT1 -> s(T17),\nT2 -> 0,\nT3 -> T18,\nX15 -> T18" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 183, 14.52/4.56 "to": 200, 14.52/4.56 "label": "EVAL-BACKTRACK" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 185, 14.52/4.56 "to": 461, 14.52/4.56 "label": "EVAL with clause\nackermann(s(X177), s(X178), X179) :- ','(ackermann(s(X177), X178, X180), ackermann(X177, X180, X179)).\nand substitutionX177 -> T94,\nT1 -> s(T94),\nX178 -> T97,\nT2 -> s(T97),\nT3 -> T96,\nX179 -> T96,\nT95 -> T97" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 185, 14.52/4.56 "to": 462, 14.52/4.56 "label": "EVAL-BACKTRACK" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 196, 14.52/4.56 "to": 266, 14.52/4.56 "label": "CASE" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 266, 14.52/4.56 "to": 267, 14.52/4.56 "label": "PARALLEL" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 266, 14.52/4.56 "to": 268, 14.52/4.56 "label": "PARALLEL" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 267, 14.52/4.56 "to": 271, 14.52/4.56 "label": "EVAL with clause\nackermann(0, X22, s(X22)).\nand substitutionT17 -> 0,\nX22 -> s(0),\nT18 -> s(s(0))" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 267, 14.52/4.56 "to": 272, 14.52/4.56 "label": "EVAL-BACKTRACK" 14.52/4.56 }, 14.52/4.56 { 14.52/4.56 "from": 268, 14.52/4.56 "to": 274, 14.52/4.56 "label": "BACKTRACK\nfor clause: ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val)because of non-unification" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 271, 14.52/4.57 "to": 273, 14.52/4.57 "label": "SUCCESS" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 274, 14.52/4.57 "to": 277, 14.52/4.57 "label": "EVAL with clause\nackermann(s(X32), s(X33), X34) :- ','(ackermann(s(X32), X33, X35), ackermann(X32, X35, X34)).\nand substitutionX32 -> T24,\nT17 -> s(T24),\nX33 -> 0,\nT18 -> T25,\nX34 -> T25" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 274, 14.52/4.57 "to": 278, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 277, 14.52/4.57 "to": 280, 14.52/4.57 "label": "SPLIT 1" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 277, 14.52/4.57 "to": 281, 14.52/4.57 "label": "SPLIT 2\nnew knowledge:\nT24 is ground\nT26 is ground\nreplacements:X35 -> T26" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 280, 14.52/4.57 "to": 282, 14.52/4.57 "label": "CASE" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 281, 14.52/4.57 "to": 443, 14.52/4.57 "label": "CASE" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 282, 14.52/4.57 "to": 283, 14.52/4.57 "label": "BACKTRACK\nfor clause: ackermann(0, N, s(N))because of non-unification" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 283, 14.52/4.57 "to": 287, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 283, 14.52/4.57 "to": 288, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 287, 14.52/4.57 "to": 289, 14.52/4.57 "label": "ONLY EVAL with clause\nackermann(s(X57), 0, X58) :- ackermann(X57, s(0), X58).\nand substitutionT24 -> T31,\nX57 -> T31,\nX35 -> X59,\nX58 -> X59" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 288, 14.52/4.57 "to": 384, 14.52/4.57 "label": "BACKTRACK\nfor clause: ackermann(s(M), s(N), Val) :- ','(ackermann(s(M), N, Val1), ackermann(M, Val1, Val))because of non-unification" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 289, 14.52/4.57 "to": 290, 14.52/4.57 "label": "CASE" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 290, 14.52/4.57 "to": 291, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 290, 14.52/4.57 "to": 292, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 291, 14.52/4.57 "to": 293, 14.52/4.57 "label": "EVAL with clause\nackermann(0, X66, s(X66)).\nand substitutionT31 -> 0,\nX66 -> s(0),\nX59 -> s(s(0))" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 291, 14.52/4.57 "to": 294, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 292, 14.52/4.57 "to": 296, 14.52/4.57 "label": "BACKTRACK\nfor clause: ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val)because of non-unification" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 293, 14.52/4.57 "to": 295, 14.52/4.57 "label": "SUCCESS" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 296, 14.52/4.57 "to": 297, 14.52/4.57 "label": "EVAL with clause\nackermann(s(X78), s(X79), X80) :- ','(ackermann(s(X78), X79, X81), ackermann(X78, X81, X80)).\nand substitutionX78 -> T35,\nT31 -> s(T35),\nX79 -> 0,\nX59 -> X82,\nX80 -> X82" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 296, 14.52/4.57 "to": 298, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 297, 14.52/4.57 "to": 299, 14.52/4.57 "label": "SPLIT 1" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 297, 14.52/4.57 "to": 300, 14.52/4.57 "label": "SPLIT 2\nnew knowledge:\nT35 is ground\nT36 is ground\nreplacements:X81 -> T36" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 299, 14.52/4.57 "to": 280, 14.52/4.57 "label": "INSTANCE with matching:\nT24 -> T35\nX35 -> X81" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 300, 14.52/4.57 "to": 365, 14.52/4.57 "label": "CASE" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 365, 14.52/4.57 "to": 366, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 365, 14.52/4.57 "to": 367, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 366, 14.52/4.57 "to": 371, 14.52/4.57 "label": "EVAL with clause\nackermann(0, X93, s(X93)).\nand substitutionT35 -> 0,\nT36 -> T43,\nX93 -> T43,\nX82 -> s(T43)" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 366, 14.52/4.57 "to": 372, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 367, 14.52/4.57 "to": 376, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 367, 14.52/4.57 "to": 377, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 371, 14.52/4.57 "to": 374, 14.52/4.57 "label": "SUCCESS" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 376, 14.52/4.57 "to": 378, 14.52/4.57 "label": "EVAL with clause\nackermann(s(X106), 0, X107) :- ackermann(X106, s(0), X107).\nand substitutionX106 -> T48,\nT35 -> s(T48),\nT36 -> 0,\nX82 -> X108,\nX107 -> X108" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 376, 14.52/4.57 "to": 379, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 377, 14.52/4.57 "to": 380, 14.52/4.57 "label": "EVAL with clause\nackermann(s(X120), s(X121), X122) :- ','(ackermann(s(X120), X121, X123), ackermann(X120, X123, X122)).\nand substitutionX120 -> T53,\nT35 -> s(T53),\nX121 -> T54,\nT36 -> s(T54),\nX82 -> X124,\nX122 -> X124" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 377, 14.52/4.57 "to": 381, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 378, 14.52/4.57 "to": 289, 14.52/4.57 "label": "INSTANCE with matching:\nT31 -> T48\nX59 -> X108" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 380, 14.52/4.57 "to": 382, 14.52/4.57 "label": "SPLIT 1" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 380, 14.52/4.57 "to": 383, 14.52/4.57 "label": "SPLIT 2\nnew knowledge:\nT53 is ground\nT54 is ground\nT55 is ground\nreplacements:X123 -> T55" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 382, 14.52/4.57 "to": 300, 14.52/4.57 "label": "INSTANCE with matching:\nT35 -> s(T53)\nT36 -> T54\nX82 -> X123" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 383, 14.52/4.57 "to": 300, 14.52/4.57 "label": "INSTANCE with matching:\nT35 -> T53\nT36 -> T55\nX82 -> X124" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 443, 14.52/4.57 "to": 445, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 443, 14.52/4.57 "to": 446, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 445, 14.52/4.57 "to": 448, 14.52/4.57 "label": "EVAL with clause\nackermann(0, X140, s(X140)).\nand substitutionT24 -> 0,\nT26 -> T65,\nX140 -> T65,\nT25 -> s(T65)" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 445, 14.52/4.57 "to": 449, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 446, 14.52/4.57 "to": 451, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 446, 14.52/4.57 "to": 452, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 448, 14.52/4.57 "to": 450, 14.52/4.57 "label": "SUCCESS" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 451, 14.52/4.57 "to": 455, 14.52/4.57 "label": "EVAL with clause\nackermann(s(X149), 0, X150) :- ackermann(X149, s(0), X150).\nand substitutionX149 -> T74,\nT24 -> s(T74),\nT26 -> 0,\nT25 -> T75,\nX150 -> T75" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 451, 14.52/4.57 "to": 456, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 452, 14.52/4.57 "to": 457, 14.52/4.57 "label": "EVAL with clause\nackermann(s(X160), s(X161), X162) :- ','(ackermann(s(X160), X161, X163), ackermann(X160, X163, X162)).\nand substitutionX160 -> T82,\nT24 -> s(T82),\nX161 -> T83,\nT26 -> s(T83),\nT25 -> T84,\nX162 -> T84" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 452, 14.52/4.57 "to": 458, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 455, 14.52/4.57 "to": 196, 14.52/4.57 "label": "INSTANCE with matching:\nT17 -> T74\nT18 -> T75" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 457, 14.52/4.57 "to": 459, 14.52/4.57 "label": "SPLIT 1" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 457, 14.52/4.57 "to": 460, 14.52/4.57 "label": "SPLIT 2\nnew knowledge:\nT82 is ground\nT83 is ground\nT85 is ground\nreplacements:X163 -> T85" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 459, 14.52/4.57 "to": 300, 14.52/4.57 "label": "INSTANCE with matching:\nT35 -> s(T82)\nT36 -> T83\nX82 -> X163" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 460, 14.52/4.57 "to": 281, 14.52/4.57 "label": "INSTANCE with matching:\nT24 -> T82\nT26 -> T85\nT25 -> T84" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 461, 14.52/4.57 "to": 463, 14.52/4.57 "label": "SPLIT 1" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 461, 14.52/4.57 "to": 464, 14.52/4.57 "label": "SPLIT 2\nnew knowledge:\nT94 is ground\nreplacements:X180 -> T98" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 463, 14.52/4.57 "to": 465, 14.52/4.57 "label": "CASE" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 464, 14.52/4.57 "to": 10, 14.52/4.57 "label": "INSTANCE with matching:\nT1 -> T94\nT2 -> T98\nT3 -> T96" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 465, 14.52/4.57 "to": 466, 14.52/4.57 "label": "BACKTRACK\nfor clause: ackermann(0, N, s(N))because of non-unification" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 466, 14.52/4.57 "to": 467, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 466, 14.52/4.57 "to": 468, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 467, 14.52/4.57 "to": 469, 14.52/4.57 "label": "EVAL with clause\nackermann(s(X198), 0, X199) :- ackermann(X198, s(0), X199).\nand substitutionT94 -> T103,\nX198 -> T103,\nT97 -> 0,\nX180 -> X200,\nX199 -> X200" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 467, 14.52/4.57 "to": 470, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 468, 14.52/4.57 "to": 471, 14.52/4.57 "label": "EVAL with clause\nackermann(s(X212), s(X213), X214) :- ','(ackermann(s(X212), X213, X215), ackermann(X212, X215, X214)).\nand substitutionT94 -> T108,\nX212 -> T108,\nX213 -> T110,\nT97 -> s(T110),\nX180 -> X216,\nX214 -> X216,\nT109 -> T110" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 468, 14.52/4.57 "to": 472, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 469, 14.52/4.57 "to": 289, 14.52/4.57 "label": "INSTANCE with matching:\nT31 -> T103\nX59 -> X200" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 471, 14.52/4.57 "to": 473, 14.52/4.57 "label": "SPLIT 1" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 471, 14.52/4.57 "to": 474, 14.52/4.57 "label": "SPLIT 2\nnew knowledge:\nT108 is ground\nreplacements:X215 -> T111" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 473, 14.52/4.57 "to": 463, 14.52/4.57 "label": "INSTANCE with matching:\nT94 -> T108\nT97 -> T110\nX180 -> X215" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 474, 14.52/4.57 "to": 478, 14.52/4.57 "label": "CASE" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 478, 14.52/4.57 "to": 482, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 478, 14.52/4.57 "to": 483, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 482, 14.52/4.57 "to": 486, 14.52/4.57 "label": "EVAL with clause\nackermann(0, X227, s(X227)).\nand substitutionT108 -> 0,\nT111 -> T118,\nX227 -> T118,\nX216 -> s(T118)" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 482, 14.52/4.57 "to": 487, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 483, 14.52/4.57 "to": 489, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 483, 14.52/4.57 "to": 490, 14.52/4.57 "label": "PARALLEL" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 486, 14.52/4.57 "to": 488, 14.52/4.57 "label": "SUCCESS" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 489, 14.52/4.57 "to": 493, 14.52/4.57 "label": "EVAL with clause\nackermann(s(X240), 0, X241) :- ackermann(X240, s(0), X241).\nand substitutionX240 -> T123,\nT108 -> s(T123),\nT111 -> 0,\nX216 -> X242,\nX241 -> X242" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 489, 14.52/4.57 "to": 494, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 490, 14.52/4.57 "to": 527, 14.52/4.57 "label": "EVAL with clause\nackermann(s(X254), s(X255), X256) :- ','(ackermann(s(X254), X255, X257), ackermann(X254, X257, X256)).\nand substitutionX254 -> T128,\nT108 -> s(T128),\nX255 -> T130,\nT111 -> s(T130),\nX216 -> X258,\nX256 -> X258,\nT129 -> T130" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 490, 14.52/4.57 "to": 528, 14.52/4.57 "label": "EVAL-BACKTRACK" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 493, 14.52/4.57 "to": 289, 14.52/4.57 "label": "INSTANCE with matching:\nT31 -> T123\nX59 -> X242" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "from": 527, 14.52/4.57 "to": 471, 14.52/4.57 "label": "INSTANCE with matching:\nT108 -> T128\nT110 -> T130\nX215 -> X257\nX216 -> X258" 14.52/4.57 } 14.52/4.57 ], 14.52/4.57 "type": "Graph" 14.52/4.57 } 14.52/4.57 } 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (118) 14.52/4.57 Obligation: 14.52/4.57 Q restricted rewrite system: 14.52/4.57 The TRS R consists of the following rules: 14.52/4.57 14.52/4.57 f10_in(0, s(T8)) -> f10_out1 14.52/4.57 f10_in(s(T17), T18) -> U1(f196_in(T17, T18), s(T17), T18) 14.52/4.57 U1(f196_out1, s(T17), T18) -> f10_out1 14.52/4.57 f10_in(s(T94), T96) -> U2(f461_in(T94, T96), s(T94), T96) 14.52/4.57 U2(f461_out1, s(T94), T96) -> f10_out1 14.52/4.57 f280_in(T31) -> U3(f289_in(T31), T31) 14.52/4.57 U3(f289_out1(X59), T31) -> f280_out1(X59) 14.52/4.57 f289_in(0) -> f289_out1(s(s(0))) 14.52/4.57 f289_in(s(T35)) -> U4(f297_in(T35), s(T35)) 14.52/4.57 U4(f297_out1(X81, X82), s(T35)) -> f289_out1(X82) 14.52/4.57 f300_in(0, T43) -> f300_out1(s(T43)) 14.52/4.57 f300_in(s(T48), 0) -> U5(f289_in(T48), s(T48), 0) 14.52/4.57 U5(f289_out1(X108), s(T48), 0) -> f300_out1(X108) 14.52/4.57 f300_in(s(T53), s(T54)) -> U6(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 U6(f380_out1(X123, X124), s(T53), s(T54)) -> f300_out1(X124) 14.52/4.57 f196_in(0, s(s(0))) -> f196_out1 14.52/4.57 f196_in(s(T24), T25) -> U7(f277_in(T24, T25), s(T24), T25) 14.52/4.57 U7(f277_out1(X35), s(T24), T25) -> f196_out1 14.52/4.57 f281_in(0, T65, s(T65)) -> f281_out1 14.52/4.57 f281_in(s(T74), 0, T75) -> U8(f196_in(T74, T75), s(T74), 0, T75) 14.52/4.57 U8(f196_out1, s(T74), 0, T75) -> f281_out1 14.52/4.57 f281_in(s(T82), s(T83), T84) -> U9(f457_in(T82, T83, T84), s(T82), s(T83), T84) 14.52/4.57 U9(f457_out1(X163), s(T82), s(T83), T84) -> f281_out1 14.52/4.57 f463_in(T103) -> U10(f289_in(T103), T103) 14.52/4.57 U10(f289_out1(X200), T103) -> f463_out1 14.52/4.57 f463_in(T108) -> U11(f471_in(T108), T108) 14.52/4.57 U11(f471_out1, T108) -> f463_out1 14.52/4.57 f474_in(0) -> f474_out1 14.52/4.57 f474_in(s(T123)) -> U12(f289_in(T123), s(T123)) 14.52/4.57 U12(f289_out1(X242), s(T123)) -> f474_out1 14.52/4.57 f474_in(s(T128)) -> U13(f471_in(T128), s(T128)) 14.52/4.57 U13(f471_out1, s(T128)) -> f474_out1 14.52/4.57 f277_in(T24, T25) -> U14(f280_in(T24), T24, T25) 14.52/4.57 U14(f280_out1(T26), T24, T25) -> U15(f281_in(T24, T26, T25), T24, T25, T26) 14.52/4.57 U15(f281_out1, T24, T25, T26) -> f277_out1(T26) 14.52/4.57 f297_in(T35) -> U16(f280_in(T35), T35) 14.52/4.57 U16(f280_out1(T36), T35) -> U17(f300_in(T35, T36), T35, T36) 14.52/4.57 U17(f300_out1(X82), T35, T36) -> f297_out1(T36, X82) 14.52/4.57 f380_in(T53, T54) -> U18(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18(f300_out1(T55), T53, T54) -> U19(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U19(f300_out1(X124), T53, T54, T55) -> f380_out1(T55, X124) 14.52/4.57 f457_in(T82, T83, T84) -> U20(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 U20(f300_out1(T85), T82, T83, T84) -> U21(f281_in(T82, T85, T84), T82, T83, T84, T85) 14.52/4.57 U21(f281_out1, T82, T83, T84, T85) -> f457_out1(T85) 14.52/4.57 f461_in(T94, T96) -> U22(f463_in(T94), T94, T96) 14.52/4.57 U22(f463_out1, T94, T96) -> U23(f10_in(T94, T96), T94, T96) 14.52/4.57 U23(f10_out1, T94, T96) -> f461_out1 14.52/4.57 f471_in(T108) -> U24(f463_in(T108), T108) 14.52/4.57 U24(f463_out1, T108) -> U25(f474_in(T108), T108) 14.52/4.57 U25(f474_out1, T108) -> f471_out1 14.52/4.57 14.52/4.57 Q is empty. 