/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern times(a,a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 38 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 200 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [EQUIVALENT, 14 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [SOUND, 2 ms] (25) QDP (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) YES (28) PiDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) PiDP (31) PiDPToQDPProof [SOUND, 0 ms] (32) QDP (33) NonTerminationLoopProof [COMPLETE, 0 ms] (34) NO (35) PiDP (36) UsableRulesProof [EQUIVALENT, 0 ms] (37) PiDP (38) PiDPToQDPProof [SOUND, 0 ms] (39) QDP (40) PiDP (41) UsableRulesProof [EQUIVALENT, 0 ms] (42) PiDP (43) PiDP (44) UsableRulesProof [EQUIVALENT, 0 ms] (45) PiDP (46) PiDP (47) UsableRulesProof [EQUIVALENT, 0 ms] (48) PiDP (49) PrologToPiTRSProof [SOUND, 21 ms] (50) PiTRS (51) DependencyPairsProof [EQUIVALENT, 204 ms] (52) PiDP (53) DependencyGraphProof [EQUIVALENT, 11 ms] (54) AND (55) PiDP (56) UsableRulesProof [EQUIVALENT, 0 ms] (57) PiDP (58) PiDPToQDPProof [EQUIVALENT, 0 ms] (59) QDP (60) QDPSizeChangeProof [EQUIVALENT, 0 ms] (61) YES (62) PiDP (63) UsableRulesProof [EQUIVALENT, 0 ms] (64) PiDP (65) PiDPToQDPProof [SOUND, 0 ms] (66) QDP (67) QDPSizeChangeProof [EQUIVALENT, 0 ms] (68) YES (69) PiDP (70) UsableRulesProof [EQUIVALENT, 0 ms] (71) PiDP (72) PiDPToQDPProof [SOUND, 2 ms] (73) QDP (74) QDPSizeChangeProof [EQUIVALENT, 0 ms] (75) YES (76) PiDP (77) UsableRulesProof [EQUIVALENT, 0 ms] (78) PiDP (79) PiDPToQDPProof [SOUND, 0 ms] (80) QDP (81) NonTerminationLoopProof [COMPLETE, 0 ms] (82) NO (83) PiDP (84) UsableRulesProof [EQUIVALENT, 0 ms] (85) PiDP (86) PiDPToQDPProof [SOUND, 0 ms] (87) QDP (88) PiDP (89) UsableRulesProof [EQUIVALENT, 0 ms] (90) PiDP (91) PiDP (92) UsableRulesProof [EQUIVALENT, 0 ms] (93) PiDP (94) PiDP (95) UsableRulesProof [EQUIVALENT, 0 ms] (96) PiDP (97) PrologToDTProblemTransformerProof [SOUND, 337 ms] (98) TRIPLES (99) TriplesToPiDPProof [SOUND, 387 ms] (100) PiDP (101) DependencyGraphProof [EQUIVALENT, 5 ms] (102) AND (103) PiDP (104) UsableRulesProof [EQUIVALENT, 0 ms] (105) PiDP (106) PiDPToQDPProof [EQUIVALENT, 0 ms] (107) QDP (108) QDPSizeChangeProof [EQUIVALENT, 0 ms] (109) YES (110) PiDP (111) UsableRulesProof [EQUIVALENT, 0 ms] (112) PiDP (113) PiDPToQDPProof [EQUIVALENT, 0 ms] (114) QDP (115) QDPSizeChangeProof [EQUIVALENT, 0 ms] (116) YES (117) PiDP (118) UsableRulesProof [EQUIVALENT, 0 ms] (119) PiDP (120) PiDPToQDPProof [EQUIVALENT, 0 ms] (121) QDP (122) QDPSizeChangeProof [EQUIVALENT, 0 ms] (123) YES (124) PiDP (125) UsableRulesProof [EQUIVALENT, 0 ms] (126) PiDP (127) PiDPToQDPProof [SOUND, 0 ms] (128) QDP (129) QDPSizeChangeProof [EQUIVALENT, 0 ms] (130) YES (131) PiDP (132) UsableRulesProof [EQUIVALENT, 0 ms] (133) PiDP (134) PiDPToQDPProof [SOUND, 0 ms] (135) QDP (136) QDPSizeChangeProof [EQUIVALENT, 0 ms] (137) YES (138) PiDP (139) UsableRulesProof [EQUIVALENT, 0 ms] (140) PiDP (141) PiDPToQDPProof [SOUND, 0 ms] (142) QDP (143) PiDP (144) UsableRulesProof [EQUIVALENT, 0 ms] (145) PiDP (146) PiDP (147) UsableRulesProof [EQUIVALENT, 0 ms] (148) PiDP (149) PiDP (150) UsableRulesProof [EQUIVALENT, 0 ms] (151) PiDP (152) PiDP (153) UsableRulesProof [EQUIVALENT, 0 ms] (154) PiDP (155) PrologToTRSTransformerProof [SOUND, 306 ms] (156) QTRS (157) DependencyPairsProof [EQUIVALENT, 0 ms] (158) QDP (159) DependencyGraphProof [EQUIVALENT, 0 ms] (160) AND (161) QDP (162) UsableRulesProof [EQUIVALENT, 0 ms] (163) QDP (164) QDPSizeChangeProof [EQUIVALENT, 0 ms] (165) YES (166) QDP (167) UsableRulesProof [EQUIVALENT, 0 ms] (168) QDP (169) QDPSizeChangeProof [EQUIVALENT, 0 ms] (170) YES (171) QDP (172) UsableRulesProof [EQUIVALENT, 0 ms] (173) QDP (174) QDPSizeChangeProof [EQUIVALENT, 0 ms] (175) YES (176) QDP (177) UsableRulesProof [EQUIVALENT, 0 ms] (178) QDP (179) QDP (180) UsableRulesProof [EQUIVALENT, 0 ms] (181) QDP (182) QDP (183) UsableRulesProof [EQUIVALENT, 0 ms] (184) QDP (185) QDP (186) UsableRulesProof [EQUIVALENT, 0 ms] (187) QDP (188) QDP (189) UsableRulesProof [EQUIVALENT, 0 ms] (190) QDP (191) QDPSizeChangeProof [EQUIVALENT, 0 ms] (192) YES (193) PrologToIRSwTTransformerProof [SOUND, 467 ms] (194) AND (195) IRSwT (196) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (197) TRUE (198) IRSwT (199) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (200) TRUE (201) IRSwT (202) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (203) TRUE (204) IRSwT (205) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (206) TRUE (207) IRSwT (208) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (209) TRUE (210) IRSwT (211) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (212) TRUE (213) IRSwT (214) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (215) IRSwT (216) IntTRSCompressionProof [EQUIVALENT, 19 ms] (217) IRSwT (218) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (219) IRSwT (220) IRSwTTerminationDigraphProof [EQUIVALENT, 5 ms] (221) IRSwT (222) FilterProof [EQUIVALENT, 0 ms] (223) IntTRS (224) IntTRSNonPeriodicNontermProof [COMPLETE, 7 ms] (225) NO (226) IRSwT (227) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (228) IRSwT (229) IntTRSCompressionProof [EQUIVALENT, 0 ms] (230) IRSwT (231) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (232) IRSwT (233) IRSwTTerminationDigraphProof [EQUIVALENT, 1 ms] (234) IRSwT (235) TempFilterProof [SOUND, 3 ms] (236) IRSwT (237) IRSwTToQDPProof [SOUND, 1 ms] (238) QDP (239) QDPSizeChangeProof [EQUIVALENT, 0 ms] (240) YES ---------------------------------------- (0) Obligation: Clauses: add(b, b, b). add(X, b, X) :- binaryZ(X). add(b, Y, Y) :- binaryZ(Y). add(X, Y, Z) :- addz(X, Y, Z). addx(one(X), b, one(X)) :- binary(X). addx(zero(X), b, zero(X)) :- binaryZ(X). addx(X, Y, Z) :- addz(X, Y, Z). addy(b, one(Y), one(Y)) :- binary(Y). addy(b, zero(Y), zero(Y)) :- binaryZ(Y). addy(X, Y, Z) :- addz(X, Y, Z). addz(zero(X), zero(Y), zero(Z)) :- addz(X, Y, Z). addz(zero(X), one(Y), one(Z)) :- addx(X, Y, Z). addz(one(X), zero(Y), one(Z)) :- addy(X, Y, Z). addz(one(X), one(Y), zero(Z)) :- addc(X, Y, Z). addc(b, b, one(b)). addc(X, b, Z) :- succZ(X, Z). addc(b, Y, Z) :- succZ(Y, Z). addc(X, Y, Z) :- addC(X, Y, Z). addX(zero(X), b, one(X)) :- binaryZ(X). addX(one(X), b, zero(Z)) :- succ(X, Z). addX(X, Y, Z) :- addC(X, Y, Z). addY(b, zero(Y), one(Y)) :- binaryZ(Y). addY(b, one(Y), zero(Z)) :- succ(Y, Z). addY(X, Y, Z) :- addC(X, Y, Z). addC(zero(X), zero(Y), one(Z)) :- addz(X, Y, Z). addC(zero(X), one(Y), zero(Z)) :- addX(X, Y, Z). addC(one(X), zero(Y), zero(Z)) :- addY(X, Y, Z). addC(one(X), one(Y), one(Z)) :- addc(X, Y, Z). binary(b). binary(zero(X)) :- binaryZ(X). binary(one(X)) :- binary(X). binaryZ(zero(X)) :- binaryZ(X). binaryZ(one(X)) :- binary(X). succ(b, one(b)). succ(zero(X), one(X)) :- binaryZ(X). succ(one(X), zero(Z)) :- succ(X, Z). succZ(zero(X), one(X)) :- binaryZ(X). succZ(one(X), zero(Z)) :- succ(X, Z). times(one(b), X, X). times(zero(R), S, zero(RS)) :- times(R, S, RS). times(one(R), S, RSS) :- ','(times(R, S, RS), add(S, zero(RS), RSS)). Query: times(a,a,g) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: times_in_3: (f,f,b) (f,f,f) add_in_3: (f,f,f) (f,f,b) binaryZ_in_1: (f) (b) binary_in_1: (f) (b) addz_in_3: (f,f,f) (f,f,b) addx_in_3: (f,f,f) (f,f,b) addy_in_3: (f,f,f) (f,f,b) addc_in_3: (f,f,f) (f,f,b) succZ_in_2: (f,f) (f,b) succ_in_2: (f,f) (f,b) addC_in_3: (f,f,f) (f,f,b) addX_in_3: (f,f,f) (f,f,b) addY_in_3: (f,f,f) (f,f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g U27_g(x1, x2) = U27_g(x2) binaryZ_out_g(x1) = binaryZ_out_g U28_g(x1, x2) = U28_g(x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x4) U11_aag(x1, x2, x3, x4) = U11_aag(x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x4) U12_aag(x1, x2, x3, x4) = U12_aag(x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x4) U13_aag(x1, x2, x3, x4) = U13_aag(x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2) U14_aag(x1, x2, x3) = U14_aag(x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1) U34_ag(x1, x2, x3) = U34_ag(x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x3) U15_aag(x1, x2, x3) = U15_aag(x3) U16_aag(x1, x2, x3, x4) = U16_aag(x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2) U24_aag(x1, x2, x3, x4) = U24_aag(x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2) U18_aag(x1, x2, x3) = U18_aag(x3) U19_aag(x1, x2, x3, x4) = U19_aag(x4) U25_aag(x1, x2, x3, x4) = U25_aag(x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2) U21_aag(x1, x2, x3) = U21_aag(x3) U22_aag(x1, x2, x3, x4) = U22_aag(x4) U26_aag(x1, x2, x3, x4) = U26_aag(x4) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g U27_g(x1, x2) = U27_g(x2) binaryZ_out_g(x1) = binaryZ_out_g U28_g(x1, x2) = U28_g(x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x4) U11_aag(x1, x2, x3, x4) = U11_aag(x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x4) U12_aag(x1, x2, x3, x4) = U12_aag(x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x4) U13_aag(x1, x2, x3, x4) = U13_aag(x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2) U14_aag(x1, x2, x3) = U14_aag(x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1) U34_ag(x1, x2, x3) = U34_ag(x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x3) U15_aag(x1, x2, x3) = U15_aag(x3) U16_aag(x1, x2, x3, x4) = U16_aag(x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2) U24_aag(x1, x2, x3, x4) = U24_aag(x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2) U18_aag(x1, x2, x3) = U18_aag(x3) U19_aag(x1, x2, x3, x4) = U19_aag(x4) U25_aag(x1, x2, x3, x4) = U25_aag(x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2) U21_aag(x1, x2, x3) = U21_aag(x3) U22_aag(x1, x2, x3, x4) = U22_aag(x4) U26_aag(x1, x2, x3, x4) = U26_aag(x4) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_AAG(zero(R), S, zero(RS)) -> U35_AAG(R, S, RS, times_in_aag(R, S, RS)) TIMES_IN_AAG(zero(R), S, zero(RS)) -> TIMES_IN_AAG(R, S, RS) TIMES_IN_AAG(one(R), S, RSS) -> U36_AAG(R, S, RSS, times_in_aaa(R, S, RS)) TIMES_IN_AAG(one(R), S, RSS) -> TIMES_IN_AAA(R, S, RS) TIMES_IN_AAA(zero(R), S, zero(RS)) -> U35_AAA(R, S, RS, times_in_aaa(R, S, RS)) TIMES_IN_AAA(zero(R), S, zero(RS)) -> TIMES_IN_AAA(R, S, RS) TIMES_IN_AAA(one(R), S, RSS) -> U36_AAA(R, S, RSS, times_in_aaa(R, S, RS)) TIMES_IN_AAA(one(R), S, RSS) -> TIMES_IN_AAA(R, S, RS) U36_AAA(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_AAA(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) U36_AAA(R, S, RSS, times_out_aaa(R, S, RS)) -> ADD_IN_AAA(S, zero(RS), RSS) ADD_IN_AAA(X, b, X) -> U1_AAA(X, binaryZ_in_a(X)) ADD_IN_AAA(X, b, X) -> BINARYZ_IN_A(X) BINARYZ_IN_A(zero(X)) -> U29_A(X, binaryZ_in_a(X)) BINARYZ_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARYZ_IN_A(one(X)) -> U30_A(X, binary_in_a(X)) BINARYZ_IN_A(one(X)) -> BINARY_IN_A(X) BINARY_IN_A(zero(X)) -> U27_A(X, binaryZ_in_a(X)) BINARY_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARY_IN_A(one(X)) -> U28_A(X, binary_in_a(X)) BINARY_IN_A(one(X)) -> BINARY_IN_A(X) ADD_IN_AAA(b, Y, Y) -> U2_AAA(Y, binaryZ_in_a(Y)) ADD_IN_AAA(b, Y, Y) -> BINARYZ_IN_A(Y) ADD_IN_AAA(X, Y, Z) -> U3_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADD_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), zero(Y), zero(Z)) -> U10_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDZ_IN_AAA(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), one(Y), one(Z)) -> U11_AAA(X, Y, Z, addx_in_aaa(X, Y, Z)) ADDZ_IN_AAA(zero(X), one(Y), one(Z)) -> ADDX_IN_AAA(X, Y, Z) ADDX_IN_AAA(one(X), b, one(X)) -> U4_AAA(X, binary_in_a(X)) ADDX_IN_AAA(one(X), b, one(X)) -> BINARY_IN_A(X) ADDX_IN_AAA(zero(X), b, zero(X)) -> U5_AAA(X, binaryZ_in_a(X)) ADDX_IN_AAA(zero(X), b, zero(X)) -> BINARYZ_IN_A(X) ADDX_IN_AAA(X, Y, Z) -> U6_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDX_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), zero(Y), one(Z)) -> U12_AAA(X, Y, Z, addy_in_aaa(X, Y, Z)) ADDZ_IN_AAA(one(X), zero(Y), one(Z)) -> ADDY_IN_AAA(X, Y, Z) ADDY_IN_AAA(b, one(Y), one(Y)) -> U7_AAA(Y, binary_in_a(Y)) ADDY_IN_AAA(b, one(Y), one(Y)) -> BINARY_IN_A(Y) ADDY_IN_AAA(b, zero(Y), zero(Y)) -> U8_AAA(Y, binaryZ_in_a(Y)) ADDY_IN_AAA(b, zero(Y), zero(Y)) -> BINARYZ_IN_A(Y) ADDY_IN_AAA(X, Y, Z) -> U9_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDY_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), one(Y), zero(Z)) -> U13_AAA(X, Y, Z, addc_in_aaa(X, Y, Z)) ADDZ_IN_AAA(one(X), one(Y), zero(Z)) -> ADDC_IN_AAA(X, Y, Z) ADDC_IN_AAA(X, b, Z) -> U14_AAA(X, Z, succZ_in_aa(X, Z)) ADDC_IN_AAA(X, b, Z) -> SUCCZ_IN_AA(X, Z) SUCCZ_IN_AA(zero(X), one(X)) -> U33_AA(X, binaryZ_in_a(X)) SUCCZ_IN_AA(zero(X), one(X)) -> BINARYZ_IN_A(X) SUCCZ_IN_AA(one(X), zero(Z)) -> U34_AA(X, Z, succ_in_aa(X, Z)) SUCCZ_IN_AA(one(X), zero(Z)) -> SUCC_IN_AA(X, Z) SUCC_IN_AA(zero(X), one(X)) -> U31_AA(X, binaryZ_in_a(X)) SUCC_IN_AA(zero(X), one(X)) -> BINARYZ_IN_A(X) SUCC_IN_AA(one(X), zero(Z)) -> U32_AA(X, Z, succ_in_aa(X, Z)) SUCC_IN_AA(one(X), zero(Z)) -> SUCC_IN_AA(X, Z) ADDC_IN_AAA(b, Y, Z) -> U15_AAA(Y, Z, succZ_in_aa(Y, Z)) ADDC_IN_AAA(b, Y, Z) -> SUCCZ_IN_AA(Y, Z) ADDC_IN_AAA(X, Y, Z) -> U16_AAA(X, Y, Z, addC_in_aaa(X, Y, Z)) ADDC_IN_AAA(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(zero(X), zero(Y), one(Z)) -> U23_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDC_IN_AAA^1(zero(X), one(Y), zero(Z)) -> U24_AAA(X, Y, Z, addX_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAA^1(X, Y, Z) ADDX_IN_AAA^1(zero(X), b, one(X)) -> U17_AAA(X, binaryZ_in_a(X)) ADDX_IN_AAA^1(zero(X), b, one(X)) -> BINARYZ_IN_A(X) ADDX_IN_AAA^1(one(X), b, zero(Z)) -> U18_AAA(X, Z, succ_in_aa(X, Z)) ADDX_IN_AAA^1(one(X), b, zero(Z)) -> SUCC_IN_AA(X, Z) ADDX_IN_AAA^1(X, Y, Z) -> U19_AAA(X, Y, Z, addC_in_aaa(X, Y, Z)) ADDX_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), zero(Y), zero(Z)) -> U25_AAA(X, Y, Z, addY_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAA^1(X, Y, Z) ADDY_IN_AAA^1(b, zero(Y), one(Y)) -> U20_AAA(Y, binaryZ_in_a(Y)) ADDY_IN_AAA^1(b, zero(Y), one(Y)) -> BINARYZ_IN_A(Y) ADDY_IN_AAA^1(b, one(Y), zero(Z)) -> U21_AAA(Y, Z, succ_in_aa(Y, Z)) ADDY_IN_AAA^1(b, one(Y), zero(Z)) -> SUCC_IN_AA(Y, Z) ADDY_IN_AAA^1(X, Y, Z) -> U22_AAA(X, Y, Z, addC_in_aaa(X, Y, Z)) ADDY_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), one(Y), one(Z)) -> U26_AAA(X, Y, Z, addc_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAA(X, Y, Z) U36_AAG(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_AAG(R, S, RSS, add_in_aag(S, zero(RS), RSS)) U36_AAG(R, S, RSS, times_out_aaa(R, S, RS)) -> ADD_IN_AAG(S, zero(RS), RSS) ADD_IN_AAG(X, b, X) -> U1_AAG(X, binaryZ_in_g(X)) ADD_IN_AAG(X, b, X) -> BINARYZ_IN_G(X) BINARYZ_IN_G(zero(X)) -> U29_G(X, binaryZ_in_g(X)) BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARYZ_IN_G(one(X)) -> U30_G(X, binary_in_g(X)) BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X) BINARY_IN_G(zero(X)) -> U27_G(X, binaryZ_in_g(X)) BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARY_IN_G(one(X)) -> U28_G(X, binary_in_g(X)) BINARY_IN_G(one(X)) -> BINARY_IN_G(X) ADD_IN_AAG(b, Y, Y) -> U2_AAG(Y, binaryZ_in_g(Y)) ADD_IN_AAG(b, Y, Y) -> BINARYZ_IN_G(Y) ADD_IN_AAG(X, Y, Z) -> U3_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADD_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> U10_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> U11_AAG(X, Y, Z, addx_in_aag(X, Y, Z)) ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> ADDX_IN_AAG(X, Y, Z) ADDX_IN_AAG(one(X), b, one(X)) -> U4_AAG(X, binary_in_g(X)) ADDX_IN_AAG(one(X), b, one(X)) -> BINARY_IN_G(X) ADDX_IN_AAG(zero(X), b, zero(X)) -> U5_AAG(X, binaryZ_in_g(X)) ADDX_IN_AAG(zero(X), b, zero(X)) -> BINARYZ_IN_G(X) ADDX_IN_AAG(X, Y, Z) -> U6_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDX_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> U12_AAG(X, Y, Z, addy_in_aag(X, Y, Z)) ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> ADDY_IN_AAG(X, Y, Z) ADDY_IN_AAG(b, one(Y), one(Y)) -> U7_AAG(Y, binary_in_g(Y)) ADDY_IN_AAG(b, one(Y), one(Y)) -> BINARY_IN_G(Y) ADDY_IN_AAG(b, zero(Y), zero(Y)) -> U8_AAG(Y, binaryZ_in_g(Y)) ADDY_IN_AAG(b, zero(Y), zero(Y)) -> BINARYZ_IN_G(Y) ADDY_IN_AAG(X, Y, Z) -> U9_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDY_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> U13_AAG(X, Y, Z, addc_in_aag(X, Y, Z)) ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> ADDC_IN_AAG(X, Y, Z) ADDC_IN_AAG(X, b, Z) -> U14_AAG(X, Z, succZ_in_ag(X, Z)) ADDC_IN_AAG(X, b, Z) -> SUCCZ_IN_AG(X, Z) SUCCZ_IN_AG(zero(X), one(X)) -> U33_AG(X, binaryZ_in_g(X)) SUCCZ_IN_AG(zero(X), one(X)) -> BINARYZ_IN_G(X) SUCCZ_IN_AG(one(X), zero(Z)) -> U34_AG(X, Z, succ_in_ag(X, Z)) SUCCZ_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z) SUCC_IN_AG(zero(X), one(X)) -> U31_AG(X, binaryZ_in_g(X)) SUCC_IN_AG(zero(X), one(X)) -> BINARYZ_IN_G(X) SUCC_IN_AG(one(X), zero(Z)) -> U32_AG(X, Z, succ_in_ag(X, Z)) SUCC_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z) ADDC_IN_AAG(b, Y, Z) -> U15_AAG(Y, Z, succZ_in_ag(Y, Z)) ADDC_IN_AAG(b, Y, Z) -> SUCCZ_IN_AG(Y, Z) ADDC_IN_AAG(X, Y, Z) -> U16_AAG(X, Y, Z, addC_in_aag(X, Y, Z)) ADDC_IN_AAG(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> U23_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> U24_AAG(X, Y, Z, addX_in_aag(X, Y, Z)) ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAG^1(X, Y, Z) ADDX_IN_AAG^1(zero(X), b, one(X)) -> U17_AAG(X, binaryZ_in_g(X)) ADDX_IN_AAG^1(zero(X), b, one(X)) -> BINARYZ_IN_G(X) ADDX_IN_AAG^1(one(X), b, zero(Z)) -> U18_AAG(X, Z, succ_in_ag(X, Z)) ADDX_IN_AAG^1(one(X), b, zero(Z)) -> SUCC_IN_AG(X, Z) ADDX_IN_AAG^1(X, Y, Z) -> U19_AAG(X, Y, Z, addC_in_aag(X, Y, Z)) ADDX_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> U25_AAG(X, Y, Z, addY_in_aag(X, Y, Z)) ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAG^1(X, Y, Z) ADDY_IN_AAG^1(b, zero(Y), one(Y)) -> U20_AAG(Y, binaryZ_in_g(Y)) ADDY_IN_AAG^1(b, zero(Y), one(Y)) -> BINARYZ_IN_G(Y) ADDY_IN_AAG^1(b, one(Y), zero(Z)) -> U21_AAG(Y, Z, succ_in_ag(Y, Z)) ADDY_IN_AAG^1(b, one(Y), zero(Z)) -> SUCC_IN_AG(Y, Z) ADDY_IN_AAG^1(X, Y, Z) -> U22_AAG(X, Y, Z, addC_in_aag(X, Y, Z)) ADDY_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> U26_AAG(X, Y, Z, addc_in_aag(X, Y, Z)) ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAG(X, Y, Z) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g U27_g(x1, x2) = U27_g(x2) binaryZ_out_g(x1) = binaryZ_out_g U28_g(x1, x2) = U28_g(x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x4) U11_aag(x1, x2, x3, x4) = U11_aag(x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x4) U12_aag(x1, x2, x3, x4) = U12_aag(x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x4) U13_aag(x1, x2, x3, x4) = U13_aag(x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2) U14_aag(x1, x2, x3) = U14_aag(x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1) U34_ag(x1, x2, x3) = U34_ag(x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x3) U15_aag(x1, x2, x3) = U15_aag(x3) U16_aag(x1, x2, x3, x4) = U16_aag(x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2) U24_aag(x1, x2, x3, x4) = U24_aag(x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2) U18_aag(x1, x2, x3) = U18_aag(x3) U19_aag(x1, x2, x3, x4) = U19_aag(x4) U25_aag(x1, x2, x3, x4) = U25_aag(x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2) U21_aag(x1, x2, x3) = U21_aag(x3) U22_aag(x1, x2, x3, x4) = U22_aag(x4) U26_aag(x1, x2, x3, x4) = U26_aag(x4) TIMES_IN_AAG(x1, x2, x3) = TIMES_IN_AAG(x3) U35_AAG(x1, x2, x3, x4) = U35_AAG(x4) U36_AAG(x1, x2, x3, x4) = U36_AAG(x3, x4) TIMES_IN_AAA(x1, x2, x3) = TIMES_IN_AAA U35_AAA(x1, x2, x3, x4) = U35_AAA(x4) U36_AAA(x1, x2, x3, x4) = U36_AAA(x4) U37_AAA(x1, x2, x3, x4) = U37_AAA(x1, x4) ADD_IN_AAA(x1, x2, x3) = ADD_IN_AAA U1_AAA(x1, x2) = U1_AAA(x2) BINARYZ_IN_A(x1) = BINARYZ_IN_A U29_A(x1, x2) = U29_A(x2) U30_A(x1, x2) = U30_A(x2) BINARY_IN_A(x1) = BINARY_IN_A U27_A(x1, x2) = U27_A(x2) U28_A(x1, x2) = U28_A(x2) U2_AAA(x1, x2) = U2_AAA(x2) U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) ADDZ_IN_AAA(x1, x2, x3) = ADDZ_IN_AAA U10_AAA(x1, x2, x3, x4) = U10_AAA(x4) U11_AAA(x1, x2, x3, x4) = U11_AAA(x4) ADDX_IN_AAA(x1, x2, x3) = ADDX_IN_AAA U4_AAA(x1, x2) = U4_AAA(x2) U5_AAA(x1, x2) = U5_AAA(x2) U6_AAA(x1, x2, x3, x4) = U6_AAA(x4) U12_AAA(x1, x2, x3, x4) = U12_AAA(x4) ADDY_IN_AAA(x1, x2, x3) = ADDY_IN_AAA U7_AAA(x1, x2) = U7_AAA(x2) U8_AAA(x1, x2) = U8_AAA(x2) U9_AAA(x1, x2, x3, x4) = U9_AAA(x4) U13_AAA(x1, x2, x3, x4) = U13_AAA(x4) ADDC_IN_AAA(x1, x2, x3) = ADDC_IN_AAA U14_AAA(x1, x2, x3) = U14_AAA(x3) SUCCZ_IN_AA(x1, x2) = SUCCZ_IN_AA U33_AA(x1, x2) = U33_AA(x2) U34_AA(x1, x2, x3) = U34_AA(x3) SUCC_IN_AA(x1, x2) = SUCC_IN_AA U31_AA(x1, x2) = U31_AA(x2) U32_AA(x1, x2, x3) = U32_AA(x3) U15_AAA(x1, x2, x3) = U15_AAA(x3) U16_AAA(x1, x2, x3, x4) = U16_AAA(x4) ADDC_IN_AAA^1(x1, x2, x3) = ADDC_IN_AAA^1 U23_AAA(x1, x2, x3, x4) = U23_AAA(x4) U24_AAA(x1, x2, x3, x4) = U24_AAA(x4) ADDX_IN_AAA^1(x1, x2, x3) = ADDX_IN_AAA^1 U17_AAA(x1, x2) = U17_AAA(x2) U18_AAA(x1, x2, x3) = U18_AAA(x3) U19_AAA(x1, x2, x3, x4) = U19_AAA(x4) U25_AAA(x1, x2, x3, x4) = U25_AAA(x4) ADDY_IN_AAA^1(x1, x2, x3) = ADDY_IN_AAA^1 U20_AAA(x1, x2) = U20_AAA(x2) U21_AAA(x1, x2, x3) = U21_AAA(x3) U22_AAA(x1, x2, x3, x4) = U22_AAA(x4) U26_AAA(x1, x2, x3, x4) = U26_AAA(x4) U37_AAG(x1, x2, x3, x4) = U37_AAG(x1, x4) ADD_IN_AAG(x1, x2, x3) = ADD_IN_AAG(x3) U1_AAG(x1, x2) = U1_AAG(x1, x2) BINARYZ_IN_G(x1) = BINARYZ_IN_G(x1) U29_G(x1, x2) = U29_G(x2) U30_G(x1, x2) = U30_G(x2) BINARY_IN_G(x1) = BINARY_IN_G(x1) U27_G(x1, x2) = U27_G(x2) U28_G(x1, x2) = U28_G(x2) U2_AAG(x1, x2) = U2_AAG(x1, x2) U3_AAG(x1, x2, x3, x4) = U3_AAG(x4) ADDZ_IN_AAG(x1, x2, x3) = ADDZ_IN_AAG(x3) U10_AAG(x1, x2, x3, x4) = U10_AAG(x4) U11_AAG(x1, x2, x3, x4) = U11_AAG(x4) ADDX_IN_AAG(x1, x2, x3) = ADDX_IN_AAG(x3) U4_AAG(x1, x2) = U4_AAG(x1, x2) U5_AAG(x1, x2) = U5_AAG(x1, x2) U6_AAG(x1, x2, x3, x4) = U6_AAG(x4) U12_AAG(x1, x2, x3, x4) = U12_AAG(x4) ADDY_IN_AAG(x1, x2, x3) = ADDY_IN_AAG(x3) U7_AAG(x1, x2) = U7_AAG(x1, x2) U8_AAG(x1, x2) = U8_AAG(x1, x2) U9_AAG(x1, x2, x3, x4) = U9_AAG(x4) U13_AAG(x1, x2, x3, x4) = U13_AAG(x4) ADDC_IN_AAG(x1, x2, x3) = ADDC_IN_AAG(x3) U14_AAG(x1, x2, x3) = U14_AAG(x3) SUCCZ_IN_AG(x1, x2) = SUCCZ_IN_AG(x2) U33_AG(x1, x2) = U33_AG(x1, x2) U34_AG(x1, x2, x3) = U34_AG(x3) SUCC_IN_AG(x1, x2) = SUCC_IN_AG(x2) U31_AG(x1, x2) = U31_AG(x1, x2) U32_AG(x1, x2, x3) = U32_AG(x3) U15_AAG(x1, x2, x3) = U15_AAG(x3) U16_AAG(x1, x2, x3, x4) = U16_AAG(x4) ADDC_IN_AAG^1(x1, x2, x3) = ADDC_IN_AAG^1(x3) U23_AAG(x1, x2, x3, x4) = U23_AAG(x4) U24_AAG(x1, x2, x3, x4) = U24_AAG(x4) ADDX_IN_AAG^1(x1, x2, x3) = ADDX_IN_AAG^1(x3) U17_AAG(x1, x2) = U17_AAG(x1, x2) U18_AAG(x1, x2, x3) = U18_AAG(x3) U19_AAG(x1, x2, x3, x4) = U19_AAG(x4) U25_AAG(x1, x2, x3, x4) = U25_AAG(x4) ADDY_IN_AAG^1(x1, x2, x3) = ADDY_IN_AAG^1(x3) U20_AAG(x1, x2) = U20_AAG(x1, x2) U21_AAG(x1, x2, x3) = U21_AAG(x3) U22_AAG(x1, x2, x3, x4) = U22_AAG(x4) U26_AAG(x1, x2, x3, x4) = U26_AAG(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_AAG(zero(R), S, zero(RS)) -> U35_AAG(R, S, RS, times_in_aag(R, S, RS)) TIMES_IN_AAG(zero(R), S, zero(RS)) -> TIMES_IN_AAG(R, S, RS) TIMES_IN_AAG(one(R), S, RSS) -> U36_AAG(R, S, RSS, times_in_aaa(R, S, RS)) TIMES_IN_AAG(one(R), S, RSS) -> TIMES_IN_AAA(R, S, RS) TIMES_IN_AAA(zero(R), S, zero(RS)) -> U35_AAA(R, S, RS, times_in_aaa(R, S, RS)) TIMES_IN_AAA(zero(R), S, zero(RS)) -> TIMES_IN_AAA(R, S, RS) TIMES_IN_AAA(one(R), S, RSS) -> U36_AAA(R, S, RSS, times_in_aaa(R, S, RS)) TIMES_IN_AAA(one(R), S, RSS) -> TIMES_IN_AAA(R, S, RS) U36_AAA(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_AAA(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) U36_AAA(R, S, RSS, times_out_aaa(R, S, RS)) -> ADD_IN_AAA(S, zero(RS), RSS) ADD_IN_AAA(X, b, X) -> U1_AAA(X, binaryZ_in_a(X)) ADD_IN_AAA(X, b, X) -> BINARYZ_IN_A(X) BINARYZ_IN_A(zero(X)) -> U29_A(X, binaryZ_in_a(X)) BINARYZ_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARYZ_IN_A(one(X)) -> U30_A(X, binary_in_a(X)) BINARYZ_IN_A(one(X)) -> BINARY_IN_A(X) BINARY_IN_A(zero(X)) -> U27_A(X, binaryZ_in_a(X)) BINARY_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARY_IN_A(one(X)) -> U28_A(X, binary_in_a(X)) BINARY_IN_A(one(X)) -> BINARY_IN_A(X) ADD_IN_AAA(b, Y, Y) -> U2_AAA(Y, binaryZ_in_a(Y)) ADD_IN_AAA(b, Y, Y) -> BINARYZ_IN_A(Y) ADD_IN_AAA(X, Y, Z) -> U3_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADD_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), zero(Y), zero(Z)) -> U10_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDZ_IN_AAA(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), one(Y), one(Z)) -> U11_AAA(X, Y, Z, addx_in_aaa(X, Y, Z)) ADDZ_IN_AAA(zero(X), one(Y), one(Z)) -> ADDX_IN_AAA(X, Y, Z) ADDX_IN_AAA(one(X), b, one(X)) -> U4_AAA(X, binary_in_a(X)) ADDX_IN_AAA(one(X), b, one(X)) -> BINARY_IN_A(X) ADDX_IN_AAA(zero(X), b, zero(X)) -> U5_AAA(X, binaryZ_in_a(X)) ADDX_IN_AAA(zero(X), b, zero(X)) -> BINARYZ_IN_A(X) ADDX_IN_AAA(X, Y, Z) -> U6_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDX_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), zero(Y), one(Z)) -> U12_AAA(X, Y, Z, addy_in_aaa(X, Y, Z)) ADDZ_IN_AAA(one(X), zero(Y), one(Z)) -> ADDY_IN_AAA(X, Y, Z) ADDY_IN_AAA(b, one(Y), one(Y)) -> U7_AAA(Y, binary_in_a(Y)) ADDY_IN_AAA(b, one(Y), one(Y)) -> BINARY_IN_A(Y) ADDY_IN_AAA(b, zero(Y), zero(Y)) -> U8_AAA(Y, binaryZ_in_a(Y)) ADDY_IN_AAA(b, zero(Y), zero(Y)) -> BINARYZ_IN_A(Y) ADDY_IN_AAA(X, Y, Z) -> U9_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDY_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), one(Y), zero(Z)) -> U13_AAA(X, Y, Z, addc_in_aaa(X, Y, Z)) ADDZ_IN_AAA(one(X), one(Y), zero(Z)) -> ADDC_IN_AAA(X, Y, Z) ADDC_IN_AAA(X, b, Z) -> U14_AAA(X, Z, succZ_in_aa(X, Z)) ADDC_IN_AAA(X, b, Z) -> SUCCZ_IN_AA(X, Z) SUCCZ_IN_AA(zero(X), one(X)) -> U33_AA(X, binaryZ_in_a(X)) SUCCZ_IN_AA(zero(X), one(X)) -> BINARYZ_IN_A(X) SUCCZ_IN_AA(one(X), zero(Z)) -> U34_AA(X, Z, succ_in_aa(X, Z)) SUCCZ_IN_AA(one(X), zero(Z)) -> SUCC_IN_AA(X, Z) SUCC_IN_AA(zero(X), one(X)) -> U31_AA(X, binaryZ_in_a(X)) SUCC_IN_AA(zero(X), one(X)) -> BINARYZ_IN_A(X) SUCC_IN_AA(one(X), zero(Z)) -> U32_AA(X, Z, succ_in_aa(X, Z)) SUCC_IN_AA(one(X), zero(Z)) -> SUCC_IN_AA(X, Z) ADDC_IN_AAA(b, Y, Z) -> U15_AAA(Y, Z, succZ_in_aa(Y, Z)) ADDC_IN_AAA(b, Y, Z) -> SUCCZ_IN_AA(Y, Z) ADDC_IN_AAA(X, Y, Z) -> U16_AAA(X, Y, Z, addC_in_aaa(X, Y, Z)) ADDC_IN_AAA(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(zero(X), zero(Y), one(Z)) -> U23_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDC_IN_AAA^1(zero(X), one(Y), zero(Z)) -> U24_AAA(X, Y, Z, addX_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAA^1(X, Y, Z) ADDX_IN_AAA^1(zero(X), b, one(X)) -> U17_AAA(X, binaryZ_in_a(X)) ADDX_IN_AAA^1(zero(X), b, one(X)) -> BINARYZ_IN_A(X) ADDX_IN_AAA^1(one(X), b, zero(Z)) -> U18_AAA(X, Z, succ_in_aa(X, Z)) ADDX_IN_AAA^1(one(X), b, zero(Z)) -> SUCC_IN_AA(X, Z) ADDX_IN_AAA^1(X, Y, Z) -> U19_AAA(X, Y, Z, addC_in_aaa(X, Y, Z)) ADDX_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), zero(Y), zero(Z)) -> U25_AAA(X, Y, Z, addY_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAA^1(X, Y, Z) ADDY_IN_AAA^1(b, zero(Y), one(Y)) -> U20_AAA(Y, binaryZ_in_a(Y)) ADDY_IN_AAA^1(b, zero(Y), one(Y)) -> BINARYZ_IN_A(Y) ADDY_IN_AAA^1(b, one(Y), zero(Z)) -> U21_AAA(Y, Z, succ_in_aa(Y, Z)) ADDY_IN_AAA^1(b, one(Y), zero(Z)) -> SUCC_IN_AA(Y, Z) ADDY_IN_AAA^1(X, Y, Z) -> U22_AAA(X, Y, Z, addC_in_aaa(X, Y, Z)) ADDY_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), one(Y), one(Z)) -> U26_AAA(X, Y, Z, addc_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAA(X, Y, Z) U36_AAG(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_AAG(R, S, RSS, add_in_aag(S, zero(RS), RSS)) U36_AAG(R, S, RSS, times_out_aaa(R, S, RS)) -> ADD_IN_AAG(S, zero(RS), RSS) ADD_IN_AAG(X, b, X) -> U1_AAG(X, binaryZ_in_g(X)) ADD_IN_AAG(X, b, X) -> BINARYZ_IN_G(X) BINARYZ_IN_G(zero(X)) -> U29_G(X, binaryZ_in_g(X)) BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARYZ_IN_G(one(X)) -> U30_G(X, binary_in_g(X)) BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X) BINARY_IN_G(zero(X)) -> U27_G(X, binaryZ_in_g(X)) BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARY_IN_G(one(X)) -> U28_G(X, binary_in_g(X)) BINARY_IN_G(one(X)) -> BINARY_IN_G(X) ADD_IN_AAG(b, Y, Y) -> U2_AAG(Y, binaryZ_in_g(Y)) ADD_IN_AAG(b, Y, Y) -> BINARYZ_IN_G(Y) ADD_IN_AAG(X, Y, Z) -> U3_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADD_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> U10_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> U11_AAG(X, Y, Z, addx_in_aag(X, Y, Z)) ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> ADDX_IN_AAG(X, Y, Z) ADDX_IN_AAG(one(X), b, one(X)) -> U4_AAG(X, binary_in_g(X)) ADDX_IN_AAG(one(X), b, one(X)) -> BINARY_IN_G(X) ADDX_IN_AAG(zero(X), b, zero(X)) -> U5_AAG(X, binaryZ_in_g(X)) ADDX_IN_AAG(zero(X), b, zero(X)) -> BINARYZ_IN_G(X) ADDX_IN_AAG(X, Y, Z) -> U6_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDX_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> U12_AAG(X, Y, Z, addy_in_aag(X, Y, Z)) ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> ADDY_IN_AAG(X, Y, Z) ADDY_IN_AAG(b, one(Y), one(Y)) -> U7_AAG(Y, binary_in_g(Y)) ADDY_IN_AAG(b, one(Y), one(Y)) -> BINARY_IN_G(Y) ADDY_IN_AAG(b, zero(Y), zero(Y)) -> U8_AAG(Y, binaryZ_in_g(Y)) ADDY_IN_AAG(b, zero(Y), zero(Y)) -> BINARYZ_IN_G(Y) ADDY_IN_AAG(X, Y, Z) -> U9_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDY_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> U13_AAG(X, Y, Z, addc_in_aag(X, Y, Z)) ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> ADDC_IN_AAG(X, Y, Z) ADDC_IN_AAG(X, b, Z) -> U14_AAG(X, Z, succZ_in_ag(X, Z)) ADDC_IN_AAG(X, b, Z) -> SUCCZ_IN_AG(X, Z) SUCCZ_IN_AG(zero(X), one(X)) -> U33_AG(X, binaryZ_in_g(X)) SUCCZ_IN_AG(zero(X), one(X)) -> BINARYZ_IN_G(X) SUCCZ_IN_AG(one(X), zero(Z)) -> U34_AG(X, Z, succ_in_ag(X, Z)) SUCCZ_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z) SUCC_IN_AG(zero(X), one(X)) -> U31_AG(X, binaryZ_in_g(X)) SUCC_IN_AG(zero(X), one(X)) -> BINARYZ_IN_G(X) SUCC_IN_AG(one(X), zero(Z)) -> U32_AG(X, Z, succ_in_ag(X, Z)) SUCC_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z) ADDC_IN_AAG(b, Y, Z) -> U15_AAG(Y, Z, succZ_in_ag(Y, Z)) ADDC_IN_AAG(b, Y, Z) -> SUCCZ_IN_AG(Y, Z) ADDC_IN_AAG(X, Y, Z) -> U16_AAG(X, Y, Z, addC_in_aag(X, Y, Z)) ADDC_IN_AAG(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> U23_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> U24_AAG(X, Y, Z, addX_in_aag(X, Y, Z)) ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAG^1(X, Y, Z) ADDX_IN_AAG^1(zero(X), b, one(X)) -> U17_AAG(X, binaryZ_in_g(X)) ADDX_IN_AAG^1(zero(X), b, one(X)) -> BINARYZ_IN_G(X) ADDX_IN_AAG^1(one(X), b, zero(Z)) -> U18_AAG(X, Z, succ_in_ag(X, Z)) ADDX_IN_AAG^1(one(X), b, zero(Z)) -> SUCC_IN_AG(X, Z) ADDX_IN_AAG^1(X, Y, Z) -> U19_AAG(X, Y, Z, addC_in_aag(X, Y, Z)) ADDX_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> U25_AAG(X, Y, Z, addY_in_aag(X, Y, Z)) ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAG^1(X, Y, Z) ADDY_IN_AAG^1(b, zero(Y), one(Y)) -> U20_AAG(Y, binaryZ_in_g(Y)) ADDY_IN_AAG^1(b, zero(Y), one(Y)) -> BINARYZ_IN_G(Y) ADDY_IN_AAG^1(b, one(Y), zero(Z)) -> U21_AAG(Y, Z, succ_in_ag(Y, Z)) ADDY_IN_AAG^1(b, one(Y), zero(Z)) -> SUCC_IN_AG(Y, Z) ADDY_IN_AAG^1(X, Y, Z) -> U22_AAG(X, Y, Z, addC_in_aag(X, Y, Z)) ADDY_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> U26_AAG(X, Y, Z, addc_in_aag(X, Y, Z)) ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAG(X, Y, Z) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g U27_g(x1, x2) = U27_g(x2) binaryZ_out_g(x1) = binaryZ_out_g U28_g(x1, x2) = U28_g(x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x4) U11_aag(x1, x2, x3, x4) = U11_aag(x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x4) U12_aag(x1, x2, x3, x4) = U12_aag(x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x4) U13_aag(x1, x2, x3, x4) = U13_aag(x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2) U14_aag(x1, x2, x3) = U14_aag(x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1) U34_ag(x1, x2, x3) = U34_ag(x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x3) U15_aag(x1, x2, x3) = U15_aag(x3) U16_aag(x1, x2, x3, x4) = U16_aag(x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2) U24_aag(x1, x2, x3, x4) = U24_aag(x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2) U18_aag(x1, x2, x3) = U18_aag(x3) U19_aag(x1, x2, x3, x4) = U19_aag(x4) U25_aag(x1, x2, x3, x4) = U25_aag(x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2) U21_aag(x1, x2, x3) = U21_aag(x3) U22_aag(x1, x2, x3, x4) = U22_aag(x4) U26_aag(x1, x2, x3, x4) = U26_aag(x4) TIMES_IN_AAG(x1, x2, x3) = TIMES_IN_AAG(x3) U35_AAG(x1, x2, x3, x4) = U35_AAG(x4) U36_AAG(x1, x2, x3, x4) = U36_AAG(x3, x4) TIMES_IN_AAA(x1, x2, x3) = TIMES_IN_AAA U35_AAA(x1, x2, x3, x4) = U35_AAA(x4) U36_AAA(x1, x2, x3, x4) = U36_AAA(x4) U37_AAA(x1, x2, x3, x4) = U37_AAA(x1, x4) ADD_IN_AAA(x1, x2, x3) = ADD_IN_AAA U1_AAA(x1, x2) = U1_AAA(x2) BINARYZ_IN_A(x1) = BINARYZ_IN_A U29_A(x1, x2) = U29_A(x2) U30_A(x1, x2) = U30_A(x2) BINARY_IN_A(x1) = BINARY_IN_A U27_A(x1, x2) = U27_A(x2) U28_A(x1, x2) = U28_A(x2) U2_AAA(x1, x2) = U2_AAA(x2) U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) ADDZ_IN_AAA(x1, x2, x3) = ADDZ_IN_AAA U10_AAA(x1, x2, x3, x4) = U10_AAA(x4) U11_AAA(x1, x2, x3, x4) = U11_AAA(x4) ADDX_IN_AAA(x1, x2, x3) = ADDX_IN_AAA U4_AAA(x1, x2) = U4_AAA(x2) U5_AAA(x1, x2) = U5_AAA(x2) U6_AAA(x1, x2, x3, x4) = U6_AAA(x4) U12_AAA(x1, x2, x3, x4) = U12_AAA(x4) ADDY_IN_AAA(x1, x2, x3) = ADDY_IN_AAA U7_AAA(x1, x2) = U7_AAA(x2) U8_AAA(x1, x2) = U8_AAA(x2) U9_AAA(x1, x2, x3, x4) = U9_AAA(x4) U13_AAA(x1, x2, x3, x4) = U13_AAA(x4) ADDC_IN_AAA(x1, x2, x3) = ADDC_IN_AAA U14_AAA(x1, x2, x3) = U14_AAA(x3) SUCCZ_IN_AA(x1, x2) = SUCCZ_IN_AA U33_AA(x1, x2) = U33_AA(x2) U34_AA(x1, x2, x3) = U34_AA(x3) SUCC_IN_AA(x1, x2) = SUCC_IN_AA U31_AA(x1, x2) = U31_AA(x2) U32_AA(x1, x2, x3) = U32_AA(x3) U15_AAA(x1, x2, x3) = U15_AAA(x3) U16_AAA(x1, x2, x3, x4) = U16_AAA(x4) ADDC_IN_AAA^1(x1, x2, x3) = ADDC_IN_AAA^1 U23_AAA(x1, x2, x3, x4) = U23_AAA(x4) U24_AAA(x1, x2, x3, x4) = U24_AAA(x4) ADDX_IN_AAA^1(x1, x2, x3) = ADDX_IN_AAA^1 U17_AAA(x1, x2) = U17_AAA(x2) U18_AAA(x1, x2, x3) = U18_AAA(x3) U19_AAA(x1, x2, x3, x4) = U19_AAA(x4) U25_AAA(x1, x2, x3, x4) = U25_AAA(x4) ADDY_IN_AAA^1(x1, x2, x3) = ADDY_IN_AAA^1 U20_AAA(x1, x2) = U20_AAA(x2) U21_AAA(x1, x2, x3) = U21_AAA(x3) U22_AAA(x1, x2, x3, x4) = U22_AAA(x4) U26_AAA(x1, x2, x3, x4) = U26_AAA(x4) U37_AAG(x1, x2, x3, x4) = U37_AAG(x1, x4) ADD_IN_AAG(x1, x2, x3) = ADD_IN_AAG(x3) U1_AAG(x1, x2) = U1_AAG(x1, x2) BINARYZ_IN_G(x1) = BINARYZ_IN_G(x1) U29_G(x1, x2) = U29_G(x2) U30_G(x1, x2) = U30_G(x2) BINARY_IN_G(x1) = BINARY_IN_G(x1) U27_G(x1, x2) = U27_G(x2) U28_G(x1, x2) = U28_G(x2) U2_AAG(x1, x2) = U2_AAG(x1, x2) U3_AAG(x1, x2, x3, x4) = U3_AAG(x4) ADDZ_IN_AAG(x1, x2, x3) = ADDZ_IN_AAG(x3) U10_AAG(x1, x2, x3, x4) = U10_AAG(x4) U11_AAG(x1, x2, x3, x4) = U11_AAG(x4) ADDX_IN_AAG(x1, x2, x3) = ADDX_IN_AAG(x3) U4_AAG(x1, x2) = U4_AAG(x1, x2) U5_AAG(x1, x2) = U5_AAG(x1, x2) U6_AAG(x1, x2, x3, x4) = U6_AAG(x4) U12_AAG(x1, x2, x3, x4) = U12_AAG(x4) ADDY_IN_AAG(x1, x2, x3) = ADDY_IN_AAG(x3) U7_AAG(x1, x2) = U7_AAG(x1, x2) U8_AAG(x1, x2) = U8_AAG(x1, x2) U9_AAG(x1, x2, x3, x4) = U9_AAG(x4) U13_AAG(x1, x2, x3, x4) = U13_AAG(x4) ADDC_IN_AAG(x1, x2, x3) = ADDC_IN_AAG(x3) U14_AAG(x1, x2, x3) = U14_AAG(x3) SUCCZ_IN_AG(x1, x2) = SUCCZ_IN_AG(x2) U33_AG(x1, x2) = U33_AG(x1, x2) U34_AG(x1, x2, x3) = U34_AG(x3) SUCC_IN_AG(x1, x2) = SUCC_IN_AG(x2) U31_AG(x1, x2) = U31_AG(x1, x2) U32_AG(x1, x2, x3) = U32_AG(x3) U15_AAG(x1, x2, x3) = U15_AAG(x3) U16_AAG(x1, x2, x3, x4) = U16_AAG(x4) ADDC_IN_AAG^1(x1, x2, x3) = ADDC_IN_AAG^1(x3) U23_AAG(x1, x2, x3, x4) = U23_AAG(x4) U24_AAG(x1, x2, x3, x4) = U24_AAG(x4) ADDX_IN_AAG^1(x1, x2, x3) = ADDX_IN_AAG^1(x3) U17_AAG(x1, x2) = U17_AAG(x1, x2) U18_AAG(x1, x2, x3) = U18_AAG(x3) U19_AAG(x1, x2, x3, x4) = U19_AAG(x4) U25_AAG(x1, x2, x3, x4) = U25_AAG(x4) ADDY_IN_AAG^1(x1, x2, x3) = ADDY_IN_AAG^1(x3) U20_AAG(x1, x2) = U20_AAG(x1, x2) U21_AAG(x1, x2, x3) = U21_AAG(x3) U22_AAG(x1, x2, x3, x4) = U22_AAG(x4) U26_AAG(x1, x2, x3, x4) = U26_AAG(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 8 SCCs with 109 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X) BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARY_IN_G(one(X)) -> BINARY_IN_G(X) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g U27_g(x1, x2) = U27_g(x2) binaryZ_out_g(x1) = binaryZ_out_g U28_g(x1, x2) = U28_g(x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x4) U11_aag(x1, x2, x3, x4) = U11_aag(x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x4) U12_aag(x1, x2, x3, x4) = U12_aag(x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x4) U13_aag(x1, x2, x3, x4) = U13_aag(x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2) U14_aag(x1, x2, x3) = U14_aag(x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1) U34_ag(x1, x2, x3) = U34_ag(x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x3) U15_aag(x1, x2, x3) = U15_aag(x3) U16_aag(x1, x2, x3, x4) = U16_aag(x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2) U24_aag(x1, x2, x3, x4) = U24_aag(x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2) U18_aag(x1, x2, x3) = U18_aag(x3) U19_aag(x1, x2, x3, x4) = U19_aag(x4) U25_aag(x1, x2, x3, x4) = U25_aag(x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2) U21_aag(x1, x2, x3) = U21_aag(x3) U22_aag(x1, x2, x3, x4) = U22_aag(x4) U26_aag(x1, x2, x3, x4) = U26_aag(x4) BINARYZ_IN_G(x1) = BINARYZ_IN_G(x1) BINARY_IN_G(x1) = BINARY_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X) BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARY_IN_G(one(X)) -> BINARY_IN_G(X) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X) BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARY_IN_G(one(X)) -> BINARY_IN_G(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X) The graph contains the following edges 1 > 1 *BINARY_IN_G(one(X)) -> BINARY_IN_G(X) The graph contains the following edges 1 > 1 *BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X) The graph contains the following edges 1 > 1 *BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCC_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g U27_g(x1, x2) = U27_g(x2) binaryZ_out_g(x1) = binaryZ_out_g U28_g(x1, x2) = U28_g(x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x4) U11_aag(x1, x2, x3, x4) = U11_aag(x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x4) U12_aag(x1, x2, x3, x4) = U12_aag(x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x4) U13_aag(x1, x2, x3, x4) = U13_aag(x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2) U14_aag(x1, x2, x3) = U14_aag(x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1) U34_ag(x1, x2, x3) = U34_ag(x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x3) U15_aag(x1, x2, x3) = U15_aag(x3) U16_aag(x1, x2, x3, x4) = U16_aag(x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2) U24_aag(x1, x2, x3, x4) = U24_aag(x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2) U18_aag(x1, x2, x3) = U18_aag(x3) U19_aag(x1, x2, x3, x4) = U19_aag(x4) U25_aag(x1, x2, x3, x4) = U25_aag(x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2) U21_aag(x1, x2, x3) = U21_aag(x3) U22_aag(x1, x2, x3, x4) = U22_aag(x4) U26_aag(x1, x2, x3, x4) = U26_aag(x4) SUCC_IN_AG(x1, x2) = SUCC_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCC_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) SUCC_IN_AG(x1, x2) = SUCC_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: SUCC_IN_AG(zero(Z)) -> SUCC_IN_AG(Z) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *SUCC_IN_AG(zero(Z)) -> SUCC_IN_AG(Z) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDX_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> ADDX_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> ADDY_IN_AAG(X, Y, Z) ADDY_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> ADDC_IN_AAG(X, Y, Z) ADDC_IN_AAG(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAG^1(X, Y, Z) ADDX_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAG^1(X, Y, Z) ADDY_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAG(X, Y, Z) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g U27_g(x1, x2) = U27_g(x2) binaryZ_out_g(x1) = binaryZ_out_g U28_g(x1, x2) = U28_g(x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x4) U11_aag(x1, x2, x3, x4) = U11_aag(x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x4) U12_aag(x1, x2, x3, x4) = U12_aag(x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x4) U13_aag(x1, x2, x3, x4) = U13_aag(x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2) U14_aag(x1, x2, x3) = U14_aag(x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1) U34_ag(x1, x2, x3) = U34_ag(x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x3) U15_aag(x1, x2, x3) = U15_aag(x3) U16_aag(x1, x2, x3, x4) = U16_aag(x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2) U24_aag(x1, x2, x3, x4) = U24_aag(x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2) U18_aag(x1, x2, x3) = U18_aag(x3) U19_aag(x1, x2, x3, x4) = U19_aag(x4) U25_aag(x1, x2, x3, x4) = U25_aag(x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2) U21_aag(x1, x2, x3) = U21_aag(x3) U22_aag(x1, x2, x3, x4) = U22_aag(x4) U26_aag(x1, x2, x3, x4) = U26_aag(x4) ADDZ_IN_AAG(x1, x2, x3) = ADDZ_IN_AAG(x3) ADDX_IN_AAG(x1, x2, x3) = ADDX_IN_AAG(x3) ADDY_IN_AAG(x1, x2, x3) = ADDY_IN_AAG(x3) ADDC_IN_AAG(x1, x2, x3) = ADDC_IN_AAG(x3) ADDC_IN_AAG^1(x1, x2, x3) = ADDC_IN_AAG^1(x3) ADDX_IN_AAG^1(x1, x2, x3) = ADDX_IN_AAG^1(x3) ADDY_IN_AAG^1(x1, x2, x3) = ADDY_IN_AAG^1(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDX_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> ADDX_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> ADDY_IN_AAG(X, Y, Z) ADDY_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> ADDC_IN_AAG(X, Y, Z) ADDC_IN_AAG(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAG^1(X, Y, Z) ADDX_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAG^1(X, Y, Z) ADDY_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAG(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) ADDZ_IN_AAG(x1, x2, x3) = ADDZ_IN_AAG(x3) ADDX_IN_AAG(x1, x2, x3) = ADDX_IN_AAG(x3) ADDY_IN_AAG(x1, x2, x3) = ADDY_IN_AAG(x3) ADDC_IN_AAG(x1, x2, x3) = ADDC_IN_AAG(x3) ADDC_IN_AAG^1(x1, x2, x3) = ADDC_IN_AAG^1(x3) ADDX_IN_AAG^1(x1, x2, x3) = ADDX_IN_AAG^1(x3) ADDY_IN_AAG^1(x1, x2, x3) = ADDY_IN_AAG^1(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: ADDX_IN_AAG(Z) -> ADDZ_IN_AAG(Z) ADDZ_IN_AAG(zero(Z)) -> ADDZ_IN_AAG(Z) ADDZ_IN_AAG(one(Z)) -> ADDX_IN_AAG(Z) ADDZ_IN_AAG(one(Z)) -> ADDY_IN_AAG(Z) ADDY_IN_AAG(Z) -> ADDZ_IN_AAG(Z) ADDZ_IN_AAG(zero(Z)) -> ADDC_IN_AAG(Z) ADDC_IN_AAG(Z) -> ADDC_IN_AAG^1(Z) ADDC_IN_AAG^1(one(Z)) -> ADDZ_IN_AAG(Z) ADDC_IN_AAG^1(zero(Z)) -> ADDX_IN_AAG^1(Z) ADDX_IN_AAG^1(Z) -> ADDC_IN_AAG^1(Z) ADDC_IN_AAG^1(zero(Z)) -> ADDY_IN_AAG^1(Z) ADDY_IN_AAG^1(Z) -> ADDC_IN_AAG^1(Z) ADDC_IN_AAG^1(one(Z)) -> ADDC_IN_AAG(Z) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *ADDZ_IN_AAG(one(Z)) -> ADDX_IN_AAG(Z) The graph contains the following edges 1 > 1 *ADDZ_IN_AAG(zero(Z)) -> ADDZ_IN_AAG(Z) The graph contains the following edges 1 > 1 *ADDX_IN_AAG(Z) -> ADDZ_IN_AAG(Z) The graph contains the following edges 1 >= 1 *ADDY_IN_AAG(Z) -> ADDZ_IN_AAG(Z) The graph contains the following edges 1 >= 1 *ADDC_IN_AAG^1(one(Z)) -> ADDZ_IN_AAG(Z) The graph contains the following edges 1 > 1 *ADDZ_IN_AAG(one(Z)) -> ADDY_IN_AAG(Z) The graph contains the following edges 1 > 1 *ADDZ_IN_AAG(zero(Z)) -> ADDC_IN_AAG(Z) The graph contains the following edges 1 > 1 *ADDC_IN_AAG(Z) -> ADDC_IN_AAG^1(Z) The graph contains the following edges 1 >= 1 *ADDC_IN_AAG^1(one(Z)) -> ADDC_IN_AAG(Z) The graph contains the following edges 1 > 1 *ADDX_IN_AAG^1(Z) -> ADDC_IN_AAG^1(Z) The graph contains the following edges 1 >= 1 *ADDY_IN_AAG^1(Z) -> ADDC_IN_AAG^1(Z) The graph contains the following edges 1 >= 1 *ADDC_IN_AAG^1(zero(Z)) -> ADDX_IN_AAG^1(Z) The graph contains the following edges 1 > 1 *ADDC_IN_AAG^1(zero(Z)) -> ADDY_IN_AAG^1(Z) The graph contains the following edges 1 > 1 ---------------------------------------- (27) YES ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: BINARYZ_IN_A(one(X)) -> BINARY_IN_A(X) BINARY_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARYZ_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARY_IN_A(one(X)) -> BINARY_IN_A(X) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g U27_g(x1, x2) = U27_g(x2) binaryZ_out_g(x1) = binaryZ_out_g U28_g(x1, x2) = U28_g(x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x4) U11_aag(x1, x2, x3, x4) = U11_aag(x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x4) U12_aag(x1, x2, x3, x4) = U12_aag(x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x4) U13_aag(x1, x2, x3, x4) = U13_aag(x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2) U14_aag(x1, x2, x3) = U14_aag(x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1) U34_ag(x1, x2, x3) = U34_ag(x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x3) U15_aag(x1, x2, x3) = U15_aag(x3) U16_aag(x1, x2, x3, x4) = U16_aag(x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2) U24_aag(x1, x2, x3, x4) = U24_aag(x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2) U18_aag(x1, x2, x3) = U18_aag(x3) U19_aag(x1, x2, x3, x4) = U19_aag(x4) U25_aag(x1, x2, x3, x4) = U25_aag(x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2) U21_aag(x1, x2, x3) = U21_aag(x3) U22_aag(x1, x2, x3, x4) = U22_aag(x4) U26_aag(x1, x2, x3, x4) = U26_aag(x4) BINARYZ_IN_A(x1) = BINARYZ_IN_A BINARY_IN_A(x1) = BINARY_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: BINARYZ_IN_A(one(X)) -> BINARY_IN_A(X) BINARY_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARYZ_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARY_IN_A(one(X)) -> BINARY_IN_A(X) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) BINARYZ_IN_A(x1) = BINARYZ_IN_A BINARY_IN_A(x1) = BINARY_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: BINARYZ_IN_A -> BINARY_IN_A BINARY_IN_A -> BINARYZ_IN_A BINARYZ_IN_A -> BINARYZ_IN_A BINARY_IN_A -> BINARY_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = BINARYZ_IN_A evaluates to t =BINARYZ_IN_A Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from BINARYZ_IN_A to BINARYZ_IN_A. ---------------------------------------- (34) NO ---------------------------------------- (35) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCC_IN_AA(one(X), zero(Z)) -> SUCC_IN_AA(X, Z) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g U27_g(x1, x2) = U27_g(x2) binaryZ_out_g(x1) = binaryZ_out_g U28_g(x1, x2) = U28_g(x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x4) U11_aag(x1, x2, x3, x4) = U11_aag(x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x4) U12_aag(x1, x2, x3, x4) = U12_aag(x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x4) U13_aag(x1, x2, x3, x4) = U13_aag(x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2) U14_aag(x1, x2, x3) = U14_aag(x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1) U34_ag(x1, x2, x3) = U34_ag(x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x3) U15_aag(x1, x2, x3) = U15_aag(x3) U16_aag(x1, x2, x3, x4) = U16_aag(x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2) U24_aag(x1, x2, x3, x4) = U24_aag(x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2) U18_aag(x1, x2, x3) = U18_aag(x3) U19_aag(x1, x2, x3, x4) = U19_aag(x4) U25_aag(x1, x2, x3, x4) = U25_aag(x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2) U21_aag(x1, x2, x3) = U21_aag(x3) U22_aag(x1, x2, x3, x4) = U22_aag(x4) U26_aag(x1, x2, x3, x4) = U26_aag(x4) SUCC_IN_AA(x1, x2) = SUCC_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (36) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (37) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCC_IN_AA(one(X), zero(Z)) -> SUCC_IN_AA(X, Z) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) SUCC_IN_AA(x1, x2) = SUCC_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (38) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: SUCC_IN_AA -> SUCC_IN_AA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (40) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDX_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), one(Y), one(Z)) -> ADDX_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), zero(Y), one(Z)) -> ADDY_IN_AAA(X, Y, Z) ADDY_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), one(Y), zero(Z)) -> ADDC_IN_AAA(X, Y, Z) ADDC_IN_AAA(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDC_IN_AAA^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAA^1(X, Y, Z) ADDX_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAA^1(X, Y, Z) ADDY_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAA(X, Y, Z) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g U27_g(x1, x2) = U27_g(x2) binaryZ_out_g(x1) = binaryZ_out_g U28_g(x1, x2) = U28_g(x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x4) U11_aag(x1, x2, x3, x4) = U11_aag(x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x4) U12_aag(x1, x2, x3, x4) = U12_aag(x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x4) U13_aag(x1, x2, x3, x4) = U13_aag(x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2) U14_aag(x1, x2, x3) = U14_aag(x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1) U34_ag(x1, x2, x3) = U34_ag(x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x3) U15_aag(x1, x2, x3) = U15_aag(x3) U16_aag(x1, x2, x3, x4) = U16_aag(x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2) U24_aag(x1, x2, x3, x4) = U24_aag(x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2) U18_aag(x1, x2, x3) = U18_aag(x3) U19_aag(x1, x2, x3, x4) = U19_aag(x4) U25_aag(x1, x2, x3, x4) = U25_aag(x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2) U21_aag(x1, x2, x3) = U21_aag(x3) U22_aag(x1, x2, x3, x4) = U22_aag(x4) U26_aag(x1, x2, x3, x4) = U26_aag(x4) ADDZ_IN_AAA(x1, x2, x3) = ADDZ_IN_AAA ADDX_IN_AAA(x1, x2, x3) = ADDX_IN_AAA ADDY_IN_AAA(x1, x2, x3) = ADDY_IN_AAA ADDC_IN_AAA(x1, x2, x3) = ADDC_IN_AAA ADDC_IN_AAA^1(x1, x2, x3) = ADDC_IN_AAA^1 ADDX_IN_AAA^1(x1, x2, x3) = ADDX_IN_AAA^1 ADDY_IN_AAA^1(x1, x2, x3) = ADDY_IN_AAA^1 We have to consider all (P,R,Pi)-chains ---------------------------------------- (41) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (42) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDX_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), one(Y), one(Z)) -> ADDX_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), zero(Y), one(Z)) -> ADDY_IN_AAA(X, Y, Z) ADDY_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), one(Y), zero(Z)) -> ADDC_IN_AAA(X, Y, Z) ADDC_IN_AAA(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDC_IN_AAA^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAA^1(X, Y, Z) ADDX_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAA^1(X, Y, Z) ADDY_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) ADDZ_IN_AAA(x1, x2, x3) = ADDZ_IN_AAA ADDX_IN_AAA(x1, x2, x3) = ADDX_IN_AAA ADDY_IN_AAA(x1, x2, x3) = ADDY_IN_AAA ADDC_IN_AAA(x1, x2, x3) = ADDC_IN_AAA ADDC_IN_AAA^1(x1, x2, x3) = ADDC_IN_AAA^1 ADDX_IN_AAA^1(x1, x2, x3) = ADDX_IN_AAA^1 ADDY_IN_AAA^1(x1, x2, x3) = ADDY_IN_AAA^1 We have to consider all (P,R,Pi)-chains ---------------------------------------- (43) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_AAA(one(R), S, RSS) -> TIMES_IN_AAA(R, S, RS) TIMES_IN_AAA(zero(R), S, zero(RS)) -> TIMES_IN_AAA(R, S, RS) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g U27_g(x1, x2) = U27_g(x2) binaryZ_out_g(x1) = binaryZ_out_g U28_g(x1, x2) = U28_g(x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x4) U11_aag(x1, x2, x3, x4) = U11_aag(x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x4) U12_aag(x1, x2, x3, x4) = U12_aag(x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x4) U13_aag(x1, x2, x3, x4) = U13_aag(x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2) U14_aag(x1, x2, x3) = U14_aag(x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1) U34_ag(x1, x2, x3) = U34_ag(x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x3) U15_aag(x1, x2, x3) = U15_aag(x3) U16_aag(x1, x2, x3, x4) = U16_aag(x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2) U24_aag(x1, x2, x3, x4) = U24_aag(x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2) U18_aag(x1, x2, x3) = U18_aag(x3) U19_aag(x1, x2, x3, x4) = U19_aag(x4) U25_aag(x1, x2, x3, x4) = U25_aag(x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2) U21_aag(x1, x2, x3) = U21_aag(x3) U22_aag(x1, x2, x3, x4) = U22_aag(x4) U26_aag(x1, x2, x3, x4) = U26_aag(x4) TIMES_IN_AAA(x1, x2, x3) = TIMES_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (44) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (45) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_AAA(one(R), S, RSS) -> TIMES_IN_AAA(R, S, RS) TIMES_IN_AAA(zero(R), S, zero(RS)) -> TIMES_IN_AAA(R, S, RS) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) TIMES_IN_AAA(x1, x2, x3) = TIMES_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (46) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_AAG(zero(R), S, zero(RS)) -> TIMES_IN_AAG(R, S, RS) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g U27_g(x1, x2) = U27_g(x2) binaryZ_out_g(x1) = binaryZ_out_g U28_g(x1, x2) = U28_g(x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x4) U11_aag(x1, x2, x3, x4) = U11_aag(x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x4) U12_aag(x1, x2, x3, x4) = U12_aag(x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x4) U13_aag(x1, x2, x3, x4) = U13_aag(x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2) U14_aag(x1, x2, x3) = U14_aag(x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1) U34_ag(x1, x2, x3) = U34_ag(x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x3) U15_aag(x1, x2, x3) = U15_aag(x3) U16_aag(x1, x2, x3, x4) = U16_aag(x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2) U24_aag(x1, x2, x3, x4) = U24_aag(x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2) U18_aag(x1, x2, x3) = U18_aag(x3) U19_aag(x1, x2, x3, x4) = U19_aag(x4) U25_aag(x1, x2, x3, x4) = U25_aag(x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2) U21_aag(x1, x2, x3) = U21_aag(x3) U22_aag(x1, x2, x3, x4) = U22_aag(x4) U26_aag(x1, x2, x3, x4) = U26_aag(x4) TIMES_IN_AAG(x1, x2, x3) = TIMES_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (47) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (48) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_AAG(zero(R), S, zero(RS)) -> TIMES_IN_AAG(R, S, RS) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) TIMES_IN_AAG(x1, x2, x3) = TIMES_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (49) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: times_in_3: (f,f,b) (f,f,f) add_in_3: (f,f,f) (f,f,b) binaryZ_in_1: (f) (b) binary_in_1: (f) (b) addz_in_3: (f,f,f) (f,f,b) addx_in_3: (f,f,f) (f,f,b) addy_in_3: (f,f,f) (f,f,b) addc_in_3: (f,f,f) (f,f,b) succZ_in_2: (f,f) (f,b) succ_in_2: (f,f) (f,b) addC_in_3: (f,f,f) (f,f,b) addX_in_3: (f,f,f) (f,f,b) addY_in_3: (f,f,f) (f,f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2, x3) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x3, x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x3, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2, x3) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x1, x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x1, x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g(x1) U27_g(x1, x2) = U27_g(x1, x2) binaryZ_out_g(x1) = binaryZ_out_g(x1) U28_g(x1, x2) = U28_g(x1, x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x3, x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x3, x4) U11_aag(x1, x2, x3, x4) = U11_aag(x3, x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2, x3) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x3, x4) U12_aag(x1, x2, x3, x4) = U12_aag(x3, x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2, x3) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x3, x4) U13_aag(x1, x2, x3, x4) = U13_aag(x3, x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2, x3) U14_aag(x1, x2, x3) = U14_aag(x2, x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1, x2) U34_ag(x1, x2, x3) = U34_ag(x2, x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1, x2) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x2, x3) U15_aag(x1, x2, x3) = U15_aag(x2, x3) U16_aag(x1, x2, x3, x4) = U16_aag(x3, x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x3, x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2, x3) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2, x3) U24_aag(x1, x2, x3, x4) = U24_aag(x3, x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2, x3) U18_aag(x1, x2, x3) = U18_aag(x2, x3) U19_aag(x1, x2, x3, x4) = U19_aag(x3, x4) U25_aag(x1, x2, x3, x4) = U25_aag(x3, x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2, x3) U21_aag(x1, x2, x3) = U21_aag(x2, x3) U22_aag(x1, x2, x3, x4) = U22_aag(x3, x4) U26_aag(x1, x2, x3, x4) = U26_aag(x3, x4) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (50) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2, x3) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x3, x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x3, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2, x3) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x1, x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x1, x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g(x1) U27_g(x1, x2) = U27_g(x1, x2) binaryZ_out_g(x1) = binaryZ_out_g(x1) U28_g(x1, x2) = U28_g(x1, x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x3, x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x3, x4) U11_aag(x1, x2, x3, x4) = U11_aag(x3, x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2, x3) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x3, x4) U12_aag(x1, x2, x3, x4) = U12_aag(x3, x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2, x3) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x3, x4) U13_aag(x1, x2, x3, x4) = U13_aag(x3, x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2, x3) U14_aag(x1, x2, x3) = U14_aag(x2, x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1, x2) U34_ag(x1, x2, x3) = U34_ag(x2, x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1, x2) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x2, x3) U15_aag(x1, x2, x3) = U15_aag(x2, x3) U16_aag(x1, x2, x3, x4) = U16_aag(x3, x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x3, x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2, x3) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2, x3) U24_aag(x1, x2, x3, x4) = U24_aag(x3, x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2, x3) U18_aag(x1, x2, x3) = U18_aag(x2, x3) U19_aag(x1, x2, x3, x4) = U19_aag(x3, x4) U25_aag(x1, x2, x3, x4) = U25_aag(x3, x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2, x3) U21_aag(x1, x2, x3) = U21_aag(x2, x3) U22_aag(x1, x2, x3, x4) = U22_aag(x3, x4) U26_aag(x1, x2, x3, x4) = U26_aag(x3, x4) ---------------------------------------- (51) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_AAG(zero(R), S, zero(RS)) -> U35_AAG(R, S, RS, times_in_aag(R, S, RS)) TIMES_IN_AAG(zero(R), S, zero(RS)) -> TIMES_IN_AAG(R, S, RS) TIMES_IN_AAG(one(R), S, RSS) -> U36_AAG(R, S, RSS, times_in_aaa(R, S, RS)) TIMES_IN_AAG(one(R), S, RSS) -> TIMES_IN_AAA(R, S, RS) TIMES_IN_AAA(zero(R), S, zero(RS)) -> U35_AAA(R, S, RS, times_in_aaa(R, S, RS)) TIMES_IN_AAA(zero(R), S, zero(RS)) -> TIMES_IN_AAA(R, S, RS) TIMES_IN_AAA(one(R), S, RSS) -> U36_AAA(R, S, RSS, times_in_aaa(R, S, RS)) TIMES_IN_AAA(one(R), S, RSS) -> TIMES_IN_AAA(R, S, RS) U36_AAA(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_AAA(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) U36_AAA(R, S, RSS, times_out_aaa(R, S, RS)) -> ADD_IN_AAA(S, zero(RS), RSS) ADD_IN_AAA(X, b, X) -> U1_AAA(X, binaryZ_in_a(X)) ADD_IN_AAA(X, b, X) -> BINARYZ_IN_A(X) BINARYZ_IN_A(zero(X)) -> U29_A(X, binaryZ_in_a(X)) BINARYZ_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARYZ_IN_A(one(X)) -> U30_A(X, binary_in_a(X)) BINARYZ_IN_A(one(X)) -> BINARY_IN_A(X) BINARY_IN_A(zero(X)) -> U27_A(X, binaryZ_in_a(X)) BINARY_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARY_IN_A(one(X)) -> U28_A(X, binary_in_a(X)) BINARY_IN_A(one(X)) -> BINARY_IN_A(X) ADD_IN_AAA(b, Y, Y) -> U2_AAA(Y, binaryZ_in_a(Y)) ADD_IN_AAA(b, Y, Y) -> BINARYZ_IN_A(Y) ADD_IN_AAA(X, Y, Z) -> U3_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADD_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), zero(Y), zero(Z)) -> U10_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDZ_IN_AAA(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), one(Y), one(Z)) -> U11_AAA(X, Y, Z, addx_in_aaa(X, Y, Z)) ADDZ_IN_AAA(zero(X), one(Y), one(Z)) -> ADDX_IN_AAA(X, Y, Z) ADDX_IN_AAA(one(X), b, one(X)) -> U4_AAA(X, binary_in_a(X)) ADDX_IN_AAA(one(X), b, one(X)) -> BINARY_IN_A(X) ADDX_IN_AAA(zero(X), b, zero(X)) -> U5_AAA(X, binaryZ_in_a(X)) ADDX_IN_AAA(zero(X), b, zero(X)) -> BINARYZ_IN_A(X) ADDX_IN_AAA(X, Y, Z) -> U6_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDX_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), zero(Y), one(Z)) -> U12_AAA(X, Y, Z, addy_in_aaa(X, Y, Z)) ADDZ_IN_AAA(one(X), zero(Y), one(Z)) -> ADDY_IN_AAA(X, Y, Z) ADDY_IN_AAA(b, one(Y), one(Y)) -> U7_AAA(Y, binary_in_a(Y)) ADDY_IN_AAA(b, one(Y), one(Y)) -> BINARY_IN_A(Y) ADDY_IN_AAA(b, zero(Y), zero(Y)) -> U8_AAA(Y, binaryZ_in_a(Y)) ADDY_IN_AAA(b, zero(Y), zero(Y)) -> BINARYZ_IN_A(Y) ADDY_IN_AAA(X, Y, Z) -> U9_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDY_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), one(Y), zero(Z)) -> U13_AAA(X, Y, Z, addc_in_aaa(X, Y, Z)) ADDZ_IN_AAA(one(X), one(Y), zero(Z)) -> ADDC_IN_AAA(X, Y, Z) ADDC_IN_AAA(X, b, Z) -> U14_AAA(X, Z, succZ_in_aa(X, Z)) ADDC_IN_AAA(X, b, Z) -> SUCCZ_IN_AA(X, Z) SUCCZ_IN_AA(zero(X), one(X)) -> U33_AA(X, binaryZ_in_a(X)) SUCCZ_IN_AA(zero(X), one(X)) -> BINARYZ_IN_A(X) SUCCZ_IN_AA(one(X), zero(Z)) -> U34_AA(X, Z, succ_in_aa(X, Z)) SUCCZ_IN_AA(one(X), zero(Z)) -> SUCC_IN_AA(X, Z) SUCC_IN_AA(zero(X), one(X)) -> U31_AA(X, binaryZ_in_a(X)) SUCC_IN_AA(zero(X), one(X)) -> BINARYZ_IN_A(X) SUCC_IN_AA(one(X), zero(Z)) -> U32_AA(X, Z, succ_in_aa(X, Z)) SUCC_IN_AA(one(X), zero(Z)) -> SUCC_IN_AA(X, Z) ADDC_IN_AAA(b, Y, Z) -> U15_AAA(Y, Z, succZ_in_aa(Y, Z)) ADDC_IN_AAA(b, Y, Z) -> SUCCZ_IN_AA(Y, Z) ADDC_IN_AAA(X, Y, Z) -> U16_AAA(X, Y, Z, addC_in_aaa(X, Y, Z)) ADDC_IN_AAA(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(zero(X), zero(Y), one(Z)) -> U23_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDC_IN_AAA^1(zero(X), one(Y), zero(Z)) -> U24_AAA(X, Y, Z, addX_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAA^1(X, Y, Z) ADDX_IN_AAA^1(zero(X), b, one(X)) -> U17_AAA(X, binaryZ_in_a(X)) ADDX_IN_AAA^1(zero(X), b, one(X)) -> BINARYZ_IN_A(X) ADDX_IN_AAA^1(one(X), b, zero(Z)) -> U18_AAA(X, Z, succ_in_aa(X, Z)) ADDX_IN_AAA^1(one(X), b, zero(Z)) -> SUCC_IN_AA(X, Z) ADDX_IN_AAA^1(X, Y, Z) -> U19_AAA(X, Y, Z, addC_in_aaa(X, Y, Z)) ADDX_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), zero(Y), zero(Z)) -> U25_AAA(X, Y, Z, addY_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAA^1(X, Y, Z) ADDY_IN_AAA^1(b, zero(Y), one(Y)) -> U20_AAA(Y, binaryZ_in_a(Y)) ADDY_IN_AAA^1(b, zero(Y), one(Y)) -> BINARYZ_IN_A(Y) ADDY_IN_AAA^1(b, one(Y), zero(Z)) -> U21_AAA(Y, Z, succ_in_aa(Y, Z)) ADDY_IN_AAA^1(b, one(Y), zero(Z)) -> SUCC_IN_AA(Y, Z) ADDY_IN_AAA^1(X, Y, Z) -> U22_AAA(X, Y, Z, addC_in_aaa(X, Y, Z)) ADDY_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), one(Y), one(Z)) -> U26_AAA(X, Y, Z, addc_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAA(X, Y, Z) U36_AAG(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_AAG(R, S, RSS, add_in_aag(S, zero(RS), RSS)) U36_AAG(R, S, RSS, times_out_aaa(R, S, RS)) -> ADD_IN_AAG(S, zero(RS), RSS) ADD_IN_AAG(X, b, X) -> U1_AAG(X, binaryZ_in_g(X)) ADD_IN_AAG(X, b, X) -> BINARYZ_IN_G(X) BINARYZ_IN_G(zero(X)) -> U29_G(X, binaryZ_in_g(X)) BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARYZ_IN_G(one(X)) -> U30_G(X, binary_in_g(X)) BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X) BINARY_IN_G(zero(X)) -> U27_G(X, binaryZ_in_g(X)) BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARY_IN_G(one(X)) -> U28_G(X, binary_in_g(X)) BINARY_IN_G(one(X)) -> BINARY_IN_G(X) ADD_IN_AAG(b, Y, Y) -> U2_AAG(Y, binaryZ_in_g(Y)) ADD_IN_AAG(b, Y, Y) -> BINARYZ_IN_G(Y) ADD_IN_AAG(X, Y, Z) -> U3_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADD_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> U10_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> U11_AAG(X, Y, Z, addx_in_aag(X, Y, Z)) ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> ADDX_IN_AAG(X, Y, Z) ADDX_IN_AAG(one(X), b, one(X)) -> U4_AAG(X, binary_in_g(X)) ADDX_IN_AAG(one(X), b, one(X)) -> BINARY_IN_G(X) ADDX_IN_AAG(zero(X), b, zero(X)) -> U5_AAG(X, binaryZ_in_g(X)) ADDX_IN_AAG(zero(X), b, zero(X)) -> BINARYZ_IN_G(X) ADDX_IN_AAG(X, Y, Z) -> U6_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDX_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> U12_AAG(X, Y, Z, addy_in_aag(X, Y, Z)) ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> ADDY_IN_AAG(X, Y, Z) ADDY_IN_AAG(b, one(Y), one(Y)) -> U7_AAG(Y, binary_in_g(Y)) ADDY_IN_AAG(b, one(Y), one(Y)) -> BINARY_IN_G(Y) ADDY_IN_AAG(b, zero(Y), zero(Y)) -> U8_AAG(Y, binaryZ_in_g(Y)) ADDY_IN_AAG(b, zero(Y), zero(Y)) -> BINARYZ_IN_G(Y) ADDY_IN_AAG(X, Y, Z) -> U9_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDY_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> U13_AAG(X, Y, Z, addc_in_aag(X, Y, Z)) ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> ADDC_IN_AAG(X, Y, Z) ADDC_IN_AAG(X, b, Z) -> U14_AAG(X, Z, succZ_in_ag(X, Z)) ADDC_IN_AAG(X, b, Z) -> SUCCZ_IN_AG(X, Z) SUCCZ_IN_AG(zero(X), one(X)) -> U33_AG(X, binaryZ_in_g(X)) SUCCZ_IN_AG(zero(X), one(X)) -> BINARYZ_IN_G(X) SUCCZ_IN_AG(one(X), zero(Z)) -> U34_AG(X, Z, succ_in_ag(X, Z)) SUCCZ_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z) SUCC_IN_AG(zero(X), one(X)) -> U31_AG(X, binaryZ_in_g(X)) SUCC_IN_AG(zero(X), one(X)) -> BINARYZ_IN_G(X) SUCC_IN_AG(one(X), zero(Z)) -> U32_AG(X, Z, succ_in_ag(X, Z)) SUCC_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z) ADDC_IN_AAG(b, Y, Z) -> U15_AAG(Y, Z, succZ_in_ag(Y, Z)) ADDC_IN_AAG(b, Y, Z) -> SUCCZ_IN_AG(Y, Z) ADDC_IN_AAG(X, Y, Z) -> U16_AAG(X, Y, Z, addC_in_aag(X, Y, Z)) ADDC_IN_AAG(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> U23_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> U24_AAG(X, Y, Z, addX_in_aag(X, Y, Z)) ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAG^1(X, Y, Z) ADDX_IN_AAG^1(zero(X), b, one(X)) -> U17_AAG(X, binaryZ_in_g(X)) ADDX_IN_AAG^1(zero(X), b, one(X)) -> BINARYZ_IN_G(X) ADDX_IN_AAG^1(one(X), b, zero(Z)) -> U18_AAG(X, Z, succ_in_ag(X, Z)) ADDX_IN_AAG^1(one(X), b, zero(Z)) -> SUCC_IN_AG(X, Z) ADDX_IN_AAG^1(X, Y, Z) -> U19_AAG(X, Y, Z, addC_in_aag(X, Y, Z)) ADDX_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> U25_AAG(X, Y, Z, addY_in_aag(X, Y, Z)) ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAG^1(X, Y, Z) ADDY_IN_AAG^1(b, zero(Y), one(Y)) -> U20_AAG(Y, binaryZ_in_g(Y)) ADDY_IN_AAG^1(b, zero(Y), one(Y)) -> BINARYZ_IN_G(Y) ADDY_IN_AAG^1(b, one(Y), zero(Z)) -> U21_AAG(Y, Z, succ_in_ag(Y, Z)) ADDY_IN_AAG^1(b, one(Y), zero(Z)) -> SUCC_IN_AG(Y, Z) ADDY_IN_AAG^1(X, Y, Z) -> U22_AAG(X, Y, Z, addC_in_aag(X, Y, Z)) ADDY_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> U26_AAG(X, Y, Z, addc_in_aag(X, Y, Z)) ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAG(X, Y, Z) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2, x3) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x3, x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x3, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2, x3) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x1, x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x1, x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g(x1) U27_g(x1, x2) = U27_g(x1, x2) binaryZ_out_g(x1) = binaryZ_out_g(x1) U28_g(x1, x2) = U28_g(x1, x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x3, x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x3, x4) U11_aag(x1, x2, x3, x4) = U11_aag(x3, x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2, x3) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x3, x4) U12_aag(x1, x2, x3, x4) = U12_aag(x3, x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2, x3) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x3, x4) U13_aag(x1, x2, x3, x4) = U13_aag(x3, x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2, x3) U14_aag(x1, x2, x3) = U14_aag(x2, x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1, x2) U34_ag(x1, x2, x3) = U34_ag(x2, x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1, x2) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x2, x3) U15_aag(x1, x2, x3) = U15_aag(x2, x3) U16_aag(x1, x2, x3, x4) = U16_aag(x3, x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x3, x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2, x3) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2, x3) U24_aag(x1, x2, x3, x4) = U24_aag(x3, x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2, x3) U18_aag(x1, x2, x3) = U18_aag(x2, x3) U19_aag(x1, x2, x3, x4) = U19_aag(x3, x4) U25_aag(x1, x2, x3, x4) = U25_aag(x3, x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2, x3) U21_aag(x1, x2, x3) = U21_aag(x2, x3) U22_aag(x1, x2, x3, x4) = U22_aag(x3, x4) U26_aag(x1, x2, x3, x4) = U26_aag(x3, x4) TIMES_IN_AAG(x1, x2, x3) = TIMES_IN_AAG(x3) U35_AAG(x1, x2, x3, x4) = U35_AAG(x3, x4) U36_AAG(x1, x2, x3, x4) = U36_AAG(x3, x4) TIMES_IN_AAA(x1, x2, x3) = TIMES_IN_AAA U35_AAA(x1, x2, x3, x4) = U35_AAA(x4) U36_AAA(x1, x2, x3, x4) = U36_AAA(x4) U37_AAA(x1, x2, x3, x4) = U37_AAA(x1, x4) ADD_IN_AAA(x1, x2, x3) = ADD_IN_AAA U1_AAA(x1, x2) = U1_AAA(x2) BINARYZ_IN_A(x1) = BINARYZ_IN_A U29_A(x1, x2) = U29_A(x2) U30_A(x1, x2) = U30_A(x2) BINARY_IN_A(x1) = BINARY_IN_A U27_A(x1, x2) = U27_A(x2) U28_A(x1, x2) = U28_A(x2) U2_AAA(x1, x2) = U2_AAA(x2) U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) ADDZ_IN_AAA(x1, x2, x3) = ADDZ_IN_AAA U10_AAA(x1, x2, x3, x4) = U10_AAA(x4) U11_AAA(x1, x2, x3, x4) = U11_AAA(x4) ADDX_IN_AAA(x1, x2, x3) = ADDX_IN_AAA U4_AAA(x1, x2) = U4_AAA(x2) U5_AAA(x1, x2) = U5_AAA(x2) U6_AAA(x1, x2, x3, x4) = U6_AAA(x4) U12_AAA(x1, x2, x3, x4) = U12_AAA(x4) ADDY_IN_AAA(x1, x2, x3) = ADDY_IN_AAA U7_AAA(x1, x2) = U7_AAA(x2) U8_AAA(x1, x2) = U8_AAA(x2) U9_AAA(x1, x2, x3, x4) = U9_AAA(x4) U13_AAA(x1, x2, x3, x4) = U13_AAA(x4) ADDC_IN_AAA(x1, x2, x3) = ADDC_IN_AAA U14_AAA(x1, x2, x3) = U14_AAA(x3) SUCCZ_IN_AA(x1, x2) = SUCCZ_IN_AA U33_AA(x1, x2) = U33_AA(x2) U34_AA(x1, x2, x3) = U34_AA(x3) SUCC_IN_AA(x1, x2) = SUCC_IN_AA U31_AA(x1, x2) = U31_AA(x2) U32_AA(x1, x2, x3) = U32_AA(x3) U15_AAA(x1, x2, x3) = U15_AAA(x3) U16_AAA(x1, x2, x3, x4) = U16_AAA(x4) ADDC_IN_AAA^1(x1, x2, x3) = ADDC_IN_AAA^1 U23_AAA(x1, x2, x3, x4) = U23_AAA(x4) U24_AAA(x1, x2, x3, x4) = U24_AAA(x4) ADDX_IN_AAA^1(x1, x2, x3) = ADDX_IN_AAA^1 U17_AAA(x1, x2) = U17_AAA(x2) U18_AAA(x1, x2, x3) = U18_AAA(x3) U19_AAA(x1, x2, x3, x4) = U19_AAA(x4) U25_AAA(x1, x2, x3, x4) = U25_AAA(x4) ADDY_IN_AAA^1(x1, x2, x3) = ADDY_IN_AAA^1 U20_AAA(x1, x2) = U20_AAA(x2) U21_AAA(x1, x2, x3) = U21_AAA(x3) U22_AAA(x1, x2, x3, x4) = U22_AAA(x4) U26_AAA(x1, x2, x3, x4) = U26_AAA(x4) U37_AAG(x1, x2, x3, x4) = U37_AAG(x1, x3, x4) ADD_IN_AAG(x1, x2, x3) = ADD_IN_AAG(x3) U1_AAG(x1, x2) = U1_AAG(x1, x2) BINARYZ_IN_G(x1) = BINARYZ_IN_G(x1) U29_G(x1, x2) = U29_G(x1, x2) U30_G(x1, x2) = U30_G(x1, x2) BINARY_IN_G(x1) = BINARY_IN_G(x1) U27_G(x1, x2) = U27_G(x1, x2) U28_G(x1, x2) = U28_G(x1, x2) U2_AAG(x1, x2) = U2_AAG(x1, x2) U3_AAG(x1, x2, x3, x4) = U3_AAG(x3, x4) ADDZ_IN_AAG(x1, x2, x3) = ADDZ_IN_AAG(x3) U10_AAG(x1, x2, x3, x4) = U10_AAG(x3, x4) U11_AAG(x1, x2, x3, x4) = U11_AAG(x3, x4) ADDX_IN_AAG(x1, x2, x3) = ADDX_IN_AAG(x3) U4_AAG(x1, x2) = U4_AAG(x1, x2) U5_AAG(x1, x2) = U5_AAG(x1, x2) U6_AAG(x1, x2, x3, x4) = U6_AAG(x3, x4) U12_AAG(x1, x2, x3, x4) = U12_AAG(x3, x4) ADDY_IN_AAG(x1, x2, x3) = ADDY_IN_AAG(x3) U7_AAG(x1, x2) = U7_AAG(x1, x2) U8_AAG(x1, x2) = U8_AAG(x1, x2) U9_AAG(x1, x2, x3, x4) = U9_AAG(x3, x4) U13_AAG(x1, x2, x3, x4) = U13_AAG(x3, x4) ADDC_IN_AAG(x1, x2, x3) = ADDC_IN_AAG(x3) U14_AAG(x1, x2, x3) = U14_AAG(x2, x3) SUCCZ_IN_AG(x1, x2) = SUCCZ_IN_AG(x2) U33_AG(x1, x2) = U33_AG(x1, x2) U34_AG(x1, x2, x3) = U34_AG(x2, x3) SUCC_IN_AG(x1, x2) = SUCC_IN_AG(x2) U31_AG(x1, x2) = U31_AG(x1, x2) U32_AG(x1, x2, x3) = U32_AG(x2, x3) U15_AAG(x1, x2, x3) = U15_AAG(x2, x3) U16_AAG(x1, x2, x3, x4) = U16_AAG(x3, x4) ADDC_IN_AAG^1(x1, x2, x3) = ADDC_IN_AAG^1(x3) U23_AAG(x1, x2, x3, x4) = U23_AAG(x3, x4) U24_AAG(x1, x2, x3, x4) = U24_AAG(x3, x4) ADDX_IN_AAG^1(x1, x2, x3) = ADDX_IN_AAG^1(x3) U17_AAG(x1, x2) = U17_AAG(x1, x2) U18_AAG(x1, x2, x3) = U18_AAG(x2, x3) U19_AAG(x1, x2, x3, x4) = U19_AAG(x3, x4) U25_AAG(x1, x2, x3, x4) = U25_AAG(x3, x4) ADDY_IN_AAG^1(x1, x2, x3) = ADDY_IN_AAG^1(x3) U20_AAG(x1, x2) = U20_AAG(x1, x2) U21_AAG(x1, x2, x3) = U21_AAG(x2, x3) U22_AAG(x1, x2, x3, x4) = U22_AAG(x3, x4) U26_AAG(x1, x2, x3, x4) = U26_AAG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (52) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_AAG(zero(R), S, zero(RS)) -> U35_AAG(R, S, RS, times_in_aag(R, S, RS)) TIMES_IN_AAG(zero(R), S, zero(RS)) -> TIMES_IN_AAG(R, S, RS) TIMES_IN_AAG(one(R), S, RSS) -> U36_AAG(R, S, RSS, times_in_aaa(R, S, RS)) TIMES_IN_AAG(one(R), S, RSS) -> TIMES_IN_AAA(R, S, RS) TIMES_IN_AAA(zero(R), S, zero(RS)) -> U35_AAA(R, S, RS, times_in_aaa(R, S, RS)) TIMES_IN_AAA(zero(R), S, zero(RS)) -> TIMES_IN_AAA(R, S, RS) TIMES_IN_AAA(one(R), S, RSS) -> U36_AAA(R, S, RSS, times_in_aaa(R, S, RS)) TIMES_IN_AAA(one(R), S, RSS) -> TIMES_IN_AAA(R, S, RS) U36_AAA(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_AAA(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) U36_AAA(R, S, RSS, times_out_aaa(R, S, RS)) -> ADD_IN_AAA(S, zero(RS), RSS) ADD_IN_AAA(X, b, X) -> U1_AAA(X, binaryZ_in_a(X)) ADD_IN_AAA(X, b, X) -> BINARYZ_IN_A(X) BINARYZ_IN_A(zero(X)) -> U29_A(X, binaryZ_in_a(X)) BINARYZ_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARYZ_IN_A(one(X)) -> U30_A(X, binary_in_a(X)) BINARYZ_IN_A(one(X)) -> BINARY_IN_A(X) BINARY_IN_A(zero(X)) -> U27_A(X, binaryZ_in_a(X)) BINARY_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARY_IN_A(one(X)) -> U28_A(X, binary_in_a(X)) BINARY_IN_A(one(X)) -> BINARY_IN_A(X) ADD_IN_AAA(b, Y, Y) -> U2_AAA(Y, binaryZ_in_a(Y)) ADD_IN_AAA(b, Y, Y) -> BINARYZ_IN_A(Y) ADD_IN_AAA(X, Y, Z) -> U3_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADD_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), zero(Y), zero(Z)) -> U10_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDZ_IN_AAA(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), one(Y), one(Z)) -> U11_AAA(X, Y, Z, addx_in_aaa(X, Y, Z)) ADDZ_IN_AAA(zero(X), one(Y), one(Z)) -> ADDX_IN_AAA(X, Y, Z) ADDX_IN_AAA(one(X), b, one(X)) -> U4_AAA(X, binary_in_a(X)) ADDX_IN_AAA(one(X), b, one(X)) -> BINARY_IN_A(X) ADDX_IN_AAA(zero(X), b, zero(X)) -> U5_AAA(X, binaryZ_in_a(X)) ADDX_IN_AAA(zero(X), b, zero(X)) -> BINARYZ_IN_A(X) ADDX_IN_AAA(X, Y, Z) -> U6_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDX_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), zero(Y), one(Z)) -> U12_AAA(X, Y, Z, addy_in_aaa(X, Y, Z)) ADDZ_IN_AAA(one(X), zero(Y), one(Z)) -> ADDY_IN_AAA(X, Y, Z) ADDY_IN_AAA(b, one(Y), one(Y)) -> U7_AAA(Y, binary_in_a(Y)) ADDY_IN_AAA(b, one(Y), one(Y)) -> BINARY_IN_A(Y) ADDY_IN_AAA(b, zero(Y), zero(Y)) -> U8_AAA(Y, binaryZ_in_a(Y)) ADDY_IN_AAA(b, zero(Y), zero(Y)) -> BINARYZ_IN_A(Y) ADDY_IN_AAA(X, Y, Z) -> U9_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDY_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), one(Y), zero(Z)) -> U13_AAA(X, Y, Z, addc_in_aaa(X, Y, Z)) ADDZ_IN_AAA(one(X), one(Y), zero(Z)) -> ADDC_IN_AAA(X, Y, Z) ADDC_IN_AAA(X, b, Z) -> U14_AAA(X, Z, succZ_in_aa(X, Z)) ADDC_IN_AAA(X, b, Z) -> SUCCZ_IN_AA(X, Z) SUCCZ_IN_AA(zero(X), one(X)) -> U33_AA(X, binaryZ_in_a(X)) SUCCZ_IN_AA(zero(X), one(X)) -> BINARYZ_IN_A(X) SUCCZ_IN_AA(one(X), zero(Z)) -> U34_AA(X, Z, succ_in_aa(X, Z)) SUCCZ_IN_AA(one(X), zero(Z)) -> SUCC_IN_AA(X, Z) SUCC_IN_AA(zero(X), one(X)) -> U31_AA(X, binaryZ_in_a(X)) SUCC_IN_AA(zero(X), one(X)) -> BINARYZ_IN_A(X) SUCC_IN_AA(one(X), zero(Z)) -> U32_AA(X, Z, succ_in_aa(X, Z)) SUCC_IN_AA(one(X), zero(Z)) -> SUCC_IN_AA(X, Z) ADDC_IN_AAA(b, Y, Z) -> U15_AAA(Y, Z, succZ_in_aa(Y, Z)) ADDC_IN_AAA(b, Y, Z) -> SUCCZ_IN_AA(Y, Z) ADDC_IN_AAA(X, Y, Z) -> U16_AAA(X, Y, Z, addC_in_aaa(X, Y, Z)) ADDC_IN_AAA(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(zero(X), zero(Y), one(Z)) -> U23_AAA(X, Y, Z, addz_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDC_IN_AAA^1(zero(X), one(Y), zero(Z)) -> U24_AAA(X, Y, Z, addX_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAA^1(X, Y, Z) ADDX_IN_AAA^1(zero(X), b, one(X)) -> U17_AAA(X, binaryZ_in_a(X)) ADDX_IN_AAA^1(zero(X), b, one(X)) -> BINARYZ_IN_A(X) ADDX_IN_AAA^1(one(X), b, zero(Z)) -> U18_AAA(X, Z, succ_in_aa(X, Z)) ADDX_IN_AAA^1(one(X), b, zero(Z)) -> SUCC_IN_AA(X, Z) ADDX_IN_AAA^1(X, Y, Z) -> U19_AAA(X, Y, Z, addC_in_aaa(X, Y, Z)) ADDX_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), zero(Y), zero(Z)) -> U25_AAA(X, Y, Z, addY_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAA^1(X, Y, Z) ADDY_IN_AAA^1(b, zero(Y), one(Y)) -> U20_AAA(Y, binaryZ_in_a(Y)) ADDY_IN_AAA^1(b, zero(Y), one(Y)) -> BINARYZ_IN_A(Y) ADDY_IN_AAA^1(b, one(Y), zero(Z)) -> U21_AAA(Y, Z, succ_in_aa(Y, Z)) ADDY_IN_AAA^1(b, one(Y), zero(Z)) -> SUCC_IN_AA(Y, Z) ADDY_IN_AAA^1(X, Y, Z) -> U22_AAA(X, Y, Z, addC_in_aaa(X, Y, Z)) ADDY_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), one(Y), one(Z)) -> U26_AAA(X, Y, Z, addc_in_aaa(X, Y, Z)) ADDC_IN_AAA^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAA(X, Y, Z) U36_AAG(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_AAG(R, S, RSS, add_in_aag(S, zero(RS), RSS)) U36_AAG(R, S, RSS, times_out_aaa(R, S, RS)) -> ADD_IN_AAG(S, zero(RS), RSS) ADD_IN_AAG(X, b, X) -> U1_AAG(X, binaryZ_in_g(X)) ADD_IN_AAG(X, b, X) -> BINARYZ_IN_G(X) BINARYZ_IN_G(zero(X)) -> U29_G(X, binaryZ_in_g(X)) BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARYZ_IN_G(one(X)) -> U30_G(X, binary_in_g(X)) BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X) BINARY_IN_G(zero(X)) -> U27_G(X, binaryZ_in_g(X)) BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARY_IN_G(one(X)) -> U28_G(X, binary_in_g(X)) BINARY_IN_G(one(X)) -> BINARY_IN_G(X) ADD_IN_AAG(b, Y, Y) -> U2_AAG(Y, binaryZ_in_g(Y)) ADD_IN_AAG(b, Y, Y) -> BINARYZ_IN_G(Y) ADD_IN_AAG(X, Y, Z) -> U3_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADD_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> U10_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> U11_AAG(X, Y, Z, addx_in_aag(X, Y, Z)) ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> ADDX_IN_AAG(X, Y, Z) ADDX_IN_AAG(one(X), b, one(X)) -> U4_AAG(X, binary_in_g(X)) ADDX_IN_AAG(one(X), b, one(X)) -> BINARY_IN_G(X) ADDX_IN_AAG(zero(X), b, zero(X)) -> U5_AAG(X, binaryZ_in_g(X)) ADDX_IN_AAG(zero(X), b, zero(X)) -> BINARYZ_IN_G(X) ADDX_IN_AAG(X, Y, Z) -> U6_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDX_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> U12_AAG(X, Y, Z, addy_in_aag(X, Y, Z)) ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> ADDY_IN_AAG(X, Y, Z) ADDY_IN_AAG(b, one(Y), one(Y)) -> U7_AAG(Y, binary_in_g(Y)) ADDY_IN_AAG(b, one(Y), one(Y)) -> BINARY_IN_G(Y) ADDY_IN_AAG(b, zero(Y), zero(Y)) -> U8_AAG(Y, binaryZ_in_g(Y)) ADDY_IN_AAG(b, zero(Y), zero(Y)) -> BINARYZ_IN_G(Y) ADDY_IN_AAG(X, Y, Z) -> U9_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDY_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> U13_AAG(X, Y, Z, addc_in_aag(X, Y, Z)) ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> ADDC_IN_AAG(X, Y, Z) ADDC_IN_AAG(X, b, Z) -> U14_AAG(X, Z, succZ_in_ag(X, Z)) ADDC_IN_AAG(X, b, Z) -> SUCCZ_IN_AG(X, Z) SUCCZ_IN_AG(zero(X), one(X)) -> U33_AG(X, binaryZ_in_g(X)) SUCCZ_IN_AG(zero(X), one(X)) -> BINARYZ_IN_G(X) SUCCZ_IN_AG(one(X), zero(Z)) -> U34_AG(X, Z, succ_in_ag(X, Z)) SUCCZ_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z) SUCC_IN_AG(zero(X), one(X)) -> U31_AG(X, binaryZ_in_g(X)) SUCC_IN_AG(zero(X), one(X)) -> BINARYZ_IN_G(X) SUCC_IN_AG(one(X), zero(Z)) -> U32_AG(X, Z, succ_in_ag(X, Z)) SUCC_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z) ADDC_IN_AAG(b, Y, Z) -> U15_AAG(Y, Z, succZ_in_ag(Y, Z)) ADDC_IN_AAG(b, Y, Z) -> SUCCZ_IN_AG(Y, Z) ADDC_IN_AAG(X, Y, Z) -> U16_AAG(X, Y, Z, addC_in_aag(X, Y, Z)) ADDC_IN_AAG(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> U23_AAG(X, Y, Z, addz_in_aag(X, Y, Z)) ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> U24_AAG(X, Y, Z, addX_in_aag(X, Y, Z)) ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAG^1(X, Y, Z) ADDX_IN_AAG^1(zero(X), b, one(X)) -> U17_AAG(X, binaryZ_in_g(X)) ADDX_IN_AAG^1(zero(X), b, one(X)) -> BINARYZ_IN_G(X) ADDX_IN_AAG^1(one(X), b, zero(Z)) -> U18_AAG(X, Z, succ_in_ag(X, Z)) ADDX_IN_AAG^1(one(X), b, zero(Z)) -> SUCC_IN_AG(X, Z) ADDX_IN_AAG^1(X, Y, Z) -> U19_AAG(X, Y, Z, addC_in_aag(X, Y, Z)) ADDX_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> U25_AAG(X, Y, Z, addY_in_aag(X, Y, Z)) ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAG^1(X, Y, Z) ADDY_IN_AAG^1(b, zero(Y), one(Y)) -> U20_AAG(Y, binaryZ_in_g(Y)) ADDY_IN_AAG^1(b, zero(Y), one(Y)) -> BINARYZ_IN_G(Y) ADDY_IN_AAG^1(b, one(Y), zero(Z)) -> U21_AAG(Y, Z, succ_in_ag(Y, Z)) ADDY_IN_AAG^1(b, one(Y), zero(Z)) -> SUCC_IN_AG(Y, Z) ADDY_IN_AAG^1(X, Y, Z) -> U22_AAG(X, Y, Z, addC_in_aag(X, Y, Z)) ADDY_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> U26_AAG(X, Y, Z, addc_in_aag(X, Y, Z)) ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAG(X, Y, Z) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2, x3) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x3, x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x3, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2, x3) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x1, x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x1, x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g(x1) U27_g(x1, x2) = U27_g(x1, x2) binaryZ_out_g(x1) = binaryZ_out_g(x1) U28_g(x1, x2) = U28_g(x1, x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x3, x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x3, x4) U11_aag(x1, x2, x3, x4) = U11_aag(x3, x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2, x3) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x3, x4) U12_aag(x1, x2, x3, x4) = U12_aag(x3, x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2, x3) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x3, x4) U13_aag(x1, x2, x3, x4) = U13_aag(x3, x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2, x3) U14_aag(x1, x2, x3) = U14_aag(x2, x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1, x2) U34_ag(x1, x2, x3) = U34_ag(x2, x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1, x2) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x2, x3) U15_aag(x1, x2, x3) = U15_aag(x2, x3) U16_aag(x1, x2, x3, x4) = U16_aag(x3, x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x3, x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2, x3) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2, x3) U24_aag(x1, x2, x3, x4) = U24_aag(x3, x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2, x3) U18_aag(x1, x2, x3) = U18_aag(x2, x3) U19_aag(x1, x2, x3, x4) = U19_aag(x3, x4) U25_aag(x1, x2, x3, x4) = U25_aag(x3, x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2, x3) U21_aag(x1, x2, x3) = U21_aag(x2, x3) U22_aag(x1, x2, x3, x4) = U22_aag(x3, x4) U26_aag(x1, x2, x3, x4) = U26_aag(x3, x4) TIMES_IN_AAG(x1, x2, x3) = TIMES_IN_AAG(x3) U35_AAG(x1, x2, x3, x4) = U35_AAG(x3, x4) U36_AAG(x1, x2, x3, x4) = U36_AAG(x3, x4) TIMES_IN_AAA(x1, x2, x3) = TIMES_IN_AAA U35_AAA(x1, x2, x3, x4) = U35_AAA(x4) U36_AAA(x1, x2, x3, x4) = U36_AAA(x4) U37_AAA(x1, x2, x3, x4) = U37_AAA(x1, x4) ADD_IN_AAA(x1, x2, x3) = ADD_IN_AAA U1_AAA(x1, x2) = U1_AAA(x2) BINARYZ_IN_A(x1) = BINARYZ_IN_A U29_A(x1, x2) = U29_A(x2) U30_A(x1, x2) = U30_A(x2) BINARY_IN_A(x1) = BINARY_IN_A U27_A(x1, x2) = U27_A(x2) U28_A(x1, x2) = U28_A(x2) U2_AAA(x1, x2) = U2_AAA(x2) U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) ADDZ_IN_AAA(x1, x2, x3) = ADDZ_IN_AAA U10_AAA(x1, x2, x3, x4) = U10_AAA(x4) U11_AAA(x1, x2, x3, x4) = U11_AAA(x4) ADDX_IN_AAA(x1, x2, x3) = ADDX_IN_AAA U4_AAA(x1, x2) = U4_AAA(x2) U5_AAA(x1, x2) = U5_AAA(x2) U6_AAA(x1, x2, x3, x4) = U6_AAA(x4) U12_AAA(x1, x2, x3, x4) = U12_AAA(x4) ADDY_IN_AAA(x1, x2, x3) = ADDY_IN_AAA U7_AAA(x1, x2) = U7_AAA(x2) U8_AAA(x1, x2) = U8_AAA(x2) U9_AAA(x1, x2, x3, x4) = U9_AAA(x4) U13_AAA(x1, x2, x3, x4) = U13_AAA(x4) ADDC_IN_AAA(x1, x2, x3) = ADDC_IN_AAA U14_AAA(x1, x2, x3) = U14_AAA(x3) SUCCZ_IN_AA(x1, x2) = SUCCZ_IN_AA U33_AA(x1, x2) = U33_AA(x2) U34_AA(x1, x2, x3) = U34_AA(x3) SUCC_IN_AA(x1, x2) = SUCC_IN_AA U31_AA(x1, x2) = U31_AA(x2) U32_AA(x1, x2, x3) = U32_AA(x3) U15_AAA(x1, x2, x3) = U15_AAA(x3) U16_AAA(x1, x2, x3, x4) = U16_AAA(x4) ADDC_IN_AAA^1(x1, x2, x3) = ADDC_IN_AAA^1 U23_AAA(x1, x2, x3, x4) = U23_AAA(x4) U24_AAA(x1, x2, x3, x4) = U24_AAA(x4) ADDX_IN_AAA^1(x1, x2, x3) = ADDX_IN_AAA^1 U17_AAA(x1, x2) = U17_AAA(x2) U18_AAA(x1, x2, x3) = U18_AAA(x3) U19_AAA(x1, x2, x3, x4) = U19_AAA(x4) U25_AAA(x1, x2, x3, x4) = U25_AAA(x4) ADDY_IN_AAA^1(x1, x2, x3) = ADDY_IN_AAA^1 U20_AAA(x1, x2) = U20_AAA(x2) U21_AAA(x1, x2, x3) = U21_AAA(x3) U22_AAA(x1, x2, x3, x4) = U22_AAA(x4) U26_AAA(x1, x2, x3, x4) = U26_AAA(x4) U37_AAG(x1, x2, x3, x4) = U37_AAG(x1, x3, x4) ADD_IN_AAG(x1, x2, x3) = ADD_IN_AAG(x3) U1_AAG(x1, x2) = U1_AAG(x1, x2) BINARYZ_IN_G(x1) = BINARYZ_IN_G(x1) U29_G(x1, x2) = U29_G(x1, x2) U30_G(x1, x2) = U30_G(x1, x2) BINARY_IN_G(x1) = BINARY_IN_G(x1) U27_G(x1, x2) = U27_G(x1, x2) U28_G(x1, x2) = U28_G(x1, x2) U2_AAG(x1, x2) = U2_AAG(x1, x2) U3_AAG(x1, x2, x3, x4) = U3_AAG(x3, x4) ADDZ_IN_AAG(x1, x2, x3) = ADDZ_IN_AAG(x3) U10_AAG(x1, x2, x3, x4) = U10_AAG(x3, x4) U11_AAG(x1, x2, x3, x4) = U11_AAG(x3, x4) ADDX_IN_AAG(x1, x2, x3) = ADDX_IN_AAG(x3) U4_AAG(x1, x2) = U4_AAG(x1, x2) U5_AAG(x1, x2) = U5_AAG(x1, x2) U6_AAG(x1, x2, x3, x4) = U6_AAG(x3, x4) U12_AAG(x1, x2, x3, x4) = U12_AAG(x3, x4) ADDY_IN_AAG(x1, x2, x3) = ADDY_IN_AAG(x3) U7_AAG(x1, x2) = U7_AAG(x1, x2) U8_AAG(x1, x2) = U8_AAG(x1, x2) U9_AAG(x1, x2, x3, x4) = U9_AAG(x3, x4) U13_AAG(x1, x2, x3, x4) = U13_AAG(x3, x4) ADDC_IN_AAG(x1, x2, x3) = ADDC_IN_AAG(x3) U14_AAG(x1, x2, x3) = U14_AAG(x2, x3) SUCCZ_IN_AG(x1, x2) = SUCCZ_IN_AG(x2) U33_AG(x1, x2) = U33_AG(x1, x2) U34_AG(x1, x2, x3) = U34_AG(x2, x3) SUCC_IN_AG(x1, x2) = SUCC_IN_AG(x2) U31_AG(x1, x2) = U31_AG(x1, x2) U32_AG(x1, x2, x3) = U32_AG(x2, x3) U15_AAG(x1, x2, x3) = U15_AAG(x2, x3) U16_AAG(x1, x2, x3, x4) = U16_AAG(x3, x4) ADDC_IN_AAG^1(x1, x2, x3) = ADDC_IN_AAG^1(x3) U23_AAG(x1, x2, x3, x4) = U23_AAG(x3, x4) U24_AAG(x1, x2, x3, x4) = U24_AAG(x3, x4) ADDX_IN_AAG^1(x1, x2, x3) = ADDX_IN_AAG^1(x3) U17_AAG(x1, x2) = U17_AAG(x1, x2) U18_AAG(x1, x2, x3) = U18_AAG(x2, x3) U19_AAG(x1, x2, x3, x4) = U19_AAG(x3, x4) U25_AAG(x1, x2, x3, x4) = U25_AAG(x3, x4) ADDY_IN_AAG^1(x1, x2, x3) = ADDY_IN_AAG^1(x3) U20_AAG(x1, x2) = U20_AAG(x1, x2) U21_AAG(x1, x2, x3) = U21_AAG(x2, x3) U22_AAG(x1, x2, x3, x4) = U22_AAG(x3, x4) U26_AAG(x1, x2, x3, x4) = U26_AAG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (53) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 8 SCCs with 109 less nodes. ---------------------------------------- (54) Complex Obligation (AND) ---------------------------------------- (55) Obligation: Pi DP problem: The TRS P consists of the following rules: BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X) BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARY_IN_G(one(X)) -> BINARY_IN_G(X) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2, x3) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x3, x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x3, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2, x3) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x1, x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x1, x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g(x1) U27_g(x1, x2) = U27_g(x1, x2) binaryZ_out_g(x1) = binaryZ_out_g(x1) U28_g(x1, x2) = U28_g(x1, x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x3, x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x3, x4) U11_aag(x1, x2, x3, x4) = U11_aag(x3, x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2, x3) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x3, x4) U12_aag(x1, x2, x3, x4) = U12_aag(x3, x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2, x3) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x3, x4) U13_aag(x1, x2, x3, x4) = U13_aag(x3, x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2, x3) U14_aag(x1, x2, x3) = U14_aag(x2, x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1, x2) U34_ag(x1, x2, x3) = U34_ag(x2, x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1, x2) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x2, x3) U15_aag(x1, x2, x3) = U15_aag(x2, x3) U16_aag(x1, x2, x3, x4) = U16_aag(x3, x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x3, x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2, x3) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2, x3) U24_aag(x1, x2, x3, x4) = U24_aag(x3, x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2, x3) U18_aag(x1, x2, x3) = U18_aag(x2, x3) U19_aag(x1, x2, x3, x4) = U19_aag(x3, x4) U25_aag(x1, x2, x3, x4) = U25_aag(x3, x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2, x3) U21_aag(x1, x2, x3) = U21_aag(x2, x3) U22_aag(x1, x2, x3, x4) = U22_aag(x3, x4) U26_aag(x1, x2, x3, x4) = U26_aag(x3, x4) BINARYZ_IN_G(x1) = BINARYZ_IN_G(x1) BINARY_IN_G(x1) = BINARY_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (56) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (57) Obligation: Pi DP problem: The TRS P consists of the following rules: BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X) BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARY_IN_G(one(X)) -> BINARY_IN_G(X) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (58) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (59) Obligation: Q DP problem: The TRS P consists of the following rules: BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X) BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X) BINARY_IN_G(one(X)) -> BINARY_IN_G(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (60) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X) The graph contains the following edges 1 > 1 *BINARY_IN_G(one(X)) -> BINARY_IN_G(X) The graph contains the following edges 1 > 1 *BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X) The graph contains the following edges 1 > 1 *BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X) The graph contains the following edges 1 > 1 ---------------------------------------- (61) YES ---------------------------------------- (62) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCC_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2, x3) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x3, x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x3, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2, x3) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x1, x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x1, x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g(x1) U27_g(x1, x2) = U27_g(x1, x2) binaryZ_out_g(x1) = binaryZ_out_g(x1) U28_g(x1, x2) = U28_g(x1, x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x3, x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x3, x4) U11_aag(x1, x2, x3, x4) = U11_aag(x3, x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2, x3) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x3, x4) U12_aag(x1, x2, x3, x4) = U12_aag(x3, x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2, x3) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x3, x4) U13_aag(x1, x2, x3, x4) = U13_aag(x3, x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2, x3) U14_aag(x1, x2, x3) = U14_aag(x2, x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1, x2) U34_ag(x1, x2, x3) = U34_ag(x2, x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1, x2) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x2, x3) U15_aag(x1, x2, x3) = U15_aag(x2, x3) U16_aag(x1, x2, x3, x4) = U16_aag(x3, x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x3, x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2, x3) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2, x3) U24_aag(x1, x2, x3, x4) = U24_aag(x3, x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2, x3) U18_aag(x1, x2, x3) = U18_aag(x2, x3) U19_aag(x1, x2, x3, x4) = U19_aag(x3, x4) U25_aag(x1, x2, x3, x4) = U25_aag(x3, x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2, x3) U21_aag(x1, x2, x3) = U21_aag(x2, x3) U22_aag(x1, x2, x3, x4) = U22_aag(x3, x4) U26_aag(x1, x2, x3, x4) = U26_aag(x3, x4) SUCC_IN_AG(x1, x2) = SUCC_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (63) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (64) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCC_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) SUCC_IN_AG(x1, x2) = SUCC_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (65) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: SUCC_IN_AG(zero(Z)) -> SUCC_IN_AG(Z) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (67) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *SUCC_IN_AG(zero(Z)) -> SUCC_IN_AG(Z) The graph contains the following edges 1 > 1 ---------------------------------------- (68) YES ---------------------------------------- (69) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDX_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> ADDX_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> ADDY_IN_AAG(X, Y, Z) ADDY_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> ADDC_IN_AAG(X, Y, Z) ADDC_IN_AAG(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAG^1(X, Y, Z) ADDX_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAG^1(X, Y, Z) ADDY_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAG(X, Y, Z) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2, x3) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x3, x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x3, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2, x3) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x1, x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x1, x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g(x1) U27_g(x1, x2) = U27_g(x1, x2) binaryZ_out_g(x1) = binaryZ_out_g(x1) U28_g(x1, x2) = U28_g(x1, x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x3, x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x3, x4) U11_aag(x1, x2, x3, x4) = U11_aag(x3, x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2, x3) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x3, x4) U12_aag(x1, x2, x3, x4) = U12_aag(x3, x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2, x3) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x3, x4) U13_aag(x1, x2, x3, x4) = U13_aag(x3, x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2, x3) U14_aag(x1, x2, x3) = U14_aag(x2, x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1, x2) U34_ag(x1, x2, x3) = U34_ag(x2, x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1, x2) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x2, x3) U15_aag(x1, x2, x3) = U15_aag(x2, x3) U16_aag(x1, x2, x3, x4) = U16_aag(x3, x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x3, x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2, x3) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2, x3) U24_aag(x1, x2, x3, x4) = U24_aag(x3, x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2, x3) U18_aag(x1, x2, x3) = U18_aag(x2, x3) U19_aag(x1, x2, x3, x4) = U19_aag(x3, x4) U25_aag(x1, x2, x3, x4) = U25_aag(x3, x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2, x3) U21_aag(x1, x2, x3) = U21_aag(x2, x3) U22_aag(x1, x2, x3, x4) = U22_aag(x3, x4) U26_aag(x1, x2, x3, x4) = U26_aag(x3, x4) ADDZ_IN_AAG(x1, x2, x3) = ADDZ_IN_AAG(x3) ADDX_IN_AAG(x1, x2, x3) = ADDX_IN_AAG(x3) ADDY_IN_AAG(x1, x2, x3) = ADDY_IN_AAG(x3) ADDC_IN_AAG(x1, x2, x3) = ADDC_IN_AAG(x3) ADDC_IN_AAG^1(x1, x2, x3) = ADDC_IN_AAG^1(x3) ADDX_IN_AAG^1(x1, x2, x3) = ADDX_IN_AAG^1(x3) ADDY_IN_AAG^1(x1, x2, x3) = ADDY_IN_AAG^1(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (70) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (71) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDX_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> ADDX_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> ADDY_IN_AAG(X, Y, Z) ADDY_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z) ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> ADDC_IN_AAG(X, Y, Z) ADDC_IN_AAG(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAG(X, Y, Z) ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAG^1(X, Y, Z) ADDX_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAG^1(X, Y, Z) ADDY_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z) ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAG(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) ADDZ_IN_AAG(x1, x2, x3) = ADDZ_IN_AAG(x3) ADDX_IN_AAG(x1, x2, x3) = ADDX_IN_AAG(x3) ADDY_IN_AAG(x1, x2, x3) = ADDY_IN_AAG(x3) ADDC_IN_AAG(x1, x2, x3) = ADDC_IN_AAG(x3) ADDC_IN_AAG^1(x1, x2, x3) = ADDC_IN_AAG^1(x3) ADDX_IN_AAG^1(x1, x2, x3) = ADDX_IN_AAG^1(x3) ADDY_IN_AAG^1(x1, x2, x3) = ADDY_IN_AAG^1(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (72) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (73) Obligation: Q DP problem: The TRS P consists of the following rules: ADDX_IN_AAG(Z) -> ADDZ_IN_AAG(Z) ADDZ_IN_AAG(zero(Z)) -> ADDZ_IN_AAG(Z) ADDZ_IN_AAG(one(Z)) -> ADDX_IN_AAG(Z) ADDZ_IN_AAG(one(Z)) -> ADDY_IN_AAG(Z) ADDY_IN_AAG(Z) -> ADDZ_IN_AAG(Z) ADDZ_IN_AAG(zero(Z)) -> ADDC_IN_AAG(Z) ADDC_IN_AAG(Z) -> ADDC_IN_AAG^1(Z) ADDC_IN_AAG^1(one(Z)) -> ADDZ_IN_AAG(Z) ADDC_IN_AAG^1(zero(Z)) -> ADDX_IN_AAG^1(Z) ADDX_IN_AAG^1(Z) -> ADDC_IN_AAG^1(Z) ADDC_IN_AAG^1(zero(Z)) -> ADDY_IN_AAG^1(Z) ADDY_IN_AAG^1(Z) -> ADDC_IN_AAG^1(Z) ADDC_IN_AAG^1(one(Z)) -> ADDC_IN_AAG(Z) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (74) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *ADDZ_IN_AAG(one(Z)) -> ADDX_IN_AAG(Z) The graph contains the following edges 1 > 1 *ADDZ_IN_AAG(zero(Z)) -> ADDZ_IN_AAG(Z) The graph contains the following edges 1 > 1 *ADDX_IN_AAG(Z) -> ADDZ_IN_AAG(Z) The graph contains the following edges 1 >= 1 *ADDY_IN_AAG(Z) -> ADDZ_IN_AAG(Z) The graph contains the following edges 1 >= 1 *ADDC_IN_AAG^1(one(Z)) -> ADDZ_IN_AAG(Z) The graph contains the following edges 1 > 1 *ADDZ_IN_AAG(one(Z)) -> ADDY_IN_AAG(Z) The graph contains the following edges 1 > 1 *ADDZ_IN_AAG(zero(Z)) -> ADDC_IN_AAG(Z) The graph contains the following edges 1 > 1 *ADDC_IN_AAG(Z) -> ADDC_IN_AAG^1(Z) The graph contains the following edges 1 >= 1 *ADDC_IN_AAG^1(one(Z)) -> ADDC_IN_AAG(Z) The graph contains the following edges 1 > 1 *ADDX_IN_AAG^1(Z) -> ADDC_IN_AAG^1(Z) The graph contains the following edges 1 >= 1 *ADDY_IN_AAG^1(Z) -> ADDC_IN_AAG^1(Z) The graph contains the following edges 1 >= 1 *ADDC_IN_AAG^1(zero(Z)) -> ADDX_IN_AAG^1(Z) The graph contains the following edges 1 > 1 *ADDC_IN_AAG^1(zero(Z)) -> ADDY_IN_AAG^1(Z) The graph contains the following edges 1 > 1 ---------------------------------------- (75) YES ---------------------------------------- (76) Obligation: Pi DP problem: The TRS P consists of the following rules: BINARYZ_IN_A(one(X)) -> BINARY_IN_A(X) BINARY_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARYZ_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARY_IN_A(one(X)) -> BINARY_IN_A(X) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2, x3) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x3, x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x3, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2, x3) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x1, x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x1, x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g(x1) U27_g(x1, x2) = U27_g(x1, x2) binaryZ_out_g(x1) = binaryZ_out_g(x1) U28_g(x1, x2) = U28_g(x1, x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x3, x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x3, x4) U11_aag(x1, x2, x3, x4) = U11_aag(x3, x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2, x3) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x3, x4) U12_aag(x1, x2, x3, x4) = U12_aag(x3, x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2, x3) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x3, x4) U13_aag(x1, x2, x3, x4) = U13_aag(x3, x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2, x3) U14_aag(x1, x2, x3) = U14_aag(x2, x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1, x2) U34_ag(x1, x2, x3) = U34_ag(x2, x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1, x2) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x2, x3) U15_aag(x1, x2, x3) = U15_aag(x2, x3) U16_aag(x1, x2, x3, x4) = U16_aag(x3, x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x3, x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2, x3) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2, x3) U24_aag(x1, x2, x3, x4) = U24_aag(x3, x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2, x3) U18_aag(x1, x2, x3) = U18_aag(x2, x3) U19_aag(x1, x2, x3, x4) = U19_aag(x3, x4) U25_aag(x1, x2, x3, x4) = U25_aag(x3, x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2, x3) U21_aag(x1, x2, x3) = U21_aag(x2, x3) U22_aag(x1, x2, x3, x4) = U22_aag(x3, x4) U26_aag(x1, x2, x3, x4) = U26_aag(x3, x4) BINARYZ_IN_A(x1) = BINARYZ_IN_A BINARY_IN_A(x1) = BINARY_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (77) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (78) Obligation: Pi DP problem: The TRS P consists of the following rules: BINARYZ_IN_A(one(X)) -> BINARY_IN_A(X) BINARY_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARYZ_IN_A(zero(X)) -> BINARYZ_IN_A(X) BINARY_IN_A(one(X)) -> BINARY_IN_A(X) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) BINARYZ_IN_A(x1) = BINARYZ_IN_A BINARY_IN_A(x1) = BINARY_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (79) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: BINARYZ_IN_A -> BINARY_IN_A BINARY_IN_A -> BINARYZ_IN_A BINARYZ_IN_A -> BINARYZ_IN_A BINARY_IN_A -> BINARY_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (81) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = BINARYZ_IN_A evaluates to t =BINARYZ_IN_A Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from BINARYZ_IN_A to BINARYZ_IN_A. ---------------------------------------- (82) NO ---------------------------------------- (83) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCC_IN_AA(one(X), zero(Z)) -> SUCC_IN_AA(X, Z) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2, x3) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x3, x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x3, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2, x3) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x1, x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x1, x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g(x1) U27_g(x1, x2) = U27_g(x1, x2) binaryZ_out_g(x1) = binaryZ_out_g(x1) U28_g(x1, x2) = U28_g(x1, x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x3, x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x3, x4) U11_aag(x1, x2, x3, x4) = U11_aag(x3, x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2, x3) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x3, x4) U12_aag(x1, x2, x3, x4) = U12_aag(x3, x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2, x3) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x3, x4) U13_aag(x1, x2, x3, x4) = U13_aag(x3, x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2, x3) U14_aag(x1, x2, x3) = U14_aag(x2, x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1, x2) U34_ag(x1, x2, x3) = U34_ag(x2, x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1, x2) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x2, x3) U15_aag(x1, x2, x3) = U15_aag(x2, x3) U16_aag(x1, x2, x3, x4) = U16_aag(x3, x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x3, x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2, x3) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2, x3) U24_aag(x1, x2, x3, x4) = U24_aag(x3, x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2, x3) U18_aag(x1, x2, x3) = U18_aag(x2, x3) U19_aag(x1, x2, x3, x4) = U19_aag(x3, x4) U25_aag(x1, x2, x3, x4) = U25_aag(x3, x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2, x3) U21_aag(x1, x2, x3) = U21_aag(x2, x3) U22_aag(x1, x2, x3, x4) = U22_aag(x3, x4) U26_aag(x1, x2, x3, x4) = U26_aag(x3, x4) SUCC_IN_AA(x1, x2) = SUCC_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (84) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (85) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCC_IN_AA(one(X), zero(Z)) -> SUCC_IN_AA(X, Z) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) SUCC_IN_AA(x1, x2) = SUCC_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (86) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (87) Obligation: Q DP problem: The TRS P consists of the following rules: SUCC_IN_AA -> SUCC_IN_AA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (88) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDX_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), one(Y), one(Z)) -> ADDX_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), zero(Y), one(Z)) -> ADDY_IN_AAA(X, Y, Z) ADDY_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), one(Y), zero(Z)) -> ADDC_IN_AAA(X, Y, Z) ADDC_IN_AAA(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDC_IN_AAA^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAA^1(X, Y, Z) ADDX_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAA^1(X, Y, Z) ADDY_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAA(X, Y, Z) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2, x3) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x3, x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x3, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2, x3) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x1, x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x1, x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g(x1) U27_g(x1, x2) = U27_g(x1, x2) binaryZ_out_g(x1) = binaryZ_out_g(x1) U28_g(x1, x2) = U28_g(x1, x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x3, x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x3, x4) U11_aag(x1, x2, x3, x4) = U11_aag(x3, x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2, x3) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x3, x4) U12_aag(x1, x2, x3, x4) = U12_aag(x3, x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2, x3) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x3, x4) U13_aag(x1, x2, x3, x4) = U13_aag(x3, x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2, x3) U14_aag(x1, x2, x3) = U14_aag(x2, x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1, x2) U34_ag(x1, x2, x3) = U34_ag(x2, x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1, x2) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x2, x3) U15_aag(x1, x2, x3) = U15_aag(x2, x3) U16_aag(x1, x2, x3, x4) = U16_aag(x3, x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x3, x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2, x3) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2, x3) U24_aag(x1, x2, x3, x4) = U24_aag(x3, x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2, x3) U18_aag(x1, x2, x3) = U18_aag(x2, x3) U19_aag(x1, x2, x3, x4) = U19_aag(x3, x4) U25_aag(x1, x2, x3, x4) = U25_aag(x3, x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2, x3) U21_aag(x1, x2, x3) = U21_aag(x2, x3) U22_aag(x1, x2, x3, x4) = U22_aag(x3, x4) U26_aag(x1, x2, x3, x4) = U26_aag(x3, x4) ADDZ_IN_AAA(x1, x2, x3) = ADDZ_IN_AAA ADDX_IN_AAA(x1, x2, x3) = ADDX_IN_AAA ADDY_IN_AAA(x1, x2, x3) = ADDY_IN_AAA ADDC_IN_AAA(x1, x2, x3) = ADDC_IN_AAA ADDC_IN_AAA^1(x1, x2, x3) = ADDC_IN_AAA^1 ADDX_IN_AAA^1(x1, x2, x3) = ADDX_IN_AAA^1 ADDY_IN_AAA^1(x1, x2, x3) = ADDY_IN_AAA^1 We have to consider all (P,R,Pi)-chains ---------------------------------------- (89) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (90) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDX_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(zero(X), one(Y), one(Z)) -> ADDX_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), zero(Y), one(Z)) -> ADDY_IN_AAA(X, Y, Z) ADDY_IN_AAA(X, Y, Z) -> ADDZ_IN_AAA(X, Y, Z) ADDZ_IN_AAA(one(X), one(Y), zero(Z)) -> ADDC_IN_AAA(X, Y, Z) ADDC_IN_AAA(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAA(X, Y, Z) ADDC_IN_AAA^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAA^1(X, Y, Z) ADDX_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAA^1(X, Y, Z) ADDY_IN_AAA^1(X, Y, Z) -> ADDC_IN_AAA^1(X, Y, Z) ADDC_IN_AAA^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) ADDZ_IN_AAA(x1, x2, x3) = ADDZ_IN_AAA