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (119) DependencyPairsProof (EQUIVALENT) 14.52/4.57 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (120) 14.52/4.57 Obligation: 14.52/4.57 Q DP problem: 14.52/4.57 The TRS P consists of the following rules: 14.52/4.57 14.52/4.57 F10_IN(s(T17), T18) -> U1^1(f196_in(T17, T18), s(T17), T18) 14.52/4.57 F10_IN(s(T17), T18) -> F196_IN(T17, T18) 14.52/4.57 F10_IN(s(T94), T96) -> U2^1(f461_in(T94, T96), s(T94), T96) 14.52/4.57 F10_IN(s(T94), T96) -> F461_IN(T94, T96) 14.52/4.57 F280_IN(T31) -> U3^1(f289_in(T31), T31) 14.52/4.57 F280_IN(T31) -> F289_IN(T31) 14.52/4.57 F289_IN(s(T35)) -> U4^1(f297_in(T35), s(T35)) 14.52/4.57 F289_IN(s(T35)) -> F297_IN(T35) 14.52/4.57 F300_IN(s(T48), 0) -> U5^1(f289_in(T48), s(T48), 0) 14.52/4.57 F300_IN(s(T48), 0) -> F289_IN(T48) 14.52/4.57 F300_IN(s(T53), s(T54)) -> U6^1(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 F300_IN(s(T53), s(T54)) -> F380_IN(T53, T54) 14.52/4.57 F196_IN(s(T24), T25) -> U7^1(f277_in(T24, T25), s(T24), T25) 14.52/4.57 F196_IN(s(T24), T25) -> F277_IN(T24, T25) 14.52/4.57 F281_IN(s(T74), 0, T75) -> U8^1(f196_in(T74, T75), s(T74), 0, T75) 14.52/4.57 F281_IN(s(T74), 0, T75) -> F196_IN(T74, T75) 14.52/4.57 F281_IN(s(T82), s(T83), T84) -> U9^1(f457_in(T82, T83, T84), s(T82), s(T83), T84) 14.52/4.57 F281_IN(s(T82), s(T83), T84) -> F457_IN(T82, T83, T84) 14.52/4.57 F463_IN(T103) -> U10^1(f289_in(T103), T103) 14.52/4.57 F463_IN(T103) -> F289_IN(T103) 14.52/4.57 F463_IN(T108) -> U11^1(f471_in(T108), T108) 14.52/4.57 F463_IN(T108) -> F471_IN(T108) 14.52/4.57 F474_IN(s(T123)) -> U12^1(f289_in(T123), s(T123)) 14.52/4.57 F474_IN(s(T123)) -> F289_IN(T123) 14.52/4.57 F474_IN(s(T128)) -> U13^1(f471_in(T128), s(T128)) 14.52/4.57 F474_IN(s(T128)) -> F471_IN(T128) 14.52/4.57 F277_IN(T24, T25) -> U14^1(f280_in(T24), T24, T25) 14.52/4.57 F277_IN(T24, T25) -> F280_IN(T24) 14.52/4.57 U14^1(f280_out1(T26), T24, T25) -> U15^1(f281_in(T24, T26, T25), T24, T25, T26) 14.52/4.57 U14^1(f280_out1(T26), T24, T25) -> F281_IN(T24, T26, T25) 14.52/4.57 F297_IN(T35) -> U16^1(f280_in(T35), T35) 14.52/4.57 F297_IN(T35) -> F280_IN(T35) 14.52/4.57 U16^1(f280_out1(T36), T35) -> U17^1(f300_in(T35, T36), T35, T36) 14.52/4.57 U16^1(f280_out1(T36), T35) -> F300_IN(T35, T36) 14.52/4.57 F380_IN(T53, T54) -> U18^1(f300_in(s(T53), T54), T53, T54) 14.52/4.57 F380_IN(T53, T54) -> F300_IN(s(T53), T54) 14.52/4.57 U18^1(f300_out1(T55), T53, T54) -> U19^1(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U18^1(f300_out1(T55), T53, T54) -> F300_IN(T53, T55) 14.52/4.57 F457_IN(T82, T83, T84) -> U20^1(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 F457_IN(T82, T83, T84) -> F300_IN(s(T82), T83) 14.52/4.57 U20^1(f300_out1(T85), T82, T83, T84) -> U21^1(f281_in(T82, T85, T84), T82, T83, T84, T85) 14.52/4.57 U20^1(f300_out1(T85), T82, T83, T84) -> F281_IN(T82, T85, T84) 14.52/4.57 F461_IN(T94, T96) -> U22^1(f463_in(T94), T94, T96) 14.52/4.57 F461_IN(T94, T96) -> F463_IN(T94) 14.52/4.57 U22^1(f463_out1, T94, T96) -> U23^1(f10_in(T94, T96), T94, T96) 14.52/4.57 U22^1(f463_out1, T94, T96) -> F10_IN(T94, T96) 14.52/4.57 F471_IN(T108) -> U24^1(f463_in(T108), T108) 14.52/4.57 F471_IN(T108) -> F463_IN(T108) 14.52/4.57 U24^1(f463_out1, T108) -> U25^1(f474_in(T108), T108) 14.52/4.57 U24^1(f463_out1, T108) -> F474_IN(T108) 14.52/4.57 14.52/4.57 The TRS R consists of the following rules: 14.52/4.57 14.52/4.57 f10_in(0, s(T8)) -> f10_out1 14.52/4.57 f10_in(s(T17), T18) -> U1(f196_in(T17, T18), s(T17), T18) 14.52/4.57 U1(f196_out1, s(T17), T18) -> f10_out1 14.52/4.57 f10_in(s(T94), T96) -> U2(f461_in(T94, T96), s(T94), T96) 14.52/4.57 U2(f461_out1, s(T94), T96) -> f10_out1 14.52/4.57 f280_in(T31) -> U3(f289_in(T31), T31) 14.52/4.57 U3(f289_out1(X59), T31) -> f280_out1(X59) 14.52/4.57 f289_in(0) -> f289_out1(s(s(0))) 14.52/4.57 f289_in(s(T35)) -> U4(f297_in(T35), s(T35)) 14.52/4.57 U4(f297_out1(X81, X82), s(T35)) -> f289_out1(X82) 14.52/4.57 f300_in(0, T43) -> f300_out1(s(T43)) 14.52/4.57 f300_in(s(T48), 0) -> U5(f289_in(T48), s(T48), 0) 14.52/4.57 U5(f289_out1(X108), s(T48), 0) -> f300_out1(X108) 14.52/4.57 f300_in(s(T53), s(T54)) -> U6(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 U6(f380_out1(X123, X124), s(T53), s(T54)) -> f300_out1(X124) 14.52/4.57 f196_in(0, s(s(0))) -> f196_out1 14.52/4.57 f196_in(s(T24), T25) -> U7(f277_in(T24, T25), s(T24), T25) 14.52/4.57 U7(f277_out1(X35), s(T24), T25) -> f196_out1 14.52/4.57 f281_in(0, T65, s(T65)) -> f281_out1 14.52/4.57 f281_in(s(T74), 0, T75) -> U8(f196_in(T74, T75), s(T74), 0, T75) 14.52/4.57 U8(f196_out1, s(T74), 0, T75) -> f281_out1 14.52/4.57 f281_in(s(T82), s(T83), T84) -> U9(f457_in(T82, T83, T84), s(T82), s(T83), T84) 14.52/4.57 U9(f457_out1(X163), s(T82), s(T83), T84) -> f281_out1 14.52/4.57 f463_in(T103) -> U10(f289_in(T103), T103) 14.52/4.57 U10(f289_out1(X200), T103) -> f463_out1 14.52/4.57 f463_in(T108) -> U11(f471_in(T108), T108) 14.52/4.57 U11(f471_out1, T108) -> f463_out1 14.52/4.57 f474_in(0) -> f474_out1 14.52/4.57 f474_in(s(T123)) -> U12(f289_in(T123), s(T123)) 14.52/4.57 U12(f289_out1(X242), s(T123)) -> f474_out1 14.52/4.57 f474_in(s(T128)) -> U13(f471_in(T128), s(T128)) 14.52/4.57 U13(f471_out1, s(T128)) -> f474_out1 14.52/4.57 f277_in(T24, T25) -> U14(f280_in(T24), T24, T25) 14.52/4.57 U14(f280_out1(T26), T24, T25) -> U15(f281_in(T24, T26, T25), T24, T25, T26) 14.52/4.57 U15(f281_out1, T24, T25, T26) -> f277_out1(T26) 14.52/4.57 f297_in(T35) -> U16(f280_in(T35), T35) 14.52/4.57 U16(f280_out1(T36), T35) -> U17(f300_in(T35, T36), T35, T36) 14.52/4.57 U17(f300_out1(X82), T35, T36) -> f297_out1(T36, X82) 14.52/4.57 f380_in(T53, T54) -> U18(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18(f300_out1(T55), T53, T54) -> U19(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U19(f300_out1(X124), T53, T54, T55) -> f380_out1(T55, X124) 14.52/4.57 f457_in(T82, T83, T84) -> U20(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 U20(f300_out1(T85), T82, T83, T84) -> U21(f281_in(T82, T85, T84), T82, T83, T84, T85) 14.52/4.57 U21(f281_out1, T82, T83, T84, T85) -> f457_out1(T85) 14.52/4.57 f461_in(T94, T96) -> U22(f463_in(T94), T94, T96) 14.52/4.57 U22(f463_out1, T94, T96) -> U23(f10_in(T94, T96), T94, T96) 14.52/4.57 U23(f10_out1, T94, T96) -> f461_out1 14.52/4.57 f471_in(T108) -> U24(f463_in(T108), T108) 14.52/4.57 U24(f463_out1, T108) -> U25(f474_in(T108), T108) 14.52/4.57 U25(f474_out1, T108) -> f471_out1 14.52/4.57 14.52/4.57 Q is empty. 14.52/4.57 We have to consider all minimal (P,Q,R)-chains. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (121) DependencyGraphProof (EQUIVALENT) 14.52/4.57 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 25 less nodes. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (122) 14.52/4.57 Complex Obligation (AND) 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (123) 14.52/4.57 Obligation: 14.52/4.57 Q DP problem: 14.52/4.57 The TRS P consists of the following rules: 14.52/4.57 14.52/4.57 F280_IN(T31) -> F289_IN(T31) 14.52/4.57 F289_IN(s(T35)) -> F297_IN(T35) 14.52/4.57 F297_IN(T35) -> U16^1(f280_in(T35), T35) 14.52/4.57 U16^1(f280_out1(T36), T35) -> F300_IN(T35, T36) 14.52/4.57 F300_IN(s(T48), 0) -> F289_IN(T48) 14.52/4.57 F300_IN(s(T53), s(T54)) -> F380_IN(T53, T54) 14.52/4.57 F380_IN(T53, T54) -> U18^1(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18^1(f300_out1(T55), T53, T54) -> F300_IN(T53, T55) 14.52/4.57 F380_IN(T53, T54) -> F300_IN(s(T53), T54) 14.52/4.57 F297_IN(T35) -> F280_IN(T35) 14.52/4.57 14.52/4.57 The TRS R consists of the following rules: 14.52/4.57 14.52/4.57 f10_in(0, s(T8)) -> f10_out1 14.52/4.57 f10_in(s(T17), T18) -> U1(f196_in(T17, T18), s(T17), T18) 14.52/4.57 U1(f196_out1, s(T17), T18) -> f10_out1 14.52/4.57 f10_in(s(T94), T96) -> U2(f461_in(T94, T96), s(T94), T96) 14.52/4.57 U2(f461_out1, s(T94), T96) -> f10_out1 14.52/4.57 f280_in(T31) -> U3(f289_in(T31), T31) 14.52/4.57 U3(f289_out1(X59), T31) -> f280_out1(X59) 14.52/4.57 f289_in(0) -> f289_out1(s(s(0))) 14.52/4.57 f289_in(s(T35)) -> U4(f297_in(T35), s(T35)) 14.52/4.57 U4(f297_out1(X81, X82), s(T35)) -> f289_out1(X82) 14.52/4.57 f300_in(0, T43) -> f300_out1(s(T43)) 14.52/4.57 f300_in(s(T48), 0) -> U5(f289_in(T48), s(T48), 0) 14.52/4.57 U5(f289_out1(X108), s(T48), 0) -> f300_out1(X108) 14.52/4.57 f300_in(s(T53), s(T54)) -> U6(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 U6(f380_out1(X123, X124), s(T53), s(T54)) -> f300_out1(X124) 14.52/4.57 f196_in(0, s(s(0))) -> f196_out1 14.52/4.57 f196_in(s(T24), T25) -> U7(f277_in(T24, T25), s(T24), T25) 14.52/4.57 U7(f277_out1(X35), s(T24), T25) -> f196_out1 14.52/4.57 f281_in(0, T65, s(T65)) -> f281_out1 14.52/4.57 f281_in(s(T74), 0, T75) -> U8(f196_in(T74, T75), s(T74), 0, T75) 14.52/4.57 U8(f196_out1, s(T74), 0, T75) -> f281_out1 14.52/4.57 f281_in(s(T82), s(T83), T84) -> U9(f457_in(T82, T83, T84), s(T82), s(T83), T84) 14.52/4.57 U9(f457_out1(X163), s(T82), s(T83), T84) -> f281_out1 14.52/4.57 f463_in(T103) -> U10(f289_in(T103), T103) 14.52/4.57 U10(f289_out1(X200), T103) -> f463_out1 14.52/4.57 f463_in(T108) -> U11(f471_in(T108), T108) 14.52/4.57 U11(f471_out1, T108) -> f463_out1 14.52/4.57 f474_in(0) -> f474_out1 14.52/4.57 f474_in(s(T123)) -> U12(f289_in(T123), s(T123)) 14.52/4.57 U12(f289_out1(X242), s(T123)) -> f474_out1 14.52/4.57 f474_in(s(T128)) -> U13(f471_in(T128), s(T128)) 14.52/4.57 U13(f471_out1, s(T128)) -> f474_out1 14.52/4.57 f277_in(T24, T25) -> U14(f280_in(T24), T24, T25) 14.52/4.57 U14(f280_out1(T26), T24, T25) -> U15(f281_in(T24, T26, T25), T24, T25, T26) 14.52/4.57 U15(f281_out1, T24, T25, T26) -> f277_out1(T26) 14.52/4.57 f297_in(T35) -> U16(f280_in(T35), T35) 14.52/4.57 U16(f280_out1(T36), T35) -> U17(f300_in(T35, T36), T35, T36) 14.52/4.57 U17(f300_out1(X82), T35, T36) -> f297_out1(T36, X82) 14.52/4.57 f380_in(T53, T54) -> U18(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18(f300_out1(T55), T53, T54) -> U19(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U19(f300_out1(X124), T53, T54, T55) -> f380_out1(T55, X124) 14.52/4.57 f457_in(T82, T83, T84) -> U20(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 U20(f300_out1(T85), T82, T83, T84) -> U21(f281_in(T82, T85, T84), T82, T83, T84, T85) 14.52/4.57 U21(f281_out1, T82, T83, T84, T85) -> f457_out1(T85) 14.52/4.57 f461_in(T94, T96) -> U22(f463_in(T94), T94, T96) 14.52/4.57 U22(f463_out1, T94, T96) -> U23(f10_in(T94, T96), T94, T96) 14.52/4.57 U23(f10_out1, T94, T96) -> f461_out1 14.52/4.57 f471_in(T108) -> U24(f463_in(T108), T108) 14.52/4.57 U24(f463_out1, T108) -> U25(f474_in(T108), T108) 14.52/4.57 U25(f474_out1, T108) -> f471_out1 14.52/4.57 14.52/4.57 Q is empty. 14.52/4.57 We have to consider all minimal (P,Q,R)-chains. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (124) QDPOrderProof (EQUIVALENT) 14.52/4.57 We use the reduction pair processor [LPAR04,JAR06]. 14.52/4.57 14.52/4.57 14.52/4.57 The following pairs can be oriented strictly and are deleted. 14.52/4.57 14.52/4.57 F280_IN(T31) -> F289_IN(T31) 14.52/4.57 F289_IN(s(T35)) -> F297_IN(T35) 14.52/4.57 U16^1(f280_out1(T36), T35) -> F300_IN(T35, T36) 14.52/4.57 F297_IN(T35) -> F280_IN(T35) 14.52/4.57 The remaining pairs can at least be oriented weakly. 14.52/4.57 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 14.52/4.57 14.52/4.57 POL( U16^1_2(x_1, x_2) ) = 2x_1 + x_2 + 2 14.52/4.57 POL( U18^1_3(x_1, ..., x_3) ) = x_2 14.52/4.57 POL( f280_in_1(x_1) ) = 0 14.52/4.57 POL( U3_2(x_1, x_2) ) = 0 14.52/4.57 POL( f289_in_1(x_1) ) = x_1 14.52/4.57 POL( f300_in_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} 14.52/4.57 POL( s_1(x_1) ) = 2x_1 + 2 14.52/4.57 POL( 0 ) = 2 14.52/4.57 POL( U5_3(x_1, ..., x_3) ) = 2x_1 14.52/4.57 POL( U6_3(x_1, ..., x_3) ) = 2 14.52/4.57 POL( f380_in_2(x_1, x_2) ) = 2x_1 + 2 14.52/4.57 POL( U4_2(x_1, x_2) ) = 2 14.52/4.57 POL( f297_in_1(x_1) ) = x_1 + 2 14.52/4.57 POL( f297_out1_2(x_1, x_2) ) = 0 14.52/4.57 POL( f289_out1_1(x_1) ) = 1 14.52/4.57 POL( U16_2(x_1, x_2) ) = 2 14.52/4.57 POL( f280_out1_1(x_1) ) = 0 14.52/4.57 POL( U17_3(x_1, ..., x_3) ) = 0 14.52/4.57 POL( U19_4(x_1, ..., x_4) ) = 0 14.52/4.57 POL( f300_out1_1(x_1) ) = 2 14.52/4.57 POL( f380_out1_2(x_1, x_2) ) = 0 14.52/4.57 POL( U18_3(x_1, ..., x_3) ) = 2x_2 14.52/4.57 POL( F280_IN_1(x_1) ) = 2x_1 + 1 14.52/4.57 POL( F289_IN_1(x_1) ) = 2x_1 14.52/4.57 POL( F297_IN_1(x_1) ) = 2x_1 + 2 14.52/4.57 POL( F300_IN_2(x_1, x_2) ) = max{0, x_1 - 2} 14.52/4.57 POL( F380_IN_2(x_1, x_2) ) = 2x_1 14.52/4.57 14.52/4.57 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 14.52/4.57 14.52/4.57 f280_in(T31) -> U3(f289_in(T31), T31) 14.52/4.57 f300_in(s(T48), 0) -> U5(f289_in(T48), s(T48), 0) 14.52/4.57 f300_in(s(T53), s(T54)) -> U6(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 f289_in(s(T35)) -> U4(f297_in(T35), s(T35)) 14.52/4.57 U4(f297_out1(X81, X82), s(T35)) -> f289_out1(X82) 14.52/4.57 f297_in(T35) -> U16(f280_in(T35), T35) 14.52/4.57 f289_in(0) -> f289_out1(s(s(0))) 14.52/4.57 U3(f289_out1(X59), T31) -> f280_out1(X59) 14.52/4.57 U16(f280_out1(T36), T35) -> U17(f300_in(T35, T36), T35, T36) 14.52/4.57 U17(f300_out1(X82), T35, T36) -> f297_out1(T36, X82) 14.52/4.57 U5(f289_out1(X108), s(T48), 0) -> f300_out1(X108) 14.52/4.57 U6(f380_out1(X123, X124), s(T53), s(T54)) -> f300_out1(X124) 14.52/4.57 f380_in(T53, T54) -> U18(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18(f300_out1(T55), T53, T54) -> U19(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U19(f300_out1(X124), T53, T54, T55) -> f380_out1(T55, X124) 14.52/4.57 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (125) 14.52/4.57 Obligation: 14.52/4.57 Q DP problem: 14.52/4.57 The TRS P consists of the following rules: 14.52/4.57 14.52/4.57 F297_IN(T35) -> U16^1(f280_in(T35), T35) 14.52/4.57 F300_IN(s(T48), 0) -> F289_IN(T48) 14.52/4.57 F300_IN(s(T53), s(T54)) -> F380_IN(T53, T54) 14.52/4.57 F380_IN(T53, T54) -> U18^1(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18^1(f300_out1(T55), T53, T54) -> F300_IN(T53, T55) 14.52/4.57 F380_IN(T53, T54) -> F300_IN(s(T53), T54) 14.52/4.57 14.52/4.57 The TRS R consists of the following rules: 14.52/4.57 14.52/4.57 f10_in(0, s(T8)) -> f10_out1 14.52/4.57 f10_in(s(T17), T18) -> U1(f196_in(T17, T18), s(T17), T18) 14.52/4.57 U1(f196_out1, s(T17), T18) -> f10_out1 14.52/4.57 f10_in(s(T94), T96) -> U2(f461_in(T94, T96), s(T94), T96) 14.52/4.57 U2(f461_out1, s(T94), T96) -> f10_out1 14.52/4.57 f280_in(T31) -> U3(f289_in(T31), T31) 14.52/4.57 U3(f289_out1(X59), T31) -> f280_out1(X59) 14.52/4.57 f289_in(0) -> f289_out1(s(s(0))) 14.52/4.57 f289_in(s(T35)) -> U4(f297_in(T35), s(T35)) 14.52/4.57 U4(f297_out1(X81, X82), s(T35)) -> f289_out1(X82) 14.52/4.57 f300_in(0, T43) -> f300_out1(s(T43)) 14.52/4.57 f300_in(s(T48), 0) -> U5(f289_in(T48), s(T48), 0) 14.52/4.57 U5(f289_out1(X108), s(T48), 0) -> f300_out1(X108) 14.52/4.57 f300_in(s(T53), s(T54)) -> U6(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 U6(f380_out1(X123, X124), s(T53), s(T54)) -> f300_out1(X124) 14.52/4.57 f196_in(0, s(s(0))) -> f196_out1 14.52/4.57 f196_in(s(T24), T25) -> U7(f277_in(T24, T25), s(T24), T25) 14.52/4.57 U7(f277_out1(X35), s(T24), T25) -> f196_out1 14.52/4.57 f281_in(0, T65, s(T65)) -> f281_out1 14.52/4.57 f281_in(s(T74), 0, T75) -> U8(f196_in(T74, T75), s(T74), 0, T75) 14.52/4.57 U8(f196_out1, s(T74), 0, T75) -> f281_out1 14.52/4.57 f281_in(s(T82), s(T83), T84) -> U9(f457_in(T82, T83, T84), s(T82), s(T83), T84) 14.52/4.57 U9(f457_out1(X163), s(T82), s(T83), T84) -> f281_out1 14.52/4.57 f463_in(T103) -> U10(f289_in(T103), T103) 14.52/4.57 U10(f289_out1(X200), T103) -> f463_out1 14.52/4.57 f463_in(T108) -> U11(f471_in(T108), T108) 14.52/4.57 U11(f471_out1, T108) -> f463_out1 14.52/4.57 f474_in(0) -> f474_out1 14.52/4.57 f474_in(s(T123)) -> U12(f289_in(T123), s(T123)) 14.52/4.57 U12(f289_out1(X242), s(T123)) -> f474_out1 14.52/4.57 f474_in(s(T128)) -> U13(f471_in(T128), s(T128)) 14.52/4.57 U13(f471_out1, s(T128)) -> f474_out1 14.52/4.57 f277_in(T24, T25) -> U14(f280_in(T24), T24, T25) 14.52/4.57 U14(f280_out1(T26), T24, T25) -> U15(f281_in(T24, T26, T25), T24, T25, T26) 14.52/4.57 U15(f281_out1, T24, T25, T26) -> f277_out1(T26) 14.52/4.57 f297_in(T35) -> U16(f280_in(T35), T35) 14.52/4.57 U16(f280_out1(T36), T35) -> U17(f300_in(T35, T36), T35, T36) 14.52/4.57 U17(f300_out1(X82), T35, T36) -> f297_out1(T36, X82) 14.52/4.57 f380_in(T53, T54) -> U18(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18(f300_out1(T55), T53, T54) -> U19(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U19(f300_out1(X124), T53, T54, T55) -> f380_out1(T55, X124) 14.52/4.57 f457_in(T82, T83, T84) -> U20(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 U20(f300_out1(T85), T82, T83, T84) -> U21(f281_in(T82, T85, T84), T82, T83, T84, T85) 14.52/4.57 U21(f281_out1, T82, T83, T84, T85) -> f457_out1(T85) 14.52/4.57 f461_in(T94, T96) -> U22(f463_in(T94), T94, T96) 14.52/4.57 U22(f463_out1, T94, T96) -> U23(f10_in(T94, T96), T94, T96) 14.52/4.57 U23(f10_out1, T94, T96) -> f461_out1 14.52/4.57 f471_in(T108) -> U24(f463_in(T108), T108) 14.52/4.57 U24(f463_out1, T108) -> U25(f474_in(T108), T108) 14.52/4.57 U25(f474_out1, T108) -> f471_out1 14.52/4.57 14.52/4.57 Q is empty. 14.52/4.57 We have to consider all minimal (P,Q,R)-chains. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (126) DependencyGraphProof (EQUIVALENT) 14.52/4.57 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (127) 14.52/4.57 Obligation: 14.52/4.57 Q DP problem: 14.52/4.57 The TRS P consists of the following rules: 14.52/4.57 14.52/4.57 F300_IN(s(T53), s(T54)) -> F380_IN(T53, T54) 14.52/4.57 F380_IN(T53, T54) -> U18^1(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18^1(f300_out1(T55), T53, T54) -> F300_IN(T53, T55) 14.52/4.57 F380_IN(T53, T54) -> F300_IN(s(T53), T54) 14.52/4.57 14.52/4.57 The TRS R consists of the following rules: 14.52/4.57 14.52/4.57 f10_in(0, s(T8)) -> f10_out1 14.52/4.57 f10_in(s(T17), T18) -> U1(f196_in(T17, T18), s(T17), T18) 14.52/4.57 U1(f196_out1, s(T17), T18) -> f10_out1 14.52/4.57 f10_in(s(T94), T96) -> U2(f461_in(T94, T96), s(T94), T96) 14.52/4.57 U2(f461_out1, s(T94), T96) -> f10_out1 14.52/4.57 f280_in(T31) -> U3(f289_in(T31), T31) 14.52/4.57 U3(f289_out1(X59), T31) -> f280_out1(X59) 14.52/4.57 f289_in(0) -> f289_out1(s(s(0))) 14.52/4.57 f289_in(s(T35)) -> U4(f297_in(T35), s(T35)) 14.52/4.57 U4(f297_out1(X81, X82), s(T35)) -> f289_out1(X82) 14.52/4.57 f300_in(0, T43) -> f300_out1(s(T43)) 14.52/4.57 f300_in(s(T48), 0) -> U5(f289_in(T48), s(T48), 0) 14.52/4.57 U5(f289_out1(X108), s(T48), 0) -> f300_out1(X108) 14.52/4.57 f300_in(s(T53), s(T54)) -> U6(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 U6(f380_out1(X123, X124), s(T53), s(T54)) -> f300_out1(X124) 14.52/4.57 f196_in(0, s(s(0))) -> f196_out1 14.52/4.57 f196_in(s(T24), T25) -> U7(f277_in(T24, T25), s(T24), T25) 14.52/4.57 U7(f277_out1(X35), s(T24), T25) -> f196_out1 14.52/4.57 f281_in(0, T65, s(T65)) -> f281_out1 14.52/4.57 f281_in(s(T74), 0, T75) -> U8(f196_in(T74, T75), s(T74), 0, T75) 14.52/4.57 U8(f196_out1, s(T74), 0, T75) -> f281_out1 14.52/4.57 f281_in(s(T82), s(T83), T84) -> U9(f457_in(T82, T83, T84), s(T82), s(T83), T84) 14.52/4.57 U9(f457_out1(X163), s(T82), s(T83), T84) -> f281_out1 14.52/4.57 f463_in(T103) -> U10(f289_in(T103), T103) 14.52/4.57 U10(f289_out1(X200), T103) -> f463_out1 14.52/4.57 f463_in(T108) -> U11(f471_in(T108), T108) 14.52/4.57 U11(f471_out1, T108) -> f463_out1 14.52/4.57 f474_in(0) -> f474_out1 14.52/4.57 f474_in(s(T123)) -> U12(f289_in(T123), s(T123)) 14.52/4.57 U12(f289_out1(X242), s(T123)) -> f474_out1 14.52/4.57 f474_in(s(T128)) -> U13(f471_in(T128), s(T128)) 14.52/4.57 U13(f471_out1, s(T128)) -> f474_out1 14.52/4.57 f277_in(T24, T25) -> U14(f280_in(T24), T24, T25) 14.52/4.57 U14(f280_out1(T26), T24, T25) -> U15(f281_in(T24, T26, T25), T24, T25, T26) 14.52/4.57 U15(f281_out1, T24, T25, T26) -> f277_out1(T26) 14.52/4.57 f297_in(T35) -> U16(f280_in(T35), T35) 14.52/4.57 U16(f280_out1(T36), T35) -> U17(f300_in(T35, T36), T35, T36) 14.52/4.57 U17(f300_out1(X82), T35, T36) -> f297_out1(T36, X82) 14.52/4.57 f380_in(T53, T54) -> U18(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18(f300_out1(T55), T53, T54) -> U19(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U19(f300_out1(X124), T53, T54, T55) -> f380_out1(T55, X124) 14.52/4.57 f457_in(T82, T83, T84) -> U20(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 U20(f300_out1(T85), T82, T83, T84) -> U21(f281_in(T82, T85, T84), T82, T83, T84, T85) 14.52/4.57 U21(f281_out1, T82, T83, T84, T85) -> f457_out1(T85) 14.52/4.57 f461_in(T94, T96) -> U22(f463_in(T94), T94, T96) 14.52/4.57 U22(f463_out1, T94, T96) -> U23(f10_in(T94, T96), T94, T96) 14.52/4.57 U23(f10_out1, T94, T96) -> f461_out1 14.52/4.57 f471_in(T108) -> U24(f463_in(T108), T108) 14.52/4.57 U24(f463_out1, T108) -> U25(f474_in(T108), T108) 14.52/4.57 U25(f474_out1, T108) -> f471_out1 14.52/4.57 14.52/4.57 Q is empty. 14.52/4.57 We have to consider all minimal (P,Q,R)-chains. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (128) QDPOrderProof (EQUIVALENT) 14.52/4.57 We use the reduction pair processor [LPAR04,JAR06]. 14.52/4.57 14.52/4.57 14.52/4.57 The following pairs can be oriented strictly and are deleted. 14.52/4.57 14.52/4.57 U18^1(f300_out1(T55), T53, T54) -> F300_IN(T53, T55) 14.52/4.57 The remaining pairs can at least be oriented weakly. 14.52/4.57 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 14.52/4.57 14.52/4.57 POL( U18^1_3(x_1, ..., x_3) ) = x_2 + 1 14.52/4.57 POL( f300_in_2(x_1, x_2) ) = 0 14.52/4.57 POL( s_1(x_1) ) = x_1 + 1 14.52/4.57 POL( 0 ) = 1 14.52/4.57 POL( U5_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} 14.52/4.57 POL( f289_in_1(x_1) ) = 0 14.52/4.57 POL( U6_3(x_1, ..., x_3) ) = max{0, x_1 - 2} 14.52/4.57 POL( f380_in_2(x_1, x_2) ) = 0 14.52/4.57 POL( U4_2(x_1, x_2) ) = max{0, 2x_1 - 2} 14.52/4.57 POL( f297_in_1(x_1) ) = 0 14.52/4.57 POL( f297_out1_2(x_1, x_2) ) = 2x_1 14.52/4.57 POL( f289_out1_1(x_1) ) = max{0, x_1 - 1} 14.52/4.57 POL( U16_2(x_1, x_2) ) = max{0, 2x_2 - 2} 14.52/4.57 POL( f280_in_1(x_1) ) = 0 14.52/4.57 POL( U3_2(x_1, x_2) ) = max{0, 2x_1 - 2} 14.52/4.57 POL( f280_out1_1(x_1) ) = 2x_1 + 2 14.52/4.57 POL( U17_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} 14.52/4.57 POL( U19_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_4 - 2} 14.52/4.57 POL( f300_out1_1(x_1) ) = max{0, x_1 - 2} 14.52/4.57 POL( f380_out1_2(x_1, x_2) ) = x_1 14.52/4.57 POL( U18_3(x_1, ..., x_3) ) = 2x_2 + 2 14.52/4.57 POL( F300_IN_2(x_1, x_2) ) = x_1 14.52/4.57 POL( F380_IN_2(x_1, x_2) ) = x_1 + 1 14.52/4.57 14.52/4.57 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 14.52/4.57 none 14.52/4.57 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (129) 14.52/4.57 Obligation: 14.52/4.57 Q DP problem: 14.52/4.57 The TRS P consists of the following rules: 14.52/4.57 14.52/4.57 F300_IN(s(T53), s(T54)) -> F380_IN(T53, T54) 14.52/4.57 F380_IN(T53, T54) -> U18^1(f300_in(s(T53), T54), T53, T54) 14.52/4.57 F380_IN(T53, T54) -> F300_IN(s(T53), T54) 14.52/4.57 14.52/4.57 The TRS R consists of the following rules: 14.52/4.57 14.52/4.57 f10_in(0, s(T8)) -> f10_out1 14.52/4.57 f10_in(s(T17), T18) -> U1(f196_in(T17, T18), s(T17), T18) 14.52/4.57 U1(f196_out1, s(T17), T18) -> f10_out1 14.52/4.57 f10_in(s(T94), T96) -> U2(f461_in(T94, T96), s(T94), T96) 14.52/4.57 U2(f461_out1, s(T94), T96) -> f10_out1 14.52/4.57 f280_in(T31) -> U3(f289_in(T31), T31) 14.52/4.57 U3(f289_out1(X59), T31) -> f280_out1(X59) 14.52/4.57 f289_in(0) -> f289_out1(s(s(0))) 14.52/4.57 f289_in(s(T35)) -> U4(f297_in(T35), s(T35)) 14.52/4.57 U4(f297_out1(X81, X82), s(T35)) -> f289_out1(X82) 14.52/4.57 f300_in(0, T43) -> f300_out1(s(T43)) 14.52/4.57 f300_in(s(T48), 0) -> U5(f289_in(T48), s(T48), 0) 14.52/4.57 U5(f289_out1(X108), s(T48), 0) -> f300_out1(X108) 14.52/4.57 f300_in(s(T53), s(T54)) -> U6(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 U6(f380_out1(X123, X124), s(T53), s(T54)) -> f300_out1(X124) 14.52/4.57 f196_in(0, s(s(0))) -> f196_out1 14.52/4.57 f196_in(s(T24), T25) -> U7(f277_in(T24, T25), s(T24), T25) 14.52/4.57 U7(f277_out1(X35), s(T24), T25) -> f196_out1 14.52/4.57 f281_in(0, T65, s(T65)) -> f281_out1 14.52/4.57 f281_in(s(T74), 0, T75) -> U8(f196_in(T74, T75), s(T74), 0, T75) 14.52/4.57 U8(f196_out1, s(T74), 0, T75) -> f281_out1 14.52/4.57 f281_in(s(T82), s(T83), T84) -> U9(f457_in(T82, T83, T84), s(T82), s(T83), T84) 14.52/4.57 U9(f457_out1(X163), s(T82), s(T83), T84) -> f281_out1 14.52/4.57 f463_in(T103) -> U10(f289_in(T103), T103) 14.52/4.57 U10(f289_out1(X200), T103) -> f463_out1 14.52/4.57 f463_in(T108) -> U11(f471_in(T108), T108) 14.52/4.57 U11(f471_out1, T108) -> f463_out1 14.52/4.57 f474_in(0) -> f474_out1 14.52/4.57 f474_in(s(T123)) -> U12(f289_in(T123), s(T123)) 14.52/4.57 U12(f289_out1(X242), s(T123)) -> f474_out1 14.52/4.57 f474_in(s(T128)) -> U13(f471_in(T128), s(T128)) 14.52/4.57 U13(f471_out1, s(T128)) -> f474_out1 14.52/4.57 f277_in(T24, T25) -> U14(f280_in(T24), T24, T25) 14.52/4.57 U14(f280_out1(T26), T24, T25) -> U15(f281_in(T24, T26, T25), T24, T25, T26) 14.52/4.57 U15(f281_out1, T24, T25, T26) -> f277_out1(T26) 14.52/4.57 f297_in(T35) -> U16(f280_in(T35), T35) 14.52/4.57 U16(f280_out1(T36), T35) -> U17(f300_in(T35, T36), T35, T36) 14.52/4.57 U17(f300_out1(X82), T35, T36) -> f297_out1(T36, X82) 14.52/4.57 f380_in(T53, T54) -> U18(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18(f300_out1(T55), T53, T54) -> U19(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U19(f300_out1(X124), T53, T54, T55) -> f380_out1(T55, X124) 14.52/4.57 f457_in(T82, T83, T84) -> U20(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 U20(f300_out1(T85), T82, T83, T84) -> U21(f281_in(T82, T85, T84), T82, T83, T84, T85) 14.52/4.57 U21(f281_out1, T82, T83, T84, T85) -> f457_out1(T85) 14.52/4.57 f461_in(T94, T96) -> U22(f463_in(T94), T94, T96) 14.52/4.57 U22(f463_out1, T94, T96) -> U23(f10_in(T94, T96), T94, T96) 14.52/4.57 U23(f10_out1, T94, T96) -> f461_out1 14.52/4.57 f471_in(T108) -> U24(f463_in(T108), T108) 14.52/4.57 U24(f463_out1, T108) -> U25(f474_in(T108), T108) 14.52/4.57 U25(f474_out1, T108) -> f471_out1 14.52/4.57 14.52/4.57 Q is empty. 14.52/4.57 We have to consider all minimal (P,Q,R)-chains. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (130) DependencyGraphProof (EQUIVALENT) 14.52/4.57 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (131) 14.52/4.57 Obligation: 14.52/4.57 Q DP problem: 14.52/4.57 The TRS P consists of the following rules: 14.52/4.57 14.52/4.57 F380_IN(T53, T54) -> F300_IN(s(T53), T54) 14.52/4.57 F300_IN(s(T53), s(T54)) -> F380_IN(T53, T54) 14.52/4.57 14.52/4.57 The TRS R consists of the following rules: 14.52/4.57 14.52/4.57 f10_in(0, s(T8)) -> f10_out1 14.52/4.57 f10_in(s(T17), T18) -> U1(f196_in(T17, T18), s(T17), T18) 14.52/4.57 U1(f196_out1, s(T17), T18) -> f10_out1 14.52/4.57 f10_in(s(T94), T96) -> U2(f461_in(T94, T96), s(T94), T96) 14.52/4.57 U2(f461_out1, s(T94), T96) -> f10_out1 14.52/4.57 f280_in(T31) -> U3(f289_in(T31), T31) 14.52/4.57 U3(f289_out1(X59), T31) -> f280_out1(X59) 14.52/4.57 f289_in(0) -> f289_out1(s(s(0))) 14.52/4.57 f289_in(s(T35)) -> U4(f297_in(T35), s(T35)) 14.52/4.57 U4(f297_out1(X81, X82), s(T35)) -> f289_out1(X82) 14.52/4.57 f300_in(0, T43) -> f300_out1(s(T43)) 14.52/4.57 f300_in(s(T48), 0) -> U5(f289_in(T48), s(T48), 0) 14.52/4.57 U5(f289_out1(X108), s(T48), 0) -> f300_out1(X108) 14.52/4.57 f300_in(s(T53), s(T54)) -> U6(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 U6(f380_out1(X123, X124), s(T53), s(T54)) -> f300_out1(X124) 14.52/4.57 f196_in(0, s(s(0))) -> f196_out1 14.52/4.57 f196_in(s(T24), T25) -> U7(f277_in(T24, T25), s(T24), T25) 14.52/4.57 U7(f277_out1(X35), s(T24), T25) -> f196_out1 14.52/4.57 f281_in(0, T65, s(T65)) -> f281_out1 14.52/4.57 f281_in(s(T74), 0, T75) -> U8(f196_in(T74, T75), s(T74), 0, T75) 14.52/4.57 U8(f196_out1, s(T74), 0, T75) -> f281_out1 14.52/4.57 f281_in(s(T82), s(T83), T84) -> U9(f457_in(T82, T83, T84), s(T82), s(T83), T84) 14.52/4.57 U9(f457_out1(X163), s(T82), s(T83), T84) -> f281_out1 14.52/4.57 f463_in(T103) -> U10(f289_in(T103), T103) 14.52/4.57 U10(f289_out1(X200), T103) -> f463_out1 14.52/4.57 f463_in(T108) -> U11(f471_in(T108), T108) 14.52/4.57 U11(f471_out1, T108) -> f463_out1 14.52/4.57 f474_in(0) -> f474_out1 14.52/4.57 f474_in(s(T123)) -> U12(f289_in(T123), s(T123)) 14.52/4.57 U12(f289_out1(X242), s(T123)) -> f474_out1 14.52/4.57 f474_in(s(T128)) -> U13(f471_in(T128), s(T128)) 14.52/4.57 U13(f471_out1, s(T128)) -> f474_out1 14.52/4.57 f277_in(T24, T25) -> U14(f280_in(T24), T24, T25) 14.52/4.57 U14(f280_out1(T26), T24, T25) -> U15(f281_in(T24, T26, T25), T24, T25, T26) 14.52/4.57 U15(f281_out1, T24, T25, T26) -> f277_out1(T26) 14.52/4.57 f297_in(T35) -> U16(f280_in(T35), T35) 14.52/4.57 U16(f280_out1(T36), T35) -> U17(f300_in(T35, T36), T35, T36) 14.52/4.57 U17(f300_out1(X82), T35, T36) -> f297_out1(T36, X82) 14.52/4.57 f380_in(T53, T54) -> U18(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18(f300_out1(T55), T53, T54) -> U19(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U19(f300_out1(X124), T53, T54, T55) -> f380_out1(T55, X124) 14.52/4.57 f457_in(T82, T83, T84) -> U20(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 U20(f300_out1(T85), T82, T83, T84) -> U21(f281_in(T82, T85, T84), T82, T83, T84, T85) 14.52/4.57 U21(f281_out1, T82, T83, T84, T85) -> f457_out1(T85) 14.52/4.57 f461_in(T94, T96) -> U22(f463_in(T94), T94, T96) 14.52/4.57 U22(f463_out1, T94, T96) -> U23(f10_in(T94, T96), T94, T96) 14.52/4.57 U23(f10_out1, T94, T96) -> f461_out1 14.52/4.57 f471_in(T108) -> U24(f463_in(T108), T108) 14.52/4.57 U24(f463_out1, T108) -> U25(f474_in(T108), T108) 14.52/4.57 U25(f474_out1, T108) -> f471_out1 14.52/4.57 14.52/4.57 Q is empty. 14.52/4.57 We have to consider all minimal (P,Q,R)-chains. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (132) UsableRulesProof (EQUIVALENT) 14.52/4.57 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (133) 14.52/4.57 Obligation: 14.52/4.57 Q DP problem: 14.52/4.57 The TRS P consists of the following rules: 14.52/4.57 14.52/4.57 F380_IN(T53, T54) -> F300_IN(s(T53), T54) 14.52/4.57 F300_IN(s(T53), s(T54)) -> F380_IN(T53, T54) 14.52/4.57 14.52/4.57 R is empty. 14.52/4.57 Q is empty. 14.52/4.57 We have to consider all minimal (P,Q,R)-chains. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (134) QDPSizeChangeProof (EQUIVALENT) 14.52/4.57 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. 14.52/4.57 14.52/4.57 From the DPs we obtained the following set of size-change graphs: 14.52/4.57 *F300_IN(s(T53), s(T54)) -> F380_IN(T53, T54) 14.52/4.57 The graph contains the following edges 1 > 1, 2 > 2 14.52/4.57 14.52/4.57 14.52/4.57 *F380_IN(T53, T54) -> F300_IN(s(T53), T54) 14.52/4.57 The graph contains the following edges 2 >= 2 14.52/4.57 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (135) 14.52/4.57 YES 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (136) 14.52/4.57 Obligation: 14.52/4.57 Q DP problem: 14.52/4.57 The TRS P consists of the following rules: 14.52/4.57 14.52/4.57 F463_IN(T108) -> F471_IN(T108) 14.52/4.57 F471_IN(T108) -> U24^1(f463_in(T108), T108) 14.52/4.57 U24^1(f463_out1, T108) -> F474_IN(T108) 14.52/4.57 F474_IN(s(T128)) -> F471_IN(T128) 14.52/4.57 F471_IN(T108) -> F463_IN(T108) 14.52/4.57 14.52/4.57 The TRS R consists of the following rules: 14.52/4.57 14.52/4.57 f10_in(0, s(T8)) -> f10_out1 14.52/4.57 f10_in(s(T17), T18) -> U1(f196_in(T17, T18), s(T17), T18) 14.52/4.57 U1(f196_out1, s(T17), T18) -> f10_out1 14.52/4.57 f10_in(s(T94), T96) -> U2(f461_in(T94, T96), s(T94), T96) 14.52/4.57 U2(f461_out1, s(T94), T96) -> f10_out1 14.52/4.57 f280_in(T31) -> U3(f289_in(T31), T31) 14.52/4.57 U3(f289_out1(X59), T31) -> f280_out1(X59) 14.52/4.57 f289_in(0) -> f289_out1(s(s(0))) 14.52/4.57 f289_in(s(T35)) -> U4(f297_in(T35), s(T35)) 14.52/4.57 U4(f297_out1(X81, X82), s(T35)) -> f289_out1(X82) 14.52/4.57 f300_in(0, T43) -> f300_out1(s(T43)) 14.52/4.57 f300_in(s(T48), 0) -> U5(f289_in(T48), s(T48), 0) 14.52/4.57 U5(f289_out1(X108), s(T48), 0) -> f300_out1(X108) 14.52/4.57 f300_in(s(T53), s(T54)) -> U6(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 U6(f380_out1(X123, X124), s(T53), s(T54)) -> f300_out1(X124) 14.52/4.57 f196_in(0, s(s(0))) -> f196_out1 14.52/4.57 f196_in(s(T24), T25) -> U7(f277_in(T24, T25), s(T24), T25) 14.52/4.57 U7(f277_out1(X35), s(T24), T25) -> f196_out1 14.52/4.57 f281_in(0, T65, s(T65)) -> f281_out1 14.52/4.57 f281_in(s(T74), 0, T75) -> U8(f196_in(T74, T75), s(T74), 0, T75) 14.52/4.57 U8(f196_out1, s(T74), 0, T75) -> f281_out1 14.52/4.57 f281_in(s(T82), s(T83), T84) -> U9(f457_in(T82, T83, T84), s(T82), s(T83), T84) 14.52/4.57 U9(f457_out1(X163), s(T82), s(T83), T84) -> f281_out1 14.52/4.57 f463_in(T103) -> U10(f289_in(T103), T103) 14.52/4.57 U10(f289_out1(X200), T103) -> f463_out1 14.52/4.57 f463_in(T108) -> U11(f471_in(T108), T108) 14.52/4.57 U11(f471_out1, T108) -> f463_out1 14.52/4.57 f474_in(0) -> f474_out1 14.52/4.57 f474_in(s(T123)) -> U12(f289_in(T123), s(T123)) 14.52/4.57 U12(f289_out1(X242), s(T123)) -> f474_out1 14.52/4.57 f474_in(s(T128)) -> U13(f471_in(T128), s(T128)) 14.52/4.57 U13(f471_out1, s(T128)) -> f474_out1 14.52/4.57 f277_in(T24, T25) -> U14(f280_in(T24), T24, T25) 14.52/4.57 U14(f280_out1(T26), T24, T25) -> U15(f281_in(T24, T26, T25), T24, T25, T26) 14.52/4.57 U15(f281_out1, T24, T25, T26) -> f277_out1(T26) 14.52/4.57 f297_in(T35) -> U16(f280_in(T35), T35) 14.52/4.57 U16(f280_out1(T36), T35) -> U17(f300_in(T35, T36), T35, T36) 14.52/4.57 U17(f300_out1(X82), T35, T36) -> f297_out1(T36, X82) 14.52/4.57 f380_in(T53, T54) -> U18(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18(f300_out1(T55), T53, T54) -> U19(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U19(f300_out1(X124), T53, T54, T55) -> f380_out1(T55, X124) 14.52/4.57 f457_in(T82, T83, T84) -> U20(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 U20(f300_out1(T85), T82, T83, T84) -> U21(f281_in(T82, T85, T84), T82, T83, T84, T85) 14.52/4.57 U21(f281_out1, T82, T83, T84, T85) -> f457_out1(T85) 14.52/4.57 f461_in(T94, T96) -> U22(f463_in(T94), T94, T96) 14.52/4.57 U22(f463_out1, T94, T96) -> U23(f10_in(T94, T96), T94, T96) 14.52/4.57 U23(f10_out1, T94, T96) -> f461_out1 14.52/4.57 f471_in(T108) -> U24(f463_in(T108), T108) 14.52/4.57 U24(f463_out1, T108) -> U25(f474_in(T108), T108) 14.52/4.57 U25(f474_out1, T108) -> f471_out1 14.52/4.57 14.52/4.57 Q is empty. 