ADDX_IN_AAA(x1, x2, x3) = ADDX_IN_AAA ADDY_IN_AAA(x1, x2, x3) = ADDY_IN_AAA ADDC_IN_AAA(x1, x2, x3) = ADDC_IN_AAA ADDC_IN_AAA^1(x1, x2, x3) = ADDC_IN_AAA^1 ADDX_IN_AAA^1(x1, x2, x3) = ADDX_IN_AAA^1 ADDY_IN_AAA^1(x1, x2, x3) = ADDY_IN_AAA^1 We have to consider all (P,R,Pi)-chains ---------------------------------------- (91) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_AAA(one(R), S, RSS) -> TIMES_IN_AAA(R, S, RS) TIMES_IN_AAA(zero(R), S, zero(RS)) -> TIMES_IN_AAA(R, S, RS) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2, x3) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x3, x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x3, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2, x3) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x1, x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x1, x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g(x1) U27_g(x1, x2) = U27_g(x1, x2) binaryZ_out_g(x1) = binaryZ_out_g(x1) U28_g(x1, x2) = U28_g(x1, x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x3, x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x3, x4) U11_aag(x1, x2, x3, x4) = U11_aag(x3, x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2, x3) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x3, x4) U12_aag(x1, x2, x3, x4) = U12_aag(x3, x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2, x3) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x3, x4) U13_aag(x1, x2, x3, x4) = U13_aag(x3, x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2, x3) U14_aag(x1, x2, x3) = U14_aag(x2, x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1, x2) U34_ag(x1, x2, x3) = U34_ag(x2, x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1, x2) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x2, x3) U15_aag(x1, x2, x3) = U15_aag(x2, x3) U16_aag(x1, x2, x3, x4) = U16_aag(x3, x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x3, x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2, x3) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2, x3) U24_aag(x1, x2, x3, x4) = U24_aag(x3, x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2, x3) U18_aag(x1, x2, x3) = U18_aag(x2, x3) U19_aag(x1, x2, x3, x4) = U19_aag(x3, x4) U25_aag(x1, x2, x3, x4) = U25_aag(x3, x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2, x3) U21_aag(x1, x2, x3) = U21_aag(x2, x3) U22_aag(x1, x2, x3, x4) = U22_aag(x3, x4) U26_aag(x1, x2, x3, x4) = U26_aag(x3, x4) TIMES_IN_AAA(x1, x2, x3) = TIMES_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (92) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (93) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_AAA(one(R), S, RSS) -> TIMES_IN_AAA(R, S, RS) TIMES_IN_AAA(zero(R), S, zero(RS)) -> TIMES_IN_AAA(R, S, RS) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) TIMES_IN_AAA(x1, x2, x3) = TIMES_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (94) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_AAG(zero(R), S, zero(RS)) -> TIMES_IN_AAG(R, S, RS) The TRS R consists of the following rules: times_in_aag(one(b), X, X) -> times_out_aag(one(b), X, X) times_in_aag(zero(R), S, zero(RS)) -> U35_aag(R, S, RS, times_in_aag(R, S, RS)) times_in_aag(one(R), S, RSS) -> U36_aag(R, S, RSS, times_in_aaa(R, S, RS)) times_in_aaa(one(b), X, X) -> times_out_aaa(one(b), X, X) times_in_aaa(zero(R), S, zero(RS)) -> U35_aaa(R, S, RS, times_in_aaa(R, S, RS)) times_in_aaa(one(R), S, RSS) -> U36_aaa(R, S, RSS, times_in_aaa(R, S, RS)) U36_aaa(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aaa(R, S, RSS, add_in_aaa(S, zero(RS), RSS)) add_in_aaa(b, b, b) -> add_out_aaa(b, b, b) add_in_aaa(X, b, X) -> U1_aaa(X, binaryZ_in_a(X)) binaryZ_in_a(zero(X)) -> U29_a(X, binaryZ_in_a(X)) binaryZ_in_a(one(X)) -> U30_a(X, binary_in_a(X)) binary_in_a(b) -> binary_out_a(b) binary_in_a(zero(X)) -> U27_a(X, binaryZ_in_a(X)) U27_a(X, binaryZ_out_a(X)) -> binary_out_a(zero(X)) binary_in_a(one(X)) -> U28_a(X, binary_in_a(X)) U28_a(X, binary_out_a(X)) -> binary_out_a(one(X)) U30_a(X, binary_out_a(X)) -> binaryZ_out_a(one(X)) U29_a(X, binaryZ_out_a(X)) -> binaryZ_out_a(zero(X)) U1_aaa(X, binaryZ_out_a(X)) -> add_out_aaa(X, b, X) add_in_aaa(b, Y, Y) -> U2_aaa(Y, binaryZ_in_a(Y)) U2_aaa(Y, binaryZ_out_a(Y)) -> add_out_aaa(b, Y, Y) add_in_aaa(X, Y, Z) -> U3_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), zero(Y), zero(Z)) -> U10_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(zero(X), one(Y), one(Z)) -> U11_aaa(X, Y, Z, addx_in_aaa(X, Y, Z)) addx_in_aaa(one(X), b, one(X)) -> U4_aaa(X, binary_in_a(X)) U4_aaa(X, binary_out_a(X)) -> addx_out_aaa(one(X), b, one(X)) addx_in_aaa(zero(X), b, zero(X)) -> U5_aaa(X, binaryZ_in_a(X)) U5_aaa(X, binaryZ_out_a(X)) -> addx_out_aaa(zero(X), b, zero(X)) addx_in_aaa(X, Y, Z) -> U6_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), zero(Y), one(Z)) -> U12_aaa(X, Y, Z, addy_in_aaa(X, Y, Z)) addy_in_aaa(b, one(Y), one(Y)) -> U7_aaa(Y, binary_in_a(Y)) U7_aaa(Y, binary_out_a(Y)) -> addy_out_aaa(b, one(Y), one(Y)) addy_in_aaa(b, zero(Y), zero(Y)) -> U8_aaa(Y, binaryZ_in_a(Y)) U8_aaa(Y, binaryZ_out_a(Y)) -> addy_out_aaa(b, zero(Y), zero(Y)) addy_in_aaa(X, Y, Z) -> U9_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) addz_in_aaa(one(X), one(Y), zero(Z)) -> U13_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) addc_in_aaa(b, b, one(b)) -> addc_out_aaa(b, b, one(b)) addc_in_aaa(X, b, Z) -> U14_aaa(X, Z, succZ_in_aa(X, Z)) succZ_in_aa(zero(X), one(X)) -> U33_aa(X, binaryZ_in_a(X)) U33_aa(X, binaryZ_out_a(X)) -> succZ_out_aa(zero(X), one(X)) succZ_in_aa(one(X), zero(Z)) -> U34_aa(X, Z, succ_in_aa(X, Z)) succ_in_aa(b, one(b)) -> succ_out_aa(b, one(b)) succ_in_aa(zero(X), one(X)) -> U31_aa(X, binaryZ_in_a(X)) U31_aa(X, binaryZ_out_a(X)) -> succ_out_aa(zero(X), one(X)) succ_in_aa(one(X), zero(Z)) -> U32_aa(X, Z, succ_in_aa(X, Z)) U32_aa(X, Z, succ_out_aa(X, Z)) -> succ_out_aa(one(X), zero(Z)) U34_aa(X, Z, succ_out_aa(X, Z)) -> succZ_out_aa(one(X), zero(Z)) U14_aaa(X, Z, succZ_out_aa(X, Z)) -> addc_out_aaa(X, b, Z) addc_in_aaa(b, Y, Z) -> U15_aaa(Y, Z, succZ_in_aa(Y, Z)) U15_aaa(Y, Z, succZ_out_aa(Y, Z)) -> addc_out_aaa(b, Y, Z) addc_in_aaa(X, Y, Z) -> U16_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(zero(X), zero(Y), one(Z)) -> U23_aaa(X, Y, Z, addz_in_aaa(X, Y, Z)) U23_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), zero(Y), one(Z)) addC_in_aaa(zero(X), one(Y), zero(Z)) -> U24_aaa(X, Y, Z, addX_in_aaa(X, Y, Z)) addX_in_aaa(zero(X), b, one(X)) -> U17_aaa(X, binaryZ_in_a(X)) U17_aaa(X, binaryZ_out_a(X)) -> addX_out_aaa(zero(X), b, one(X)) addX_in_aaa(one(X), b, zero(Z)) -> U18_aaa(X, Z, succ_in_aa(X, Z)) U18_aaa(X, Z, succ_out_aa(X, Z)) -> addX_out_aaa(one(X), b, zero(Z)) addX_in_aaa(X, Y, Z) -> U19_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), zero(Y), zero(Z)) -> U25_aaa(X, Y, Z, addY_in_aaa(X, Y, Z)) addY_in_aaa(b, zero(Y), one(Y)) -> U20_aaa(Y, binaryZ_in_a(Y)) U20_aaa(Y, binaryZ_out_a(Y)) -> addY_out_aaa(b, zero(Y), one(Y)) addY_in_aaa(b, one(Y), zero(Z)) -> U21_aaa(Y, Z, succ_in_aa(Y, Z)) U21_aaa(Y, Z, succ_out_aa(Y, Z)) -> addY_out_aaa(b, one(Y), zero(Z)) addY_in_aaa(X, Y, Z) -> U22_aaa(X, Y, Z, addC_in_aaa(X, Y, Z)) addC_in_aaa(one(X), one(Y), one(Z)) -> U26_aaa(X, Y, Z, addc_in_aaa(X, Y, Z)) U26_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), one(Y), one(Z)) U22_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addY_out_aaa(X, Y, Z) U25_aaa(X, Y, Z, addY_out_aaa(X, Y, Z)) -> addC_out_aaa(one(X), zero(Y), zero(Z)) U19_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addX_out_aaa(X, Y, Z) U24_aaa(X, Y, Z, addX_out_aaa(X, Y, Z)) -> addC_out_aaa(zero(X), one(Y), zero(Z)) U16_aaa(X, Y, Z, addC_out_aaa(X, Y, Z)) -> addc_out_aaa(X, Y, Z) U13_aaa(X, Y, Z, addc_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), one(Y), zero(Z)) U9_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addy_out_aaa(X, Y, Z) U12_aaa(X, Y, Z, addy_out_aaa(X, Y, Z)) -> addz_out_aaa(one(X), zero(Y), one(Z)) U6_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addx_out_aaa(X, Y, Z) U11_aaa(X, Y, Z, addx_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), one(Y), one(Z)) U10_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> addz_out_aaa(zero(X), zero(Y), zero(Z)) U3_aaa(X, Y, Z, addz_out_aaa(X, Y, Z)) -> add_out_aaa(X, Y, Z) U37_aaa(R, S, RSS, add_out_aaa(S, zero(RS), RSS)) -> times_out_aaa(one(R), S, RSS) U35_aaa(R, S, RS, times_out_aaa(R, S, RS)) -> times_out_aaa(zero(R), S, zero(RS)) U36_aag(R, S, RSS, times_out_aaa(R, S, RS)) -> U37_aag(R, S, RSS, add_in_aag(S, zero(RS), RSS)) add_in_aag(b, b, b) -> add_out_aag(b, b, b) add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X)) binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X)) binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X)) binary_in_g(b) -> binary_out_g(b) binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X)) U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X)) binary_in_g(one(X)) -> U28_g(X, binary_in_g(X)) U28_g(X, binary_out_g(X)) -> binary_out_g(one(X)) U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X)) U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X)) U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X) add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y)) U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y) add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z)) addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X)) U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X)) addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X)) U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X)) addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z)) addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y)) U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y)) addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y)) U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y)) addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z)) addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z)) addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b)) addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z)) succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X)) U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X)) succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z)) succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b)) succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X)) U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X)) succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z)) U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z)) U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z)) U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z) addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z)) U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z) addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z)) U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z)) addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z)) addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X)) U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X)) addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z)) U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z)) addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z)) addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y)) U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y)) addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z)) U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z)) addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z)) addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z)) U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z)) U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z) U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z)) U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z) U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z)) U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z) U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z)) U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z) U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z)) U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z) U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z)) U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z)) U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z) U37_aag(R, S, RSS, add_out_aag(S, zero(RS), RSS)) -> times_out_aag(one(R), S, RSS) U35_aag(R, S, RS, times_out_aag(R, S, RS)) -> times_out_aag(zero(R), S, zero(RS)) The argument filtering Pi contains the following mapping: times_in_aag(x1, x2, x3) = times_in_aag(x3) times_out_aag(x1, x2, x3) = times_out_aag(x1, x2, x3) zero(x1) = zero(x1) U35_aag(x1, x2, x3, x4) = U35_aag(x3, x4) U36_aag(x1, x2, x3, x4) = U36_aag(x3, x4) times_in_aaa(x1, x2, x3) = times_in_aaa times_out_aaa(x1, x2, x3) = times_out_aaa(x1) U35_aaa(x1, x2, x3, x4) = U35_aaa(x4) U36_aaa(x1, x2, x3, x4) = U36_aaa(x4) U37_aaa(x1, x2, x3, x4) = U37_aaa(x1, x4) add_in_aaa(x1, x2, x3) = add_in_aaa add_out_aaa(x1, x2, x3) = add_out_aaa(x1, x2, x3) U1_aaa(x1, x2) = U1_aaa(x2) binaryZ_in_a(x1) = binaryZ_in_a U29_a(x1, x2) = U29_a(x2) U30_a(x1, x2) = U30_a(x2) binary_in_a(x1) = binary_in_a binary_out_a(x1) = binary_out_a(x1) U27_a(x1, x2) = U27_a(x2) binaryZ_out_a(x1) = binaryZ_out_a(x1) U28_a(x1, x2) = U28_a(x2) U2_aaa(x1, x2) = U2_aaa(x2) U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) addz_in_aaa(x1, x2, x3) = addz_in_aaa U10_aaa(x1, x2, x3, x4) = U10_aaa(x4) U11_aaa(x1, x2, x3, x4) = U11_aaa(x4) addx_in_aaa(x1, x2, x3) = addx_in_aaa U4_aaa(x1, x2) = U4_aaa(x2) addx_out_aaa(x1, x2, x3) = addx_out_aaa(x1, x2, x3) U5_aaa(x1, x2) = U5_aaa(x2) U6_aaa(x1, x2, x3, x4) = U6_aaa(x4) U12_aaa(x1, x2, x3, x4) = U12_aaa(x4) addy_in_aaa(x1, x2, x3) = addy_in_aaa U7_aaa(x1, x2) = U7_aaa(x2) addy_out_aaa(x1, x2, x3) = addy_out_aaa(x1, x2, x3) U8_aaa(x1, x2) = U8_aaa(x2) U9_aaa(x1, x2, x3, x4) = U9_aaa(x4) U13_aaa(x1, x2, x3, x4) = U13_aaa(x4) addc_in_aaa(x1, x2, x3) = addc_in_aaa addc_out_aaa(x1, x2, x3) = addc_out_aaa(x1, x2, x3) U14_aaa(x1, x2, x3) = U14_aaa(x3) succZ_in_aa(x1, x2) = succZ_in_aa U33_aa(x1, x2) = U33_aa(x2) succZ_out_aa(x1, x2) = succZ_out_aa(x1, x2) U34_aa(x1, x2, x3) = U34_aa(x3) succ_in_aa(x1, x2) = succ_in_aa succ_out_aa(x1, x2) = succ_out_aa(x1, x2) U31_aa(x1, x2) = U31_aa(x2) U32_aa(x1, x2, x3) = U32_aa(x3) U15_aaa(x1, x2, x3) = U15_aaa(x3) U16_aaa(x1, x2, x3, x4) = U16_aaa(x4) addC_in_aaa(x1, x2, x3) = addC_in_aaa U23_aaa(x1, x2, x3, x4) = U23_aaa(x4) addz_out_aaa(x1, x2, x3) = addz_out_aaa(x1, x2, x3) addC_out_aaa(x1, x2, x3) = addC_out_aaa(x1, x2, x3) U24_aaa(x1, x2, x3, x4) = U24_aaa(x4) addX_in_aaa(x1, x2, x3) = addX_in_aaa U17_aaa(x1, x2) = U17_aaa(x2) addX_out_aaa(x1, x2, x3) = addX_out_aaa(x1, x2, x3) U18_aaa(x1, x2, x3) = U18_aaa(x3) U19_aaa(x1, x2, x3, x4) = U19_aaa(x4) U25_aaa(x1, x2, x3, x4) = U25_aaa(x4) addY_in_aaa(x1, x2, x3) = addY_in_aaa U20_aaa(x1, x2) = U20_aaa(x2) addY_out_aaa(x1, x2, x3) = addY_out_aaa(x1, x2, x3) U21_aaa(x1, x2, x3) = U21_aaa(x3) U22_aaa(x1, x2, x3, x4) = U22_aaa(x4) U26_aaa(x1, x2, x3, x4) = U26_aaa(x4) U37_aag(x1, x2, x3, x4) = U37_aag(x1, x3, x4) add_in_aag(x1, x2, x3) = add_in_aag(x3) b = b add_out_aag(x1, x2, x3) = add_out_aag(x1, x2, x3) U1_aag(x1, x2) = U1_aag(x1, x2) binaryZ_in_g(x1) = binaryZ_in_g(x1) U29_g(x1, x2) = U29_g(x1, x2) one(x1) = one(x1) U30_g(x1, x2) = U30_g(x1, x2) binary_in_g(x1) = binary_in_g(x1) binary_out_g(x1) = binary_out_g(x1) U27_g(x1, x2) = U27_g(x1, x2) binaryZ_out_g(x1) = binaryZ_out_g(x1) U28_g(x1, x2) = U28_g(x1, x2) U2_aag(x1, x2) = U2_aag(x1, x2) U3_aag(x1, x2, x3, x4) = U3_aag(x3, x4) addz_in_aag(x1, x2, x3) = addz_in_aag(x3) U10_aag(x1, x2, x3, x4) = U10_aag(x3, x4) U11_aag(x1, x2, x3, x4) = U11_aag(x3, x4) addx_in_aag(x1, x2, x3) = addx_in_aag(x3) U4_aag(x1, x2) = U4_aag(x1, x2) addx_out_aag(x1, x2, x3) = addx_out_aag(x1, x2, x3) U5_aag(x1, x2) = U5_aag(x1, x2) U6_aag(x1, x2, x3, x4) = U6_aag(x3, x4) U12_aag(x1, x2, x3, x4) = U12_aag(x3, x4) addy_in_aag(x1, x2, x3) = addy_in_aag(x3) U7_aag(x1, x2) = U7_aag(x1, x2) addy_out_aag(x1, x2, x3) = addy_out_aag(x1, x2, x3) U8_aag(x1, x2) = U8_aag(x1, x2) U9_aag(x1, x2, x3, x4) = U9_aag(x3, x4) U13_aag(x1, x2, x3, x4) = U13_aag(x3, x4) addc_in_aag(x1, x2, x3) = addc_in_aag(x3) addc_out_aag(x1, x2, x3) = addc_out_aag(x1, x2, x3) U14_aag(x1, x2, x3) = U14_aag(x2, x3) succZ_in_ag(x1, x2) = succZ_in_ag(x2) U33_ag(x1, x2) = U33_ag(x1, x2) succZ_out_ag(x1, x2) = succZ_out_ag(x1, x2) U34_ag(x1, x2, x3) = U34_ag(x2, x3) succ_in_ag(x1, x2) = succ_in_ag(x2) succ_out_ag(x1, x2) = succ_out_ag(x1, x2) U31_ag(x1, x2) = U31_ag(x1, x2) U32_ag(x1, x2, x3) = U32_ag(x2, x3) U15_aag(x1, x2, x3) = U15_aag(x2, x3) U16_aag(x1, x2, x3, x4) = U16_aag(x3, x4) addC_in_aag(x1, x2, x3) = addC_in_aag(x3) U23_aag(x1, x2, x3, x4) = U23_aag(x3, x4) addz_out_aag(x1, x2, x3) = addz_out_aag(x1, x2, x3) addC_out_aag(x1, x2, x3) = addC_out_aag(x1, x2, x3) U24_aag(x1, x2, x3, x4) = U24_aag(x3, x4) addX_in_aag(x1, x2, x3) = addX_in_aag(x3) U17_aag(x1, x2) = U17_aag(x1, x2) addX_out_aag(x1, x2, x3) = addX_out_aag(x1, x2, x3) U18_aag(x1, x2, x3) = U18_aag(x2, x3) U19_aag(x1, x2, x3, x4) = U19_aag(x3, x4) U25_aag(x1, x2, x3, x4) = U25_aag(x3, x4) addY_in_aag(x1, x2, x3) = addY_in_aag(x3) U20_aag(x1, x2) = U20_aag(x1, x2) addY_out_aag(x1, x2, x3) = addY_out_aag(x1, x2, x3) U21_aag(x1, x2, x3) = U21_aag(x2, x3) U22_aag(x1, x2, x3, x4) = U22_aag(x3, x4) U26_aag(x1, x2, x3, x4) = U26_aag(x3, x4) TIMES_IN_AAG(x1, x2, x3) = TIMES_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (95) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (96) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_AAG(zero(R), S, zero(RS)) -> TIMES_IN_AAG(R, S, RS) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) TIMES_IN_AAG(x1, x2, x3) = TIMES_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (97) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(add (b) (b) (b))", null ], [ "(add X (b) X)", "(binaryZ X)" ], [ "(add (b) Y Y)", "(binaryZ Y)" ], [ "(add X Y Z)", "(addz X Y Z)" ], [ "(addx (one X) (b) (one X))", "(binary X)" ], [ "(addx (zero X) (b) (zero X))", "(binaryZ X)" ], [ "(addx X Y Z)", "(addz X Y Z)" ], [ "(addy (b) (one Y) (one Y))", "(binary Y)" ], [ "(addy (b) (zero Y) (zero Y))", "(binaryZ Y)" ], [ "(addy X Y Z)", "(addz X Y Z)" ], [ "(addz (zero X) (zero Y) (zero Z))", "(addz X Y Z)" ], [ "(addz (zero X) (one Y) (one Z))", "(addx X Y Z)" ], [ "(addz (one X) (zero Y) (one Z))", "(addy X Y Z)" ], [ "(addz (one X) (one Y) (zero Z))", "(addc X Y Z)" ], [ "(addc (b) (b) 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X2)" ], [ "(times T1 T2 T969)", "(times (zero X1010) X1011 (zero X1012))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T969"], "free": [ "X2", "X1010", "X1011", "X1012" ], "exprvars": [] } }, "225": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2532": { "goal": [{ "clause": -1, "scope": -1, "term": "(add T859 (zero T859) T851)" }], "kb": { "nonunifying": [[ "(times T1 T2 T851)", "(times (zero X6) X7 (zero X8))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T851"], "free": [ "X6", "X7", "X8" ], "exprvars": [] } }, "2653": { "goal": [ { "clause": 39, "scope": 31, "term": "(',' (times T970 T971 X1074) (add T971 (zero X1074) T969))" }, { "clause": 40, "scope": 31, "term": "(',' (times T970 T971 X1074) (add T971 (zero X1074) T969))" } ], "kb": { "nonunifying": [ [ "(times T1 T2 T969)", "(times (one (b)) X2 X2)" ], [ "(times T1 T2 T969)", "(times (zero X1010) X1011 (zero X1012))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T969"], "free": [ "X2", "X1010", "X1011", "X1012", "X1074" ], "exprvars": [] } }, "2531": { "goal": [ { "clause": 39, "scope": 29, "term": "(',' (times T852 T853 X959) (add T853 (zero X959) T851))" }, { "clause": 40, "scope": 29, "term": "(',' (times T852 T853 X959) (add T853 (zero X959) T851))" } ], "kb": { "nonunifying": [[ "(times T1 T2 T851)", "(times (zero X6) X7 (zero X8))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T851"], "free": [ "X6", "X7", "X8", "X959" ], "exprvars": [] } }, "2652": { "goal": [{ "clause": 38, "scope": 31, "term": "(',' (times T970 T971 X1074) (add T971 (zero X1074) T969))" }], "kb": { "nonunifying": [ [ "(times T1 T2 T969)", "(times (one (b)) X2 X2)" ], [ "(times T1 T2 T969)", "(times (zero X1010) X1011 (zero X1012))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T969"], "free": [ "X2", "X1010", "X1011", "X1012", "X1074" ], "exprvars": [] } }, "2530": { "goal": [{ "clause": 38, "scope": 29, "term": "(',' (times T852 T853 X959) (add T853 (zero X959) T851))" }], "kb": { "nonunifying": [[ "(times T1 T2 T851)", "(times (zero X6) X7 (zero X8))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T851"], "free": [ "X6", "X7", "X8", "X959" ], "exprvars": [] } }, "2651": { "goal": [ { "clause": 38, "scope": 31, "term": "(',' (times T970 T971 X1074) (add T971 (zero X1074) T969))" }, { "clause": 39, "scope": 31, "term": "(',' (times T970 T971 X1074) (add T971 (zero X1074) T969))" }, { "clause": 40, "scope": 31, "term": "(',' (times T970 T971 X1074) (add T971 (zero X1074) T969))" } ], "kb": { "nonunifying": [ [ "(times T1 T2 T969)", "(times (one (b)) X2 X2)" ], [ "(times T1 T2 T969)", "(times (zero X1010) X1011 (zero X1012))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T969"], "free": [ "X2", "X1010", "X1011", "X1012", "X1074" ], "exprvars": [] } }, "2419": { "goal": [{ "clause": 26, "scope": 26, "term": "(addC T669 T670 T668)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T668"], "free": [], "exprvars": [] } }, "1207": { "goal": [ { "clause": 28, "scope": 7, "term": "(binary T123)" }, { "clause": 29, "scope": 7, "term": "(binary T123)" }, { "clause": 30, "scope": 7, "term": "(binary T123)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1449": { "goal": [{ "clause": 10, "scope": 8, "term": "(addz T145 (zero T146) X147)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X147"], "exprvars": [] } }, "2659": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 7, "label": "CASE" }, { "from": 7, "to": 25, "label": "EVAL with clause\ntimes(one(b), X2, X2).\nand substitutionT1 -> one(b),\nT2 -> T5,\nX2 -> T5,\nT3 -> T5" }, { "from": 7, "to": 26, "label": "EVAL-BACKTRACK" }, { "from": 25, "to": 27, "label": "SUCCESS" }, { "from": 26, "to": 2610, "label": "EVAL with clause\ntimes(zero(X1010), X1011, zero(X1012)) :- times(X1010, X1011, X1012).\nand substitutionX1010 -> T906,\nT1 -> zero(T906),\nT2 -> T907,\nX1011 -> T907,\nX1012 -> T905,\nT3 -> zero(T905),\nT903 -> T906,\nT904 -> T907" }, { "from": 26, "to": 2611, "label": "EVAL-BACKTRACK" }, { "from": 27, "to": 28, "label": "EVAL with clause\ntimes(zero(X6), X7, zero(X8)) :- times(X6, X7, X8).\nand substitutionX6 -> T12,\nT1 -> zero(T12),\nT2 -> T13,\nX7 -> T13,\nX8 -> T11,\nT5 -> zero(T11),\nT9 -> T12,\nT10 -> T13" }, { "from": 27, "to": 29, "label": "EVAL-BACKTRACK" }, { "from": 28, "to": 30, "label": "CASE" }, { "from": 29, "to": 2525, "label": "EVAL with clause\ntimes(one(X956), X957, X958) :- ','(times(X956, X957, X959), add(X957, zero(X959), X958)).\nand substitutionX956 -> T852,\nT1 -> one(T852),\nT2 -> T853,\nX957 -> T853,\nT5 -> T851,\nX958 -> T851,\nT849 -> T852,\nT850 -> T853" }, { "from": 29, "to": 2526, "label": "EVAL-BACKTRACK" }, { "from": 30, "to": 31, "label": "PARALLEL" }, { "from": 30, "to": 32, "label": "PARALLEL" }, { "from": 31, "to": 33, "label": "EVAL with clause\ntimes(one(b), X13, X13).\nand substitutionT12 -> one(b),\nT13 -> T18,\nX13 -> T18,\nT11 -> T18" }, { "from": 31, "to": 34, "label": "EVAL-BACKTRACK" }, { "from": 32, "to": 36, "label": "PARALLEL" }, { "from": 32, "to": 37, "label": "PARALLEL" }, { "from": 33, "to": 35, "label": "SUCCESS" }, { "from": 36, "to": 38, "label": "EVAL with clause\ntimes(zero(X26), X27, zero(X28)) :- times(X26, X27, X28).\nand substitutionX26 -> T34,\nT12 -> zero(T34),\nT13 -> T35,\nX27 -> T35,\nX28 -> T33,\nT11 -> zero(T33),\nT31 -> T34,\nT32 -> T35" }, { "from": 36, "to": 39, "label": "EVAL-BACKTRACK" }, { "from": 37, "to": 214, "label": "PARALLEL" }, { "from": 37, "to": 215, "label": "PARALLEL" }, { "from": 38, "to": 1, "label": "INSTANCE with matching:\nT1 -> T34\nT2 -> T35\nT3 -> T33" }, { "from": 214, "to": 216, "label": "EVAL with clause\ntimes(one(X45), X46, X47) :- ','(times(X45, X46, X48), add(X46, zero(X48), X47)).\nand substitutionX45 -> T53,\nT12 -> one(T53),\nT13 -> T54,\nX46 -> T54,\nT11 -> T52,\nX47 -> T52,\nT50 -> T53,\nT51 -> T54" }, { "from": 214, "to": 217, "label": "EVAL-BACKTRACK" }, { "from": 215, "to": 2500, "label": "FAILURE" }, { "from": 216, "to": 218, "label": "SPLIT 1" }, { "from": 216, "to": 219, "label": "SPLIT 2\nnew knowledge:\nT53 is ground\nreplacements:X48 -> T57,\nT54 -> T58" }, { "from": 218, "to": 220, "label": "CASE" }, { "from": 219, "to": 2184, "label": "CASE" }, { "from": 220, "to": 221, "label": "PARALLEL" }, { "from": 220, "to": 222, "label": "PARALLEL" }, { "from": 221, "to": 223, "label": "EVAL with clause\ntimes(one(b), X57, X57).\nand substitutionT53 -> one(b),\nT54 -> T65,\nX57 -> T65,\nX48 -> T65" }, { "from": 221, "to": 224, "label": "EVAL-BACKTRACK" }, { "from": 222, "to": 610, "label": "PARALLEL" }, { "from": 222, "to": 612, "label": "PARALLEL" }, { "from": 223, "to": 225, "label": "SUCCESS" }, { "from": 610, "to": 620, "label": "EVAL with clause\ntimes(zero(X74), X75, zero(X76)) :- times(X74, X75, X76).\nand substitutionX74 -> T76,\nT53 -> zero(T76),\nT54 -> T77,\nX75 -> T77,\nX76 -> X77,\nX48 -> zero(X77),\nT74 -> T76,\nT75 -> T77" }, { "from": 610, "to": 623, "label": "EVAL-BACKTRACK" }, { "from": 612, "to": 630, "label": "EVAL with clause\ntimes(one(X89), X90, X91) :- ','(times(X89, X90, X92), add(X90, zero(X92), X91)).\nand substitutionX89 -> T86,\nT53 -> one(T86),\nT54 -> T87,\nX90 -> T87,\nX48 -> X93,\nX91 -> X93,\nT84 -> T86,\nT85 -> T87" }, { "from": 612, "to": 631, "label": "EVAL-BACKTRACK" }, { "from": 620, "to": 218, "label": "INSTANCE with matching:\nT53 -> T76\nT54 -> T77\nX48 -> X77" }, { "from": 630, "to": 637, "label": "SPLIT 1" }, { "from": 630, "to": 638, "label": "SPLIT 2\nnew knowledge:\nT86 is ground\nreplacements:X92 -> T90,\nT87 -> T91" }, { "from": 637, "to": 218, "label": "INSTANCE with matching:\nT53 -> T86\nT54 -> T87\nX48 -> X92" }, { "from": 638, "to": 641, "label": "CASE" }, { "from": 641, "to": 642, "label": "BACKTRACK\nfor clause: add(b, b, b)because of non-unification" }, { "from": 642, "to": 645, "label": "BACKTRACK\nfor clause: add(X, b, X) :- binaryZ(X)because of non-unification" }, { "from": 645, "to": 656, "label": "PARALLEL" }, { "from": 645, "to": 658, "label": "PARALLEL" }, { "from": 656, "to": 671, "label": "EVAL with clause\nadd(b, X103, X103) :- binaryZ(X103).