14.52/4.57 We have to consider all minimal (P,Q,R)-chains. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (137) QDPOrderProof (EQUIVALENT) 14.52/4.57 We use the reduction pair processor [LPAR04,JAR06]. 14.52/4.57 14.52/4.57 14.52/4.57 The following pairs can be oriented strictly and are deleted. 14.52/4.57 14.52/4.57 U24^1(f463_out1, T108) -> F474_IN(T108) 14.52/4.57 F474_IN(s(T128)) -> F471_IN(T128) 14.52/4.57 The remaining pairs can at least be oriented weakly. 14.52/4.57 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 14.52/4.57 14.52/4.57 POL( U24^1_2(x_1, x_2) ) = 2x_2 + 2 14.52/4.57 POL( f463_in_1(x_1) ) = 2 14.52/4.57 POL( U10_2(x_1, x_2) ) = max{0, 2x_2 - 2} 14.52/4.57 POL( f289_in_1(x_1) ) = 2x_1 14.52/4.57 POL( U11_2(x_1, x_2) ) = max{0, 2x_2 - 2} 14.52/4.57 POL( f471_in_1(x_1) ) = 0 14.52/4.57 POL( f471_out1 ) = 2 14.52/4.57 POL( f463_out1 ) = 2 14.52/4.57 POL( U24_2(x_1, x_2) ) = max{0, 2x_2 - 2} 14.52/4.57 POL( U25_2(x_1, x_2) ) = max{0, 2x_1 - 2} 14.52/4.57 POL( f474_in_1(x_1) ) = 2 14.52/4.57 POL( 0 ) = 0 14.52/4.57 POL( f474_out1 ) = 1 14.52/4.57 POL( s_1(x_1) ) = x_1 + 2 14.52/4.57 POL( U12_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} 14.52/4.57 POL( U13_2(x_1, x_2) ) = max{0, 2x_1 - 2} 14.52/4.57 POL( f289_out1_1(x_1) ) = max{0, x_1 - 2} 14.52/4.57 POL( U4_2(x_1, x_2) ) = 2x_1 + 2 14.52/4.57 POL( f297_in_1(x_1) ) = 0 14.52/4.57 POL( f297_out1_2(x_1, x_2) ) = x_2 14.52/4.57 POL( U16_2(x_1, x_2) ) = 2x_2 + 2 14.52/4.57 POL( f280_in_1(x_1) ) = 2x_1 + 1 14.52/4.57 POL( U3_2(x_1, x_2) ) = max{0, 2x_1 - 2} 14.52/4.57 POL( f280_out1_1(x_1) ) = 2x_1 + 2 14.52/4.57 POL( U17_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2} 14.52/4.57 POL( f300_in_2(x_1, x_2) ) = max{0, 2x_2 - 2} 14.52/4.57 POL( U19_4(x_1, ..., x_4) ) = max{0, 2x_1 - 2} 14.52/4.57 POL( f300_out1_1(x_1) ) = max{0, 2x_1 - 2} 14.52/4.57 POL( U5_3(x_1, ..., x_3) ) = 2x_3 + 2 14.52/4.57 POL( U6_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2 14.52/4.57 POL( f380_in_2(x_1, x_2) ) = x_1 + 2 14.52/4.57 POL( f380_out1_2(x_1, x_2) ) = 2x_1 + 1 14.52/4.57 POL( U18_3(x_1, ..., x_3) ) = 2 14.52/4.57 POL( F463_IN_1(x_1) ) = 2x_1 + 2 14.52/4.57 POL( F471_IN_1(x_1) ) = 2x_1 + 2 14.52/4.57 POL( F474_IN_1(x_1) ) = 2x_1 + 1 14.52/4.57 14.52/4.57 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 14.52/4.57 none 14.52/4.57 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (138) 14.52/4.57 Obligation: 14.52/4.57 Q DP problem: 14.52/4.57 The TRS P consists of the following rules: 14.52/4.57 14.52/4.57 F463_IN(T108) -> F471_IN(T108) 14.52/4.57 F471_IN(T108) -> U24^1(f463_in(T108), T108) 14.52/4.57 F471_IN(T108) -> F463_IN(T108) 14.52/4.57 14.52/4.57 The TRS R consists of the following rules: 14.52/4.57 14.52/4.57 f10_in(0, s(T8)) -> f10_out1 14.52/4.57 f10_in(s(T17), T18) -> U1(f196_in(T17, T18), s(T17), T18) 14.52/4.57 U1(f196_out1, s(T17), T18) -> f10_out1 14.52/4.57 f10_in(s(T94), T96) -> U2(f461_in(T94, T96), s(T94), T96) 14.52/4.57 U2(f461_out1, s(T94), T96) -> f10_out1 14.52/4.57 f280_in(T31) -> U3(f289_in(T31), T31) 14.52/4.57 U3(f289_out1(X59), T31) -> f280_out1(X59) 14.52/4.57 f289_in(0) -> f289_out1(s(s(0))) 14.52/4.57 f289_in(s(T35)) -> U4(f297_in(T35), s(T35)) 14.52/4.57 U4(f297_out1(X81, X82), s(T35)) -> f289_out1(X82) 14.52/4.57 f300_in(0, T43) -> f300_out1(s(T43)) 14.52/4.57 f300_in(s(T48), 0) -> U5(f289_in(T48), s(T48), 0) 14.52/4.57 U5(f289_out1(X108), s(T48), 0) -> f300_out1(X108) 14.52/4.57 f300_in(s(T53), s(T54)) -> U6(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 U6(f380_out1(X123, X124), s(T53), s(T54)) -> f300_out1(X124) 14.52/4.57 f196_in(0, s(s(0))) -> f196_out1 14.52/4.57 f196_in(s(T24), T25) -> U7(f277_in(T24, T25), s(T24), T25) 14.52/4.57 U7(f277_out1(X35), s(T24), T25) -> f196_out1 14.52/4.57 f281_in(0, T65, s(T65)) -> f281_out1 14.52/4.57 f281_in(s(T74), 0, T75) -> U8(f196_in(T74, T75), s(T74), 0, T75) 14.52/4.57 U8(f196_out1, s(T74), 0, T75) -> f281_out1 14.52/4.57 f281_in(s(T82), s(T83), T84) -> U9(f457_in(T82, T83, T84), s(T82), s(T83), T84) 14.52/4.57 U9(f457_out1(X163), s(T82), s(T83), T84) -> f281_out1 14.52/4.57 f463_in(T103) -> U10(f289_in(T103), T103) 14.52/4.57 U10(f289_out1(X200), T103) -> f463_out1 14.52/4.57 f463_in(T108) -> U11(f471_in(T108), T108) 14.52/4.57 U11(f471_out1, T108) -> f463_out1 14.52/4.57 f474_in(0) -> f474_out1 14.52/4.57 f474_in(s(T123)) -> U12(f289_in(T123), s(T123)) 14.52/4.57 U12(f289_out1(X242), s(T123)) -> f474_out1 14.52/4.57 f474_in(s(T128)) -> U13(f471_in(T128), s(T128)) 14.52/4.57 U13(f471_out1, s(T128)) -> f474_out1 14.52/4.57 f277_in(T24, T25) -> U14(f280_in(T24), T24, T25) 14.52/4.57 U14(f280_out1(T26), T24, T25) -> U15(f281_in(T24, T26, T25), T24, T25, T26) 14.52/4.57 U15(f281_out1, T24, T25, T26) -> f277_out1(T26) 14.52/4.57 f297_in(T35) -> U16(f280_in(T35), T35) 14.52/4.57 U16(f280_out1(T36), T35) -> U17(f300_in(T35, T36), T35, T36) 14.52/4.57 U17(f300_out1(X82), T35, T36) -> f297_out1(T36, X82) 14.52/4.57 f380_in(T53, T54) -> U18(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18(f300_out1(T55), T53, T54) -> U19(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U19(f300_out1(X124), T53, T54, T55) -> f380_out1(T55, X124) 14.52/4.57 f457_in(T82, T83, T84) -> U20(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 U20(f300_out1(T85), T82, T83, T84) -> U21(f281_in(T82, T85, T84), T82, T83, T84, T85) 14.52/4.57 U21(f281_out1, T82, T83, T84, T85) -> f457_out1(T85) 14.52/4.57 f461_in(T94, T96) -> U22(f463_in(T94), T94, T96) 14.52/4.57 U22(f463_out1, T94, T96) -> U23(f10_in(T94, T96), T94, T96) 14.52/4.57 U23(f10_out1, T94, T96) -> f461_out1 14.52/4.57 f471_in(T108) -> U24(f463_in(T108), T108) 14.52/4.57 U24(f463_out1, T108) -> U25(f474_in(T108), T108) 14.52/4.57 U25(f474_out1, T108) -> f471_out1 14.52/4.57 14.52/4.57 Q is empty. 14.52/4.57 We have to consider all minimal (P,Q,R)-chains. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (139) DependencyGraphProof (EQUIVALENT) 14.52/4.57 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (140) 14.52/4.57 Obligation: 14.52/4.57 Q DP problem: 14.52/4.57 The TRS P consists of the following rules: 14.52/4.57 14.52/4.57 F471_IN(T108) -> F463_IN(T108) 14.52/4.57 F463_IN(T108) -> F471_IN(T108) 14.52/4.57 14.52/4.57 The TRS R consists of the following rules: 14.52/4.57 14.52/4.57 f10_in(0, s(T8)) -> f10_out1 14.52/4.57 f10_in(s(T17), T18) -> U1(f196_in(T17, T18), s(T17), T18) 14.52/4.57 U1(f196_out1, s(T17), T18) -> f10_out1 14.52/4.57 f10_in(s(T94), T96) -> U2(f461_in(T94, T96), s(T94), T96) 14.52/4.57 U2(f461_out1, s(T94), T96) -> f10_out1 14.52/4.57 f280_in(T31) -> U3(f289_in(T31), T31) 14.52/4.57 U3(f289_out1(X59), T31) -> f280_out1(X59) 14.52/4.57 f289_in(0) -> f289_out1(s(s(0))) 14.52/4.57 f289_in(s(T35)) -> U4(f297_in(T35), s(T35)) 14.52/4.57 U4(f297_out1(X81, X82), s(T35)) -> f289_out1(X82) 14.52/4.57 f300_in(0, T43) -> f300_out1(s(T43)) 14.52/4.57 f300_in(s(T48), 0) -> U5(f289_in(T48), s(T48), 0) 14.52/4.57 U5(f289_out1(X108), s(T48), 0) -> f300_out1(X108) 14.52/4.57 f300_in(s(T53), s(T54)) -> U6(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 U6(f380_out1(X123, X124), s(T53), s(T54)) -> f300_out1(X124) 14.52/4.57 f196_in(0, s(s(0))) -> f196_out1 14.52/4.57 f196_in(s(T24), T25) -> U7(f277_in(T24, T25), s(T24), T25) 14.52/4.57 U7(f277_out1(X35), s(T24), T25) -> f196_out1 14.52/4.57 f281_in(0, T65, s(T65)) -> f281_out1 14.52/4.57 f281_in(s(T74), 0, T75) -> U8(f196_in(T74, T75), s(T74), 0, T75) 14.52/4.57 U8(f196_out1, s(T74), 0, T75) -> f281_out1 14.52/4.57 f281_in(s(T82), s(T83), T84) -> U9(f457_in(T82, T83, T84), s(T82), s(T83), T84) 14.52/4.57 U9(f457_out1(X163), s(T82), s(T83), T84) -> f281_out1 14.52/4.57 f463_in(T103) -> U10(f289_in(T103), T103) 14.52/4.57 U10(f289_out1(X200), T103) -> f463_out1 14.52/4.57 f463_in(T108) -> U11(f471_in(T108), T108) 14.52/4.57 U11(f471_out1, T108) -> f463_out1 14.52/4.57 f474_in(0) -> f474_out1 14.52/4.57 f474_in(s(T123)) -> U12(f289_in(T123), s(T123)) 14.52/4.57 U12(f289_out1(X242), s(T123)) -> f474_out1 14.52/4.57 f474_in(s(T128)) -> U13(f471_in(T128), s(T128)) 14.52/4.57 U13(f471_out1, s(T128)) -> f474_out1 14.52/4.57 f277_in(T24, T25) -> U14(f280_in(T24), T24, T25) 14.52/4.57 U14(f280_out1(T26), T24, T25) -> U15(f281_in(T24, T26, T25), T24, T25, T26) 14.52/4.57 U15(f281_out1, T24, T25, T26) -> f277_out1(T26) 14.52/4.57 f297_in(T35) -> U16(f280_in(T35), T35) 14.52/4.57 U16(f280_out1(T36), T35) -> U17(f300_in(T35, T36), T35, T36) 14.52/4.57 U17(f300_out1(X82), T35, T36) -> f297_out1(T36, X82) 14.52/4.57 f380_in(T53, T54) -> U18(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18(f300_out1(T55), T53, T54) -> U19(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U19(f300_out1(X124), T53, T54, T55) -> f380_out1(T55, X124) 14.52/4.57 f457_in(T82, T83, T84) -> U20(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 U20(f300_out1(T85), T82, T83, T84) -> U21(f281_in(T82, T85, T84), T82, T83, T84, T85) 14.52/4.57 U21(f281_out1, T82, T83, T84, T85) -> f457_out1(T85) 14.52/4.57 f461_in(T94, T96) -> U22(f463_in(T94), T94, T96) 14.52/4.57 U22(f463_out1, T94, T96) -> U23(f10_in(T94, T96), T94, T96) 14.52/4.57 U23(f10_out1, T94, T96) -> f461_out1 14.52/4.57 f471_in(T108) -> U24(f463_in(T108), T108) 14.52/4.57 U24(f463_out1, T108) -> U25(f474_in(T108), T108) 14.52/4.57 U25(f474_out1, T108) -> f471_out1 14.52/4.57 14.52/4.57 Q is empty. 14.52/4.57 We have to consider all minimal (P,Q,R)-chains. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (141) UsableRulesProof (EQUIVALENT) 14.52/4.57 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (142) 14.52/4.57 Obligation: 14.52/4.57 Q DP problem: 14.52/4.57 The TRS P consists of the following rules: 14.52/4.57 14.52/4.57 F471_IN(T108) -> F463_IN(T108) 14.52/4.57 F463_IN(T108) -> F471_IN(T108) 14.52/4.57 14.52/4.57 R is empty. 14.52/4.57 Q is empty. 14.52/4.57 We have to consider all minimal (P,Q,R)-chains. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (143) NonTerminationLoopProof (COMPLETE) 14.52/4.57 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 14.52/4.57 Found a loop by narrowing to the left: 14.52/4.57 14.52/4.57 s = F463_IN(T108') evaluates to t =F463_IN(T108') 14.52/4.57 14.52/4.57 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 14.52/4.57 * Matcher: [ ] 14.52/4.57 * Semiunifier: [ ] 14.52/4.57 14.52/4.57 -------------------------------------------------------------------------------- 14.52/4.57 Rewriting sequence 14.52/4.57 14.52/4.57 F463_IN(T108') -> F471_IN(T108') 14.52/4.57 with rule F463_IN(T108'') -> F471_IN(T108'') at position [] and matcher [T108'' / T108'] 14.52/4.57 14.52/4.57 F471_IN(T108') -> F463_IN(T108') 14.52/4.57 with rule F471_IN(T108) -> F463_IN(T108) 14.52/4.57 14.52/4.57 Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence 14.52/4.57 14.52/4.57 14.52/4.57 All these steps are and every following step will be a correct step w.r.t to Q. 14.52/4.57 14.52/4.57 14.52/4.57 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (144) 14.52/4.57 NO 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (145) 14.52/4.57 Obligation: 14.52/4.57 Q DP problem: 14.52/4.57 The TRS P consists of the following rules: 14.52/4.57 14.52/4.57 F277_IN(T24, T25) -> U14^1(f280_in(T24), T24, T25) 14.52/4.57 U14^1(f280_out1(T26), T24, T25) -> F281_IN(T24, T26, T25) 14.52/4.57 F281_IN(s(T74), 0, T75) -> F196_IN(T74, T75) 14.52/4.57 F196_IN(s(T24), T25) -> F277_IN(T24, T25) 14.52/4.57 F281_IN(s(T82), s(T83), T84) -> F457_IN(T82, T83, T84) 14.52/4.57 F457_IN(T82, T83, T84) -> U20^1(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 U20^1(f300_out1(T85), T82, T83, T84) -> F281_IN(T82, T85, T84) 14.52/4.57 14.52/4.57 The TRS R consists of the following rules: 14.52/4.57 14.52/4.57 f10_in(0, s(T8)) -> f10_out1 14.52/4.57 f10_in(s(T17), T18) -> U1(f196_in(T17, T18), s(T17), T18) 14.52/4.57 U1(f196_out1, s(T17), T18) -> f10_out1 14.52/4.57 f10_in(s(T94), T96) -> U2(f461_in(T94, T96), s(T94), T96) 14.52/4.57 U2(f461_out1, s(T94), T96) -> f10_out1 14.52/4.57 f280_in(T31) -> U3(f289_in(T31), T31) 14.52/4.57 U3(f289_out1(X59), T31) -> f280_out1(X59) 14.52/4.57 f289_in(0) -> f289_out1(s(s(0))) 14.52/4.57 f289_in(s(T35)) -> U4(f297_in(T35), s(T35)) 14.52/4.57 U4(f297_out1(X81, X82), s(T35)) -> f289_out1(X82) 14.52/4.57 f300_in(0, T43) -> f300_out1(s(T43)) 14.52/4.57 f300_in(s(T48), 0) -> U5(f289_in(T48), s(T48), 0) 14.52/4.57 U5(f289_out1(X108), s(T48), 0) -> f300_out1(X108) 14.52/4.57 f300_in(s(T53), s(T54)) -> U6(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 U6(f380_out1(X123, X124), s(T53), s(T54)) -> f300_out1(X124) 14.52/4.57 f196_in(0, s(s(0))) -> f196_out1 14.52/4.57 f196_in(s(T24), T25) -> U7(f277_in(T24, T25), s(T24), T25) 14.52/4.57 U7(f277_out1(X35), s(T24), T25) -> f196_out1 14.52/4.57 f281_in(0, T65, s(T65)) -> f281_out1 14.52/4.57 f281_in(s(T74), 0, T75) -> U8(f196_in(T74, T75), s(T74), 0, T75) 14.52/4.57 U8(f196_out1, s(T74), 0, T75) -> f281_out1 14.52/4.57 f281_in(s(T82), s(T83), T84) -> U9(f457_in(T82, T83, T84), s(T82), s(T83), T84) 14.52/4.57 U9(f457_out1(X163), s(T82), s(T83), T84) -> f281_out1 14.52/4.57 f463_in(T103) -> U10(f289_in(T103), T103) 14.52/4.57 U10(f289_out1(X200), T103) -> f463_out1 14.52/4.57 f463_in(T108) -> U11(f471_in(T108), T108) 14.52/4.57 U11(f471_out1, T108) -> f463_out1 14.52/4.57 f474_in(0) -> f474_out1 14.52/4.57 f474_in(s(T123)) -> U12(f289_in(T123), s(T123)) 14.52/4.57 U12(f289_out1(X242), s(T123)) -> f474_out1 14.52/4.57 f474_in(s(T128)) -> U13(f471_in(T128), s(T128)) 14.52/4.57 U13(f471_out1, s(T128)) -> f474_out1 14.52/4.57 f277_in(T24, T25) -> U14(f280_in(T24), T24, T25) 14.52/4.57 U14(f280_out1(T26), T24, T25) -> U15(f281_in(T24, T26, T25), T24, T25, T26) 14.52/4.57 U15(f281_out1, T24, T25, T26) -> f277_out1(T26) 14.52/4.57 f297_in(T35) -> U16(f280_in(T35), T35) 14.52/4.57 U16(f280_out1(T36), T35) -> U17(f300_in(T35, T36), T35, T36) 14.52/4.57 U17(f300_out1(X82), T35, T36) -> f297_out1(T36, X82) 14.52/4.57 f380_in(T53, T54) -> U18(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18(f300_out1(T55), T53, T54) -> U19(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U19(f300_out1(X124), T53, T54, T55) -> f380_out1(T55, X124) 14.52/4.57 f457_in(T82, T83, T84) -> U20(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 U20(f300_out1(T85), T82, T83, T84) -> U21(f281_in(T82, T85, T84), T82, T83, T84, T85) 14.52/4.57 U21(f281_out1, T82, T83, T84, T85) -> f457_out1(T85) 14.52/4.57 f461_in(T94, T96) -> U22(f463_in(T94), T94, T96) 14.52/4.57 U22(f463_out1, T94, T96) -> U23(f10_in(T94, T96), T94, T96) 14.52/4.57 U23(f10_out1, T94, T96) -> f461_out1 14.52/4.57 f471_in(T108) -> U24(f463_in(T108), T108) 14.52/4.57 U24(f463_out1, T108) -> U25(f474_in(T108), T108) 14.52/4.57 U25(f474_out1, T108) -> f471_out1 14.52/4.57 14.52/4.57 Q is empty. 14.52/4.57 We have to consider all minimal (P,Q,R)-chains. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (146) QDPSizeChangeProof (EQUIVALENT) 14.52/4.57 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. 