\nand substitutionT91 -> b,\nT90 -> T100,\nX103 -> zero(T100),\nX93 -> zero(T100),\nT99 -> T100" }, { "from": 656, "to": 673, "label": "EVAL-BACKTRACK" }, { "from": 658, "to": 1416, "label": "ONLY EVAL with clause\nadd(X144, X145, X146) :- addz(X144, X145, X146).\nand substitutionT91 -> T145,\nX144 -> T145,\nT90 -> T146,\nX145 -> zero(T146),\nX93 -> X147,\nX146 -> X147,\nT143 -> T145,\nT144 -> T146" }, { "from": 671, "to": 678, "label": "CASE" }, { "from": 678, "to": 1138, "label": "PARALLEL" }, { "from": 678, "to": 1139, "label": "PARALLEL" }, { "from": 1138, "to": 1153, "label": "ONLY EVAL with clause\nbinaryZ(zero(X111)) :- binaryZ(X111).\nand substitutionT100 -> T111,\nX111 -> T111,\nT110 -> T111" }, { "from": 1139, "to": 1272, "label": "BACKTRACK\nfor clause: binaryZ(one(X)) :- binary(X)because of non-unification" }, { "from": 1153, "to": 1163, "label": "CASE" }, { "from": 1163, "to": 1171, "label": "PARALLEL" }, { "from": 1163, "to": 1172, "label": "PARALLEL" }, { "from": 1171, "to": 1180, "label": "EVAL with clause\nbinaryZ(zero(X117)) :- binaryZ(X117).\nand substitutionX117 -> T118,\nT111 -> zero(T118),\nT117 -> T118" }, { "from": 1171, "to": 1182, "label": "EVAL-BACKTRACK" }, { "from": 1172, "to": 1197, "label": "EVAL with clause\nbinaryZ(one(X121)) :- binary(X121).\nand substitutionX121 -> T123,\nT111 -> one(T123),\nT122 -> T123" }, { "from": 1172, "to": 1199, "label": "EVAL-BACKTRACK" }, { "from": 1180, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T118" }, { "from": 1197, "to": 1207, "label": "CASE" }, { "from": 1207, "to": 1210, "label": "PARALLEL" }, { "from": 1207, "to": 1211, "label": "PARALLEL" }, { "from": 1210, "to": 1215, "label": "EVAL with clause\nbinary(b).\nand substitutionT123 -> b" }, { "from": 1210, "to": 1217, "label": "EVAL-BACKTRACK" }, { "from": 1211, "to": 1225, "label": "PARALLEL" }, { "from": 1211, "to": 1226, "label": "PARALLEL" }, { "from": 1215, "to": 1219, "label": "SUCCESS" }, { "from": 1225, "to": 1235, "label": "EVAL with clause\nbinary(zero(X126)) :- binaryZ(X126).\nand substitutionX126 -> T129,\nT123 -> zero(T129),\nT128 -> T129" }, { "from": 1225, "to": 1236, "label": "EVAL-BACKTRACK" }, { "from": 1226, "to": 1270, "label": "EVAL with clause\nbinary(one(X130)) :- binary(X130).\nand substitutionX130 -> T134,\nT123 -> one(T134),\nT133 -> T134" }, { "from": 1226, "to": 1271, "label": "EVAL-BACKTRACK" }, { "from": 1235, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T129" }, { "from": 1270, "to": 1197, "label": "INSTANCE with matching:\nT123 -> T134" }, { "from": 1416, "to": 1432, "label": "CASE" }, { "from": 1432, "to": 1449, "label": "PARALLEL" }, { "from": 1432, "to": 1450, "label": "PARALLEL" }, { "from": 1449, "to": 1478, "label": "EVAL with clause\naddz(zero(X168), zero(X169), zero(X170)) :- addz(X168, X169, X170).\nand substitutionX168 -> T159,\nT145 -> zero(T159),\nT146 -> T160,\nX169 -> T160,\nX170 -> X171,\nX147 -> zero(X171),\nT157 -> T159,\nT158 -> T160" }, { "from": 1449, "to": 1480, "label": "EVAL-BACKTRACK" }, { "from": 1450, "to": 2175, "label": "BACKTRACK\nfor clause: addz(zero(X), one(Y), one(Z)) :- addx(X, Y, Z)because of non-unification" }, { "from": 1478, "to": 1497, "label": "CASE" }, { "from": 1497, "to": 1516, "label": "PARALLEL" }, { "from": 1497, "to": 1517, "label": "PARALLEL" }, { "from": 1516, "to": 1527, "label": "EVAL with clause\naddz(zero(X192), zero(X193), zero(X194)) :- addz(X192, X193, X194).\nand substitutionX192 -> T173,\nT159 -> zero(T173),\nX193 -> T174,\nT160 -> zero(T174),\nX194 -> X195,\nX171 -> zero(X195),\nT171 -> T173,\nT172 -> T174" }, { "from": 1516, "to": 1528, "label": "EVAL-BACKTRACK" }, { "from": 1517, "to": 1550, "label": "PARALLEL" }, { "from": 1517, "to": 1552, "label": "PARALLEL" }, { "from": 1527, "to": 1478, "label": "INSTANCE with matching:\nT159 -> T173\nT160 -> T174\nX171 -> X195" }, { "from": 1550, "to": 1588, "label": "EVAL with clause\naddz(zero(X216), one(X217), one(X218)) :- addx(X216, X217, X218).\nand substitutionX216 -> T187,\nT159 -> zero(T187),\nX217 -> T188,\nT160 -> one(T188),\nX218 -> X219,\nX171 -> one(X219),\nT185 -> T187,\nT186 -> T188" }, { "from": 1550, "to": 1589, "label": "EVAL-BACKTRACK" }, { "from": 1552, "to": 1613, "label": "PARALLEL" }, { "from": 1552, "to": 1614, "label": "PARALLEL" }, { "from": 1588, "to": 1590, "label": "CASE" }, { "from": 1590, "to": 1591, "label": "PARALLEL" }, { "from": 1590, "to": 1592, "label": "PARALLEL" }, { "from": 1591, "to": 1593, "label": "EVAL with clause\naddx(one(X225), b, one(X225)) :- binary(X225).\nand substitutionX225 -> T195,\nT187 -> one(T195),\nT188 -> b,\nX219 -> one(T195),\nT194 -> T195" }, { "from": 1591, "to": 1594, "label": "EVAL-BACKTRACK" }, { "from": 1592, "to": 1596, "label": "PARALLEL" }, { "from": 1592, "to": 1597, "label": "PARALLEL" }, { "from": 1593, "to": 1197, "label": "INSTANCE with matching:\nT123 -> T195" }, { "from": 1596, "to": 1600, "label": "EVAL with clause\naddx(zero(X230), b, zero(X230)) :- binaryZ(X230).\nand substitutionX230 -> T201,\nT187 -> zero(T201),\nT188 -> b,\nX219 -> zero(T201),\nT200 -> T201" }, { "from": 1596, "to": 1601, "label": "EVAL-BACKTRACK" }, { "from": 1597, "to": 1607, "label": "ONLY EVAL with clause\naddx(X244, X245, X246) :- addz(X244, X245, X246).\nand substitutionT187 -> T213,\nX244 -> T213,\nT188 -> T214,\nX245 -> T214,\nX219 -> X247,\nX246 -> X247,\nT211 -> T213,\nT212 -> T214" }, { "from": 1600, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T201" }, { "from": 1607, "to": 1478, "label": "INSTANCE with matching:\nT159 -> T213\nT160 -> T214\nX171 -> X247" }, { "from": 1613, "to": 1630, "label": "EVAL with clause\naddz(one(X268), zero(X269), one(X270)) :- addy(X268, X269, X270).\nand substitutionX268 -> T227,\nT159 -> one(T227),\nX269 -> T228,\nT160 -> zero(T228),\nX270 -> X271,\nX171 -> one(X271),\nT225 -> T227,\nT226 -> T228" }, { "from": 1613, "to": 1631, "label": "EVAL-BACKTRACK" }, { "from": 1614, "to": 1864, "label": "EVAL with clause\naddz(one(X312), one(X313), zero(X314)) :- addc(X312, X313, X314).\nand substitutionX312 -> T263,\nT159 -> one(T263),\nX313 -> T264,\nT160 -> one(T264),\nX314 -> X315,\nX171 -> zero(X315),\nT261 -> T263,\nT262 -> T264" }, { "from": 1614, "to": 1865, "label": "EVAL-BACKTRACK" }, { "from": 1630, "to": 1634, "label": "CASE" }, { "from": 1634, "to": 1655, "label": "PARALLEL" }, { "from": 1634, "to": 1656, "label": "PARALLEL" }, { "from": 1655, "to": 1659, "label": "EVAL with clause\naddy(b, one(X277), one(X277)) :- binary(X277).\nand substitutionT227 -> b,\nX277 -> T235,\nT228 -> one(T235),\nX271 -> one(T235),\nT234 -> T235" }, { "from": 1655, "to": 1660, "label": "EVAL-BACKTRACK" }, { "from": 1656, "to": 1661, "label": "PARALLEL" }, { "from": 1656, "to": 1662, "label": "PARALLEL" }, { "from": 1659, "to": 1197, "label": "INSTANCE with matching:\nT123 -> T235" }, { "from": 1661, "to": 1663, "label": "EVAL with clause\naddy(b, zero(X282), zero(X282)) :- binaryZ(X282).\nand substitutionT227 -> b,\nX282 -> T241,\nT228 -> zero(T241),\nX271 -> zero(T241),\nT240 -> T241" }, { "from": 1661, "to": 1664, "label": "EVAL-BACKTRACK" }, { "from": 1662, "to": 1856, "label": "ONLY EVAL with clause\naddy(X296, X297, X298) :- addz(X296, X297, X298).\nand substitutionT227 -> T253,\nX296 -> T253,\nT228 -> T254,\nX297 -> T254,\nX271 -> X299,\nX298 -> X299,\nT251 -> T253,\nT252 -> T254" }, { "from": 1663, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T241" }, { "from": 1856, "to": 1478, "label": "INSTANCE with matching:\nT159 -> T253\nT160 -> T254\nX171 -> X299" }, { "from": 1864, "to": 1866, "label": "CASE" }, { "from": 1866, "to": 1867, "label": "PARALLEL" }, { "from": 1866, "to": 1868, "label": "PARALLEL" }, { "from": 1867, "to": 1869, "label": "EVAL with clause\naddc(b, b, one(b)).\nand substitutionT263 -> b,\nT264 -> b,\nX315 -> one(b)" }, { "from": 1867, "to": 1870, "label": "EVAL-BACKTRACK" }, { "from": 1868, "to": 1877, "label": "PARALLEL" }, { "from": 1868, "to": 1878, "label": "PARALLEL" }, { "from": 1869, "to": 1871, "label": "SUCCESS" }, { "from": 1877, "to": 1884, "label": "EVAL with clause\naddc(X328, b, X329) :- succZ(X328, X329).\nand substitutionT263 -> T270,\nX328 -> T270,\nT264 -> b,\nX315 -> X330,\nX329 -> X330,\nT269 -> T270" }, { "from": 1877, "to": 1885, "label": "EVAL-BACKTRACK" }, { "from": 1878, "to": 2064, "label": "PARALLEL" }, { "from": 1878, "to": 2065, "label": "PARALLEL" }, { "from": 1884, "to": 2036, "label": "CASE" }, { "from": 2036, "to": 2037, "label": "PARALLEL" }, { "from": 2036, "to": 2038, "label": "PARALLEL" }, { "from": 2037, "to": 2039, "label": "EVAL with clause\nsuccZ(zero(X336), one(X336)) :- binaryZ(X336).\nand substitutionX336 -> T277,\nT270 -> zero(T277),\nX330 -> one(T277),\nT276 -> T277" }, { "from": 2037, "to": 2040, "label": "EVAL-BACKTRACK" }, { "from": 2038, "to": 2045, "label": "EVAL with clause\nsuccZ(one(X344), zero(X345)) :- succ(X344, X345).\nand substitutionX344 -> T282,\nT270 -> one(T282),\nX345 -> X346,\nX330 -> zero(X346),\nT281 -> T282" }, { "from": 2038, "to": 2046, "label": "EVAL-BACKTRACK" }, { "from": 2039, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T277" }, { "from": 2045, "to": 2047, "label": "CASE" }, { "from": 2047, "to": 2048, "label": "PARALLEL" }, { "from": 2047, "to": 2049, "label": "PARALLEL" }, { "from": 2048, "to": 2050, "label": "EVAL with clause\nsucc(b, one(b)).\nand substitutionT282 -> b,\nX346 -> one(b)" }, { "from": 2048, "to": 2051, "label": "EVAL-BACKTRACK" }, { "from": 2049, "to": 2055, "label": "PARALLEL" }, { "from": 2049, "to": 2056, "label": "PARALLEL" }, { "from": 2050, "to": 2052, "label": "SUCCESS" }, { "from": 2055, "to": 2057, "label": "EVAL with clause\nsucc(zero(X351), one(X351)) :- binaryZ(X351).\nand substitutionX351 -> T288,\nT282 -> zero(T288),\nX346 -> one(T288),\nT287 -> T288" }, { "from": 2055, "to": 2058, "label": "EVAL-BACKTRACK" }, { "from": 2056, "to": 2059, "label": "EVAL with clause\nsucc(one(X359), zero(X360)) :- succ(X359, X360).\nand substitutionX359 -> T293,\nT282 -> one(T293),\nX360 -> X361,\nX346 -> zero(X361),\nT292 -> T293" }, { "from": 2056, "to": 2060, "label": "EVAL-BACKTRACK" }, { "from": 2057, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T288" }, { "from": 2059, "to": 2045, "label": "INSTANCE with matching:\nT282 -> T293\nX346 -> X361" }, { "from": 2064, "to": 2067, "label": "EVAL with clause\naddc(b, X374, X375) :- succZ(X374, X375).\nand substitutionT263 -> b,\nT264 -> T299,\nX374 -> T299,\nX315 -> X376,\nX375 -> X376,\nT298 -> T299" }, { "from": 2064, "to": 2068, "label": "EVAL-BACKTRACK" }, { "from": 2065, "to": 2075, "label": "ONLY EVAL with clause\naddc(X390, X391, X392) :- addC(X390, X391, X392).\nand substitutionT263 -> T311,\nX390 -> T311,\nT264 -> T312,\nX391 -> T312,\nX315 -> X393,\nX392 -> X393,\nT309 -> T311,\nT310 -> T312" }, { "from": 2067, "to": 1884, "label": "INSTANCE with matching:\nT270 -> T299\nX330 -> X376" }, { "from": 2075, "to": 2076, "label": "CASE" }, { "from": 2076, "to": 2077, "label": "PARALLEL" }, { "from": 2076, "to": 2078, "label": "PARALLEL" }, { "from": 2077, "to": 2079, "label": "EVAL with clause\naddC(zero(X414), zero(X415), one(X416)) :- addz(X414, X415, X416).\nand substitutionX414 -> T325,\nT311 -> zero(T325),\nX415 -> T326,\nT312 -> zero(T326),\nX416 -> X417,\nX393 -> one(X417),\nT323 -> T325,\nT324 -> T326" }, { "from": 2077, "to": 2080, "label": "EVAL-BACKTRACK" }, { "from": 2078, "to": 2081, "label": "PARALLEL" }, { "from": 2078, "to": 2082, "label": "PARALLEL" }, { "from": 2079, "to": 1478, "label": "INSTANCE with matching:\nT159 -> T325\nT160 -> T326\nX171 -> X417" }, { "from": 2081, "to": 2092, "label": "EVAL with clause\naddC(zero(X438), one(X439), zero(X440)) :- addX(X438, X439, X440).\nand substitutionX438 -> T339,\nT311 -> zero(T339),\nX439 -> T340,\nT312 -> one(T340),\nX440 -> X441,\nX393 -> zero(X441),\nT337 -> T339,\nT338 -> T340" }, { "from": 2081, "to": 2093, "label": "EVAL-BACKTRACK" }, { "from": 2082, "to": 2125, "label": "PARALLEL" }, { "from": 2082, "to": 2126, "label": "PARALLEL" }, { "from": 2092, "to": 2094, "label": "CASE" }, { "from": 2094, "to": 2101, "label": "PARALLEL" }, { "from": 2094, "to": 2102, "label": "PARALLEL" }, { "from": 2101, "to": 2103, "label": "EVAL with clause\naddX(zero(X447), b, one(X447)) :- binaryZ(X447).\nand substitutionX447 -> T347,\nT339 -> zero(T347),\nT340 -> b,\nX441 -> one(T347),\nT346 -> T347" }, { "from": 2101, "to": 2104, "label": "EVAL-BACKTRACK" }, { "from": 2102, "to": 2105, "label": "PARALLEL" }, { "from": 2102, "to": 2106, "label": "PARALLEL" }, { "from": 2103, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T347" }, { "from": 2105, "to": 2107, "label": "EVAL with clause\naddX(one(X461), b, zero(X462)) :- succ(X461, X462).\nand substitutionX461 -> T354,\nT339 -> one(T354),\nT340 -> b,\nX462 -> X463,\nX441 -> zero(X463),\nT353 -> T354" }, { "from": 2105, "to": 2108, "label": "EVAL-BACKTRACK" }, { "from": 2106, "to": 2124, "label": "ONLY EVAL with clause\naddX(X476, X477, X478) :- addC(X476, X477, X478).\nand substitutionT339 -> T365,\nX476 -> T365,\nT340 -> T366,\nX477 -> T366,\nX441 -> X479,\nX478 -> X479,\nT363 -> T365,\nT364 -> T366" }, { "from": 2107, "to": 2045, "label": "INSTANCE with matching:\nT282 -> T354\nX346 -> X463" }, { "from": 2124, "to": 2075, "label": "INSTANCE with matching:\nT311 -> T365\nT312 -> T366\nX393 -> X479" }, { "from": 2125, "to": 2127, "label": "EVAL with clause\naddC(one(X500), zero(X501), zero(X502)) :- addY(X500, X501, X502).\nand substitutionX500 -> T379,\nT311 -> one(T379),\nX501 -> T380,\nT312 -> zero(T380),\nX502 -> X503,\nX393 -> zero(X503),\nT377 -> T379,\nT378 -> T380" }, { "from": 2125, "to": 2128, "label": "EVAL-BACKTRACK" }, { "from": 2126, "to": 2171, "label": "EVAL with clause\naddC(one(X554), one(X555), one(X556)) :- addc(X554, X555, X556).\nand substitutionX554 -> T415,\nT311 -> one(T415),\nX555 -> T416,\nT312 -> one(T416),\nX556 -> X557,\nX393 -> one(X557),\nT413 -> T415,\nT414 -> T416" }, { "from": 2126, "to": 2172, "label": "EVAL-BACKTRACK" }, { "from": 2127, "to": 2129, "label": "CASE" }, { "from": 2129, "to": 2130, "label": "PARALLEL" }, { "from": 2129, "to": 2131, "label": "PARALLEL" }, { "from": 2130, "to": 2147, "label": "EVAL with clause\naddY(b, zero(X509), one(X509)) :- binaryZ(X509).\nand substitutionT379 -> b,\nX509 -> T387,\nT380 -> zero(T387),\nX503 -> one(T387),\nT386 -> T387" }, { "from": 2130, "to": 2148, "label": "EVAL-BACKTRACK" }, { "from": 2131, "to": 2151, "label": "PARALLEL" }, { "from": 2131, "to": 2152, "label": "PARALLEL" }, { "from": 2147, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T387" }, { "from": 2151, "to": 2156, "label": "EVAL with clause\naddY(b, one(X523), zero(X524)) :- succ(X523, X524).\nand substitutionT379 -> b,\nX523 -> T394,\nT380 -> one(T394),\nX524 -> X525,\nX503 -> zero(X525),\nT393 -> T394" }, { "from": 2151, "to": 2157, "label": "EVAL-BACKTRACK" }, { "from": 2152, "to": 2160, "label": "ONLY EVAL with clause\naddY(X538, X539, X540) :- addC(X538, X539, X540).\nand substitutionT379 -> T405,\nX538 -> T405,\nT380 -> T406,\nX539 -> T406,\nX503 -> X541,\nX540 -> X541,\nT403 -> T405,\nT404 -> T406" }, { "from": 2156, "to": 2045, "label": "INSTANCE with matching:\nT282 -> T394\nX346 -> X525" }, { "from": 2160, "to": 2075, "label": "INSTANCE with matching:\nT311 -> T405\nT312 -> T406\nX393 -> X541" }, { "from": 2171, "to": 1864, "label": "INSTANCE with matching:\nT263 -> T415\nT264 -> T416\nX315 -> X557" }, { "from": 2175, "to": 2178, "label": "PARALLEL" }, { "from": 2175, "to": 2179, "label": "PARALLEL" }, { "from": 2178, "to": 2180, "label": "EVAL with clause\naddz(one(X577), zero(X578), one(X579)) :- addy(X577, X578, X579).\nand substitutionX577 -> T428,\nT145 -> one(T428),\nT146 -> T429,\nX578 -> T429,\nX579 -> X580,\nX147 -> one(X580),\nT426 -> T428,\nT427 -> T429" }, { "from": 2178, "to": 2181, "label": "EVAL-BACKTRACK" }, { "from": 2179, "to": 2183, "label": "BACKTRACK\nfor clause: addz(one(X), one(Y), zero(Z)) :- addc(X, Y, Z)because of non-unification" }, { "from": 2180, "to": 1630, "label": "INSTANCE with matching:\nT227 -> T428\nT228 -> T429\nX271 -> X580" }, { "from": 2184, "to": 2185, "label": "BACKTRACK\nfor clause: add(b, b, b)because of non-unification" }, { "from": 2185, "to": 2186, "label": "BACKTRACK\nfor clause: add(X, b, X) :- binaryZ(X)because of non-unification" }, { "from": 2186, "to": 2187, "label": "PARALLEL" }, { "from": 2186, "to": 2188, "label": "PARALLEL" }, { "from": 2187, "to": 2201, "label": "EVAL with clause\nadd(b, X590, X590) :- binaryZ(X590).\nand substitutionT58 -> b,\nT57 -> T437,\nX590 -> zero(T437),\nT52 -> zero(T437)" }, { "from": 2187, "to": 2202, "label": "EVAL-BACKTRACK" }, { "from": 2188, "to": 2203, "label": "ONLY EVAL with clause\nadd(X601, X602, X603) :- addz(X601, X602, X603).\nand substitutionT58 -> T453,\nX601 -> T453,\nT57 -> T454,\nX602 -> zero(T454),\nT52 -> T452,\nX603 -> T452,\nT450 -> T453,\nT451 -> T454" }, { "from": 2201, "to": 671, "label": "INSTANCE with matching:\nT100 -> T437" }, { "from": 2203, "to": 2204, "label": "CASE" }, { "from": 2204, "to": 2206, "label": "PARALLEL" }, { "from": 2204, "to": 2207, "label": "PARALLEL" }, { "from": 2206, "to": 2210, "label": "EVAL with clause\naddz(zero(X619), zero(X620), zero(X621)) :- addz(X619, X620, X621).\nand substitutionX619 -> T473,\nT453 -> zero(T473),\nT454 -> T474,\nX620 -> T474,\nX621 -> T472,\nT452 -> zero(T472),\nT470 -> T473,\nT471 -> T474" }, { "from": 2206, "to": 2211, "label": "EVAL-BACKTRACK" }, { "from": 2207, "to": 2480, "label": "BACKTRACK\nfor clause: addz(zero(X), one(Y), one(Z)) :- addx(X, Y, Z)because of non-unification" }, { "from": 2210, "to": 2212, "label": "CASE" }, { "from": 2212, "to": 2213, "label": "PARALLEL" }, { "from": 2212, "to": 2214, "label": "PARALLEL" }, { "from": 2213, "to": 2222, "label": "EVAL with clause\naddz(zero(X637), zero(X638), zero(X639)) :- addz(X637, X638, X639).\nand substitutionX637 -> T493,\nT473 -> zero(T493),\nX638 -> T494,\nT474 -> zero(T494),\nX639 -> T492,\nT472 -> zero(T492),\nT490 -> T493,\nT491 -> T494" }, { "from": 2213, "to": 2223, "label": "EVAL-BACKTRACK" }, { "from": 2214, "to": 2228, "label": "PARALLEL" }, { "from": 2214, "to": 2229, "label": "PARALLEL" }, { "from": 2222, "to": 2210, "label": "INSTANCE with matching:\nT473 -> T493\nT474 -> T494\nT472 -> T492" }, { "from": 2228, "to": 2235, "label": "EVAL with clause\naddz(zero(X655), one(X656), one(X657)) :- addx(X655, X656, X657).\nand substitutionX655 -> T513,\nT473 -> zero(T513),\nX656 -> T514,\nT474 -> one(T514),\nX657 -> T512,\nT472 -> one(T512),\nT510 -> T513,\nT511 -> T514" }, { "from": 2228, "to": 2236, "label": "EVAL-BACKTRACK" }, { "from": 2229, "to": 2272, "label": "PARALLEL" }, { "from": 2229, "to": 2273, "label": "PARALLEL" }, { "from": 2235, "to": 2239, "label": "CASE" }, { "from": 2239, "to": 2241, "label": "PARALLEL" }, { "from": 2239, "to": 2242, "label": "PARALLEL" }, { "from": 2241, "to": 2244, "label": "EVAL with clause\naddx(one(X663), b, one(X663)) :- binary(X663).\nand substitutionX663 -> T520,\nT513 -> one(T520),\nT514 -> b,\nT512 -> one(T520)" }, { "from": 2241, "to": 2245, "label": "EVAL-BACKTRACK" }, { "from": 2242, "to": 2250, "label": "PARALLEL" }, { "from": 2242, "to": 2251, "label": "PARALLEL" }, { "from": 2244, "to": 1197, "label": "INSTANCE with matching:\nT123 -> T520" }, { "from": 2250, "to": 2252, "label": "EVAL with clause\naddx(zero(X668), b, zero(X668)) :- binaryZ(X668).\nand substitutionX668 -> T525,\nT513 -> zero(T525),\nT514 -> b,\nT512 -> zero(T525)" }, { "from": 2250, "to": 2253, "label": "EVAL-BACKTRACK" }, { "from": 2251, "to": 2266, "label": "ONLY EVAL with clause\naddx(X679, X680, X681) :- addz(X679, X680, X681).\nand substitutionT513 -> T541,\nX679 -> T541,\nT514 -> T542,\nX680 -> T542,\nT512 -> T540,\nX681 -> T540,\nT538 -> T541,\nT539 -> T542" }, { "from": 2252, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T525" }, { "from": 2266, "to": 2210, "label": "INSTANCE with matching:\nT473 -> T541\nT474 -> T542\nT472 -> T540" }, { "from": 2272, "to": 2284, "label": "EVAL with clause\naddz(one(X697), zero(X698), one(X699)) :- addy(X697, X698, X699).\nand substitutionX697 -> T561,\nT473 -> one(T561),\nX698 -> T562,\nT474 -> zero(T562),\nX699 -> T560,\nT472 -> one(T560),\nT558 -> T561,\nT559 -> T562" }, { "from": 2272, "to": 2285, "label": "EVAL-BACKTRACK" }, { "from": 2273, "to": 2306, "label": "EVAL with clause\naddz(one(X733), one(X734), zero(X735)) :- addc(X733, X734, X735).\nand substitutionX733 -> T603,\nT473 -> one(T603),\nX734 -> T604,\nT474 -> one(T604),\nX735 -> T602,\nT472 -> zero(T602),\nT600 -> T603,\nT601 -> T604" }, { "from": 2273, "to": 2307, "label": "EVAL-BACKTRACK" }, { "from": 2284, "to": 2286, "label": "CASE" }, { "from": 2286, "to": 2289, "label": "PARALLEL" }, { "from": 2286, "to": 2290, "label": "PARALLEL" }, { "from": 2289, "to": 2292, "label": "EVAL with clause\naddy(b, one(X705), one(X705)) :- binary(X705).\nand substitutionT561 -> b,\nX705 -> T568,\nT562 -> one(T568),\nT560 -> one(T568)" }, { "from": 2289, "to": 2293, "label": "EVAL-BACKTRACK" }, { "from": 2290, "to": 2294, "label": "PARALLEL" }, { "from": 2290, "to": 2295, "label": "PARALLEL" }, { "from": 2292, "to": 1197, "label": "INSTANCE with matching:\nT123 -> T568" }, { "from": 2294, "to": 2297, "label": "EVAL with clause\naddy(b, zero(X710), zero(X710)) :- binaryZ(X710).\nand substitutionT561 -> b,\nX710 -> T573,\nT562 -> zero(T573),\nT560 -> zero(T573)" }, { "from": 2294, "to": 2298, "label": "EVAL-BACKTRACK" }, { "from": 2295, "to": 2302, "label": "ONLY EVAL with clause\naddy(X721, X722, X723) :- addz(X721, X722, X723).\nand substitutionT561 -> T589,\nX721 -> T589,\nT562 -> T590,\nX722 -> T590,\nT560 -> T588,\nX723 -> T588,\nT586 -> T589,\nT587 -> T590" }, { "from": 2297, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T573" }, { "from": 2302, "to": 2210, "label": "INSTANCE with matching:\nT473 -> T589\nT474 -> T590\nT472 -> T588" }, { "from": 2306, "to": 2308, "label": "CASE" }, { "from": 2308, "to": 2309, "label": "PARALLEL" }, { "from": 2308, "to": 2310, "label": "PARALLEL" }, { "from": 2309, "to": 2311, "label": "EVAL with clause\naddc(b, b, one(b)).\nand substitutionT603 -> b,\nT604 -> b,\nT602 -> one(b)" }, { "from": 2309, "to": 2312, "label": "EVAL-BACKTRACK" }, { "from": 2310, "to": 2314, "label": "PARALLEL" }, { "from": 2310, "to": 2315, "label": "PARALLEL" }, { "from": 2311, "to": 2313, "label": "SUCCESS" }, { "from": 2314, "to": 2319, "label": "EVAL with clause\naddc(X744, b, X745) :- succZ(X744, X745).\nand substitutionT603 -> T615,\nX744 -> T615,\nT604 -> b,\nT602 -> T614,\nX745 -> T614,\nT613 -> T615" }, { "from": 2314, "to": 2320, "label": "EVAL-BACKTRACK" }, { "from": 2315, "to": 2365, "label": "PARALLEL" }, { "from": 2315, "to": 2366, "label": "PARALLEL" }, { "from": 2319, "to": 2324, "label": "CASE" }, { "from": 2324, "to": 2325, "label": "PARALLEL" }, { "from": 2324, "to": 2326, "label": "PARALLEL" }, { "from": 2325, "to": 2329, "label": "EVAL with clause\nsuccZ(zero(X751), one(X751)) :- binaryZ(X751).\nand substitutionX751 -> T621,\nT615 -> zero(T621),\nT614 -> one(T621)" }, { "from": 2325, "to": 2330, "label": "EVAL-BACKTRACK" }, { "from": 2326, "to": 2336, "label": "EVAL with clause\nsuccZ(one(X757), zero(X758)) :- succ(X757, X758).\nand substitutionX757 -> T629,\nT615 -> one(T629),\nX758 -> T628,\nT614 -> zero(T628),\nT627 -> T629" }, { "from": 2326, "to": 2337, "label": "EVAL-BACKTRACK" }, { "from": 2329, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T621" }, { "from": 2336, "to": 2340, "label": "CASE" }, { "from": 2340, "to": 2341, "label": "PARALLEL" }, { "from": 2340, "to": 2342, "label": "PARALLEL" }, { "from": 2341, "to": 2344, "label": "EVAL with clause\nsucc(b, one(b)).\nand substitutionT629 -> b,\nT628 -> one(b)" }, { "from": 2341, "to": 2346, "label": "EVAL-BACKTRACK" }, { "from": 2342, "to": 2349, "label": "PARALLEL" }, { "from": 2342, "to": 2350, "label": "PARALLEL" }, { "from": 2344, "to": 2347, "label": "SUCCESS" }, { "from": 2349, "to": 2354, "label": "EVAL with clause\nsucc(zero(X763), one(X763)) :- binaryZ(X763).\nand substitutionX763 -> T634,\nT629 -> zero(T634),\nT628 -> one(T634)" }, { "from": 2349, "to": 2355, "label": "EVAL-BACKTRACK" }, { "from": 2350, "to": 2363, "label": "EVAL with clause\nsucc(one(X769), zero(X770)) :- succ(X769, X770).\nand substitutionX769 -> T642,\nT629 -> one(T642),\nX770 -> T641,\nT628 -> zero(T641),\nT640 -> T642" }, { "from": 2350, "to": 2364, "label": "EVAL-BACKTRACK" }, { "from": 2354, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T634" }, { "from": 2363, "to": 2336, "label": "INSTANCE with matching:\nT629 -> T642\nT628 -> T641" }, { "from": 2365, "to": 2367, "label": "EVAL with clause\naddc(b, X779, X780) :- succZ(X779, X780).\nand substitutionT603 -> b,\nT604 -> T653,\nX779 -> T653,\nT602 -> T652,\nX780 -> T652,\nT651 -> T653" }, { "from": 2365, "to": 2368, "label": "EVAL-BACKTRACK" }, { "from": 2366, "to": 2374, "label": "ONLY EVAL with clause\naddc(X791, X792, X793) :- addC(X791, X792, X793).\nand substitutionT603 -> T669,\nX791 -> T669,\nT604 -> T670,\nX792 -> T670,\nT602 -> T668,\nX793 -> T668,\nT666 -> T669,\nT667 -> T670" }, { "from": 2367, "to": 2319, "label": "INSTANCE with matching:\nT615 -> T653\nT614 -> T652" }, { "from": 2374, "to": 2377, "label": "CASE" }, { "from": 2377, "to": 2378, "label": "PARALLEL" }, { "from": 2377, "to": 2379, "label": "PARALLEL" }, { "from": 2378, "to": 2382, "label": "EVAL with clause\naddC(zero(X809), zero(X810), one(X811)) :- addz(X809, X810, X811).\nand substitutionX809 -> T689,\nT669 -> zero(T689),\nX810 -> T690,\nT670 -> zero(T690),\nX811 -> T688,\nT668 -> one(T688),\nT686 -> T689,\nT687 -> T690" }, { "from": 2378, "to": 2383, "label": "EVAL-BACKTRACK" }, { "from": 2379, "to": 2384, "label": "PARALLEL" }, { "from": 2379, "to": 2385, "label": "PARALLEL" }, { "from": 2382, "to": 2210, "label": "INSTANCE with matching:\nT473 -> T689\nT474 -> T690\nT472 -> T688" }, { "from": 2384, "to": 2386, "label": "EVAL with clause\naddC(zero(X827), one(X828), zero(X829)) :- addX(X827, X828, X829).\nand substitutionX827 -> T709,\nT669 -> zero(T709),\nX828 -> T710,\nT670 -> one(T710),\nX829 -> T708,\nT668 -> zero(T708),\nT706 -> T709,\nT707 -> T710" }, { "from": 2384, "to": 2387, "label": "EVAL-BACKTRACK" }, { "from": 2385, "to": 2419, "label": "PARALLEL" }, { "from": 2385, "to": 2420, "label": "PARALLEL" }, { "from": 2386, "to": 2388, "label": "CASE" }, { "from": 2388, "to": 2389, "label": "PARALLEL" }, { "from": 2388, "to": 2390, "label": "PARALLEL" }, { "from": 2389, "to": 2391, "label": "EVAL with clause\naddX(zero(X835), b, one(X835)) :- binaryZ(X835).\nand substitutionX835 -> T716,\nT709 -> zero(T716),\nT710 -> b,\nT708 -> one(T716)" }, { "from": 2389, "to": 2392, "label": "EVAL-BACKTRACK" }, { "from": 2390, "to": 2396, "label": "PARALLEL" }, { "from": 2390, "to": 2397, "label": "PARALLEL" }, { "from": 2391, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T716" }, { "from": 2396, "to": 2400, "label": "EVAL with clause\naddX(one(X845), b, zero(X846)) :- succ(X845, X846).\nand substitutionX845 -> T728,\nT709 -> one(T728),\nT710 -> b,\nX846 -> T727,\nT708 -> zero(T727),\nT726 -> T728" }, { "from": 2396, "to": 2401, "label": "EVAL-BACKTRACK" }, { "from": 2397, "to": 2414, "label": "ONLY EVAL with clause\naddX(X856, X857, X858) :- addC(X856, X857, X858).\nand substitutionT709 -> T743,\nX856 -> T743,\nT710 -> T744,\nX857 -> T744,\nT708 -> T742,\nX858 -> T742,\nT740 -> T743,\nT741 -> T744" }, { "from": 2400, "to": 2336, "label": "INSTANCE with matching:\nT629 -> T728\nT628 -> T727" }, { "from": 2414, "to": 2374, "label": "INSTANCE with matching:\nT669 -> T743\nT670 -> T744\nT668 -> T742" }, { "from": 2419, "to": 2423, "label": "EVAL with clause\naddC(one(X874), zero(X875), zero(X876)) :- addY(X874, X875, X876).\nand substitutionX874 -> T763,\nT669 -> one(T763),\nX875 -> T764,\nT670 -> zero(T764),\nX876 -> T762,\nT668 -> zero(T762),\nT760 -> T763,\nT761 -> T764" }, { "from": 2419, "to": 2424, "label": "EVAL-BACKTRACK" }, { "from": 2420, "to": 2476, "label": "EVAL with clause\naddC(one(X915), one(X916), one(X917)) :- addc(X915, X916, X917).\nand substitutionX915 -> T811,\nT669 -> one(T811),\nX916 -> T812,\nT670 -> one(T812),\nX917 -> T810,\nT668 -> one(T810),\nT808 -> T811,\nT809 -> T812" }, { "from": 2420, "to": 2477, "label": "EVAL-BACKTRACK" }, { "from": 2423, "to": 2428, "label": "CASE" }, { "from": 2428, "to": 2430, "label": "PARALLEL" }, { "from": 2428, "to": 2431, "label": "PARALLEL" }, { "from": 2430, "to": 2434, "label": "EVAL with clause\naddY(b, zero(X882), one(X882)) :- binaryZ(X882).\nand substitutionT763 -> b,\nX882 -> T770,\nT764 -> zero(T770),\nT762 -> one(T770)" }, { "from": 2430, "to": 2435, "label": "EVAL-BACKTRACK" }, { "from": 2431, "to": 2440, "label": "PARALLEL" }, { "from": 2431, "to": 2441, "label": "PARALLEL" }, { "from": 2434, "to": 1153, "label": "INSTANCE with matching:\nT111 -> T770" }, { "from": 2440, "to": 2450, "label": "EVAL with clause\naddY(b, one(X892), zero(X893)) :- succ(X892, X893).\nand substitutionT763 -> b,\nX892 -> T782,\nT764 -> one(T782),\nX893 -> T781,\nT762 -> zero(T781),\nT780 -> T782" }, { "from": 2440, "to": 2451, "label": "EVAL-BACKTRACK" }, { "from": 2441, "to": 2460, "label": "ONLY EVAL with clause\naddY(X903, X904, X905) :- addC(X903, X904, X905).\nand substitutionT763 -> T797,\nX903 -> T797,\nT764 -> T798,\nX904 -> T798,\nT762 -> T796,\nX905 -> T796,\nT794 -> T797,\nT795 -> T798" }, { "from": 2450, "to": 2336, "label": "INSTANCE with matching:\nT629 -> T782\nT628 -> T781" }, { "from": 2460, "to": 2374, "label": "INSTANCE with matching:\nT669 -> T797\nT670 -> T798\nT668 -> T796" }, { "from": 2476, "to": 2306, "label": "INSTANCE with matching:\nT603 -> T811\nT604 -> T812\nT602 -> T810" }, { "from": 2480, "to": 2483, "label": "PARALLEL" }, { "from": 2480, "to": 2484, "label": "PARALLEL" }, { "from": 2483, "to": 2495, "label": "EVAL with clause\naddz(one(X933), zero(X934), one(X935)) :- addy(X933, X934, X935).\nand substitutionX933 -> T829,\nT453 -> one(T829),\nT454 -> T830,\nX934 -> T830,\nX935 -> T828,\nT452 -> one(T828),\nT826 -> T829,\nT827 -> T830" }, { "from": 2483, "to": 2496, "label": "EVAL-BACKTRACK" }, { "from": 2484, "to": 2499, "label": "BACKTRACK\nfor clause: addz(one(X), one(Y), zero(Z)) :- addc(X, Y, Z)because of non-unification" }, { "from": 2495, "to": 2284, "label": "INSTANCE with matching:\nT561 -> T829\nT562 -> T830\nT560 -> T828" }, { "from": 2500, "to": 2505, "label": "EVAL with clause\ntimes(one(X947), X948, X949) :- ','(times(X947, X948, X950), add(X948, zero(X950), X949)).\nand substitutionX947 -> T842,\nT1 -> one(T842),\nT2 -> T843,\nX948 -> T843,\nT11 -> T841,\nX949 -> zero(T841),\nT839 -> T842,\nT840 -> T843" }, { "from": 2500, "to": 2506, "label": "EVAL-BACKTRACK" }, { "from": 2505, "to": 216, "label": "INSTANCE with matching:\nT53 -> T842\nT54 -> T843\nX48 -> X950\nT52 -> zero(T841)" }, { "from": 2525, "to": 2527, "label": "CASE" }, { "from": 2527, "to": 2530, "label": "PARALLEL" }, { "from": 2527, "to": 2531, "label": "PARALLEL" }, { "from": 2530, "to": 2532, "label": "EVAL with clause\ntimes(one(b), X964, X964).\nand substitutionT852 -> one(b),\nT853 -> T859,\nX964 -> T859,\nX959 -> T859,\nT858 -> T859" }, { "from": 2530, "to": 2533, "label": "EVAL-BACKTRACK" }, { "from": 2531, "to": 2553, "label": "PARALLEL" }, { "from": 2531, "to": 2554, "label": "PARALLEL" }, { "from": 2532, "to": 219, "label": "INSTANCE with matching:\nT58 -> T859\nT57 -> T859\nT52 -> T851" }, { "from": 2553, "to": 2559, "label": "EVAL with clause\ntimes(zero(X981), X982, zero(X983)) :- times(X981, X982, X983).\nand substitutionX981 -> T870,\nT852 -> zero(T870),\nT853 -> T871,\nX982 -> T871,\nX983 -> X984,\nX959 -> zero(X984),\nT868 -> T870,\nT869 -> T871" }, { "from": 2553, "to": 2560, "label": "EVAL-BACKTRACK" }, { "from": 2554, "to": 2563, "label": "EVAL with clause\ntimes(one(X998), X999, X1000) :- ','(times(X998, X999, X1001), add(X999, zero(X1001), X1000)).\nand substitutionX998 -> T886,\nT852 -> one(T886),\nT853 -> T887,\nX999 -> T887,\nX959 -> X1002,\nX1000 -> X1002,\nT884 -> T886,\nT885 -> T887" }, { "from": 2554, "to": 2564, "label": "EVAL-BACKTRACK" }, { "from": 2559, "to": 2561, "label": "SPLIT 1" }, { "from": 2559, "to": 2562, "label": "SPLIT 2\nnew knowledge:\nT870 is ground\nreplacements:X984 -> T874,\nT871 -> T875,\nT1 -> T876,\nT2 -> T877" }, { "from": 2561, "to": 218, "label": "INSTANCE with matching:\nT53 -> T870\nT54 -> T871\nX48 -> X984" }, { "from": 2562, "to": 219, "label": "INSTANCE with matching:\nT58 -> T875\nT57 -> zero(T874)\nT52 -> T851" }, { "from": 2563, "to": 2565, "label": "SPLIT 1" }, { "from": 2563, "to": 2566, "label": "SPLIT 2\nnew knowledge:\nT886 is ground\nreplacements:X1001 -> T890,\nT887 -> T891,\nT1 -> T892,\nT2 -> T893" }, { "from": 2565, "to": 218, "label": "INSTANCE with matching:\nT53 -> T886\nT54 -> T887\nX48 -> X1001" }, { "from": 2566, "to": 2567, "label": "SPLIT 1" }, { "from": 2566, "to": 2568, "label": "SPLIT 2\nnew knowledge:\nT897 is ground\nT890 is ground\nT896 is ground\nreplacements:X1002 -> T896,\nT891 -> T897,\nT892 -> T898,\nT893 -> T899" }, { "from": 2567, "to": 638, "label": "INSTANCE with matching:\nT91 -> T891\nT90 -> T890\nX93 -> X1002" }, { "from": 2568, "to": 219, "label": "INSTANCE with matching:\nT58 -> T897\nT57 -> T896\nT52 -> T851" }, { "from": 2610, "to": 2612, "label": "CASE" }, { "from": 2611, "to": 2649, "label": "EVAL with clause\ntimes(one(X1071), X1072, X1073) :- ','(times(X1071, X1072, X1074), add(X1072, zero(X1074), X1073)).\nand substitutionX1071 -> T970,\nT1 -> one(T970),\nT2 -> T971,\nX1072 -> T971,\nT3 -> T969,\nX1073 -> T969,\nT967 -> T970,\nT968 -> T971" }, { "from": 2611, "to": 2650, "label": "EVAL-BACKTRACK" }, { "from": 2612, "to": 2614, "label": "PARALLEL" }, { "from": 2612, "to": 2615, "label": "PARALLEL" }, { "from": 2614, "to": 2616, "label": "EVAL with clause\ntimes(one(b), X1017, X1017).\nand substitutionT906 -> one(b),\nT907 -> T912,\nX1017 -> T912,\nT905 -> T912" }, { "from": 2614, "to": 2617, "label": "EVAL-BACKTRACK" }, { "from": 2615, "to": 2619, "label": "PARALLEL" }, { "from": 2615, "to": 2620, "label": "PARALLEL" }, { "from": 2616, "to": 2618, "label": "SUCCESS" }, { "from": 2619, "to": 2626, "label": "EVAL with clause\ntimes(zero(X1030), X1031, zero(X1032)) :- times(X1030, X1031, X1032).\nand substitutionX1030 -> T928,\nT906 -> zero(T928),\nT907 -> T929,\nX1031 -> T929,\nX1032 -> T927,\nT905 -> zero(T927),\nT925 -> T928,\nT926 -> T929" }, { "from": 2619, "to": 2627, "label": "EVAL-BACKTRACK" }, { "from": 2620, "to": 2634, "label": "PARALLEL" }, { "from": 2620, "to": 2635, "label": "PARALLEL" }, { "from": 2626, "to": 1, "label": "INSTANCE with matching:\nT1 -> T928\nT2 -> T929\nT3 -> T927" }, { "from": 2634, "to": 2636, "label": "EVAL with clause\ntimes(one(X1049), X1050, X1051) :- ','(times(X1049, X1050, X1052), add(X1050, zero(X1052), X1051)).\nand substitutionX1049 -> T947,\nT906 -> one(T947),\nT907 -> T948,\nX1050 -> T948,\nT905 -> T946,\nX1051 -> T946,\nT944 -> T947,\nT945 -> T948" }, { "from": 2634, "to": 2637, "label": "EVAL-BACKTRACK" }, { "from": 2635, "to": 2640, "label": "FAILURE" }, { "from": 2636, "to": 216, "label": "INSTANCE with matching:\nT53 -> T947\nT54 -> T948\nX48 -> X1052\nT52 -> T946" }, { "from": 2640, "to": 2644, "label": "EVAL with clause\ntimes(one(X1062), X1063, X1064) :- ','(times(X1062, X1063, X1065), add(X1063, zero(X1065), X1064)).\nand substitutionX1062 -> T960,\nT1 -> one(T960),\nT2 -> T961,\nX1063 -> T961,\nT905 -> T959,\nX1064 -> zero(T959),\nT957 -> T960,\nT958 -> T961" }, { "from": 2640, "to": 2645, "label": "EVAL-BACKTRACK" }, { "from": 2644, "to": 216, "label": "INSTANCE with matching:\nT53 -> T960\nT54 -> T961\nX48 -> X1065\nT52 -> zero(T959)" }, { "from": 2649, "to": 2651, "label": "CASE" }, { "from": 2651, "to": 2652, "label": "PARALLEL" }, { "from": 2651, "to": 2653, "label": "PARALLEL" }, { "from": 2652, "to": 2654, "label": "EVAL with clause\ntimes(one(b), X1079, X1079).\nand substitutionT970 -> one(b),\nT971 -> T977,\nX1079 -> T977,\nX1074 -> T977,\nT976 -> T977" }, { "from": 2652, "to": 2655, "label": "EVAL-BACKTRACK" }, { "from": 2653, "to": 2656, "label": "PARALLEL" }, { "from": 2653, "to": 2657, "label": "PARALLEL" }, { "from": 2654, "to": 219, "label": "INSTANCE with matching:\nT58 -> T977\nT57 -> T977\nT52 -> T969" }, { "from": 2656, "to": 2658, "label": "EVAL with clause\ntimes(zero(X1096), X1097, zero(X1098)) :- times(X1096, X1097, X1098).\nand substitutionX1096 -> T988,\nT970 -> zero(T988),\nT971 -> T989,\nX1097 -> T989,\nX1098 -> X1099,\nX1074 -> zero(X1099),\nT986 -> T988,\nT987 -> T989" }, { "from": 2656, "to": 2659, "label": "EVAL-BACKTRACK" }, { "from": 2657, "to": 2660, "label": "EVAL with clause\ntimes(one(X1111), X1112, X1113) :- ','(times(X1111, X1112, X1114), add(X1112, zero(X1114), X1113)).\nand substitutionX1111 -> T998,\nT970 -> one(T998),\nT971 -> T999,\nX1112 -> T999,\nX1074 -> X1115,\nX1113 -> X1115,\nT996 -> T998,\nT997 -> T999" }, { "from": 2657, "to": 2661, "label": "EVAL-BACKTRACK" }, { "from": 2658, "to": 2559, "label": "INSTANCE with matching:\nT870 -> T988\nT871 -> T989\nX984 -> X1099\nT851 -> T969\nX6 -> X1010\nX7 -> X1011\nX8 -> X1012" }, { "from": 2660, "to": 2563, "label": "INSTANCE with matching:\nT886 -> T998\nT887 -> T999\nX1001 -> X1114\nX1002 -> X1115\nT851 -> T969\nX6 -> X1010\nX7 -> X1011\nX8 -> X1012" } ], "type": "Graph" } } ---------------------------------------- (98) Obligation: Triples: timesF(zero(X1), X2, zero(X3)) :- timesF(X1, X2, X3). timesF(one(X1), X2, X3) :- timesF(X1, X2, X4). timesF(one(X1), X2, X3) :- ','(timescF(X1, X2, X4), addG(X2, X4, X3)). binaryZH(zero(X1)) :- binaryZH(X1). binaryZH(one(X1)) :- binaryI(X1). binaryI(zero(X1)) :- binaryZH(X1). binaryI(one(X1)) :- binaryI(X1). addzJ(zero(X1), zero(X2), zero(X3)) :- addzJ(X1, X2, X3). addzJ(zero(one(X1)), one(b), one(one(X1))) :- binaryI(X1). addzJ(zero(zero(X1)), one(b), one(zero(X1))) :- binaryZH(X1). addzJ(zero(X1), one(X2), one(X3)) :- addzJ(X1, X2, X3). addzJ(one(X1), zero(X2), one(X3)) :- addyK(X1, X2, X3). addzJ(one(X1), one(X2), zero(X3)) :- addcL(X1, X2, X3). succM(zero(X1), one(X1)) :- binaryZH(X1). succM(one(X1), zero(X2)) :- succM(X1, X2). succZN(zero(X1), one(X1)) :- binaryZH(X1). succZN(one(X1), zero(X2)) :- succM(X1, X2). addCO(zero(X1), zero(X2), one(X3)) :- addzJ(X1, X2, X3). addCO(zero(zero(X1)), one(b), zero(one(X1))) :- binaryZH(X1). addCO(zero(one(X1)), one(b), zero(zero(X2))) :- succM(X1, X2). addCO(zero(X1), one(X2), zero(X3)) :- addCO(X1, X2, X3). addCO(one(b), zero(zero(X1)), zero(one(X1))) :- binaryZH(X1). addCO(one(b), zero(one(X1)), zero(zero(X2))) :- succM(X1, X2). addCO(one(X1), zero(X2), zero(X3)) :- addCO(X1, X2, X3). addCO(one(X1), one(X2), one(X3)) :- addcL(X1, X2, X3). addcL(X1, b, X2) :- succZN(X1, X2). addcL(b, X1, X2) :- succZN(X1, X2). addcL(X1, X2, X3) :- addCO(X1, X2, X3). addyK(b, one(X1), one(X1)) :- binaryI(X1). addyK(b, zero(X1), zero(X1)) :- binaryZH(X1). addyK(X1, X2, X3) :- addzJ(X1, X2, X3). binaryZP(X1) :- binaryZH(X1). addzQ(zero(X1), zero(X2), zero(X3)) :- addzQ(X1, X2, X3). addzQ(zero(one(X1)), one(b), one(one(X1))) :- binaryI(X1). addzQ(zero(zero(X1)), one(b), one(zero(X1))) :- binaryZH(X1). addzQ(zero(X1), one(X2), one(X3)) :- addzQ(X1, X2, X3). addzQ(one(X1), zero(X2), one(X3)) :- addyR(X1, X2, X3). addzQ(one(X1), one(X2), zero(X3)) :- addcS(X1, X2, X3). succT(zero(X1), one(X1)) :- binaryZH(X1). succT(one(X1), zero(X2)) :- succT(X1, X2). succZU(zero(X1), one(X1)) :- binaryZH(X1). succZU(one(X1), zero(X2)) :- succT(X1, X2). addCV(zero(X1), zero(X2), one(X3)) :- addzQ(X1, X2, X3). addCV(zero(zero(X1)), one(b), zero(one(X1))) :- binaryZH(X1). addCV(zero(one(X1)), one(b), zero(zero(X2))) :- succT(X1, X2). addCV(zero(X1), one(X2), zero(X3)) :- addCV(X1, X2, X3). addCV(one(b), zero(zero(X1)), zero(one(X1))) :- binaryZH(X1). addCV(one(b), zero(one(X1)), zero(zero(X2))) :- succT(X1, X2). addCV(one(X1), zero(X2), zero(X3)) :- addCV(X1, X2, X3). addCV(one(X1), one(X2), one(X3)) :- addcS(X1, X2, X3). addcS(X1, b, X2) :- succZU(X1, X2). addcS(b, X1, X2) :- succZU(X1, X2). addcS(X1, X2, X3) :- addCV(X1, X2, X3). addyR(b, one(X1), one(X1)) :- binaryI(X1). addyR(b, zero(X1), zero(X1)) :- binaryZH(X1). addyR(X1, X2, X3) :- addzQ(X1, X2, X3). pB(X1, X2, X3, X4) :- timesF(X1, X2, X3). pB(X1, X2, X3, X4) :- ','(timescF(X1, X2, X3), addC1(X2, X3, X4)). addC1(b, X1, zero(X1)) :- binaryZP(X1). addC1(zero(X1), X2, zero(X3)) :- addzQ(X1, X2, X3). addC1(one(X1), X2, one(X3)) :- addyR(X1, X2, X3). addG(b, X1, zero(X1)) :- binaryZP(X1). addG(zero(X1), X2, zero(X3)) :- addzJ(X1, X2, X3). addG(one(X1), X2, one(X3)) :- addyK(X1, X2, X3). pD(X1, X2, X3, X4) :- timesF(X1, X2, X3). pD(X1, X2, X3, X4) :- ','(timescF(X1, X2, X3), addC1(X2, zero(X3), X4)). pE(X1, X2, X3, X4, X5) :- timesF(X1, X2, X3). pE(X1, X2, X3, X4, X5) :- ','(timescF(X1, X2, X3), addG(X2, X3, X4)). pE(X1, X2, X3, X4, X5) :- ','(timescF(X1, X2, X3), ','(addcG(X2, X3, X4), addC1(X2, X4, X5))). timesA(zero(zero(X1)), X2, zero(zero(X3))) :- timesA(X1, X2, X3). timesA(zero(one(X1)), X2, zero(X3)) :- pB(X1, X2, X4, X3). timesA(one(X1), X2, zero(X3)) :- pB(X1, X2, X4, zero(X3)). timesA(one(one(b)), X1, X2) :- addC1(X1, X1, X2). timesA(one(zero(X1)), X2, X3) :- pD(X1, X2, X4, X3). timesA(one(one(X1)), X2, X3) :- pE(X1, X2, X4, X5, X3). timesA(zero(zero(X1)), X2, zero(zero(X3))) :- timesA(X1, X2, X3). timesA(zero(one(X1)), X2, zero(X3)) :- pB(X1, X2, X4, X3). timesA(one(X1), X2, zero(X3)) :- pB(X1, X2, X4, zero(X3)). timesA(one(one(b)), X1, X2) :- addC1(X1, X1, X2). timesA(one(zero(X1)), X2, X3) :- pD(X1, X2, X4, X3). timesA(one(one(X1)), X2, X3) :- pE(X1, X2, X4, X5, X3). Clauses: timescA(one(b), X1, X1). timescA(zero(one(b)), X1, zero(X1)). timescA(zero(zero(X1)), X2, zero(zero(X3))) :- timescA(X1, X2, X3). timescA(zero(one(X1)), X2, zero(X3)) :- qcB(X1, X2, X4, X3). timescA(one(X1), X2, zero(X3)) :- qcB(X1, X2, X4, zero(X3)). timescA(one(one(b)), X1, X2) :- addcC(X1, X1, X2). timescA(one(zero(X1)), X2, X3) :- qcD(X1, X2, X4, X3). timescA(one(one(X1)), X2, X3) :- qcE(X1, X2, X4, X5, X3). timescA(zero(one(b)), X1, zero(X1)). timescA(zero(zero(X1)), X2, zero(zero(X3))) :- timescA(X1, X2, X3). timescA(zero(one(X1)), X2, zero(X3)) :- qcB(X1, X2, X4, X3). timescA(one(X1), X2, zero(X3)) :- qcB(X1, X2, X4, zero(X3)). timescA(one(one(b)), X1, X2) :- addcC(X1, X1, X2). timescA(one(zero(X1)), X2, X3) :- qcD(X1, X2, X4, X3). timescA(one(one(X1)), X2, X3) :- qcE(X1, X2, X4, X5, X3). timescF(one(b), X1, X1). timescF(zero(X1), X2, zero(X3)) :- timescF(X1, X2, X3). timescF(one(X1), X2, X3) :- ','(timescF(X1, X2, X4), addcG(X2, X4, X3)). binaryZcH(zero(X1)) :- binaryZcH(X1). binaryZcH(one(X1)) :- binarycI(X1). binarycI(b). binarycI(zero(X1)) :- binaryZcH(X1). binarycI(one(X1)) :- binarycI(X1). addzcJ(zero(X1), zero(X2), zero(X3)) :- addzcJ(X1, X2, X3). addzcJ(zero(one(X1)), one(b), one(one(X1))) :- binarycI(X1). addzcJ(zero(zero(X1)), one(b), one(zero(X1))) :- binaryZcH(X1). addzcJ(zero(X1), one(X2), one(X3)) :- addzcJ(X1, X2, X3). addzcJ(one(X1), zero(X2), one(X3)) :- addycK(X1, X2, X3). addzcJ(one(X1), one(X2), zero(X3)) :- addccL(X1, X2, X3). succcM(b, one(b)). succcM(zero(X1), one(X1)) :- binaryZcH(X1). succcM(one(X1), zero(X2)) :- succcM(X1, X2). succZcN(zero(X1), one(X1)) :- binaryZcH(X1). succZcN(one(X1), zero(X2)) :- succcM(X1, X2). addCcO(zero(X1), zero(X2), one(X3)) :- addzcJ(X1, X2, X3). addCcO(zero(zero(X1)), one(b), zero(one(X1))) :- binaryZcH(X1). addCcO(zero(one(X1)), one(b), zero(zero(X2))) :- succcM(X1, X2). addCcO(zero(X1), one(X2), zero(X3)) :- addCcO(X1, X2, X3). addCcO(one(b), zero(zero(X1)), zero(one(X1))) :- binaryZcH(X1). addCcO(one(b), zero(one(X1)), zero(zero(X2))) :- succcM(X1, X2). addCcO(one(X1), zero(X2), zero(X3)) :- addCcO(X1, X2, X3). addCcO(one(X1), one(X2), one(X3)) :- addccL(X1, X2, X3). addccL(b, b, one(b)). addccL(X1, b, X2) :- succZcN(X1, X2). addccL(b, X1, X2) :- succZcN(X1, X2). addccL(X1, X2, X3) :- addCcO(X1, X2, X3). addycK(b, one(X1), one(X1)) :- binarycI(X1). addycK(b, zero(X1), zero(X1)) :- binaryZcH(X1). addycK(X1, X2, X3) :- addzcJ(X1, X2, X3). binaryZcP(X1) :- binaryZcH(X1). addzcQ(zero(X1), zero(X2), zero(X3)) :- addzcQ(X1, X2, X3). addzcQ(zero(one(X1)), one(b), one(one(X1))) :- binarycI(X1). addzcQ(zero(zero(X1)), one(b), one(zero(X1))) :- binaryZcH(X1). addzcQ(zero(X1), one(X2), one(X3)) :- addzcQ(X1, X2, X3). addzcQ(one(X1), zero(X2), one(X3)) :- addycR(X1, X2, X3). addzcQ(one(X1), one(X2), zero(X3)) :- addccS(X1, X2, X3). succcT(b, one(b)). succcT(zero(X1), one(X1)) :- binaryZcH(X1). succcT(one(X1), zero(X2)) :- succcT(X1, X2). succZcU(zero(X1), one(X1)) :- binaryZcH(X1). succZcU(one(X1), zero(X2)) :- succcT(X1, X2). addCcV(zero(X1), zero(X2), one(X3)) :- addzcQ(X1, X2, X3). addCcV(zero(zero(X1)), one(b), zero(one(X1))) :- binaryZcH(X1). addCcV(zero(one(X1)), one(b), zero(zero(X2))) :- succcT(X1, X2). addCcV(zero(X1), one(X2), zero(X3)) :- addCcV(X1, X2, X3). addCcV(one(b), zero(zero(X1)), zero(one(X1))) :- binaryZcH(X1). addCcV(one(b), zero(one(X1)), zero(zero(X2))) :- succcT(X1, X2). addCcV(one(X1), zero(X2), zero(X3)) :- addCcV(X1, X2, X3). addCcV(one(X1), one(X2), one(X3)) :- addccS(X1, X2, X3). addccS(b, b, one(b)). addccS(X1, b, X2) :- succZcU(X1, X2). addccS(b, X1, X2) :- succZcU(X1, X2). addccS(X1, X2, X3) :- addCcV(X1, X2, X3). addycR(b, one(X1), one(X1)) :- binarycI(X1). addycR(b, zero(X1), zero(X1)) :- binaryZcH(X1). addycR(X1, X2, X3) :- addzcQ(X1, X2, X3). qcB(X1, X2, X3, X4) :- ','(timescF(X1, X2, X3), addcC(X2, X3, X4)). addcC(b, X1, zero(X1)) :- binaryZcP(X1). addcC(zero(X1), X2, zero(X3)) :- addzcQ(X1, X2, X3). addcC(one(X1), X2, one(X3)) :- addycR(X1, X2, X3). addcG(b, X1, zero(X1)) :- binaryZcP(X1). addcG(zero(X1), X2, zero(X3)) :- addzcJ(X1, X2, X3). addcG(one(X1), X2, one(X3)) :- addycK(X1, X2, X3). qcD(X1, X2, X3, X4) :- ','(timescF(X1, X2, X3), addcC(X2, zero(X3), X4)). qcE(X1, X2, X3, X4, X5) :- ','(timescF(X1, X2, X3), ','(addcG(X2, X3, X4), addcC(X2, X4, X5))). Afs: timesA(x1, x2, x3) = timesA(x3) ---------------------------------------- (99) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: timesA_in_3: (f,f,b) pB_in_4: (f,f,f,b) timesF_in_3: (f,f,f) timescF_in_3: (f,f,f) addcG_in_3: (f,f,f) binaryZcP_in_1: (f) binaryZcH_in_1: (f) binarycI_in_1: (f) addzcJ_in_3: (f,f,f) addycK_in_3: (f,f,f) addccL_in_3: (f,f,f) succZcN_in_2: (f,f) succcM_in_2: (f,f) addCcO_in_3: (f,f,f) addG_in_3: (f,f,f) binaryZP_in_1: (f) (b) binaryZH_in_1: (f) (b) binaryI_in_1: (f) (b) addzJ_in_3: (f,f,f) addyK_in_3: (f,f,f) addcL_in_3: (f,f,f) succZN_in_2: (f,f) succM_in_2: (f,f) addCO_in_3: (f,f,f) addC1_in_3: (f,f,b) (b,b,b) addzQ_in_3: (f,f,b) (b,b,b) addyR_in_3: (f,f,b) (b,b,b) addcS_in_3: (f,f,b) (b,b,b) succZU_in_2: (f,b) (b,b) succT_in_2: (f,b) (b,b) addCV_in_3: (f,f,b) (b,b,b) pD_in_4: (f,f,f,b) pE_in_5: (f,f,f,f,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: TIMESA_IN_AAG(zero(zero(X1)), X2, zero(zero(X3))) -> U75_AAG(X1, X2, X3, timesA_in_aag(X1, X2, X3)) TIMESA_IN_AAG(zero(zero(X1)), X2, zero(zero(X3))) -> TIMESA_IN_AAG(X1, X2, X3) TIMESA_IN_AAG(zero(one(X1)), X2, zero(X3)) -> U76_AAG(X1, X2, X3, pB_in_aaag(X1, X2, X4, X3)) TIMESA_IN_AAG(zero(one(X1)), X2, zero(X3)) -> PB_IN_AAAG(X1, X2, X4, X3) PB_IN_AAAG(X1, X2, X3, X4) -> U58_AAAG(X1, X2, X3, X4, timesF_in_aaa(X1, X2, X3)) PB_IN_AAAG(X1, X2, X3, X4) -> TIMESF_IN_AAA(X1, X2, X3) TIMESF_IN_AAA(zero(X1), X2, zero(X3)) -> U1_AAA(X1, X2, X3, timesF_in_aaa(X1, X2, X3)) TIMESF_IN_AAA(zero(X1), X2, zero(X3)) -> TIMESF_IN_AAA(X1, X2, X3) TIMESF_IN_AAA(one(X1), X2, X3) -> U2_AAA(X1, X2, X3, timesF_in_aaa(X1, X2, X4)) TIMESF_IN_AAA(one(X1), X2, X3) -> TIMESF_IN_AAA(X1, X2, X4) TIMESF_IN_AAA(one(X1), X2, X3) -> U3_AAA(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U3_AAA(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U4_AAA(X1, X2, X3, addG_in_aaa(X2, X4, X3)) U3_AAA(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> ADDG_IN_AAA(X2, X4, X3) ADDG_IN_AAA(b, X1, zero(X1)) -> U64_AAA(X1, binaryZP_in_a(X1)) ADDG_IN_AAA(b, X1, zero(X1)) -> BINARYZP_IN_A(X1) BINARYZP_IN_A(X1) -> U33_A(X1, binaryZH_in_a(X1)) BINARYZP_IN_A(X1) -> BINARYZH_IN_A(X1) BINARYZH_IN_A(zero(X1)) -> U5_A(X1, binaryZH_in_a(X1)) BINARYZH_IN_A(zero(X1)) -> BINARYZH_IN_A(X1) BINARYZH_IN_A(one(X1)) -> U6_A(X1, binaryI_in_a(X1)) BINARYZH_IN_A(one(X1)) -> BINARYI_IN_A(X1) BINARYI_IN_A(zero(X1)) -> U7_A(X1, binaryZH_in_a(X1)) BINARYI_IN_A(zero(X1)) -> BINARYZH_IN_A(X1) BINARYI_IN_A(one(X1)) -> U8_A(X1, binaryI_in_a(X1)) BINARYI_IN_A(one(X1)) -> BINARYI_IN_A(X1) ADDG_IN_AAA(zero(X1), X2, zero(X3)) -> U65_AAA(X1, X2, X3, addzJ_in_aaa(X1, X2, X3)) ADDG_IN_AAA(zero(X1), X2, zero(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(zero(X1), zero(X2), zero(X3)) -> U9_AAA(X1, X2, X3, addzJ_in_aaa(X1, X2, X3)) ADDZJ_IN_AAA(zero(X1), zero(X2), zero(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(zero(one(X1)), one(b), one(one(X1))) -> U10_AAA(X1, binaryI_in_a(X1)) ADDZJ_IN_AAA(zero(one(X1)), one(b), one(one(X1))) -> BINARYI_IN_A(X1) ADDZJ_IN_AAA(zero(zero(X1)), one(b), one(zero(X1))) -> U11_AAA(X1, binaryZH_in_a(X1)) ADDZJ_IN_AAA(zero(zero(X1)), one(b), one(zero(X1))) -> BINARYZH_IN_A(X1) ADDZJ_IN_AAA(zero(X1), one(X2), one(X3)) -> U12_AAA(X1, X2, X3, addzJ_in_aaa(X1, X2, X3)) ADDZJ_IN_AAA(zero(X1), one(X2), one(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(one(X1), zero(X2), one(X3)) -> U13_AAA(X1, X2, X3, addyK_in_aaa(X1, X2, X3)) ADDZJ_IN_AAA(one(X1), zero(X2), one(X3)) -> ADDYK_IN_AAA(X1, X2, X3) ADDYK_IN_AAA(b, one(X1), one(X1)) -> U30_AAA(X1, binaryI_in_a(X1)) ADDYK_IN_AAA(b, one(X1), one(X1)) -> BINARYI_IN_A(X1) ADDYK_IN_AAA(b, zero(X1), zero(X1)) -> U31_AAA(X1, binaryZH_in_a(X1)) ADDYK_IN_AAA(b, zero(X1), zero(X1)) -> BINARYZH_IN_A(X1) ADDYK_IN_AAA(X1, X2, X3) -> U32_AAA(X1, X2, X3, addzJ_in_aaa(X1, X2, X3)) ADDYK_IN_AAA(X1, X2, X3) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(one(X1), one(X2), zero(X3)) -> U14_AAA(X1, X2, X3, addcL_in_aaa(X1, X2, X3)) ADDZJ_IN_AAA(one(X1), one(X2), zero(X3)) -> ADDCL_IN_AAA(X1, X2, X3) ADDCL_IN_AAA(X1, b, X2) -> U27_AAA(X1, X2, succZN_in_aa(X1, X2)) ADDCL_IN_AAA(X1, b, X2) -> SUCCZN_IN_AA(X1, X2) SUCCZN_IN_AA(zero(X1), one(X1)) -> U17_AA(X1, binaryZH_in_a(X1)) SUCCZN_IN_AA(zero(X1), one(X1)) -> BINARYZH_IN_A(X1) SUCCZN_IN_AA(one(X1), zero(X2)) -> U18_AA(X1, X2, succM_in_aa(X1, X2)) SUCCZN_IN_AA(one(X1), zero(X2)) -> SUCCM_IN_AA(X1, X2) SUCCM_IN_AA(zero(X1), one(X1)) -> U15_AA(X1, binaryZH_in_a(X1)) SUCCM_IN_AA(zero(X1), one(X1)) -> BINARYZH_IN_A(X1) SUCCM_IN_AA(one(X1), zero(X2)) -> U16_AA(X1, X2, succM_in_aa(X1, X2)) SUCCM_IN_AA(one(X1), zero(X2)) -> SUCCM_IN_AA(X1, X2) ADDCL_IN_AAA(b, X1, X2) -> U28_AAA(X1, X2, succZN_in_aa(X1, X2)) ADDCL_IN_AAA(b, X1, X2) -> SUCCZN_IN_AA(X1, X2) ADDCL_IN_AAA(X1, X2, X3) -> U29_AAA(X1, X2, X3, addCO_in_aaa(X1, X2, X3)) ADDCL_IN_AAA(X1, X2, X3) -> ADDCO_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(zero(X1), zero(X2), one(X3)) -> U19_AAA(X1, X2, X3, addzJ_in_aaa(X1, X2, X3)) ADDCO_IN_AAA(zero(X1), zero(X2), one(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(zero(zero(X1)), one(b), zero(one(X1))) -> U20_AAA(X1, binaryZH_in_a(X1)) ADDCO_IN_AAA(zero(zero(X1)), one(b), zero(one(X1))) -> BINARYZH_IN_A(X1) ADDCO_IN_AAA(zero(one(X1)), one(b), zero(zero(X2))) -> U21_AAA(X1, X2, succM_in_aa(X1, X2)) ADDCO_IN_AAA(zero(one(X1)), one(b), zero(zero(X2))) -> SUCCM_IN_AA(X1, X2) ADDCO_IN_AAA(zero(X1), one(X2), zero(X3)) -> U22_AAA(X1, X2, X3, addCO_in_aaa(X1, X2, X3)) ADDCO_IN_AAA(zero(X1), one(X2), zero(X3)) -> ADDCO_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(one(b), zero(zero(X1)), zero(one(X1))) -> U23_AAA(X1, binaryZH_in_a(X1)) ADDCO_IN_AAA(one(b), zero(zero(X1)), zero(one(X1))) -> BINARYZH_IN_A(X1) ADDCO_IN_AAA(one(b), zero(one(X1)), zero(zero(X2))) -> U24_AAA(X1, X2, succM_in_aa(X1, X2)) ADDCO_IN_AAA(one(b), zero(one(X1)), zero(zero(X2))) -> SUCCM_IN_AA(X1, X2) ADDCO_IN_AAA(one(X1), zero(X2), zero(X3)) -> U25_AAA(X1, X2, X3, addCO_in_aaa(X1, X2, X3)) ADDCO_IN_AAA(one(X1), zero(X2), zero(X3)) -> ADDCO_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(one(X1), one(X2), one(X3)) -> U26_AAA(X1, X2, X3, addcL_in_aaa(X1, X2, X3)) ADDCO_IN_AAA(one(X1), one(X2), one(X3)) -> ADDCL_IN_AAA(X1, X2, X3) ADDG_IN_AAA(one(X1), X2, one(X3)) -> U66_AAA(X1, X2, X3, addyK_in_aaa(X1, X2, X3)) ADDG_IN_AAA(one(X1), X2, one(X3)) -> ADDYK_IN_AAA(X1, X2, X3) PB_IN_AAAG(X1, X2, X3, X4) -> U59_AAAG(X1, X2, X3, X4, timescF_in_aaa(X1, X2, X3)) U59_AAAG(X1, X2, X3, X4, timescF_out_aaa(X1, X2, X3)) -> U60_AAAG(X1, X2, X3, X4, addC1_in_aag(X2, X3, X4)) U59_AAAG(X1, X2, X3, X4, timescF_out_aaa(X1, X2, X3)) -> ADDC1_IN_AAG(X2, X3, X4) ADDC1_IN_AAG(b, X1, zero(X1)) -> U61_AAG(X1, binaryZP_in_g(X1)) ADDC1_IN_AAG(b, X1, zero(X1)) -> BINARYZP_IN_G(X1) BINARYZP_IN_G(X1) -> U33_G(X1, binaryZH_in_g(X1)) BINARYZP_IN_G(X1) -> BINARYZH_IN_G(X1) BINARYZH_IN_G(zero(X1)) -> U5_G(X1, binaryZH_in_g(X1)) BINARYZH_IN_G(zero(X1)) -> BINARYZH_IN_G(X1) BINARYZH_IN_G(one(X1)) -> U6_G(X1, binaryI_in_g(X1)) BINARYZH_IN_G(one(X1)) -> BINARYI_IN_G(X1) BINARYI_IN_G(zero(X1)) -> U7_G(X1, binaryZH_in_g(X1)) BINARYI_IN_G(zero(X1)) -> BINARYZH_IN_G(X1) BINARYI_IN_G(one(X1)) -> U8_G(X1, binaryI_in_g(X1)) BINARYI_IN_G(one(X1)) -> BINARYI_IN_G(X1) ADDC1_IN_AAG(zero(X1), X2, zero(X3)) -> U62_AAG(X1, X2, X3, addzQ_in_aag(X1, X2, X3)) ADDC1_IN_AAG(zero(X1), X2, zero(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(zero(X1), zero(X2), zero(X3)) -> U34_AAG(X1, X2, X3, addzQ_in_aag(X1, X2, X3)) ADDZQ_IN_AAG(zero(X1), zero(X2), zero(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(zero(one(X1)), one(b), one(one(X1))) -> U35_AAG(X1, binaryI_in_g(X1)) ADDZQ_IN_AAG(zero(one(X1)), one(b), one(one(X1))) -> BINARYI_IN_G(X1) ADDZQ_IN_AAG(zero(zero(X1)), one(b), one(zero(X1))) -> U36_AAG(X1, binaryZH_in_g(X1)) ADDZQ_IN_AAG(zero(zero(X1)), one(b), one(zero(X1))) -> BINARYZH_IN_G(X1) ADDZQ_IN_AAG(zero(X1), one(X2), one(X3)) -> U37_AAG(X1, X2, X3, addzQ_in_aag(X1, X2, X3)) ADDZQ_IN_AAG(zero(X1), one(X2), one(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(one(X1), zero(X2), one(X3)) -> U38_AAG(X1, X2, X3, addyR_in_aag(X1, X2, X3)) ADDZQ_IN_AAG(one(X1), zero(X2), one(X3)) -> ADDYR_IN_AAG(X1, X2, X3) ADDYR_IN_AAG(b, one(X1), one(X1)) -> U55_AAG(X1, binaryI_in_g(X1)) ADDYR_IN_AAG(b, one(X1), one(X1)) -> BINARYI_IN_G(X1) ADDYR_IN_AAG(b, zero(X1), zero(X1)) -> U56_AAG(X1, binaryZH_in_g(X