14.52/4.57 14.52/4.57 From the DPs we obtained the following set of size-change graphs: 14.52/4.57 *U14^1(f280_out1(T26), T24, T25) -> F281_IN(T24, T26, T25) 14.52/4.57 The graph contains the following edges 2 >= 1, 1 > 2, 3 >= 3 14.52/4.57 14.52/4.57 14.52/4.57 *F196_IN(s(T24), T25) -> F277_IN(T24, T25) 14.52/4.57 The graph contains the following edges 1 > 1, 2 >= 2 14.52/4.57 14.52/4.57 14.52/4.57 *F277_IN(T24, T25) -> U14^1(f280_in(T24), T24, T25) 14.52/4.57 The graph contains the following edges 1 >= 2, 2 >= 3 14.52/4.57 14.52/4.57 14.52/4.57 *U20^1(f300_out1(T85), T82, T83, T84) -> F281_IN(T82, T85, T84) 14.52/4.57 The graph contains the following edges 2 >= 1, 1 > 2, 4 >= 3 14.52/4.57 14.52/4.57 14.52/4.57 *F457_IN(T82, T83, T84) -> U20^1(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4 14.52/4.57 14.52/4.57 14.52/4.57 *F281_IN(s(T74), 0, T75) -> F196_IN(T74, T75) 14.52/4.57 The graph contains the following edges 1 > 1, 3 >= 2 14.52/4.57 14.52/4.57 14.52/4.57 *F281_IN(s(T82), s(T83), T84) -> F457_IN(T82, T83, T84) 14.52/4.57 The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 14.52/4.57 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (147) 14.52/4.57 YES 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (148) 14.52/4.57 Obligation: 14.52/4.57 Q DP problem: 14.52/4.57 The TRS P consists of the following rules: 14.52/4.57 14.52/4.57 F10_IN(s(T94), T96) -> F461_IN(T94, T96) 14.52/4.57 F461_IN(T94, T96) -> U22^1(f463_in(T94), T94, T96) 14.52/4.57 U22^1(f463_out1, T94, T96) -> F10_IN(T94, T96) 14.52/4.57 14.52/4.57 The TRS R consists of the following rules: 14.52/4.57 14.52/4.57 f10_in(0, s(T8)) -> f10_out1 14.52/4.57 f10_in(s(T17), T18) -> U1(f196_in(T17, T18), s(T17), T18) 14.52/4.57 U1(f196_out1, s(T17), T18) -> f10_out1 14.52/4.57 f10_in(s(T94), T96) -> U2(f461_in(T94, T96), s(T94), T96) 14.52/4.57 U2(f461_out1, s(T94), T96) -> f10_out1 14.52/4.57 f280_in(T31) -> U3(f289_in(T31), T31) 14.52/4.57 U3(f289_out1(X59), T31) -> f280_out1(X59) 14.52/4.57 f289_in(0) -> f289_out1(s(s(0))) 14.52/4.57 f289_in(s(T35)) -> U4(f297_in(T35), s(T35)) 14.52/4.57 U4(f297_out1(X81, X82), s(T35)) -> f289_out1(X82) 14.52/4.57 f300_in(0, T43) -> f300_out1(s(T43)) 14.52/4.57 f300_in(s(T48), 0) -> U5(f289_in(T48), s(T48), 0) 14.52/4.57 U5(f289_out1(X108), s(T48), 0) -> f300_out1(X108) 14.52/4.57 f300_in(s(T53), s(T54)) -> U6(f380_in(T53, T54), s(T53), s(T54)) 14.52/4.57 U6(f380_out1(X123, X124), s(T53), s(T54)) -> f300_out1(X124) 14.52/4.57 f196_in(0, s(s(0))) -> f196_out1 14.52/4.57 f196_in(s(T24), T25) -> U7(f277_in(T24, T25), s(T24), T25) 14.52/4.57 U7(f277_out1(X35), s(T24), T25) -> f196_out1 14.52/4.57 f281_in(0, T65, s(T65)) -> f281_out1 14.52/4.57 f281_in(s(T74), 0, T75) -> U8(f196_in(T74, T75), s(T74), 0, T75) 14.52/4.57 U8(f196_out1, s(T74), 0, T75) -> f281_out1 14.52/4.57 f281_in(s(T82), s(T83), T84) -> U9(f457_in(T82, T83, T84), s(T82), s(T83), T84) 14.52/4.57 U9(f457_out1(X163), s(T82), s(T83), T84) -> f281_out1 14.52/4.57 f463_in(T103) -> U10(f289_in(T103), T103) 14.52/4.57 U10(f289_out1(X200), T103) -> f463_out1 14.52/4.57 f463_in(T108) -> U11(f471_in(T108), T108) 14.52/4.57 U11(f471_out1, T108) -> f463_out1 14.52/4.57 f474_in(0) -> f474_out1 14.52/4.57 f474_in(s(T123)) -> U12(f289_in(T123), s(T123)) 14.52/4.57 U12(f289_out1(X242), s(T123)) -> f474_out1 14.52/4.57 f474_in(s(T128)) -> U13(f471_in(T128), s(T128)) 14.52/4.57 U13(f471_out1, s(T128)) -> f474_out1 14.52/4.57 f277_in(T24, T25) -> U14(f280_in(T24), T24, T25) 14.52/4.57 U14(f280_out1(T26), T24, T25) -> U15(f281_in(T24, T26, T25), T24, T25, T26) 14.52/4.57 U15(f281_out1, T24, T25, T26) -> f277_out1(T26) 14.52/4.57 f297_in(T35) -> U16(f280_in(T35), T35) 14.52/4.57 U16(f280_out1(T36), T35) -> U17(f300_in(T35, T36), T35, T36) 14.52/4.57 U17(f300_out1(X82), T35, T36) -> f297_out1(T36, X82) 14.52/4.57 f380_in(T53, T54) -> U18(f300_in(s(T53), T54), T53, T54) 14.52/4.57 U18(f300_out1(T55), T53, T54) -> U19(f300_in(T53, T55), T53, T54, T55) 14.52/4.57 U19(f300_out1(X124), T53, T54, T55) -> f380_out1(T55, X124) 14.52/4.57 f457_in(T82, T83, T84) -> U20(f300_in(s(T82), T83), T82, T83, T84) 14.52/4.57 U20(f300_out1(T85), T82, T83, T84) -> U21(f281_in(T82, T85, T84), T82, T83, T84, T85) 14.52/4.57 U21(f281_out1, T82, T83, T84, T85) -> f457_out1(T85) 14.52/4.57 f461_in(T94, T96) -> U22(f463_in(T94), T94, T96) 14.52/4.57 U22(f463_out1, T94, T96) -> U23(f10_in(T94, T96), T94, T96) 14.52/4.57 U23(f10_out1, T94, T96) -> f461_out1 14.52/4.57 f471_in(T108) -> U24(f463_in(T108), T108) 14.52/4.57 U24(f463_out1, T108) -> U25(f474_in(T108), T108) 14.52/4.57 U25(f474_out1, T108) -> f471_out1 14.52/4.57 14.52/4.57 Q is empty. 14.52/4.57 We have to consider all minimal (P,Q,R)-chains. 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (149) QDPSizeChangeProof (EQUIVALENT) 14.52/4.57 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. 14.52/4.57 14.52/4.57 From the DPs we obtained the following set of size-change graphs: 14.52/4.57 *F461_IN(T94, T96) -> U22^1(f463_in(T94), T94, T96) 14.52/4.57 The graph contains the following edges 1 >= 2, 2 >= 3 14.52/4.57 14.52/4.57 14.52/4.57 *U22^1(f463_out1, T94, T96) -> F10_IN(T94, T96) 14.52/4.57 The graph contains the following edges 2 >= 1, 3 >= 2 14.52/4.57 14.52/4.57 14.52/4.57 *F10_IN(s(T94), T96) -> F461_IN(T94, T96) 14.52/4.57 The graph contains the following edges 1 > 1, 2 >= 2 14.52/4.57 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (150) 14.52/4.57 YES 14.52/4.57 14.52/4.57 ---------------------------------------- 14.52/4.57 14.52/4.57 (151) PrologToIRSwTTransformerProof (SOUND) 14.52/4.57 Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert 14.52/4.57 14.52/4.57 { 14.52/4.57 "root": 53, 14.52/4.57 "program": { 14.52/4.57 "directives": [], 14.52/4.57 "clauses": [ 14.52/4.57 [ 14.52/4.57 "(ackermann (0) N (s N))", 14.52/4.57 null 14.52/4.57 ], 14.52/4.57 [ 14.52/4.57 "(ackermann (s M) (0) Val)", 14.52/4.57 "(ackermann M (s (0)) Val)" 14.52/4.57 ], 14.52/4.57 [ 14.52/4.57 "(ackermann (s M) (s N) Val)", 14.52/4.57 "(',' (ackermann (s M) N Val1) (ackermann M Val1 Val))" 14.52/4.57 ] 14.52/4.57 ] 14.52/4.57 }, 14.52/4.57 "graph": { 14.52/4.57 "nodes": { 14.52/4.57 "type": "Nodes", 14.52/4.57 "195": { 14.52/4.57 "goal": [{ 14.52/4.57 "clause": -1, 14.52/4.57 "scope": -1, 14.52/4.57 "term": "(ackermann T17 (s (0)) T18)" 14.52/4.57 }], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": [ 14.52/4.57 "T17", 14.52/4.57 "T18" 14.52/4.57 ], 14.52/4.57 "free": [], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "350": { 14.52/4.57 "goal": [], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": [], 14.52/4.57 "free": [], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "352": { 14.52/4.57 "goal": [{ 14.52/4.57 "clause": -1, 14.52/4.57 "scope": -1, 14.52/4.57 "term": "(',' (ackermann (s T53) T54 X123) (ackermann T53 X123 X124))" 14.52/4.57 }], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": [ 14.52/4.57 "T53", 14.52/4.57 "T54" 14.52/4.57 ], 14.52/4.57 "free": [ 14.52/4.57 "X124", 14.52/4.57 "X123" 14.52/4.57 ], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "199": { 14.52/4.57 "goal": [], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": [], 14.52/4.57 "free": [], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "353": { 14.52/4.57 "goal": [], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": [], 14.52/4.57 "free": [], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "234": { 14.52/4.57 "goal": [{ 14.52/4.57 "clause": -1, 14.52/4.57 "scope": -1, 14.52/4.57 "term": "(true)" 14.52/4.57 }], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": [], 14.52/4.57 "free": [], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "355": { 14.52/4.57 "goal": [{ 14.52/4.57 "clause": -1, 14.52/4.57 "scope": -1, 14.52/4.57 "term": "(ackermann (s T53) T54 X123)" 14.52/4.57 }], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": [ 14.52/4.57 "T53", 14.52/4.57 "T54" 14.52/4.57 ], 14.52/4.57 "free": ["X123"], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "356": { 14.52/4.57 "goal": [{ 14.52/4.57 "clause": -1, 14.52/4.57 "scope": -1, 14.52/4.57 "term": "(ackermann T53 T55 X124)" 14.52/4.57 }], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": [ 14.52/4.57 "T53", 14.52/4.57 "T55" 14.52/4.57 ], 14.52/4.57 "free": ["X124"], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "510": { 14.52/4.57 "goal": [{ 14.52/4.57 "clause": -1, 14.52/4.57 "scope": -1, 14.52/4.57 "term": "(ackermann T94 T98 T96)" 14.52/4.57 }], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": [ 14.52/4.57 "T94", 14.52/4.57 "T96" 14.52/4.57 ], 14.52/4.57 "free": [], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "511": { 14.52/4.57 "goal": [ 14.52/4.57 { 14.52/4.57 "clause": 0, 14.52/4.57 "scope": 7, 14.52/4.57 "term": "(ackermann (s T94) T97 X180)" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "clause": 1, 14.52/4.57 "scope": 7, 14.52/4.57 "term": "(ackermann (s T94) T97 X180)" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "clause": 2, 14.52/4.57 "scope": 7, 14.52/4.57 "term": "(ackermann (s T94) T97 X180)" 14.52/4.57 } 14.52/4.57 ], 14.52/4.57 "kb": { 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14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": ["T94"], 14.52/4.57 "free": ["X180"], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "315": { 14.52/4.57 "goal": [{ 14.52/4.57 "clause": -1, 14.52/4.57 "scope": -1, 14.52/4.57 "term": "(ackermann T31 (s (0)) X59)" 14.52/4.57 }], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": ["T31"], 14.52/4.57 "free": ["X59"], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "239": { 14.52/4.57 "goal": [], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": [], 14.52/4.57 "free": [], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "316": { 14.52/4.57 "goal": [ 14.52/4.57 { 14.52/4.57 "clause": 0, 14.52/4.57 "scope": 4, 14.52/4.57 "term": "(ackermann T31 (s (0)) X59)" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "clause": 1, 14.52/4.57 "scope": 4, 14.52/4.57 "term": "(ackermann T31 (s (0)) X59)" 14.52/4.57 }, 14.52/4.57 { 14.52/4.57 "clause": 2, 14.52/4.57 "scope": 4, 14.52/4.57 "term": "(ackermann T31 (s (0)) X59)" 14.52/4.57 } 14.52/4.57 ], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": ["T31"], 14.52/4.57 "free": ["X59"], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "317": { 14.52/4.57 "goal": [{ 14.52/4.57 "clause": 0, 14.52/4.57 "scope": 4, 14.52/4.57 "term": "(ackermann T31 (s (0)) X59)" 14.52/4.57 }], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 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14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": ["T103"], 14.52/4.57 "free": ["X200"], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "518": { 14.52/4.57 "goal": [], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": [], 14.52/4.57 "free": [], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "519": { 14.52/4.57 "goal": [{ 14.52/4.57 "clause": -1, 14.52/4.57 "scope": -1, 14.52/4.57 "term": "(',' (ackermann (s T108) T110 X215) (ackermann T108 X215 X216))" 14.52/4.57 }], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.57 "intvars": {}, 14.52/4.57 "arithmetic": { 14.52/4.57 "type": "PlainIntegerRelationState", 14.52/4.57 "relations": [] 14.52/4.57 }, 14.52/4.57 "ground": ["T108"], 14.52/4.57 "free": [ 14.52/4.57 "X216", 14.52/4.57 "X215" 14.52/4.57 ], 14.52/4.57 "exprvars": [] 14.52/4.57 } 14.52/4.57 }, 14.52/4.57 "53": { 14.52/4.57 "goal": [{ 14.52/4.57 "clause": -1, 14.52/4.57 "scope": -1, 14.52/4.57 "term": "(ackermann T1 T2 T3)" 14.52/4.57 }], 14.52/4.57 "kb": { 14.52/4.57 "nonunifying": [], 14.52/4.58 "intvars": {}, 14.52/4.58 "arithmetic": { 14.52/4.58 "type": "PlainIntegerRelationState", 14.52/4.58 "relations": [] 14.52/4.58 }, 14.52/4.58 "ground": [ 14.52/4.58 "T1", 14.52/4.58 "T3" 14.52/4.58 ], 14.52/4.58 "free": [], 14.52/4.58 "exprvars": [] 14.52/4.58 } 14.52/4.58 }, 14.52/4.58 "54": { 14.52/4.58 "goal": [ 14.52/4.58 { 14.52/4.58 "clause": 0, 14.52/4.58 "scope": 1, 14.52/4.58 "term": "(ackermann T1 T2 T3)" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "clause": 1, 14.52/4.58 "scope": 1, 14.52/4.58 "term": "(ackermann T1 T2 T3)" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "clause": 2, 14.52/4.58 "scope": 1, 14.52/4.58 "term": "(ackermann T1 T2 T3)" 14.52/4.58 } 14.52/4.58 ], 14.52/4.58 "kb": { 14.52/4.58 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"PlainIntegerRelationState", 14.52/4.58 "relations": [] 14.52/4.58 }, 14.52/4.58 "ground": ["T48"], 14.52/4.58 "free": ["X108"], 14.52/4.58 "exprvars": [] 14.52/4.58 } 14.52/4.58 }, 14.52/4.58 "306": { 14.52/4.58 "goal": [{ 14.52/4.58 "clause": 2, 14.52/4.58 "scope": 3, 14.52/4.58 "term": "(ackermann (s T24) (0) X35)" 14.52/4.58 }], 14.52/4.58 "kb": { 14.52/4.58 "nonunifying": [], 14.52/4.58 "intvars": {}, 14.52/4.58 "arithmetic": { 14.52/4.58 "type": "PlainIntegerRelationState", 14.52/4.58 "relations": [] 14.52/4.58 }, 14.52/4.58 "ground": ["T24"], 14.52/4.58 "free": ["X35"], 14.52/4.58 "exprvars": [] 14.52/4.58 } 14.52/4.58 }, 14.52/4.58 "504": { 14.52/4.58 "goal": [{ 14.52/4.58 "clause": -1, 14.52/4.58 "scope": -1, 14.52/4.58 "term": "(',' (ackermann (s T94) T97 X180) (ackermann T94 X180 T96))" 14.52/4.58 }], 14.52/4.58 "kb": { 14.52/4.58 "nonunifying": [], 14.52/4.58 "intvars": {}, 14.52/4.58 "arithmetic": { 14.52/4.58 "type": "PlainIntegerRelationState", 14.52/4.58 "relations": [] 14.52/4.58 }, 14.52/4.58 "ground": [ 14.52/4.58 "T94", 14.52/4.58 "T96" 14.52/4.58 ], 14.52/4.58 "free": ["X180"], 14.52/4.58 "exprvars": [] 14.52/4.58 } 14.52/4.58 }, 14.52/4.58 "505": { 14.52/4.58 "goal": [], 14.52/4.58 "kb": { 14.52/4.58 "nonunifying": [], 14.52/4.58 "intvars": {}, 14.52/4.58 "arithmetic": { 14.52/4.58 "type": "PlainIntegerRelationState", 14.52/4.58 "relations": [] 14.52/4.58 }, 14.52/4.58 "ground": [], 14.52/4.58 "free": [], 14.52/4.58 "exprvars": [] 14.52/4.58 } 14.52/4.58 }, 14.52/4.58 "509": { 14.52/4.58 "goal": [{ 14.52/4.58 "clause": -1, 14.52/4.58 "scope": -1, 14.52/4.58 "term": "(ackermann (s T94) T97 X180)" 14.52/4.58 }], 14.52/4.58 "kb": { 14.52/4.58 "nonunifying": [], 14.52/4.58 "intvars": {}, 14.52/4.58 "arithmetic": { 14.52/4.58 "type": "PlainIntegerRelationState", 14.52/4.58 "relations": [] 14.52/4.58 }, 14.52/4.58 "ground": ["T94"], 14.52/4.58 "free": ["X180"], 14.52/4.58 "exprvars": [] 14.52/4.58 } 14.52/4.58 } 14.52/4.58 }, 14.52/4.58 "edges": [ 14.52/4.58 { 14.52/4.58 "from": 53, 14.52/4.58 "to": 54, 14.52/4.58 "label": "CASE" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 54, 14.52/4.58 "to": 55, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 54, 14.52/4.58 "to": 56, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 55, 14.52/4.58 "to": 168, 14.52/4.58 "label": "EVAL with clause\nackermann(0, X5, s(X5)).\nand substitutionT1 -> 0,\nT2 -> T8,\nX5 -> T8,\nT3 -> s(T8)" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 55, 14.52/4.58 "to": 173, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 56, 14.52/4.58 "to": 182, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 56, 14.52/4.58 "to": 184, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 168, 14.52/4.58 "to": 175, 14.52/4.58 "label": "SUCCESS" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 182, 14.52/4.58 "to": 195, 14.52/4.58 "label": "EVAL with clause\nackermann(s(X14), 0, X15) :- ackermann(X14, s(0), X15).\nand substitutionX14 -> T17,\nT1 -> s(T17),\nT2 -> 0,\nT3 -> T18,\nX15 -> T18" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 182, 14.52/4.58 "to": 199, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 184, 14.52/4.58 "to": 504, 14.52/4.58 "label": "EVAL with clause\nackermann(s(X177), s(X178), X179) :- ','(ackermann(s(X177), X178, X180), ackermann(X177, X180, X179)).\nand substitutionX177 -> T94,\nT1 -> s(T94),\nX178 -> T97,\nT2 -> s(T97),\nT3 -> T96,\nX179 -> T96,\nT95 -> T97" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 184, 14.52/4.58 "to": 505, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 195, 14.52/4.58 "to": 222, 14.52/4.58 "label": "CASE" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 222, 14.52/4.58 "to": 226, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 222, 14.52/4.58 "to": 228, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 226, 14.52/4.58 "to": 234, 14.52/4.58 "label": "EVAL with clause\nackermann(0, X22, s(X22)).\nand substitutionT17 -> 0,\nX22 -> s(0),\nT18 -> s(s(0))" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 226, 14.52/4.58 "to": 237, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 228, 14.52/4.58 "to": 242, 14.52/4.58 "label": "BACKTRACK\nfor clause: ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val)because of non-unification" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 234, 14.52/4.58 "to": 239, 14.52/4.58 "label": "SUCCESS" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 242, 14.52/4.58 "to": 254, 14.52/4.58 "label": "EVAL with clause\nackermann(s(X32), s(X33), X34) :- ','(ackermann(s(X32), X33, X35), ackermann(X32, X35, X34)).\nand substitutionX32 -> T24,\nT17 -> s(T24),\nX33 -> 0,\nT18 -> T25,\nX34 -> T25" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 242, 14.52/4.58 "to": 257, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 254, 14.52/4.58 "to": 301, 14.52/4.58 "label": "SPLIT 1" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 254, 14.52/4.58 "to": 302, 14.52/4.58 "label": "SPLIT 2\nnew knowledge:\nT24 is ground\nT26 is ground\nreplacements:X35 -> T26" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 301, 14.52/4.58 "to": 303, 14.52/4.58 "label": "CASE" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 302, 14.52/4.58 "to": 362, 14.52/4.58 "label": "CASE" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 303, 14.52/4.58 "to": 304, 14.52/4.58 "label": "BACKTRACK\nfor clause: ackermann(0, N, s(N))because of non-unification" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 304, 14.52/4.58 "to": 305, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 304, 14.52/4.58 "to": 306, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 305, 14.52/4.58 "to": 315, 14.52/4.58 "label": "ONLY EVAL with clause\nackermann(s(X57), 0, X58) :- ackermann(X57, s(0), X58).\nand substitutionT24 -> T31,\nX57 -> T31,\nX35 -> X59,\nX58 -> X59" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 306, 14.52/4.58 "to": 361, 14.52/4.58 "label": "BACKTRACK\nfor clause: ackermann(s(M), s(N), Val) :- ','(ackermann(s(M), N, Val1), ackermann(M, Val1, Val))because of non-unification" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 315, 14.52/4.58 "to": 316, 14.52/4.58 "label": "CASE" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 316, 14.52/4.58 "to": 317, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 316, 14.52/4.58 "to": 318, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 317, 14.52/4.58 "to": 320, 14.52/4.58 "label": "EVAL with clause\nackermann(0, X66, s(X66)).\nand substitutionT31 -> 0,\nX66 -> s(0),\nX59 -> s(s(0))" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 317, 14.52/4.58 "to": 323, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 318, 14.52/4.58 "to": 325, 14.52/4.58 "label": "BACKTRACK\nfor clause: ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val)because of non-unification" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 320, 14.52/4.58 "to": 324, 14.52/4.58 "label": "SUCCESS" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 325, 14.52/4.58 "to": 329, 14.52/4.58 "label": "EVAL with clause\nackermann(s(X78), s(X79), X80) :- ','(ackermann(s(X78), X79, X81), ackermann(X78, X81, X80)).\nand substitutionX78 -> T35,\nT31 -> s(T35),\nX79 -> 0,\nX59 -> X82,\nX80 -> X82" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 325, 14.52/4.58 "to": 330, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 329, 14.52/4.58 "to": 333, 14.52/4.58 "label": "SPLIT 1" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 329, 14.52/4.58 "to": 334, 14.52/4.58 "label": "SPLIT 2\nnew knowledge:\nT35 is ground\nT36 is ground\nreplacements:X81 -> T36" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 333, 14.52/4.58 "to": 301, 14.52/4.58 "label": "INSTANCE with matching:\nT24 -> T35\nX35 -> X81" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 334, 14.52/4.58 "to": 337, 14.52/4.58 "label": "CASE" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 337, 14.52/4.58 "to": 338, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 337, 14.52/4.58 "to": 339, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 338, 14.52/4.58 "to": 340, 14.52/4.58 "label": "EVAL with clause\nackermann(0, X93, s(X93)).\nand substitutionT35 -> 0,\nT36 -> T43,\nX93 -> T43,\nX82 -> s(T43)" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 338, 14.52/4.58 "to": 342, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 339, 14.52/4.58 "to": 345, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 339, 14.52/4.58 "to": 346, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 340, 14.52/4.58 "to": 344, 14.52/4.58 "label": "SUCCESS" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 345, 14.52/4.58 "to": 349, 14.52/4.58 "label": "EVAL with clause\nackermann(s(X106), 0, X107) :- ackermann(X106, s(0), X107).\nand substitutionX106 -> T48,\nT35 -> s(T48),\nT36 -> 0,\nX82 -> X108,\nX107 -> X108" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 345, 14.52/4.58 "to": 350, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 346, 14.52/4.58 "to": 352, 14.52/4.58 "label": "EVAL with clause\nackermann(s(X120), s(X121), X122) :- ','(ackermann(s(X120), X121, X123), ackermann(X120, X123, X122)).\nand substitutionX120 -> T53,\nT35 -> s(T53),\nX121 -> T54,\nT36 -> s(T54),\nX82 -> X124,\nX122 -> X124" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 346, 14.52/4.58 "to": 353, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 349, 14.52/4.58 "to": 315, 14.52/4.58 "label": "INSTANCE with matching:\nT31 -> T48\nX59 -> X108" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 352, 14.52/4.58 "to": 355, 14.52/4.58 "label": "SPLIT 1" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 352, 14.52/4.58 "to": 356, 14.52/4.58 "label": "SPLIT 2\nnew knowledge:\nT53 is ground\nT54 is ground\nT55 is ground\nreplacements:X123 -> T55" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 355, 14.52/4.58 "to": 334, 14.52/4.58 "label": "INSTANCE with matching:\nT35 -> s(T53)\nT36 -> T54\nX82 -> X123" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 356, 14.52/4.58 "to": 334, 14.52/4.58 "label": "INSTANCE with matching:\nT35 -> T53\nT36 -> T55\nX82 -> X124" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 362, 14.52/4.58 "to": 363, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 362, 14.52/4.58 "to": 364, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 363, 14.52/4.58 "to": 368, 14.52/4.58 "label": "EVAL with clause\nackermann(0, X140, s(X140)).\nand substitutionT24 -> 0,\nT26 -> T65,\nX140 -> T65,\nT25 -> s(T65)" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 363, 14.52/4.58 "to": 369, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 364, 14.52/4.58 "to": 373, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 364, 14.52/4.58 "to": 375, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 368, 14.52/4.58 "to": 370, 14.52/4.58 "label": "SUCCESS" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 373, 14.52/4.58 "to": 444, 14.52/4.58 "label": "EVAL with clause\nackermann(s(X149), 0, X150) :- ackermann(X149, s(0), X150).\nand substitutionX149 -> T74,\nT24 -> s(T74),\nT26 -> 0,\nT25 -> T75,\nX150 -> T75" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 373, 14.52/4.58 "to": 447, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 375, 14.52/4.58 "to": 453, 14.52/4.58 "label": "EVAL with clause\nackermann(s(X160), s(X161), X162) :- ','(ackermann(s(X160), X161, X163), ackermann(X160, X163, X162)).\nand substitutionX160 -> T82,\nT24 -> s(T82),\nX161 -> T83,\nT26 -> s(T83),\nT25 -> T84,\nX162 -> T84" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 375, 14.52/4.58 "to": 454, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 444, 14.52/4.58 "to": 195, 14.52/4.58 "label": "INSTANCE with matching:\nT17 -> T74\nT18 -> T75" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 453, 14.52/4.58 "to": 497, 14.52/4.58 "label": "SPLIT 1" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 453, 14.52/4.58 "to": 498, 14.52/4.58 "label": "SPLIT 2\nnew knowledge:\nT82 is ground\nT83 is ground\nT85 is ground\nreplacements:X163 -> T85" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 497, 14.52/4.58 "to": 334, 14.52/4.58 "label": "INSTANCE with matching:\nT35 -> s(T82)\nT36 -> T83\nX82 -> X163" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 498, 14.52/4.58 "to": 302, 14.52/4.58 "label": "INSTANCE with matching:\nT24 -> T82\nT26 -> T85\nT25 -> T84" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 504, 14.52/4.58 "to": 509, 14.52/4.58 "label": "SPLIT 1" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 504, 14.52/4.58 "to": 510, 14.52/4.58 "label": "SPLIT 2\nnew knowledge:\nT94 is ground\nreplacements:X180 -> T98" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 509, 14.52/4.58 "to": 511, 14.52/4.58 "label": "CASE" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 510, 14.52/4.58 "to": 53, 14.52/4.58 "label": "INSTANCE with matching:\nT1 -> T94\nT2 -> T98\nT3 -> T96" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 511, 14.52/4.58 "to": 512, 14.52/4.58 "label": "BACKTRACK\nfor clause: ackermann(0, N, s(N))because of non-unification" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 512, 14.52/4.58 "to": 515, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 512, 14.52/4.58 "to": 516, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 515, 14.52/4.58 "to": 517, 14.52/4.58 "label": "EVAL with clause\nackermann(s(X198), 0, X199) :- ackermann(X198, s(0), X199).\nand substitutionT94 -> T103,\nX198 -> T103,\nT97 -> 0,\nX180 -> X200,\nX199 -> X200" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 515, 14.52/4.58 "to": 518, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 516, 14.52/4.58 "to": 519, 14.52/4.58 "label": "EVAL with clause\nackermann(s(X212), s(X213), X214) :- ','(ackermann(s(X212), X213, X215), ackermann(X212, X215, X214)).\nand substitutionT94 -> T108,\nX212 -> T108,\nX213 -> T110,\nT97 -> s(T110),\nX180 -> X216,\nX214 -> X216,\nT109 -> T110" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 516, 14.52/4.58 "to": 526, 14.52/4.58 "label": "EVAL-BACKTRACK" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 517, 14.52/4.58 "to": 315, 14.52/4.58 "label": "INSTANCE with matching:\nT31 -> T103\nX59 -> X200" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 519, 14.52/4.58 "to": 529, 14.52/4.58 "label": "SPLIT 1" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 519, 14.52/4.58 "to": 530, 14.52/4.58 "label": "SPLIT 2\nnew knowledge:\nT108 is ground\nreplacements:X215 -> T111" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 529, 14.52/4.58 "to": 509, 14.52/4.58 "label": "INSTANCE with matching:\nT94 -> T108\nT97 -> T110\nX180 -> X215" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 530, 14.52/4.58 "to": 531, 14.52/4.58 "label": "CASE" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 531, 14.52/4.58 "to": 532, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 531, 14.52/4.58 "to": 533, 14.52/4.58 "label": "PARALLEL" 14.52/4.58 }, 14.52/4.58 { 14.52/4.58 "from": 532, 14.57/4.58 "to": 534, 14.57/4.58 "label": "EVAL with clause\nackermann(0, X227, s(X227)).\nand substitutionT108 -> 0,\nT111 -> T118,\nX227 -> T118,\nX216 -> s(T118)" 14.57/4.58 }, 14.57/4.58 { 14.57/4.58 "from": 532, 14.57/4.58 "to": 535, 14.57/4.58 "label": "EVAL-BACKTRACK" 14.57/4.58 }, 14.57/4.58 { 14.57/4.58 "from": 533, 14.57/4.58 "to": 537, 14.57/4.58 "label": "PARALLEL" 14.57/4.58 }, 14.57/4.58 { 14.57/4.58 "from": 533, 14.57/4.58 "to": 538, 14.57/4.58 "label": "PARALLEL" 14.57/4.58 }, 14.57/4.58 { 14.57/4.58 "from": 534, 14.57/4.58 "to": 536, 14.57/4.58 "label": "SUCCESS" 14.57/4.58 }, 14.57/4.58 { 14.57/4.58 "from": 537, 14.57/4.58 "to": 539, 14.57/4.58 "label": "EVAL with clause\nackermann(s(X240), 0, X241) :- ackermann(X240, s(0), X241).\nand substitutionX240 -> T123,\nT108 -> s(T123),\nT111 -> 0,\nX216 -> X242,\nX241 -> X242" 14.57/4.58 }, 14.57/4.58 { 14.57/4.58 "from": 537, 14.57/4.58 "to": 540, 14.57/4.58 "label": "EVAL-BACKTRACK" 14.57/4.58 }, 14.57/4.58 { 14.57/4.58 "from": 538, 14.57/4.58 "to": 541, 14.57/4.58 "label": "EVAL with clause\nackermann(s(X254), s(X255), X256) :- ','(ackermann(s(X254), X255, X257), ackermann(X254, X257, X256)).\nand substitutionX254 -> T128,\nT108 -> s(T128),\nX255 -> T130,\nT111 -> s(T130),\nX216 -> X258,\nX256 -> X258,\nT129 -> T130" 14.57/4.58 }, 14.57/4.58 { 14.57/4.58 "from": 538, 14.57/4.58 "to": 542, 14.57/4.58 "label": "EVAL-BACKTRACK" 14.57/4.58 }, 14.57/4.58 { 14.57/4.58 "from": 539, 14.57/4.58 "to": 315, 14.57/4.58 "label": "INSTANCE with matching:\nT31 -> T123\nX59 -> X242" 14.57/4.58 }, 14.57/4.58 { 14.57/4.58 "from": 541, 14.57/4.58 "to": 519, 14.57/4.58 "label": "INSTANCE with matching:\nT108 -> T128\nT110 -> T130\nX215 -> X257\nX216 -> X258" 14.57/4.58 } 14.57/4.58 ], 14.57/4.58 "type": "Graph" 14.57/4.58 } 14.57/4.58 } 14.57/4.58 14.57/4.58 ---------------------------------------- 14.57/4.58 14.57/4.58 (152) 14.57/4.58 Complex Obligation (AND) 14.57/4.58 14.57/4.58 ---------------------------------------- 14.57/4.58 14.57/4.58 (153) 14.57/4.58 Obligation: 14.57/4.58 Rules: 14.57/4.58 f303_out(T24) -> f301_out(T24) :|: TRUE 14.57/4.58 f301_in(x) -> f303_in(x) :|: TRUE 14.57/4.58 f345_in(s(T48), 0) -> f349_in(T48) :|: TRUE 14.57/4.58 f350_out -> f345_out(T35, T36) :|: TRUE 14.57/4.58 f345_in(x1, x2) -> f350_in :|: TRUE 14.57/4.58 f349_out(x3) -> f345_out(s(x3), 0) :|: TRUE 14.57/4.58 f353_out -> f346_out(x4, x5) :|: TRUE 14.57/4.58 f352_out(T53, T54) -> f346_out(s(T53), s(T54)) :|: TRUE 14.57/4.58 f346_in(x6, x7) -> f353_in :|: TRUE 14.57/4.58 f346_in(s(x8), s(x9)) -> f352_in(x8, x9) :|: TRUE 14.57/4.58 f329_in(x10) -> f333_in(x10) :|: TRUE 14.57/4.58 f334_out(x11, x12) -> f329_out(x11) :|: TRUE 14.57/4.58 f333_out(x13) -> f334_in(x13, x14) :|: TRUE 14.57/4.58 f317_out(T31) -> f316_out(T31) :|: TRUE 14.57/4.58 f318_out(x15) -> f316_out(x15) :|: TRUE 14.57/4.58 f316_in(x16) -> f318_in(x16) :|: TRUE 14.57/4.58 f316_in(x17) -> f317_in(x17) :|: TRUE 14.57/4.58 f355_out(x18, x19) -> f356_in(x18, x20) :|: TRUE 14.57/4.58 f356_out(x21, x22) -> f352_out(x21, x23) :|: TRUE 14.57/4.58 f352_in(x24, x25) -> f355_in(x24, x25) :|: TRUE 14.57/4.58 f301_out(x26) -> f333_out(x26) :|: TRUE 14.57/4.58 f333_in(x27) -> f301_in(x27) :|: TRUE 14.57/4.58 f303_in(x28) -> f304_in(x28) :|: TRUE 14.57/4.58 f304_out(x29) -> f303_out(x29) :|: TRUE 14.57/4.58 f325_out(x30) -> f318_out(x30) :|: TRUE 14.57/4.58 f318_in(x31) -> f325_in(x31) :|: TRUE 14.57/4.58 f337_in(x32, x33) -> f339_in(x32, x33) :|: TRUE 14.57/4.58 f338_out(x34, x35) -> f337_out(x34, x35) :|: TRUE 14.57/4.58 f339_out(x36, x37) -> f337_out(x36, x37) :|: TRUE 14.57/4.58 f337_in(x38, x39) -> f338_in(x38, x39) :|: TRUE 14.57/4.58 f315_out(x40) -> f349_out(x40) :|: TRUE 14.57/4.58 f349_in(x41) -> f315_in(x41) :|: TRUE 14.57/4.58 f337_out(x42, x43) -> f334_out(x42, x43) :|: TRUE 14.57/4.58 f334_in(x44, x45) -> f337_in(x44, x45) :|: TRUE 14.57/4.58 f355_in(x46, x47) -> f334_in(s(x46), x47) :|: TRUE 14.57/4.58 f334_out(s(x48), x49) -> f355_out(x48, x49) :|: TRUE 14.57/4.58 f305_in(x50) -> f315_in(x50) :|: TRUE 14.57/4.58 f315_out(x51) -> f305_out(x51) :|: TRUE 14.57/4.58 f304_in(x52) -> f305_in(x52) :|: TRUE 14.57/4.58 f306_out(x53) -> f304_out(x53) :|: TRUE 14.57/4.58 f304_in(x54) -> f306_in(x54) :|: TRUE 14.57/4.58 f305_out(x55) -> f304_out(x55) :|: TRUE 14.57/4.58 f339_in(x56, x57) -> f346_in(x56, x57) :|: TRUE 14.57/4.58 f346_out(x58, x59) -> f339_out(x58, x59) :|: TRUE 14.57/4.58 f345_out(x60, x61) -> f339_out(x60, x61) :|: TRUE 14.57/4.58 f339_in(x62, x63) -> f345_in(x62, x63) :|: TRUE 14.57/4.58 f356_in(x64, x65) -> f334_in(x64, x65) :|: TRUE 14.57/4.58 f334_out(x66, x67) -> f356_out(x66, x67) :|: TRUE 14.57/4.58 f325_in(x68) -> f330_in :|: TRUE 14.57/4.58 f330_out -> f325_out(x69) :|: TRUE 14.57/4.58 f329_out(x70) -> f325_out(s(x70)) :|: TRUE 14.57/4.58 f325_in(s(x71)) -> f329_in(x71) :|: TRUE 14.57/4.58 f316_out(x72) -> f315_out(x72) :|: TRUE 14.57/4.58 f315_in(x73) -> f316_in(x73) :|: TRUE 14.57/4.58 f54_out(T1, T3) -> f53_out(T1, T3) :|: TRUE 14.57/4.58 f53_in(x74, x75) -> f54_in(x74, x75) :|: TRUE 14.57/4.58 f54_in(x76, x77) -> f56_in(x76, x77) :|: TRUE 14.57/4.58 f55_out(x78, x79) -> f54_out(x78, x79) :|: TRUE 14.57/4.58 f54_in(x80, x81) -> f55_in(x80, x81) :|: TRUE 14.57/4.58 f56_out(x82, x83) -> f54_out(x82, x83) :|: TRUE 14.57/4.58 f56_in(x84, x85) -> f184_in(x84, x85) :|: TRUE 14.57/4.58 f182_out(x86, x87) -> f56_out(x86, x87) :|: TRUE 14.57/4.58 f56_in(x88, x89) -> f182_in(x88, x89) :|: TRUE 14.57/4.58 f184_out(x90, x91) -> f56_out(x90, x91) :|: TRUE 14.57/4.58 f195_out(T17, T18) -> f182_out(s(T17), T18) :|: TRUE 14.57/4.58 f182_in(x92, x93) -> f199_in :|: TRUE 14.57/4.58 f182_in(s(x94), x95) -> f195_in(x94, x95) :|: TRUE 14.57/4.58 f199_out -> f182_out(x96, x97) :|: TRUE 14.57/4.58 f222_out(x98, x99) -> f195_out(x98, x99) :|: TRUE 14.57/4.58 f195_in(x100, x101) -> f222_in(x100, x101) :|: TRUE 14.57/4.58 f226_out(x102, x103) -> f222_out(x102, x103) :|: TRUE 14.57/4.58 f228_out(x104, x105) -> f222_out(x104, x105) :|: TRUE 14.57/4.58 f222_in(x106, x107) -> f226_in(x106, x107) :|: TRUE 14.57/4.58 f222_in(x108, x109) -> f228_in(x108, x109) :|: TRUE 14.57/4.58 f228_in(x110, x111) -> f242_in(x110, x111) :|: TRUE 14.57/4.58 f242_out(x112, x113) -> f228_out(x112, x113) :|: TRUE 14.57/4.58 f242_in(x114, x115) -> f257_in :|: TRUE 14.57/4.58 f257_out -> f242_out(x116, x117) :|: TRUE 14.57/4.58 f242_in(s(x118), x119) -> f254_in(x118, x119) :|: TRUE 14.57/4.58 f254_out(x120, x121) -> f242_out(s(x120), x121) :|: TRUE 14.57/4.58 f254_in(x122, x123) -> f301_in(x122) :|: TRUE 14.57/4.58 f301_out(x124) -> f302_in(x124, x125, x126) :|: TRUE 14.57/4.58 f302_out(x127, x128, x129) -> f254_out(x127, x129) :|: TRUE 14.57/4.58 f362_out(x130, x131, x132) -> f302_out(x130, x131, x132) :|: TRUE 14.57/4.58 f302_in(x133, x134, x135) -> f362_in(x133, x134, x135) :|: TRUE 14.57/4.58 f363_out(x136, x137, x138) -> f362_out(x136, x137, x138) :|: TRUE 14.57/4.58 f362_in(x139, x140, x141) -> f364_in(x139, x140, x141) :|: TRUE 14.57/4.58 f362_in(x142, x143, x144) -> f363_in(x142, x143, x144) :|: TRUE 14.57/4.59 f364_out(x145, x146, x147) -> f362_out(x145, x146, x147) :|: TRUE 14.57/4.59 f364_in(x148, x149, x150) -> f373_in(x148, x149, x150) :|: TRUE 14.57/4.59 f364_in(x151, x152, x153) -> f375_in(x151, x152, x153) :|: TRUE 14.57/4.59 f375_out(x154, x155, x156) -> f364_out(x154, x155, x156) :|: TRUE 14.57/4.59 f373_out(x157, x158, x159) -> f364_out(x157, x158, x159) :|: TRUE 14.57/4.59 f375_in(s(T82), s(T83), T84) -> f453_in(T82, T83, T84) :|: TRUE 14.57/4.59 f454_out -> f375_out(x160, x161, x162) :|: TRUE 14.57/4.59 f375_in(x163, x164, x165) -> f454_in :|: TRUE 14.57/4.59 f453_out(x166, x167, x168) -> f375_out(s(x166), s(x167), x168) :|: TRUE 14.57/4.59 f497_out(x169, x170) -> f498_in(x169, x171, x172) :|: TRUE 14.57/4.59 f453_in(x173, x174, x175) -> f497_in(x173, x174) :|: TRUE 14.57/4.59 f498_out(x176, x177, x178) -> f453_out(x176, x179, x178) :|: TRUE 14.57/4.59 f334_out(s(x180), x181) -> f497_out(x180, x181) :|: TRUE 14.57/4.59 f497_in(x182, x183) -> f334_in(s(x182), x183) :|: TRUE 14.57/4.59 f504_out(T94, T96) -> f184_out(s(T94), T96) :|: TRUE 14.57/4.59 f184_in(s(x184), x185) -> f504_in(x184, x185) :|: TRUE 14.57/4.59 f184_in(x186, x187) -> f505_in :|: TRUE 14.57/4.59 f505_out -> f184_out(x188, x189) :|: TRUE 14.57/4.59 f510_out(x190, x191) -> f504_out(x190, x191) :|: TRUE 14.57/4.59 f504_in(x192, x193) -> f509_in(x192) :|: TRUE 14.57/4.59 f509_out(x194) -> f510_in(x194, x195) :|: TRUE 14.57/4.59 f509_in(x196) -> f511_in(x196) :|: TRUE 14.57/4.59 f511_out(x197) -> f509_out(x197) :|: TRUE 14.57/4.59 f511_in(x198) -> f512_in(x198) :|: TRUE 14.57/4.59 f512_out(x199) -> f511_out(x199) :|: TRUE 14.57/4.59 f516_out(x200) -> f512_out(x200) :|: TRUE 14.57/4.59 f515_out(x201) -> f512_out(x201) :|: TRUE 14.57/4.59 f512_in(x202) -> f515_in(x202) :|: TRUE 14.57/4.59 f512_in(x203) -> f516_in(x203) :|: TRUE 14.57/4.59 f515_in(x204) -> f518_in :|: TRUE 14.57/4.59 f517_out(T103) -> f515_out(T103) :|: TRUE 14.57/4.59 f515_in(x205) -> f517_in(x205) :|: TRUE 14.57/4.59 f518_out -> f515_out(x206) :|: TRUE 14.57/4.59 f315_out(x207) -> f517_out(x207) :|: TRUE 14.57/4.59 f517_in(x208) -> f315_in(x208) :|: TRUE 14.57/4.59 f519_out(T108) -> f516_out(T108) :|: TRUE 14.57/4.59 f526_out -> f516_out(x209) :|: TRUE 14.57/4.59 f516_in(x210) -> f526_in :|: TRUE 14.57/4.59 f516_in(x211) -> f519_in(x211) :|: TRUE 14.57/4.59 f519_in(x212) -> f529_in(x212) :|: TRUE 14.57/4.59 f529_out(x213) -> f530_in(x213) :|: TRUE 14.57/4.59 f530_out(x214) -> f519_out(x214) :|: TRUE 14.57/4.59 f530_in(x215) -> f531_in(x215) :|: TRUE 14.57/4.59 f531_out(x216) -> f530_out(x216) :|: TRUE 14.57/4.59 f533_out(x217) -> f531_out(x217) :|: TRUE 14.57/4.59 f532_out(x218) -> f531_out(x218) :|: TRUE 14.57/4.59 f531_in(x219) -> f532_in(x219) :|: TRUE 14.57/4.59 f531_in(x220) -> f533_in(x220) :|: TRUE 14.57/4.59 f533_in(x221) -> f537_in(x221) :|: TRUE 14.57/4.59 f538_out(x222) -> f533_out(x222) :|: TRUE 14.57/4.59 f537_out(x223) -> f533_out(x223) :|: TRUE 14.57/4.59 f533_in(x224) -> f538_in(x224) :|: TRUE 14.57/4.59 f537_in(s(T123)) -> f539_in(T123) :|: TRUE 14.57/4.59 f537_in(x225) -> f540_in :|: TRUE 14.57/4.59 f539_out(x226) -> f537_out(s(x226)) :|: TRUE 14.57/4.59 f540_out -> f537_out(x227) :|: TRUE 14.57/4.59 f539_in(x228) -> f315_in(x228) :|: TRUE 14.57/4.59 f315_out(x229) -> f539_out(x229) :|: TRUE 14.57/4.59 Start term: f53_in(T1, T3) 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (154) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 14.57/4.59 Constructed simple dependency graph. 14.57/4.59 14.57/4.59 Simplified to the following IRSwTs: 14.57/4.59 14.57/4.59 intTRSProblem: 14.57/4.59 f301_in(x) -> f303_in(x) :|: TRUE 14.57/4.59 f329_in(x10) -> f333_in(x10) :|: TRUE 14.57/4.59 f316_in(x16) -> f318_in(x16) :|: TRUE 14.57/4.59 f333_in(x27) -> f301_in(x27) :|: TRUE 14.57/4.59 f303_in(x28) -> f304_in(x28) :|: TRUE 14.57/4.59 f318_in(x31) -> f325_in(x31) :|: TRUE 14.57/4.59 f305_in(x50) -> f315_in(x50) :|: TRUE 14.57/4.59 f304_in(x52) -> f305_in(x52) :|: TRUE 14.57/4.59 f325_in(s(x71)) -> f329_in(x71) :|: TRUE 14.57/4.59 f315_in(x73) -> f316_in(x73) :|: TRUE 14.57/4.59 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (155) 14.57/4.59 Obligation: 14.57/4.59 Rules: 14.57/4.59 f301_in(x) -> f303_in(x) :|: TRUE 14.57/4.59 f329_in(x10) -> f333_in(x10) :|: TRUE 14.57/4.59 f316_in(x16) -> f318_in(x16) :|: TRUE 14.57/4.59 f333_in(x27) -> f301_in(x27) :|: TRUE 14.57/4.59 f303_in(x28) -> f304_in(x28) :|: TRUE 14.57/4.59 f318_in(x31) -> f325_in(x31) :|: TRUE 14.57/4.59 f305_in(x50) -> f315_in(x50) :|: TRUE 14.57/4.59 f304_in(x52) -> f305_in(x52) :|: TRUE 14.57/4.59 f325_in(s(x71)) -> f329_in(x71) :|: TRUE 14.57/4.59 f315_in(x73) -> f316_in(x73) :|: TRUE 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (156) IntTRSCompressionProof (EQUIVALENT) 14.57/4.59 Compressed rules. 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (157) 14.57/4.59 Obligation: 14.57/4.59 Rules: 14.57/4.59 f316_in(s(x71:0)) -> f316_in(x71:0) :|: TRUE 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (158) IRSFormatTransformerProof (EQUIVALENT) 14.57/4.59 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (159) 14.57/4.59 Obligation: 14.57/4.59 Rules: 14.57/4.59 f316_in(s(x71:0)) -> f316_in(x71:0) :|: TRUE 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (160) IRSwTTerminationDigraphProof (EQUIVALENT) 14.57/4.59 Constructed termination digraph! 14.57/4.59 Nodes: 14.57/4.59 (1) f316_in(s(x71:0)) -> f316_in(x71:0) :|: TRUE 14.57/4.59 14.57/4.59 Arcs: 14.57/4.59 (1) -> (1) 14.57/4.59 14.57/4.59 This digraph is fully evaluated! 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (161) 14.57/4.59 Obligation: 14.57/4.59 14.57/4.59 Termination digraph: 14.57/4.59 Nodes: 14.57/4.59 (1) f316_in(s(x71:0)) -> f316_in(x71:0) :|: TRUE 14.57/4.59 14.57/4.59 Arcs: 14.57/4.59 (1) -> (1) 14.57/4.59 14.57/4.59 This digraph is fully evaluated! 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (162) TempFilterProof (SOUND) 14.57/4.59 Used the following sort dictionary for filtering: 14.57/4.59 f316_in(VARIABLE) 14.57/4.59 s(VARIABLE) 14.57/4.59 Removed predefined arithmetic. 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (163) 14.57/4.59 Obligation: 14.57/4.59 Rules: 14.57/4.59 f316_in(s(x71:0)) -> f316_in(x71:0) 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (164) IRSwTToQDPProof (SOUND) 14.57/4.59 Removed the integers and created a QDP-Problem. 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (165) 14.57/4.59 Obligation: 14.57/4.59 Q DP problem: 14.57/4.59 The TRS P consists of the following rules: 14.57/4.59 14.57/4.59 f316_in(s(x71:0)) -> f316_in(x71:0) 14.57/4.59 14.57/4.59 R is empty. 14.57/4.59 Q is empty. 14.57/4.59 We have to consider all (P,Q,R)-chains. 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (166) QDPSizeChangeProof (EQUIVALENT) 14.57/4.59 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. 14.57/4.59 14.57/4.59 From the DPs we obtained the following set of size-change graphs: 14.57/4.59 *f316_in(s(x71:0)) -> f316_in(x71:0) 14.57/4.59 The graph contains the following edges 1 > 1 14.57/4.59 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (167) 14.57/4.59 YES 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (168) 14.57/4.59 Obligation: 14.57/4.59 Rules: 14.57/4.59 f516_out(T94) -> f512_out(T94) :|: TRUE 14.57/4.59 f515_out(x) -> f512_out(x) :|: TRUE 14.57/4.59 f512_in(x1) -> f515_in(x1) :|: TRUE 14.57/4.59 f512_in(x2) -> f516_in(x2) :|: TRUE 14.57/4.59 f530_in(T108) -> f531_in(T108) :|: TRUE 14.57/4.59 f531_out(x3) -> f530_out(x3) :|: TRUE 14.57/4.59 f541_in(T128) -> f519_in(T128) :|: TRUE 14.57/4.59 f519_out(x4) -> f541_out(x4) :|: TRUE 14.57/4.59 f533_out(x5) -> f531_out(x5) :|: TRUE 14.57/4.59 f532_out(x6) -> f531_out(x6) :|: TRUE 14.57/4.59 f531_in(x7) -> f532_in(x7) :|: TRUE 14.57/4.59 f531_in(x8) -> f533_in(x8) :|: TRUE 14.57/4.59 f511_in(x9) -> f512_in(x9) :|: TRUE 14.57/4.59 f512_out(x10) -> f511_out(x10) :|: TRUE 14.57/4.59 f538_in(x11) -> f542_in :|: TRUE 14.57/4.59 f542_out -> f538_out(x12) :|: TRUE 14.57/4.59 f541_out(x13) -> f538_out(s(x13)) :|: TRUE 14.57/4.59 f538_in(s(x14)) -> f541_in(x14) :|: TRUE 14.57/4.59 f509_out(x15) -> f529_out(x15) :|: TRUE 14.57/4.59 f529_in(x16) -> f509_in(x16) :|: TRUE 14.57/4.59 f519_out(x17) -> f516_out(x17) :|: TRUE 14.57/4.59 f526_out -> f516_out(x18) :|: TRUE 14.57/4.59 f516_in(x19) -> f526_in :|: TRUE 14.57/4.59 f516_in(x20) -> f519_in(x20) :|: TRUE 14.57/4.59 f533_in(x21) -> f537_in(x21) :|: TRUE 14.57/4.59 f538_out(x22) -> f533_out(x22) :|: TRUE 14.57/4.59 f537_out(x23) -> f533_out(x23) :|: TRUE 14.57/4.59 f533_in(x24) -> f538_in(x24) :|: TRUE 14.57/4.59 f519_in(x25) -> f529_in(x25) :|: TRUE 14.57/4.59 f529_out(x26) -> f530_in(x26) :|: TRUE 14.57/4.59 f530_out(x27) -> f519_out(x27) :|: TRUE 14.57/4.59 f509_in(x28) -> f511_in(x28) :|: TRUE 14.57/4.59 f511_out(x29) -> f509_out(x29) :|: TRUE 14.57/4.59 f54_out(T1, T3) -> f53_out(T1, T3) :|: TRUE 14.57/4.59 f53_in(x30, x31) -> f54_in(x30, x31) :|: TRUE 14.57/4.59 f54_in(x32, x33) -> f56_in(x32, x33) :|: TRUE 14.57/4.59 f55_out(x34, x35) -> f54_out(x34, x35) :|: TRUE 14.57/4.59 f54_in(x36, x37) -> f55_in(x36, x37) :|: TRUE 14.57/4.59 f56_out(x38, x39) -> f54_out(x38, x39) :|: TRUE 14.57/4.59 f56_in(x40, x41) -> f184_in(x40, x41) :|: TRUE 14.57/4.59 f182_out(x42, x43) -> f56_out(x42, x43) :|: TRUE 14.57/4.59 f56_in(x44, x45) -> f182_in(x44, x45) :|: TRUE 14.57/4.59 f184_out(x46, x47) -> f56_out(x46, x47) :|: TRUE 14.57/4.59 f504_out(x48, x49) -> f184_out(s(x48), x49) :|: TRUE 14.57/4.59 f184_in(s(x50), x51) -> f504_in(x50, x51) :|: TRUE 14.57/4.59 f184_in(x52, x53) -> f505_in :|: TRUE 14.57/4.59 f505_out -> f184_out(x54, x55) :|: TRUE 14.57/4.59 f510_out(x56, x57) -> f504_out(x56, x57) :|: TRUE 14.57/4.59 f504_in(x58, x59) -> f509_in(x58) :|: TRUE 14.57/4.59 f509_out(x60) -> f510_in(x60, x61) :|: TRUE 14.57/4.59 Start term: f53_in(T1, T3) 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (169) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 14.57/4.59 Constructed simple dependency graph. 14.57/4.59 14.57/4.59 Simplified to the following IRSwTs: 14.57/4.59 14.57/4.59 intTRSProblem: 14.57/4.59 f512_in(x2) -> f516_in(x2) :|: TRUE 14.57/4.59 f511_in(x9) -> f512_in(x9) :|: TRUE 14.57/4.59 f529_in(x16) -> f509_in(x16) :|: TRUE 14.57/4.59 f516_in(x20) -> f519_in(x20) :|: TRUE 14.57/4.59 f519_in(x25) -> f529_in(x25) :|: TRUE 14.57/4.59 f509_in(x28) -> f511_in(x28) :|: TRUE 14.57/4.59 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (170) 14.57/4.59 Obligation: 14.57/4.59 Rules: 14.57/4.59 f512_in(x2) -> f516_in(x2) :|: TRUE 14.57/4.59 f511_in(x9) -> f512_in(x9) :|: TRUE 14.57/4.59 f529_in(x16) -> f509_in(x16) :|: TRUE 14.57/4.59 f516_in(x20) -> f519_in(x20) :|: TRUE 14.57/4.59 f519_in(x25) -> f529_in(x25) :|: TRUE 14.57/4.59 f509_in(x28) -> f511_in(x28) :|: TRUE 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (171) IntTRSCompressionProof (EQUIVALENT) 14.57/4.59 Compressed rules. 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (172) 14.57/4.59 Obligation: 14.57/4.59 Rules: 14.57/4.59 f511_in(x9:0) -> f511_in(x9:0) :|: TRUE 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (173) IRSFormatTransformerProof (EQUIVALENT) 14.57/4.59 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (174) 14.57/4.59 Obligation: 14.57/4.59 Rules: 14.57/4.59 f511_in(x9:0) -> f511_in(x9:0) :|: TRUE 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (175) IRSwTTerminationDigraphProof (EQUIVALENT) 14.57/4.59 Constructed termination digraph! 14.57/4.59 Nodes: 14.57/4.59 (1) f511_in(x9:0) -> f511_in(x9:0) :|: TRUE 14.57/4.59 14.57/4.59 Arcs: 14.57/4.59 (1) -> (1) 14.57/4.59 14.57/4.59 This digraph is fully evaluated! 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (176) 14.57/4.59 Obligation: 14.57/4.59 14.57/4.59 Termination digraph: 14.57/4.59 Nodes: 14.57/4.59 (1) f511_in(x9:0) -> f511_in(x9:0) :|: TRUE 14.57/4.59 14.57/4.59 Arcs: 14.57/4.59 (1) -> (1) 14.57/4.59 14.57/4.59 This digraph is fully evaluated! 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (177) FilterProof (EQUIVALENT) 14.57/4.59 Used the following sort dictionary for filtering: 14.57/4.59 f511_in(VARIABLE) 14.57/4.59 Replaced non-predefined constructor symbols by 0. 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (178) 14.57/4.59 Obligation: 14.57/4.59 Rules: 14.57/4.59 f511_in(x9:0) -> f511_in(x9:0) :|: TRUE 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (179) IntTRSNonPeriodicNontermProof (COMPLETE) 14.57/4.59 Normalized system to the following form: 14.57/4.59 f(pc, x9:0) -> f(1, x9:0) :|: pc = 1 && TRUE 14.57/4.59 Proved unsatisfiability of the following formula, indicating that the system is never left after entering: 14.57/4.59 (((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T)) 14.57/4.59 Proved satisfiability of the following formula, indicating that the system is entered at least once: 14.57/4.59 ((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and T)) 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (180) 14.57/4.59 NO 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (181) 14.57/4.59 Obligation: 14.57/4.59 Rules: 14.57/4.59 f303_out(T24) -> f301_out(T24) :|: TRUE 14.57/4.59 f301_in(x) -> f303_in(x) :|: TRUE 14.57/4.59 f320_in -> f320_out :|: TRUE 14.57/4.59 f195_out(T74, T75) -> f444_out(T74, T75) :|: TRUE 14.57/4.59 f444_in(x1, x2) -> f195_in(x1, x2) :|: TRUE 14.57/4.59 f364_in(x3, x4, x5) -> f373_in(x3, x4, x5) :|: TRUE 14.57/4.59 f364_in(x6, x7, x8) -> f375_in(x6, x7, x8) :|: TRUE 14.57/4.59 f375_out(x9, x10, x11) -> f364_out(x9, x10, x11) :|: TRUE 14.57/4.59 f373_out(x12, x13, x14) -> f364_out(x12, x13, x14) :|: TRUE 14.57/4.59 f444_out(x15, x16) -> f373_out(s(x15), 0, x16) :|: TRUE 14.57/4.59 f373_in(s(x17), 0, x18) -> f444_in(x17, x18) :|: TRUE 14.57/4.59 f373_in(x19, x20, x21) -> f447_in :|: TRUE 14.57/4.59 f447_out -> f373_out(x22, x23, x24) :|: TRUE 14.57/4.59 f303_in(x25) -> f304_in(x25) :|: TRUE 14.57/4.59 f304_out(x26) -> f303_out(x26) :|: TRUE 14.57/4.59 f375_in(s(T82), s(T83), T84) -> f453_in(T82, T83, T84) :|: TRUE 14.57/4.59 f454_out -> f375_out(x27, x28, x29) :|: TRUE 14.57/4.59 f375_in(x30, x31, x32) -> f454_in :|: TRUE 14.57/4.59 f453_out(x33, x34, x35) -> f375_out(s(x33), s(x34), x35) :|: TRUE 14.57/4.59 f337_in(T35, T36) -> f339_in(T35, T36) :|: TRUE 14.57/4.59 f338_out(x36, x37) -> f337_out(x36, x37) :|: TRUE 14.57/4.59 f339_out(x38, x39) -> f337_out(x38, x39) :|: TRUE 14.57/4.59 f337_in(x40, x41) -> f338_in(x40, x41) :|: TRUE 14.57/4.59 f315_out(T48) -> f349_out(T48) :|: TRUE 14.57/4.59 f349_in(x42) -> f315_in(x42) :|: TRUE 14.57/4.59 f337_out(x43, x44) -> f334_out(x43, x44) :|: TRUE 14.57/4.59 f334_in(x45, x46) -> f337_in(x45, x46) :|: TRUE 14.57/4.59 f340_in -> f340_out :|: TRUE 14.57/4.59 f355_in(T53, T54) -> f334_in(s(T53), T54) :|: TRUE 14.57/4.59 f334_out(s(x47), x48) -> f355_out(x47, x48) :|: TRUE 14.57/4.59 f304_in(x49) -> f305_in(x49) :|: TRUE 14.57/4.59 f306_out(x50) -> f304_out(x50) :|: TRUE 14.57/4.59 f304_in(x51) -> f306_in(x51) :|: TRUE 14.57/4.59 f305_out(x52) -> f304_out(x52) :|: TRUE 14.57/4.59 f339_in(x53, x54) -> f346_in(x53, x54) :|: TRUE 14.57/4.59 f346_out(x55, x56) -> f339_out(x55, x56) :|: TRUE 14.57/4.59 f345_out(x57, x58) -> f339_out(x57, x58) :|: TRUE 14.57/4.59 f339_in(x59, x60) -> f345_in(x59, x60) :|: TRUE 14.57/4.59 f340_out -> f338_out(0, T43) :|: TRUE 14.57/4.59 f338_in(0, x61) -> f340_in :|: TRUE 14.57/4.59 f342_out -> f338_out(x62, x63) :|: TRUE 14.57/4.59 f338_in(x64, x65) -> f342_in :|: TRUE 14.57/4.59 f242_in(T17, T18) -> f257_in :|: TRUE 14.57/4.59 f257_out -> f242_out(x66, x67) :|: TRUE 14.57/4.59 f242_in(s(x68), x69) -> f254_in(x68, x69) :|: TRUE 14.57/4.59 f254_out(x70, x71) -> f242_out(s(x70), x71) :|: TRUE 14.57/4.59 f254_in(x72, x73) -> f301_in(x72) :|: TRUE 14.57/4.59 f301_out(x74) -> f302_in(x74, x75, x76) :|: TRUE 14.57/4.59 f302_out(x77, x78, x79) -> f254_out(x77, x79) :|: TRUE 14.57/4.59 f498_in(x80, x81, x82) -> f302_in(x80, x81, x82) :|: TRUE 14.57/4.59 f302_out(x83, x84, x85) -> f498_out(x83, x84, x85) :|: TRUE 14.57/4.59 f222_out(x86, x87) -> f195_out(x86, x87) :|: TRUE 14.57/4.59 f195_in(x88, x89) -> f222_in(x88, x89) :|: TRUE 14.57/4.59 f361_out -> f306_out(x90) :|: TRUE 14.57/4.59 f306_in(x91) -> f361_in :|: TRUE 14.57/4.59 f325_in(T31) -> f330_in :|: TRUE 14.57/4.59 f330_out -> f325_out(x92) :|: TRUE 14.57/4.59 f329_out(x93) -> f325_out(s(x93)) :|: TRUE 14.57/4.59 f325_in(s(x94)) -> f329_in(x94) :|: TRUE 14.57/4.59 f316_out(x95) -> f315_out(x95) :|: TRUE 14.57/4.59 f315_in(x96) -> f316_in(x96) :|: TRUE 14.57/4.59 f362_out(x97, x98, x99) -> f302_out(x97, x98, x99) :|: TRUE 14.57/4.59 f302_in(x100, x101, x102) -> f362_in(x100, x101, x102) :|: TRUE 14.57/4.59 f345_in(s(x103), 0) -> f349_in(x103) :|: TRUE 14.57/4.59 f350_out -> f345_out(x104, x105) :|: TRUE 14.57/4.59 f345_in(x106, x107) -> f350_in :|: TRUE 14.57/4.59 f349_out(x108) -> f345_out(s(x108), 0) :|: TRUE 14.57/4.59 f353_out -> f346_out(x109, x110) :|: TRUE 14.57/4.59 f352_out(x111, x112) -> f346_out(s(x111), s(x112)) :|: TRUE 14.57/4.59 f346_in(x113, x114) -> f353_in :|: TRUE 14.57/4.59 f346_in(s(x115), s(x116)) -> f352_in(x115, x116) :|: TRUE 14.57/4.59 f329_in(x117) -> f333_in(x117) :|: TRUE 14.57/4.59 f334_out(x118, x119) -> f329_out(x118) :|: TRUE 14.57/4.59 f333_out(x120) -> f334_in(x120, x121) :|: TRUE 14.57/4.59 f317_out(x122) -> f316_out(x122) :|: TRUE 14.57/4.59 f318_out(x123) -> f316_out(x123) :|: TRUE 14.57/4.59 f316_in(x124) -> f318_in(x124) :|: TRUE 14.57/4.59 f316_in(x125) -> f317_in(x125) :|: TRUE 14.57/4.59 f334_out(s(x126), x127) -> f497_out(x126, x127) :|: TRUE 14.57/4.59 f497_in(x128, x129) -> f334_in(s(x128), x129) :|: TRUE 14.57/4.59 f228_in(x130, x131) -> f242_in(x130, x131) :|: TRUE 14.57/4.59 f242_out(x132, x133) -> f228_out(x132, x133) :|: TRUE 14.57/4.59 f355_out(x134, x135) -> f356_in(x134, x136) :|: TRUE 14.57/4.59 f356_out(x137, x138) -> f352_out(x137, x139) :|: TRUE 14.57/4.59 f352_in(x140, x141) -> f355_in(x140, x141) :|: TRUE 14.57/4.59 f317_in(x142) -> f323_in :|: TRUE 14.57/4.59 f323_out -> f317_out(x143) :|: TRUE 14.57/4.59 f320_out -> f317_out(0) :|: TRUE 14.57/4.59 f317_in(0) -> f320_in :|: TRUE 14.57/4.59 f301_out(x144) -> f333_out(x144) :|: TRUE 14.57/4.59 f333_in(x145) -> f301_in(x145) :|: TRUE 14.57/4.59 f325_out(x146) -> f318_out(x146) :|: TRUE 14.57/4.59 f318_in(x147) -> f325_in(x147) :|: TRUE 14.57/4.59 f305_in(x148) -> f315_in(x148) :|: TRUE 14.57/4.59 f315_out(x149) -> f305_out(x149) :|: TRUE 14.57/4.59 f226_out(x150, x151) -> f222_out(x150, x151) :|: TRUE 14.57/4.59 f228_out(x152, x153) -> f222_out(x152, x153) :|: TRUE 14.57/4.59 f222_in(x154, x155) -> f226_in(x154, x155) :|: TRUE 14.57/4.59 f222_in(x156, x157) -> f228_in(x156, x157) :|: TRUE 14.57/4.59 f356_in(x158, x159) -> f334_in(x158, x159) :|: TRUE 14.57/4.59 f334_out(x160, x161) -> f356_out(x160, x161) :|: TRUE 14.57/4.59 f497_out(x162, x163) -> f498_in(x162, x164, x165) :|: TRUE 14.57/4.59 f453_in(x166, x167, x168) -> f497_in(x166, x167) :|: TRUE 14.57/4.59 f498_out(x169, x170, x171) -> f453_out(x169, x172, x171) :|: TRUE 14.57/4.59 f363_out(x173, x174, x175) -> f362_out(x173, x174, x175) :|: TRUE 14.57/4.59 f362_in(x176, x177, x178) -> f364_in(x176, x177, x178) :|: TRUE 14.57/4.59 f362_in(x179, x180, x181) -> f363_in(x179, x180, x181) :|: TRUE 14.57/4.59 f364_out(x182, x183, x184) -> f362_out(x182, x183, x184) :|: TRUE 14.57/4.59 f54_out(T1, T3) -> f53_out(T1, T3) :|: TRUE 14.57/4.59 f53_in(x185, x186) -> f54_in(x185, x186) :|: TRUE 14.57/4.59 f54_in(x187, x188) -> f56_in(x187, x188) :|: TRUE 14.57/4.59 f55_out(x189, x190) -> f54_out(x189, x190) :|: TRUE 14.57/4.59 f54_in(x191, x192) -> f55_in(x191, x192) :|: TRUE 14.57/4.59 f56_out(x193, x194) -> f54_out(x193, x194) :|: TRUE 14.57/4.59 f56_in(x195, x196) -> f184_in(x195, x196) :|: TRUE 14.57/4.59 f182_out(x197, x198) -> f56_out(x197, x198) :|: TRUE 14.57/4.59 f56_in(x199, x200) -> f182_in(x199, x200) :|: TRUE 14.57/4.59 f184_out(x201, x202) -> f56_out(x201, x202) :|: TRUE 14.57/4.59 f195_out(x203, x204) -> f182_out(s(x203), x204) :|: TRUE 14.57/4.59 f182_in(x205, x206) -> f199_in :|: TRUE 14.57/4.59 f182_in(s(x207), x208) -> f195_in(x207, x208) :|: TRUE 14.57/4.59 f199_out -> f182_out(x209, x210) :|: TRUE 14.57/4.59 Start term: f53_in(T1, T3) 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (182) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 14.57/4.59 Constructed simple dependency graph. 14.57/4.59 14.57/4.59 Simplified to the following IRSwTs: 14.57/4.59 14.57/4.59 intTRSProblem: 14.57/4.59 f303_out(T24) -> f301_out(T24) :|: TRUE 14.57/4.59 f301_in(x) -> f303_in(x) :|: TRUE 14.57/4.59 f320_in -> f320_out :|: TRUE 14.57/4.59 f444_in(x1, x2) -> f195_in(x1, x2) :|: TRUE 14.57/4.59 f364_in(x3, x4, x5) -> f373_in(x3, x4, x5) :|: TRUE 14.57/4.59 f364_in(x6, x7, x8) -> f375_in(x6, x7, x8) :|: TRUE 14.57/4.59 f373_in(s(x17), 0, x18) -> f444_in(x17, x18) :|: TRUE 14.57/4.59 f303_in(x25) -> f304_in(x25) :|: TRUE 14.57/4.59 f304_out(x26) -> f303_out(x26) :|: TRUE 14.57/4.59 f375_in(s(T82), s(T83), T84) -> f453_in(T82, T83, T84) :|: TRUE 14.57/4.59 f337_in(T35, T36) -> f339_in(T35, T36) :|: TRUE 14.57/4.59 f338_out(x36, x37) -> f337_out(x36, x37) :|: TRUE 14.57/4.59 f339_out(x38, x39) -> f337_out(x38, x39) :|: TRUE 14.57/4.59 f337_in(x40, x41) -> f338_in(x40, x41) :|: TRUE 14.57/4.59 f315_out(T48) -> f349_out(T48) :|: TRUE 14.57/4.59 f349_in(x42) -> f315_in(x42) :|: TRUE 14.57/4.59 f337_out(x43, x44) -> f334_out(x43, x44) :|: TRUE 14.57/4.59 f334_in(x45, x46) -> f337_in(x45, x46) :|: TRUE 14.57/4.59 f340_in -> f340_out :|: TRUE 14.57/4.59 f355_in(T53, T54) -> f334_in(s(T53), T54) :|: TRUE 14.57/4.59 f334_out(s(x47), x48) -> f355_out(x47, x48) :|: TRUE 14.57/4.59 f304_in(x49) -> f305_in(x49) :|: TRUE 14.57/4.59 f305_out(x52) -> f304_out(x52) :|: TRUE 14.57/4.59 f339_in(x53, x54) -> f346_in(x53, x54) :|: TRUE 14.57/4.59 f346_out(x55, x56) -> f339_out(x55, x56) :|: TRUE 14.57/4.59 f345_out(x57, x58) -> f339_out(x57, x58) :|: TRUE 14.57/4.59 f339_in(x59, x60) -> f345_in(x59, x60) :|: TRUE 14.57/4.59 f340_out -> f338_out(0, T43) :|: TRUE 14.57/4.59 f338_in(0, x61) -> f340_in :|: TRUE 14.57/4.59 f242_in(s(x68), x69) -> f254_in(x68, x69) :|: TRUE 14.57/4.59 f254_in(x72, x73) -> f301_in(x72) :|: TRUE 14.57/4.59 f301_out(x74) -> f302_in(x74, x75, x76) :|: TRUE 14.57/4.59 f498_in(x80, x81, x82) -> f302_in(x80, x81, x82) :|: TRUE 14.57/4.59 f195_in(x88, x89) -> f222_in(x88, x89) :|: TRUE 14.57/4.59 f329_out(x93) -> f325_out(s(x93)) :|: TRUE 14.57/4.59 f325_in(s(x94)) -> f329_in(x94) :|: TRUE 14.57/4.59 f316_out(x95) -> f315_out(x95) :|: TRUE 14.57/4.59 f315_in(x96) -> f316_in(x96) :|: TRUE 14.57/4.59 f302_in(x100, x101, x102) -> f362_in(x100, x101, x102) :|: TRUE 14.57/4.59 f345_in(s(x103), 0) -> f349_in(x103) :|: TRUE 14.57/4.59 f349_out(x108) -> f345_out(s(x108), 0) :|: TRUE 14.57/4.59 f352_out(x111, x112) -> f346_out(s(x111), s(x112)) :|: TRUE 14.57/4.59 f346_in(s(x115), s(x116)) -> f352_in(x115, x116) :|: TRUE 14.57/4.59 f329_in(x117) -> f333_in(x117) :|: TRUE 14.57/4.59 f334_out(x118, x119) -> f329_out(x118) :|: TRUE 14.57/4.59 f333_out(x120) -> f334_in(x120, x121) :|: TRUE 14.57/4.59 f317_out(x122) -> f316_out(x122) :|: TRUE 14.57/4.59 f318_out(x123) -> f316_out(x123) :|: TRUE 14.57/4.59 f316_in(x124) -> f318_in(x124) :|: TRUE 14.57/4.59 f316_in(x125) -> f317_in(x125) :|: TRUE 14.57/4.59 f334_out(s(x126), x127) -> f497_out(x126, x127) :|: TRUE 14.57/4.59 f497_in(x128, x129) -> f334_in(s(x128), x129) :|: TRUE 14.57/4.59 f228_in(x130, x131) -> f242_in(x130, x131) :|: TRUE 14.57/4.59 f355_out(x134, x135) -> f356_in(x134, x136) :|: TRUE 14.57/4.59 f356_out(x137, x138) -> f352_out(x137, x139) :|: TRUE 14.57/4.59 f352_in(x140, x141) -> f355_in(x140, x141) :|: TRUE 14.57/4.59 f320_out -> f317_out(0) :|: TRUE 14.57/4.59 f317_in(0) -> f320_in :|: TRUE 14.57/4.59 f301_out(x144) -> f333_out(x144) :|: TRUE 14.57/4.59 f333_in(x145) -> f301_in(x145) :|: TRUE 14.57/4.59 f325_out(x146) -> f318_out(x146) :|: TRUE 14.57/4.59 f318_in(x147) -> f325_in(x147) :|: TRUE 14.57/4.59 f305_in(x148) -> f315_in(x148) :|: TRUE 14.57/4.59 f315_out(x149) -> f305_out(x149) :|: TRUE 14.57/4.59 f222_in(x156, x157) -> f228_in(x156, x157) :|: TRUE 14.57/4.59 f356_in(x158, x159) -> f334_in(x158, x159) :|: TRUE 14.57/4.59 f334_out(x160, x161) -> f356_out(x160, x161) :|: TRUE 14.57/4.59 f497_out(x162, x163) -> f498_in(x162, x164, x165) :|: TRUE 14.57/4.59 f453_in(x166, x167, x168) -> f497_in(x166, x167) :|: TRUE 14.57/4.59 f362_in(x176, x177, x178) -> f364_in(x176, x177, x178) :|: TRUE 14.57/4.59 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (183) 14.57/4.59 Obligation: 14.57/4.59 Rules: 14.57/4.59 f303_out(T24) -> f301_out(T24) :|: TRUE 14.57/4.59 f301_in(x) -> f303_in(x) :|: TRUE 14.57/4.59 f320_in -> f320_out :|: TRUE 14.57/4.59 f444_in(x1, x2) -> f195_in(x1, x2) :|: TRUE 14.57/4.59 f364_in(x3, x4, x5) -> f373_in(x3, x4, x5) :|: TRUE 14.57/4.59 f364_in(x6, x7, x8) -> f375_in(x6, x7, x8) :|: TRUE 14.57/4.59 f373_in(s(x17), 0, x18) -> f444_in(x17, x18) :|: TRUE 14.57/4.59 f303_in(x25) -> f304_in(x25) :|: TRUE 14.57/4.59 f304_out(x26) -> f303_out(x26) :|: TRUE 14.57/4.59 f375_in(s(T82), s(T83), T84) -> f453_in(T82, T83, T84) :|: TRUE 14.57/4.59 f337_in(T35, T36) -> f339_in(T35, T36) :|: TRUE 14.57/4.59 f338_out(x36, x37) -> f337_out(x36, x37) :|: TRUE 14.57/4.59 f339_out(x38, x39) -> f337_out(x38, x39) :|: TRUE 14.57/4.59 f337_in(x40, x41) -> f338_in(x40, x41) :|: TRUE 14.57/4.59 f315_out(T48) -> f349_out(T48) :|: TRUE 14.57/4.59 f349_in(x42) -> f315_in(x42) :|: TRUE 14.57/4.59 f337_out(x43, x44) -> f334_out(x43, x44) :|: TRUE 14.57/4.59 f334_in(x45, x46) -> f337_in(x45, x46) :|: TRUE 14.57/4.59 f340_in -> f340_out :|: TRUE 14.57/4.59 f355_in(T53, T54) -> f334_in(s(T53), T54) :|: TRUE 14.57/4.59 f334_out(s(x47), x48) -> f355_out(x47, x48) :|: TRUE 14.57/4.59 f304_in(x49) -> f305_in(x49) :|: TRUE 14.57/4.59 f305_out(x52) -> f304_out(x52) :|: TRUE 14.57/4.59 f339_in(x53, x54) -> f346_in(x53, x54) :|: TRUE 14.57/4.59 f346_out(x55, x56) -> f339_out(x55, x56) :|: TRUE 14.57/4.59 f345_out(x57, x58) -> f339_out(x57, x58) :|: TRUE 14.57/4.59 f339_in(x59, x60) -> f345_in(x59, x60) :|: TRUE 14.57/4.59 f340_out -> f338_out(0, T43) :|: TRUE 14.57/4.59 f338_in(0, x61) -> f340_in :|: TRUE 14.57/4.59 f242_in(s(x68), x69) -> f254_in(x68, x69) :|: TRUE 14.57/4.59 f254_in(x72, x73) -> f301_in(x72) :|: TRUE 14.57/4.59 f301_out(x74) -> f302_in(x74, x75, x76) :|: TRUE 14.57/4.59 f498_in(x80, x81, x82) -> f302_in(x80, x81, x82) :|: TRUE 14.57/4.59 f195_in(x88, x89) -> f222_in(x88, x89) :|: TRUE 14.57/4.59 f329_out(x93) -> f325_out(s(x93)) :|: TRUE 14.57/4.59 f325_in(s(x94)) -> f329_in(x94) :|: TRUE 14.57/4.59 f316_out(x95) -> f315_out(x95) :|: TRUE 14.57/4.59 f315_in(x96) -> f316_in(x96) :|: TRUE 14.57/4.59 f302_in(x100, x101, x102) -> f362_in(x100, x101, x102) :|: TRUE 14.57/4.59 f345_in(s(x103), 0) -> f349_in(x103) :|: TRUE 14.57/4.59 f349_out(x108) -> f345_out(s(x108), 0) :|: TRUE 14.57/4.59 f352_out(x111, x112) -> f346_out(s(x111), s(x112)) :|: TRUE 14.57/4.59 f346_in(s(x115), s(x116)) -> f352_in(x115, x116) :|: TRUE 14.57/4.59 f329_in(x117) -> f333_in(x117) :|: TRUE 14.57/4.59 f334_out(x118, x119) -> f329_out(x118) :|: TRUE 14.57/4.59 f333_out(x120) -> f334_in(x120, x121) :|: TRUE 14.57/4.59 f317_out(x122) -> f316_out(x122) :|: TRUE 14.57/4.59 f318_out(x123) -> f316_out(x123) :|: TRUE 14.57/4.59 f316_in(x124) -> f318_in(x124) :|: TRUE 14.57/4.59 f316_in(x125) -> f317_in(x125) :|: TRUE 14.57/4.59 f334_out(s(x126), x127) -> f497_out(x126, x127) :|: TRUE 14.57/4.59 f497_in(x128, x129) -> f334_in(s(x128), x129) :|: TRUE 14.57/4.59 f228_in(x130, x131) -> f242_in(x130, x131) :|: TRUE 14.57/4.59 f355_out(x134, x135) -> f356_in(x134, x136) :|: TRUE 14.57/4.59 f356_out(x137, x138) -> f352_out(x137, x139) :|: TRUE 14.57/4.59 f352_in(x140, x141) -> f355_in(x140, x141) :|: TRUE 14.57/4.59 f320_out -> f317_out(0) :|: TRUE 14.57/4.59 f317_in(0) -> f320_in :|: TRUE 14.57/4.59 f301_out(x144) -> f333_out(x144) :|: TRUE 14.57/4.59 f333_in(x145) -> f301_in(x145) :|: TRUE 14.57/4.59 f325_out(x146) -> f318_out(x146) :|: TRUE 14.57/4.59 f318_in(x147) -> f325_in(x147) :|: TRUE 14.57/4.59 f305_in(x148) -> f315_in(x148) :|: TRUE 14.57/4.59 f315_out(x149) -> f305_out(x149) :|: TRUE 14.57/4.59 f222_in(x156, x157) -> f228_in(x156, x157) :|: TRUE 14.57/4.59 f356_in(x158, x159) -> f334_in(x158, x159) :|: TRUE 14.57/4.59 f334_out(x160, x161) -> f356_out(x160, x161) :|: TRUE 14.57/4.59 f497_out(x162, x163) -> f498_in(x162, x164, x165) :|: TRUE 14.57/4.59 f453_in(x166, x167, x168) -> f497_in(x166, x167) :|: TRUE 14.57/4.59 f362_in(x176, x177, x178) -> f364_in(x176, x177, x178) :|: TRUE 14.57/4.59 14.57/4.59 ---------------------------------------- 14.57/4.59 14.57/4.59 (184) 14.57/4.59 Obligation: 14.57/4.59 Rules: 14.57/4.59 f303_out(T24) -> f301_out(T24) :|: TRUE 14.57/4.59 f301_in(x) -> f303_in(x) :|: TRUE 14.57/4.59 f320_in -> f320_out :|: TRUE 14.57/4.59 f533_out(T108) -> f531_out(T108) :|: TRUE 14.57/4.59 f532_out(x1) -> f531_out(x1) :|: TRUE 14.57/4.59 f531_in(x2) -> f532_in(x2) :|: TRUE 14.57/4.59 f531_in(x3) -> f533_in(x3) :|: TRUE 14.57/4.59 f510_in(T94, T96) -> f53_in(T94, T96) :|: TRUE 14.57/4.59 f53_out(x4, x5) -> f510_out(x4, x5) :|: TRUE 14.57/4.59 f303_in(x6) -> f304_in(x6) :|: TRUE 14.57/4.59 f304_out(x7) -> f303_out(x7) :|: TRUE 14.57/4.59 f337_in(T35, T36) -> f339_in(T35, T36) :|: TRUE 14.57/4.59 f338_out(x8, x9) -> f337_out(x8, x9) :|: TRUE 14.57/4.59 f339_out(x10, x11) -> f337_out(x10, x11) :|: TRUE 14.57/4.59 f337_in(x12, x13) -> f338_in(x12, x13) :|: TRUE 14.57/4.59 f56_in(T1, T3) -> f184_in(T1, T3) :|: TRUE 14.57/4.59 f182_out(x14, x15) -> f56_out(x14, x15) :|: TRUE 14.57/4.59 f56_in(x16, x17) -> f182_in(x16, x17) :|: TRUE 14.57/4.59 f184_out(x18, x19) -> f56_out(x18, x19) :|: TRUE 14.57/4.59 f54_in(x20, x21) -> f56_in(x20, x21) :|: TRUE 14.57/4.59 f55_out(x22, x23) -> f54_out(x22, x23) :|: TRUE 14.57/4.59 f54_in(x24, x25) -> f55_in(x24, x25) :|: TRUE 14.57/4.59 f56_out(x26, x27) -> f54_out(x26, x27) :|: TRUE 14.57/4.59 f539_in(T123) -> f315_in(T123) :|: TRUE 14.57/4.59 f315_out(x28) -> f539_out(x28) :|: TRUE 14.57/4.59 f339_in(x29, x30) -> f346_in(x29, x30) :|: TRUE 14.57/4.59 f346_out(x31, x32) -> f339_out(x31, x32) :|: TRUE 14.57/4.59 f345_out(x33, x34) -> f339_out(x33, x34) :|: TRUE 14.57/4.59 f339_in(x35, x36) -> f345_in(x35, x36) :|: TRUE 14.57/4.59 f340_out -> f338_out(0, T43) :|: TRUE 14.57/4.59 f338_in(0, x37) -> f340_in :|: TRUE 14.57/4.59 f342_out -> f338_out(x38, x39) :|: TRUE 14.57/4.59 f338_in(x40, x41) -> f342_in :|: TRUE 14.57/4.59 f534_in -> f534_out :|: TRUE 14.57/4.59 f361_out -> f306_out(x42) :|: TRUE 14.57/4.59 f306_in(x43) -> f361_in :|: TRUE 14.57/4.59 f315_out(T103) -> f517_out(T103) :|: TRUE 14.57/4.59 f517_in(x44) -> f315_in(x44) :|: TRUE 14.57/4.59 f325_in(T31) -> f330_in :|: TRUE 14.57/4.59 f330_out -> f325_out(x45) :|: TRUE 14.57/4.59 f329_out(x46) -> f325_out(s(x46)) :|: TRUE 14.57/4.59 f325_in(s(x47)) -> f329_in(x47) :|: TRUE 14.57/4.59 f519_in(x48) -> f529_in(x48) :|: TRUE 14.57/4.59 f529_out(x49) -> f530_in(x49) :|: TRUE 14.57/4.59 f530_out(x50) -> f519_out(x50) :|: TRUE 14.57/4.59 f345_in(s(T48), 0) -> f349_in(T48) :|: TRUE 14.57/4.59 f350_out -> f345_out(x51, x52) :|: TRUE 14.57/4.59 f345_in(x53, x54) -> f350_in :|: TRUE 14.57/4.59 f349_out(x55) -> f345_out(s(x55), 0) :|: TRUE 14.57/4.59 f353_out -> f346_out(x56, x57) :|: TRUE 14.57/4.59 f352_out(T53, T54) -> f346_out(s(T53), s(T54)) :|: TRUE 14.57/4.59 f346_in(x58, x59) -> f353_in :|: TRUE 14.57/4.59 f346_in(s(x60), s(x61)) -> f352_in(x60, x61) :|: TRUE 14.57/4.59 f329_in(x62) -> f333_in(x62) :|: TRUE 14.57/4.59 f334_out(x63, x64) -> f329_out(x63) :|: TRUE 14.57/4.59 f333_out(x65) -> f334_in(x65, x66) :|: TRUE 14.57/4.59 f509_out(x67) -> f529_out(x67) :|: TRUE 14.57/4.59 f529_in(x68) -> f509_in(x68) :|: TRUE 14.57/4.59 f317_out(x69) -> f316_out(x69) :|: TRUE 14.57/4.59 f318_out(x70) -> f316_out(x70) :|: TRUE 14.57/4.59 f316_in(x71) -> f318_in(x71) :|: TRUE 14.57/4.59 f316_in(x72) -> f317_in(x72) :|: TRUE 14.57/4.59 f510_out(x73, x74) -> f504_out(x73, x74) :|: TRUE 14.57/4.59 f504_in(x75, x76) -> f509_in(x75) :|: TRUE 14.57/4.59 f509_out(x77) -> f510_in(x77, x78) :|: TRUE 14.57/4.59 f355_out(x79, x80) -> f356_in(x79, x81) :|: TRUE 14.57/4.59 f356_out(x82, x83) -> f352_out(x82, x84) :|: TRUE 14.57/4.59 f352_in(x85, x86) -> f355_in(x85, x86) :|: TRUE 14.57/4.59 f54_out(x87, x88) -> f53_out(x87, x88) :|: TRUE 14.57/4.59 f53_in(x89, x90) -> f54_in(x89, x90) :|: TRUE 14.57/4.59 f317_in(x91) -> f323_in :|: TRUE 14.57/4.59 f323_out -> f317_out(x92) :|: TRUE 14.57/4.59 f320_out -> f317_out(0) :|: TRUE 14.57/4.59 f317_in(0) -> f320_in :|: TRUE 14.57/4.59 f325_out(x93) -> f318_out(x93) :|: TRUE 14.57/4.59 f318_in(x94) -> f325_in(x94) :|: TRUE 14.57/4.59 f538_in(x95) -> f542_in :|: TRUE 14.57/4.59 f542_out -> f538_out(x96) :|: TRUE 14.57/4.59 f541_out(T128) -> f538_out(s(T128)) :|: TRUE 14.57/4.59 f538_in(s(x97)) -> f541_in(x97) :|: TRUE 14.57/4.59 f305_in(x98) -> f315_in(x98) :|: TRUE 14.57/4.59 f315_out(x99) -> f305_out(x99) :|: TRUE 14.57/4.59 f504_out(x100, x101) -> f184_out(s(x100), x101) :|: TRUE 14.57/4.59 f184_in(s(x102), x103) -> f504_in(x102, x103) :|: TRUE 14.57/4.59 f184_in(x104, x105) -> f505_in :|: TRUE 14.57/4.59 f505_out -> f184_out(x106, x107) :|: TRUE 14.57/4.59 f515_in(x108) -> f518_in :|: TRUE 14.57/4.59 f517_out(x109) -> f515_out(x109) :|: TRUE 14.57/4.59 f515_in(x110) -> f517_in(x110) :|: TRUE 14.57/4.59 f518_out -> f515_out(x111) :|: TRUE 14.57/4.59 f530_in(x112) -> f531_in(x112) :|: TRUE 14.57/4.59 f531_out(x113) -> f530_out(x113) :|: TRUE 14.57/4.59 f315_out(x114) -> f349_out(x114) :|: TRUE 14.57/4.59 f349_in(x115) -> f315_in(x115) :|: TRUE 14.57/4.59 f337_out(x116, x117) -> f334_out(x116, x117) :|: TRUE 14.57/4.59 f334_in(x118, x119) -> f337_in(x118, x119) :|: TRUE 14.57/4.59 f340_in -> f340_out :|: TRUE 14.57/4.59 f355_in(x120, x121) -> f334_in(s(x120), x121) :|: TRUE 14.57/4.59 f334_out(s(x122), x123) -> f355_out(x122, x123) :|: TRUE 14.57/4.59 f304_in(x124) -> f305_in(x124) :|: TRUE 14.57/4.59 f306_out(x125) -> f304_out(x125) :|: TRUE 14.57/4.59 f304_in(x126) -> f306_in(x126) :|: TRUE 14.57/4.59 f305_out(x127) -> f304_out(x127) :|: TRUE 14.57/4.59 f519_out(x128) -> f516_out(x128) :|: TRUE 14.57/4.59 f526_out -> f516_out(x129) :|: TRUE 14.57/4.59 f516_in(x130) -> f526_in :|: TRUE 14.57/4.59 f516_in(x131) -> f519_in(x131) :|: TRUE 14.57/4.59 f316_out(x132) -> f315_out(x132) :|: TRUE 14.57/4.59 f315_in(x133) -> f316_in(x133) :|: TRUE 14.57/4.59 f509_in(x134) -> f511_in(x134) :|: TRUE 14.57/4.59 f511_out(x135) -> f509_out(x135) :|: TRUE 14.57/4.59 f541_in(x136) -> f519_in(x136) :|: TRUE 14.57/4.59 f519_out(x137) -> f541_out(x137) :|: TRUE 14.57/4.59 f535_out -> f532_out(x138) :|: TRUE 14.57/4.59 f534_out -> f532_out(0) :|: TRUE 14.57/4.59 f532_in(0) -> f534_in :|: TRUE 14.57/4.59 f532_in(x139) -> f535_in :|: TRUE 14.57/4.59 f533_in(x140) -> f537_in(x140) :|: TRUE 14.57/4.59 f538_out(x141) -> f533_out(x141) :|: TRUE 14.57/4.59 f537_out(x142) -> f533_out(x142) :|: TRUE 14.57/4.59 f533_in(x143) -> f538_in(x143) :|: TRUE 14.57/4.59 f301_out(x144) -> f333_out(x144) :|: TRUE 14.57/4.59 f333_in(x145) -> f301_in(x145) :|: TRUE 14.57/4.59 f516_out(x146) -> f512_out(x146) :|: TRUE 14.57/4.59 f515_out(x147) -> f512_out(x147) :|: TRUE 14.57/4.59 f512_in(x148) -> f515_in(x148) :|: TRUE 14.57/4.59 f512_in(x149) -> f516_in(x149) :|: TRUE 14.57/4.59 f511_in(x150) -> f512_in(x150) :|: TRUE 14.57/4.59 f512_out(x151) -> f511_out(x151) :|: TRUE 14.57/4.59 f537_in(s(x152)) -> f539_in(x152) :|: TRUE 14.57/4.59 f537_in(x153) -> f540_in :|: TRUE 14.57/4.59 f539_out(x154) -> f537_out(s(x154)) :|: TRUE 14.57/4.59 f540_out -> f537_out(x155) :|: TRUE 14.57/4.59 f356_in(x156, x157) -> f334_in(x156, x157) :|: TRUE 14.57/4.59 f334_out(x158, x159) -> f356_out(x158, x159) :|: TRUE 14.57/4.59 Start term: f53_in(T1, T3) 14.59/4.63 EOF