1)) ADDYR_IN_AAG(b, zero(X1), zero(X1)) -> BINARYZH_IN_G(X1) ADDYR_IN_AAG(X1, X2, X3) -> U57_AAG(X1, X2, X3, addzQ_in_aag(X1, X2, X3)) ADDYR_IN_AAG(X1, X2, X3) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(one(X1), one(X2), zero(X3)) -> U39_AAG(X1, X2, X3, addcS_in_aag(X1, X2, X3)) ADDZQ_IN_AAG(one(X1), one(X2), zero(X3)) -> ADDCS_IN_AAG(X1, X2, X3) ADDCS_IN_AAG(X1, b, X2) -> U52_AAG(X1, X2, succZU_in_ag(X1, X2)) ADDCS_IN_AAG(X1, b, X2) -> SUCCZU_IN_AG(X1, X2) SUCCZU_IN_AG(zero(X1), one(X1)) -> U42_AG(X1, binaryZH_in_g(X1)) SUCCZU_IN_AG(zero(X1), one(X1)) -> BINARYZH_IN_G(X1) SUCCZU_IN_AG(one(X1), zero(X2)) -> U43_AG(X1, X2, succT_in_ag(X1, X2)) SUCCZU_IN_AG(one(X1), zero(X2)) -> SUCCT_IN_AG(X1, X2) SUCCT_IN_AG(zero(X1), one(X1)) -> U40_AG(X1, binaryZH_in_g(X1)) SUCCT_IN_AG(zero(X1), one(X1)) -> BINARYZH_IN_G(X1) SUCCT_IN_AG(one(X1), zero(X2)) -> U41_AG(X1, X2, succT_in_ag(X1, X2)) SUCCT_IN_AG(one(X1), zero(X2)) -> SUCCT_IN_AG(X1, X2) ADDCS_IN_AAG(b, X1, X2) -> U53_AAG(X1, X2, succZU_in_ag(X1, X2)) ADDCS_IN_AAG(b, X1, X2) -> SUCCZU_IN_AG(X1, X2) ADDCS_IN_AAG(X1, X2, X3) -> U54_AAG(X1, X2, X3, addCV_in_aag(X1, X2, X3)) ADDCS_IN_AAG(X1, X2, X3) -> ADDCV_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(zero(X1), zero(X2), one(X3)) -> U44_AAG(X1, X2, X3, addzQ_in_aag(X1, X2, X3)) ADDCV_IN_AAG(zero(X1), zero(X2), one(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(zero(zero(X1)), one(b), zero(one(X1))) -> U45_AAG(X1, binaryZH_in_g(X1)) ADDCV_IN_AAG(zero(zero(X1)), one(b), zero(one(X1))) -> BINARYZH_IN_G(X1) ADDCV_IN_AAG(zero(one(X1)), one(b), zero(zero(X2))) -> U46_AAG(X1, X2, succT_in_ag(X1, X2)) ADDCV_IN_AAG(zero(one(X1)), one(b), zero(zero(X2))) -> SUCCT_IN_AG(X1, X2) ADDCV_IN_AAG(zero(X1), one(X2), zero(X3)) -> U47_AAG(X1, X2, X3, addCV_in_aag(X1, X2, X3)) ADDCV_IN_AAG(zero(X1), one(X2), zero(X3)) -> ADDCV_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(one(b), zero(zero(X1)), zero(one(X1))) -> U48_AAG(X1, binaryZH_in_g(X1)) ADDCV_IN_AAG(one(b), zero(zero(X1)), zero(one(X1))) -> BINARYZH_IN_G(X1) ADDCV_IN_AAG(one(b), zero(one(X1)), zero(zero(X2))) -> U49_AAG(X1, X2, succT_in_ag(X1, X2)) ADDCV_IN_AAG(one(b), zero(one(X1)), zero(zero(X2))) -> SUCCT_IN_AG(X1, X2) ADDCV_IN_AAG(one(X1), zero(X2), zero(X3)) -> U50_AAG(X1, X2, X3, addCV_in_aag(X1, X2, X3)) ADDCV_IN_AAG(one(X1), zero(X2), zero(X3)) -> ADDCV_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(one(X1), one(X2), one(X3)) -> U51_AAG(X1, X2, X3, addcS_in_aag(X1, X2, X3)) ADDCV_IN_AAG(one(X1), one(X2), one(X3)) -> ADDCS_IN_AAG(X1, X2, X3) ADDC1_IN_AAG(one(X1), X2, one(X3)) -> U63_AAG(X1, X2, X3, addyR_in_aag(X1, X2, X3)) ADDC1_IN_AAG(one(X1), X2, one(X3)) -> ADDYR_IN_AAG(X1, X2, X3) TIMESA_IN_AAG(one(X1), X2, zero(X3)) -> U77_AAG(X1, X2, X3, pB_in_aaag(X1, X2, X4, zero(X3))) TIMESA_IN_AAG(one(X1), X2, zero(X3)) -> PB_IN_AAAG(X1, X2, X4, zero(X3)) TIMESA_IN_AAG(one(one(b)), X1, X2) -> U78_AAG(X1, X2, addC1_in_aag(X1, X1, X2)) TIMESA_IN_AAG(one(one(b)), X1, X2) -> ADDC1_IN_AAG(X1, X1, X2) TIMESA_IN_AAG(one(zero(X1)), X2, X3) -> U79_AAG(X1, X2, X3, pD_in_aaag(X1, X2, X4, X3)) TIMESA_IN_AAG(one(zero(X1)), X2, X3) -> PD_IN_AAAG(X1, X2, X4, X3) PD_IN_AAAG(X1, X2, X3, X4) -> U67_AAAG(X1, X2, X3, X4, timesF_in_aaa(X1, X2, X3)) PD_IN_AAAG(X1, X2, X3, X4) -> TIMESF_IN_AAA(X1, X2, X3) PD_IN_AAAG(X1, X2, X3, X4) -> U68_AAAG(X1, X2, X3, X4, timescF_in_aaa(X1, X2, X3)) U68_AAAG(X1, X2, X3, X4, timescF_out_aaa(X1, X2, X3)) -> U69_AAAG(X1, X2, X3, X4, addC1_in_aag(X2, zero(X3), X4)) U68_AAAG(X1, X2, X3, X4, timescF_out_aaa(X1, X2, X3)) -> ADDC1_IN_AAG(X2, zero(X3), X4) TIMESA_IN_AAG(one(one(X1)), X2, X3) -> U80_AAG(X1, X2, X3, pE_in_aaaag(X1, X2, X4, X5, X3)) TIMESA_IN_AAG(one(one(X1)), X2, X3) -> PE_IN_AAAAG(X1, X2, X4, X5, X3) PE_IN_AAAAG(X1, X2, X3, X4, X5) -> U70_AAAAG(X1, X2, X3, X4, X5, timesF_in_aaa(X1, X2, X3)) PE_IN_AAAAG(X1, X2, X3, X4, X5) -> TIMESF_IN_AAA(X1, X2, X3) PE_IN_AAAAG(X1, X2, X3, X4, X5) -> U71_AAAAG(X1, X2, X3, X4, X5, timescF_in_aaa(X1, X2, X3)) U71_AAAAG(X1, X2, X3, X4, X5, timescF_out_aaa(X1, X2, X3)) -> U72_AAAAG(X1, X2, X3, X4, X5, addG_in_aaa(X2, X3, X4)) U71_AAAAG(X1, X2, X3, X4, X5, timescF_out_aaa(X1, X2, X3)) -> ADDG_IN_AAA(X2, X3, X4) U71_AAAAG(X1, X2, X3, X4, X5, timescF_out_aaa(X1, X2, X3)) -> U73_AAAAG(X1, X2, X3, X4, X5, addcG_in_aaa(X2, X3, X4)) U73_AAAAG(X1, X2, X3, X4, X5, addcG_out_aaa(X2, X3, X4)) -> U74_AAAAG(X1, X2, X3, X4, X5, addC1_in_ggg(X2, X4, X5)) U73_AAAAG(X1, X2, X3, X4, X5, addcG_out_aaa(X2, X3, X4)) -> ADDC1_IN_GGG(X2, X4, X5) ADDC1_IN_GGG(b, X1, zero(X1)) -> U61_GGG(X1, binaryZP_in_g(X1)) ADDC1_IN_GGG(b, X1, zero(X1)) -> BINARYZP_IN_G(X1) ADDC1_IN_GGG(zero(X1), X2, zero(X3)) -> U62_GGG(X1, X2, X3, addzQ_in_ggg(X1, X2, X3)) ADDC1_IN_GGG(zero(X1), X2, zero(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(zero(X1), zero(X2), zero(X3)) -> U34_GGG(X1, X2, X3, addzQ_in_ggg(X1, X2, X3)) ADDZQ_IN_GGG(zero(X1), zero(X2), zero(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(zero(one(X1)), one(b), one(one(X1))) -> U35_GGG(X1, binaryI_in_g(X1)) ADDZQ_IN_GGG(zero(one(X1)), one(b), one(one(X1))) -> BINARYI_IN_G(X1) ADDZQ_IN_GGG(zero(zero(X1)), one(b), one(zero(X1))) -> U36_GGG(X1, binaryZH_in_g(X1)) ADDZQ_IN_GGG(zero(zero(X1)), one(b), one(zero(X1))) -> BINARYZH_IN_G(X1) ADDZQ_IN_GGG(zero(X1), one(X2), one(X3)) -> U37_GGG(X1, X2, X3, addzQ_in_ggg(X1, X2, X3)) ADDZQ_IN_GGG(zero(X1), one(X2), one(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(one(X1), zero(X2), one(X3)) -> U38_GGG(X1, X2, X3, addyR_in_ggg(X1, X2, X3)) ADDZQ_IN_GGG(one(X1), zero(X2), one(X3)) -> ADDYR_IN_GGG(X1, X2, X3) ADDYR_IN_GGG(b, one(X1), one(X1)) -> U55_GGG(X1, binaryI_in_g(X1)) ADDYR_IN_GGG(b, one(X1), one(X1)) -> BINARYI_IN_G(X1) ADDYR_IN_GGG(b, zero(X1), zero(X1)) -> U56_GGG(X1, binaryZH_in_g(X1)) ADDYR_IN_GGG(b, zero(X1), zero(X1)) -> BINARYZH_IN_G(X1) ADDYR_IN_GGG(X1, X2, X3) -> U57_GGG(X1, X2, X3, addzQ_in_ggg(X1, X2, X3)) ADDYR_IN_GGG(X1, X2, X3) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(one(X1), one(X2), zero(X3)) -> U39_GGG(X1, X2, X3, addcS_in_ggg(X1, X2, X3)) ADDZQ_IN_GGG(one(X1), one(X2), zero(X3)) -> ADDCS_IN_GGG(X1, X2, X3) ADDCS_IN_GGG(X1, b, X2) -> U52_GGG(X1, X2, succZU_in_gg(X1, X2)) ADDCS_IN_GGG(X1, b, X2) -> SUCCZU_IN_GG(X1, X2) SUCCZU_IN_GG(zero(X1), one(X1)) -> U42_GG(X1, binaryZH_in_g(X1)) SUCCZU_IN_GG(zero(X1), one(X1)) -> BINARYZH_IN_G(X1) SUCCZU_IN_GG(one(X1), zero(X2)) -> U43_GG(X1, X2, succT_in_gg(X1, X2)) SUCCZU_IN_GG(one(X1), zero(X2)) -> SUCCT_IN_GG(X1, X2) SUCCT_IN_GG(zero(X1), one(X1)) -> U40_GG(X1, binaryZH_in_g(X1)) SUCCT_IN_GG(zero(X1), one(X1)) -> BINARYZH_IN_G(X1) SUCCT_IN_GG(one(X1), zero(X2)) -> U41_GG(X1, X2, succT_in_gg(X1, X2)) SUCCT_IN_GG(one(X1), zero(X2)) -> SUCCT_IN_GG(X1, X2) ADDCS_IN_GGG(b, X1, X2) -> U53_GGG(X1, X2, succZU_in_gg(X1, X2)) ADDCS_IN_GGG(b, X1, X2) -> SUCCZU_IN_GG(X1, X2) ADDCS_IN_GGG(X1, X2, X3) -> U54_GGG(X1, X2, X3, addCV_in_ggg(X1, X2, X3)) ADDCS_IN_GGG(X1, X2, X3) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(zero(X1), zero(X2), one(X3)) -> U44_GGG(X1, X2, X3, addzQ_in_ggg(X1, X2, X3)) ADDCV_IN_GGG(zero(X1), zero(X2), one(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(zero(zero(X1)), one(b), zero(one(X1))) -> U45_GGG(X1, binaryZH_in_g(X1)) ADDCV_IN_GGG(zero(zero(X1)), one(b), zero(one(X1))) -> BINARYZH_IN_G(X1) ADDCV_IN_GGG(zero(one(X1)), one(b), zero(zero(X2))) -> U46_GGG(X1, X2, succT_in_gg(X1, X2)) ADDCV_IN_GGG(zero(one(X1)), one(b), zero(zero(X2))) -> SUCCT_IN_GG(X1, X2) ADDCV_IN_GGG(zero(X1), one(X2), zero(X3)) -> U47_GGG(X1, X2, X3, addCV_in_ggg(X1, X2, X3)) ADDCV_IN_GGG(zero(X1), one(X2), zero(X3)) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(one(b), zero(zero(X1)), zero(one(X1))) -> U48_GGG(X1, binaryZH_in_g(X1)) ADDCV_IN_GGG(one(b), zero(zero(X1)), zero(one(X1))) -> BINARYZH_IN_G(X1) ADDCV_IN_GGG(one(b), zero(one(X1)), zero(zero(X2))) -> U49_GGG(X1, X2, succT_in_gg(X1, X2)) ADDCV_IN_GGG(one(b), zero(one(X1)), zero(zero(X2))) -> SUCCT_IN_GG(X1, X2) ADDCV_IN_GGG(one(X1), zero(X2), zero(X3)) -> U50_GGG(X1, X2, X3, addCV_in_ggg(X1, X2, X3)) ADDCV_IN_GGG(one(X1), zero(X2), zero(X3)) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(one(X1), one(X2), one(X3)) -> U51_GGG(X1, X2, X3, addcS_in_ggg(X1, X2, X3)) ADDCV_IN_GGG(one(X1), one(X2), one(X3)) -> ADDCS_IN_GGG(X1, X2, X3) ADDC1_IN_GGG(one(X1), X2, one(X3)) -> U63_GGG(X1, X2, X3, addyR_in_ggg(X1, X2, X3)) ADDC1_IN_GGG(one(X1), X2, one(X3)) -> ADDYR_IN_GGG(X1, X2, X3) The TRS R consists of the following rules: timescF_in_aaa(one(b), X1, X1) -> timescF_out_aaa(one(b), X1, X1) timescF_in_aaa(zero(X1), X2, zero(X3)) -> U88_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X3)) timescF_in_aaa(one(X1), X2, X3) -> U89_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U89_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U90_aaa(X1, X2, X3, addcG_in_aaa(X2, X4, X3)) addcG_in_aaa(b, X1, zero(X1)) -> U149_aaa(X1, binaryZcP_in_a(X1)) binaryZcP_in_a(X1) -> U119_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(zero(X1)) -> U91_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(one(X1)) -> U92_a(X1, binarycI_in_a(X1)) binarycI_in_a(b) -> binarycI_out_a(b) binarycI_in_a(zero(X1)) -> U93_a(X1, binaryZcH_in_a(X1)) U93_a(X1, binaryZcH_out_a(X1)) -> binarycI_out_a(zero(X1)) binarycI_in_a(one(X1)) -> U94_a(X1, binarycI_in_a(X1)) U94_a(X1, binarycI_out_a(X1)) -> binarycI_out_a(one(X1)) U92_a(X1, binarycI_out_a(X1)) -> binaryZcH_out_a(one(X1)) U91_a(X1, binaryZcH_out_a(X1)) -> binaryZcH_out_a(zero(X1)) U119_a(X1, binaryZcH_out_a(X1)) -> binaryZcP_out_a(X1) U149_aaa(X1, binaryZcP_out_a(X1)) -> addcG_out_aaa(b, X1, zero(X1)) addcG_in_aaa(zero(X1), X2, zero(X3)) -> U150_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(X1), zero(X2), zero(X3)) -> U95_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(one(X1)), one(b), one(one(X1))) -> U96_aaa(X1, binarycI_in_a(X1)) U96_aaa(X1, binarycI_out_a(X1)) -> addzcJ_out_aaa(zero(one(X1)), one(b), one(one(X1))) addzcJ_in_aaa(zero(zero(X1)), one(b), one(zero(X1))) -> U97_aaa(X1, binaryZcH_in_a(X1)) U97_aaa(X1, binaryZcH_out_a(X1)) -> addzcJ_out_aaa(zero(zero(X1)), one(b), one(zero(X1))) addzcJ_in_aaa(zero(X1), one(X2), one(X3)) -> U98_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), zero(X2), one(X3)) -> U99_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) addycK_in_aaa(b, one(X1), one(X1)) -> U116_aaa(X1, binarycI_in_a(X1)) U116_aaa(X1, binarycI_out_a(X1)) -> addycK_out_aaa(b, one(X1), one(X1)) addycK_in_aaa(b, zero(X1), zero(X1)) -> U117_aaa(X1, binaryZcH_in_a(X1)) U117_aaa(X1, binaryZcH_out_a(X1)) -> addycK_out_aaa(b, zero(X1), zero(X1)) addycK_in_aaa(X1, X2, X3) -> U118_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), one(X2), zero(X3)) -> U100_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) addccL_in_aaa(b, b, one(b)) -> addccL_out_aaa(b, b, one(b)) addccL_in_aaa(X1, b, X2) -> U113_aaa(X1, X2, succZcN_in_aa(X1, X2)) succZcN_in_aa(zero(X1), one(X1)) -> U103_aa(X1, binaryZcH_in_a(X1)) U103_aa(X1, binaryZcH_out_a(X1)) -> succZcN_out_aa(zero(X1), one(X1)) succZcN_in_aa(one(X1), zero(X2)) -> U104_aa(X1, X2, succcM_in_aa(X1, X2)) succcM_in_aa(b, one(b)) -> succcM_out_aa(b, one(b)) succcM_in_aa(zero(X1), one(X1)) -> U101_aa(X1, binaryZcH_in_a(X1)) U101_aa(X1, binaryZcH_out_a(X1)) -> succcM_out_aa(zero(X1), one(X1)) succcM_in_aa(one(X1), zero(X2)) -> U102_aa(X1, X2, succcM_in_aa(X1, X2)) U102_aa(X1, X2, succcM_out_aa(X1, X2)) -> succcM_out_aa(one(X1), zero(X2)) U104_aa(X1, X2, succcM_out_aa(X1, X2)) -> succZcN_out_aa(one(X1), zero(X2)) U113_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(X1, b, X2) addccL_in_aaa(b, X1, X2) -> U114_aaa(X1, X2, succZcN_in_aa(X1, X2)) U114_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(b, X1, X2) addccL_in_aaa(X1, X2, X3) -> U115_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(zero(X1), zero(X2), one(X3)) -> U105_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) U105_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), zero(X2), one(X3)) addCcO_in_aaa(zero(zero(X1)), one(b), zero(one(X1))) -> U106_aaa(X1, binaryZcH_in_a(X1)) U106_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(zero(zero(X1)), one(b), zero(one(X1))) addCcO_in_aaa(zero(one(X1)), one(b), zero(zero(X2))) -> U107_aaa(X1, X2, succcM_in_aa(X1, X2)) U107_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(zero(one(X1)), one(b), zero(zero(X2))) addCcO_in_aaa(zero(X1), one(X2), zero(X3)) -> U108_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(b), zero(zero(X1)), zero(one(X1))) -> U109_aaa(X1, binaryZcH_in_a(X1)) U109_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(one(b), zero(zero(X1)), zero(one(X1))) addCcO_in_aaa(one(b), zero(one(X1)), zero(zero(X2))) -> U110_aaa(X1, X2, succcM_in_aa(X1, X2)) U110_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(one(b), zero(one(X1)), zero(zero(X2))) addCcO_in_aaa(one(X1), zero(X2), zero(X3)) -> U111_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(X1), one(X2), one(X3)) -> U112_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) U112_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), one(X2), one(X3)) U111_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), zero(X2), zero(X3)) U108_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), one(X2), zero(X3)) U115_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addccL_out_aaa(X1, X2, X3) U100_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), one(X2), zero(X3)) U118_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addycK_out_aaa(X1, X2, X3) U99_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), zero(X2), one(X3)) U98_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), one(X2), one(X3)) U95_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), zero(X2), zero(X3)) U150_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addcG_out_aaa(zero(X1), X2, zero(X3)) addcG_in_aaa(one(X1), X2, one(X3)) -> U151_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) U151_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addcG_out_aaa(one(X1), X2, one(X3)) U90_aaa(X1, X2, X3, addcG_out_aaa(X2, X4, X3)) -> timescF_out_aaa(one(X1), X2, X3) U88_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X3)) -> timescF_out_aaa(zero(X1), X2, zero(X3)) The argument filtering Pi contains the following mapping: timesA_in_aag(x1, x2, x3) = timesA_in_aag(x3) zero(x1) = zero(x1) pB_in_aaag(x1, x2, x3, x4) = pB_in_aaag(x4) timesF_in_aaa(x1, x2, x3) = timesF_in_aaa timescF_in_aaa(x1, x2, x3) = timescF_in_aaa timescF_out_aaa(x1, x2, x3) = timescF_out_aaa(x1) U88_aaa(x1, x2, x3, x4) = U88_aaa(x4) U89_aaa(x1, x2, x3, x4) = U89_aaa(x4) U90_aaa(x1, x2, x3, x4) = U90_aaa(x1, x4) addcG_in_aaa(x1, x2, x3) = addcG_in_aaa U149_aaa(x1, x2) = U149_aaa(x2) binaryZcP_in_a(x1) = binaryZcP_in_a U119_a(x1, x2) = U119_a(x2) binaryZcH_in_a(x1) = binaryZcH_in_a U91_a(x1, x2) = U91_a(x2) U92_a(x1, x2) = U92_a(x2) binarycI_in_a(x1) = binarycI_in_a binarycI_out_a(x1) = binarycI_out_a(x1) U93_a(x1, x2) = U93_a(x2) binaryZcH_out_a(x1) = binaryZcH_out_a(x1) U94_a(x1, x2) = U94_a(x2) binaryZcP_out_a(x1) = binaryZcP_out_a(x1) addcG_out_aaa(x1, x2, x3) = addcG_out_aaa(x1, x2, x3) U150_aaa(x1, x2, x3, x4) = U150_aaa(x4) addzcJ_in_aaa(x1, x2, x3) = addzcJ_in_aaa U95_aaa(x1, x2, x3, x4) = U95_aaa(x4) U96_aaa(x1, x2) = U96_aaa(x2) addzcJ_out_aaa(x1, x2, x3) = addzcJ_out_aaa(x1, x2, x3) U97_aaa(x1, x2) = U97_aaa(x2) U98_aaa(x1, x2, x3, x4) = U98_aaa(x4) U99_aaa(x1, x2, x3, x4) = U99_aaa(x4) addycK_in_aaa(x1, x2, x3) = addycK_in_aaa U116_aaa(x1, x2) = U116_aaa(x2) addycK_out_aaa(x1, x2, x3) = addycK_out_aaa(x1, x2, x3) U117_aaa(x1, x2) = U117_aaa(x2) U118_aaa(x1, x2, x3, x4) = U118_aaa(x4) U100_aaa(x1, x2, x3, x4) = U100_aaa(x4) addccL_in_aaa(x1, x2, x3) = addccL_in_aaa addccL_out_aaa(x1, x2, x3) = addccL_out_aaa(x1, x2, x3) U113_aaa(x1, x2, x3) = U113_aaa(x3) succZcN_in_aa(x1, x2) = succZcN_in_aa U103_aa(x1, x2) = U103_aa(x2) succZcN_out_aa(x1, x2) = succZcN_out_aa(x1, x2) U104_aa(x1, x2, x3) = U104_aa(x3) succcM_in_aa(x1, x2) = succcM_in_aa succcM_out_aa(x1, x2) = succcM_out_aa(x1, x2) U101_aa(x1, x2) = U101_aa(x2) U102_aa(x1, x2, x3) = U102_aa(x3) U114_aaa(x1, x2, x3) = U114_aaa(x3) U115_aaa(x1, x2, x3, x4) = U115_aaa(x4) addCcO_in_aaa(x1, x2, x3) = addCcO_in_aaa U105_aaa(x1, x2, x3, x4) = U105_aaa(x4) addCcO_out_aaa(x1, x2, x3) = addCcO_out_aaa(x1, x2, x3) U106_aaa(x1, x2) = U106_aaa(x2) U107_aaa(x1, x2, x3) = U107_aaa(x3) U108_aaa(x1, x2, x3, x4) = U108_aaa(x4) U109_aaa(x1, x2) = U109_aaa(x2) U110_aaa(x1, x2, x3) = U110_aaa(x3) U111_aaa(x1, x2, x3, x4) = U111_aaa(x4) U112_aaa(x1, x2, x3, x4) = U112_aaa(x4) U151_aaa(x1, x2, x3, x4) = U151_aaa(x4) addG_in_aaa(x1, x2, x3) = addG_in_aaa binaryZP_in_a(x1) = binaryZP_in_a binaryZH_in_a(x1) = binaryZH_in_a binaryI_in_a(x1) = binaryI_in_a addzJ_in_aaa(x1, x2, x3) = addzJ_in_aaa addyK_in_aaa(x1, x2, x3) = addyK_in_aaa addcL_in_aaa(x1, x2, x3) = addcL_in_aaa succZN_in_aa(x1, x2) = succZN_in_aa succM_in_aa(x1, x2) = succM_in_aa addCO_in_aaa(x1, x2, x3) = addCO_in_aaa addC1_in_aag(x1, x2, x3) = addC1_in_aag(x3) binaryZP_in_g(x1) = binaryZP_in_g(x1) binaryZH_in_g(x1) = binaryZH_in_g(x1) one(x1) = one(x1) binaryI_in_g(x1) = binaryI_in_g(x1) addzQ_in_aag(x1, x2, x3) = addzQ_in_aag(x3) addyR_in_aag(x1, x2, x3) = addyR_in_aag(x3) addcS_in_aag(x1, x2, x3) = addcS_in_aag(x3) succZU_in_ag(x1, x2) = succZU_in_ag(x2) succT_in_ag(x1, x2) = succT_in_ag(x2) addCV_in_aag(x1, x2, x3) = addCV_in_aag(x3) pD_in_aaag(x1, x2, x3, x4) = pD_in_aaag(x4) pE_in_aaaag(x1, x2, x3, x4, x5) = pE_in_aaaag(x5) addC1_in_ggg(x1, x2, x3) = addC1_in_ggg(x1, x2, x3) b = b addzQ_in_ggg(x1, x2, x3) = addzQ_in_ggg(x1, x2, x3) addyR_in_ggg(x1, x2, x3) = addyR_in_ggg(x1, x2, x3) addcS_in_ggg(x1, x2, x3) = addcS_in_ggg(x1, x2, x3) succZU_in_gg(x1, x2) = succZU_in_gg(x1, x2) succT_in_gg(x1, x2) = succT_in_gg(x1, x2) addCV_in_ggg(x1, x2, x3) = addCV_in_ggg(x1, x2, x3) TIMESA_IN_AAG(x1, x2, x3) = TIMESA_IN_AAG(x3) U75_AAG(x1, x2, x3, x4) = U75_AAG(x3, x4) U76_AAG(x1, x2, x3, x4) = U76_AAG(x3, x4) PB_IN_AAAG(x1, x2, x3, x4) = PB_IN_AAAG(x4) U58_AAAG(x1, x2, x3, x4, x5) = U58_AAAG(x4, x5) TIMESF_IN_AAA(x1, x2, x3) = TIMESF_IN_AAA U1_AAA(x1, x2, x3, x4) = U1_AAA(x4) U2_AAA(x1, x2, x3, x4) = U2_AAA(x4) U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) U4_AAA(x1, x2, x3, x4) = U4_AAA(x1, x4) ADDG_IN_AAA(x1, x2, x3) = ADDG_IN_AAA U64_AAA(x1, x2) = U64_AAA(x2) BINARYZP_IN_A(x1) = BINARYZP_IN_A U33_A(x1, x2) = U33_A(x2) BINARYZH_IN_A(x1) = BINARYZH_IN_A U5_A(x1, x2) = U5_A(x2) U6_A(x1, x2) = U6_A(x2) BINARYI_IN_A(x1) = BINARYI_IN_A U7_A(x1, x2) = U7_A(x2) U8_A(x1, x2) = U8_A(x2) U65_AAA(x1, x2, x3, x4) = U65_AAA(x4) ADDZJ_IN_AAA(x1, x2, x3) = ADDZJ_IN_AAA U9_AAA(x1, x2, x3, x4) = U9_AAA(x4) U10_AAA(x1, x2) = U10_AAA(x2) U11_AAA(x1, x2) = U11_AAA(x2) U12_AAA(x1, x2, x3, x4) = U12_AAA(x4) U13_AAA(x1, x2, x3, x4) = U13_AAA(x4) ADDYK_IN_AAA(x1, x2, x3) = ADDYK_IN_AAA U30_AAA(x1, x2) = U30_AAA(x2) U31_AAA(x1, x2) = U31_AAA(x2) U32_AAA(x1, x2, x3, x4) = U32_AAA(x4) U14_AAA(x1, x2, x3, x4) = U14_AAA(x4) ADDCL_IN_AAA(x1, x2, x3) = ADDCL_IN_AAA U27_AAA(x1, x2, x3) = U27_AAA(x3) SUCCZN_IN_AA(x1, x2) = SUCCZN_IN_AA U17_AA(x1, x2) = U17_AA(x2) U18_AA(x1, x2, x3) = U18_AA(x3) SUCCM_IN_AA(x1, x2) = SUCCM_IN_AA U15_AA(x1, x2) = U15_AA(x2) U16_AA(x1, x2, x3) = U16_AA(x3) U28_AAA(x1, x2, x3) = U28_AAA(x3) U29_AAA(x1, x2, x3, x4) = U29_AAA(x4) ADDCO_IN_AAA(x1, x2, x3) = ADDCO_IN_AAA U19_AAA(x1, x2, x3, x4) = U19_AAA(x4) U20_AAA(x1, x2) = U20_AAA(x2) U21_AAA(x1, x2, x3) = U21_AAA(x3) U22_AAA(x1, x2, x3, x4) = U22_AAA(x4) U23_AAA(x1, x2) = U23_AAA(x2) U24_AAA(x1, x2, x3) = U24_AAA(x3) U25_AAA(x1, x2, x3, x4) = U25_AAA(x4) U26_AAA(x1, x2, x3, x4) = U26_AAA(x4) U66_AAA(x1, x2, x3, x4) = U66_AAA(x4) U59_AAAG(x1, x2, x3, x4, x5) = U59_AAAG(x4, x5) U60_AAAG(x1, x2, x3, x4, x5) = U60_AAAG(x1, x4, x5) ADDC1_IN_AAG(x1, x2, x3) = ADDC1_IN_AAG(x3) U61_AAG(x1, x2) = U61_AAG(x1, x2) BINARYZP_IN_G(x1) = BINARYZP_IN_G(x1) U33_G(x1, x2) = U33_G(x1, x2) BINARYZH_IN_G(x1) = BINARYZH_IN_G(x1) U5_G(x1, x2) = U5_G(x1, x2) U6_G(x1, x2) = U6_G(x1, x2) BINARYI_IN_G(x1) = BINARYI_IN_G(x1) U7_G(x1, x2) = U7_G(x1, x2) U8_G(x1, x2) = U8_G(x1, x2) U62_AAG(x1, x2, x3, x4) = U62_AAG(x3, x4) ADDZQ_IN_AAG(x1, x2, x3) = ADDZQ_IN_AAG(x3) U34_AAG(x1, x2, x3, x4) = U34_AAG(x3, x4) U35_AAG(x1, x2) = U35_AAG(x1, x2) U36_AAG(x1, x2) = U36_AAG(x1, x2) U37_AAG(x1, x2, x3, x4) = U37_AAG(x3, x4) U38_AAG(x1, x2, x3, x4) = U38_AAG(x3, x4) ADDYR_IN_AAG(x1, x2, x3) = ADDYR_IN_AAG(x3) U55_AAG(x1, x2) = U55_AAG(x1, x2) U56_AAG(x1, x2) = U56_AAG(x1, x2) U57_AAG(x1, x2, x3, x4) = U57_AAG(x3, x4) U39_AAG(x1, x2, x3, x4) = U39_AAG(x3, x4) ADDCS_IN_AAG(x1, x2, x3) = ADDCS_IN_AAG(x3) U52_AAG(x1, x2, x3) = U52_AAG(x2, x3) SUCCZU_IN_AG(x1, x2) = SUCCZU_IN_AG(x2) U42_AG(x1, x2) = U42_AG(x1, x2) U43_AG(x1, x2, x3) = U43_AG(x2, x3) SUCCT_IN_AG(x1, x2) = SUCCT_IN_AG(x2) U40_AG(x1, x2) = U40_AG(x1, x2) U41_AG(x1, x2, x3) = U41_AG(x2, x3) U53_AAG(x1, x2, x3) = U53_AAG(x2, x3) U54_AAG(x1, x2, x3, x4) = U54_AAG(x3, x4) ADDCV_IN_AAG(x1, x2, x3) = ADDCV_IN_AAG(x3) U44_AAG(x1, x2, x3, x4) = U44_AAG(x3, x4) U45_AAG(x1, x2) = U45_AAG(x1, x2) U46_AAG(x1, x2, x3) = U46_AAG(x2, x3) U47_AAG(x1, x2, x3, x4) = U47_AAG(x3, x4) U48_AAG(x1, x2) = U48_AAG(x1, x2) U49_AAG(x1, x2, x3) = U49_AAG(x2, x3) U50_AAG(x1, x2, x3, x4) = U50_AAG(x3, x4) U51_AAG(x1, x2, x3, x4) = U51_AAG(x3, x4) U63_AAG(x1, x2, x3, x4) = U63_AAG(x3, x4) U77_AAG(x1, x2, x3, x4) = U77_AAG(x3, x4) U78_AAG(x1, x2, x3) = U78_AAG(x2, x3) U79_AAG(x1, x2, x3, x4) = U79_AAG(x3, x4) PD_IN_AAAG(x1, x2, x3, x4) = PD_IN_AAAG(x4) U67_AAAG(x1, x2, x3, x4, x5) = U67_AAAG(x4, x5) U68_AAAG(x1, x2, x3, x4, x5) = U68_AAAG(x4, x5) U69_AAAG(x1, x2, x3, x4, x5) = U69_AAAG(x1, x4, x5) U80_AAG(x1, x2, x3, x4) = U80_AAG(x3, x4) PE_IN_AAAAG(x1, x2, x3, x4, x5) = PE_IN_AAAAG(x5) U70_AAAAG(x1, x2, x3, x4, x5, x6) = U70_AAAAG(x5, x6) U71_AAAAG(x1, x2, x3, x4, x5, x6) = U71_AAAAG(x5, x6) U72_AAAAG(x1, x2, x3, x4, x5, x6) = U72_AAAAG(x1, x5, x6) U73_AAAAG(x1, x2, x3, x4, x5, x6) = U73_AAAAG(x1, x5, x6) U74_AAAAG(x1, x2, x3, x4, x5, x6) = U74_AAAAG(x1, x2, x5, x6) ADDC1_IN_GGG(x1, x2, x3) = ADDC1_IN_GGG(x1, x2, x3) U61_GGG(x1, x2) = U61_GGG(x1, x2) U62_GGG(x1, x2, x3, x4) = U62_GGG(x1, x2, x3, x4) ADDZQ_IN_GGG(x1, x2, x3) = ADDZQ_IN_GGG(x1, x2, x3) U34_GGG(x1, x2, x3, x4) = U34_GGG(x1, x2, x3, x4) U35_GGG(x1, x2) = U35_GGG(x1, x2) U36_GGG(x1, x2) = U36_GGG(x1, x2) U37_GGG(x1, x2, x3, x4) = U37_GGG(x1, x2, x3, x4) U38_GGG(x1, x2, x3, x4) = U38_GGG(x1, x2, x3, x4) ADDYR_IN_GGG(x1, x2, x3) = ADDYR_IN_GGG(x1, x2, x3) U55_GGG(x1, x2) = U55_GGG(x1, x2) U56_GGG(x1, x2) = U56_GGG(x1, x2) U57_GGG(x1, x2, x3, x4) = U57_GGG(x1, x2, x3, x4) U39_GGG(x1, x2, x3, x4) = U39_GGG(x1, x2, x3, x4) ADDCS_IN_GGG(x1, x2, x3) = ADDCS_IN_GGG(x1, x2, x3) U52_GGG(x1, x2, x3) = U52_GGG(x1, x2, x3) SUCCZU_IN_GG(x1, x2) = SUCCZU_IN_GG(x1, x2) U42_GG(x1, x2) = U42_GG(x1, x2) U43_GG(x1, x2, x3) = U43_GG(x1, x2, x3) SUCCT_IN_GG(x1, x2) = SUCCT_IN_GG(x1, x2) U40_GG(x1, x2) = U40_GG(x1, x2) U41_GG(x1, x2, x3) = U41_GG(x1, x2, x3) U53_GGG(x1, x2, x3) = U53_GGG(x1, x2, x3) U54_GGG(x1, x2, x3, x4) = U54_GGG(x1, x2, x3, x4) ADDCV_IN_GGG(x1, x2, x3) = ADDCV_IN_GGG(x1, x2, x3) U44_GGG(x1, x2, x3, x4) = U44_GGG(x1, x2, x3, x4) U45_GGG(x1, x2) = U45_GGG(x1, x2) U46_GGG(x1, x2, x3) = U46_GGG(x1, x2, x3) U47_GGG(x1, x2, x3, x4) = U47_GGG(x1, x2, x3, x4) U48_GGG(x1, x2) = U48_GGG(x1, x2) U49_GGG(x1, x2, x3) = U49_GGG(x1, x2, x3) U50_GGG(x1, x2, x3, x4) = U50_GGG(x1, x2, x3, x4) U51_GGG(x1, x2, x3, x4) = U51_GGG(x1, x2, x3, x4) U63_GGG(x1, x2, x3, x4) = U63_GGG(x1, x2, x3, x4) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (100) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMESA_IN_AAG(zero(zero(X1)), X2, zero(zero(X3))) -> U75_AAG(X1, X2, X3, timesA_in_aag(X1, X2, X3)) TIMESA_IN_AAG(zero(zero(X1)), X2, zero(zero(X3))) -> TIMESA_IN_AAG(X1, X2, X3) TIMESA_IN_AAG(zero(one(X1)), X2, zero(X3)) -> U76_AAG(X1, X2, X3, pB_in_aaag(X1, X2, X4, X3)) TIMESA_IN_AAG(zero(one(X1)), X2, zero(X3)) -> PB_IN_AAAG(X1, X2, X4, X3) PB_IN_AAAG(X1, X2, X3, X4) -> U58_AAAG(X1, X2, X3, X4, timesF_in_aaa(X1, X2, X3)) PB_IN_AAAG(X1, X2, X3, X4) -> TIMESF_IN_AAA(X1, X2, X3) TIMESF_IN_AAA(zero(X1), X2, zero(X3)) -> U1_AAA(X1, X2, X3, timesF_in_aaa(X1, X2, X3)) TIMESF_IN_AAA(zero(X1), X2, zero(X3)) -> TIMESF_IN_AAA(X1, X2, X3) TIMESF_IN_AAA(one(X1), X2, X3) -> U2_AAA(X1, X2, X3, timesF_in_aaa(X1, X2, X4)) TIMESF_IN_AAA(one(X1), X2, X3) -> TIMESF_IN_AAA(X1, X2, X4) TIMESF_IN_AAA(one(X1), X2, X3) -> U3_AAA(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U3_AAA(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U4_AAA(X1, X2, X3, addG_in_aaa(X2, X4, X3)) U3_AAA(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> ADDG_IN_AAA(X2, X4, X3) ADDG_IN_AAA(b, X1, zero(X1)) -> U64_AAA(X1, binaryZP_in_a(X1)) ADDG_IN_AAA(b, X1, zero(X1)) -> BINARYZP_IN_A(X1) BINARYZP_IN_A(X1) -> U33_A(X1, binaryZH_in_a(X1)) BINARYZP_IN_A(X1) -> BINARYZH_IN_A(X1) BINARYZH_IN_A(zero(X1)) -> U5_A(X1, binaryZH_in_a(X1)) BINARYZH_IN_A(zero(X1)) -> BINARYZH_IN_A(X1) BINARYZH_IN_A(one(X1)) -> U6_A(X1, binaryI_in_a(X1)) BINARYZH_IN_A(one(X1)) -> BINARYI_IN_A(X1) BINARYI_IN_A(zero(X1)) -> U7_A(X1, binaryZH_in_a(X1)) BINARYI_IN_A(zero(X1)) -> BINARYZH_IN_A(X1) BINARYI_IN_A(one(X1)) -> U8_A(X1, binaryI_in_a(X1)) BINARYI_IN_A(one(X1)) -> BINARYI_IN_A(X1) ADDG_IN_AAA(zero(X1), X2, zero(X3)) -> U65_AAA(X1, X2, X3, addzJ_in_aaa(X1, X2, X3)) ADDG_IN_AAA(zero(X1), X2, zero(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(zero(X1), zero(X2), zero(X3)) -> U9_AAA(X1, X2, X3, addzJ_in_aaa(X1, X2, X3)) ADDZJ_IN_AAA(zero(X1), zero(X2), zero(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(zero(one(X1)), one(b), one(one(X1))) -> U10_AAA(X1, binaryI_in_a(X1)) ADDZJ_IN_AAA(zero(one(X1)), one(b), one(one(X1))) -> BINARYI_IN_A(X1) ADDZJ_IN_AAA(zero(zero(X1)), one(b), one(zero(X1))) -> U11_AAA(X1, binaryZH_in_a(X1)) ADDZJ_IN_AAA(zero(zero(X1)), one(b), one(zero(X1))) -> BINARYZH_IN_A(X1) ADDZJ_IN_AAA(zero(X1), one(X2), one(X3)) -> U12_AAA(X1, X2, X3, addzJ_in_aaa(X1, X2, X3)) ADDZJ_IN_AAA(zero(X1), one(X2), one(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(one(X1), zero(X2), one(X3)) -> U13_AAA(X1, X2, X3, addyK_in_aaa(X1, X2, X3)) ADDZJ_IN_AAA(one(X1), zero(X2), one(X3)) -> ADDYK_IN_AAA(X1, X2, X3) ADDYK_IN_AAA(b, one(X1), one(X1)) -> U30_AAA(X1, binaryI_in_a(X1)) ADDYK_IN_AAA(b, one(X1), one(X1)) -> BINARYI_IN_A(X1) ADDYK_IN_AAA(b, zero(X1), zero(X1)) -> U31_AAA(X1, binaryZH_in_a(X1)) ADDYK_IN_AAA(b, zero(X1), zero(X1)) -> BINARYZH_IN_A(X1) ADDYK_IN_AAA(X1, X2, X3) -> U32_AAA(X1, X2, X3, addzJ_in_aaa(X1, X2, X3)) ADDYK_IN_AAA(X1, X2, X3) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(one(X1), one(X2), zero(X3)) -> U14_AAA(X1, X2, X3, addcL_in_aaa(X1, X2, X3)) ADDZJ_IN_AAA(one(X1), one(X2), zero(X3)) -> ADDCL_IN_AAA(X1, X2, X3) ADDCL_IN_AAA(X1, b, X2) -> U27_AAA(X1, X2, succZN_in_aa(X1, X2)) ADDCL_IN_AAA(X1, b, X2) -> SUCCZN_IN_AA(X1, X2) SUCCZN_IN_AA(zero(X1), one(X1)) -> U17_AA(X1, binaryZH_in_a(X1)) SUCCZN_IN_AA(zero(X1), one(X1)) -> BINARYZH_IN_A(X1) SUCCZN_IN_AA(one(X1), zero(X2)) -> U18_AA(X1, X2, succM_in_aa(X1, X2)) SUCCZN_IN_AA(one(X1), zero(X2)) -> SUCCM_IN_AA(X1, X2) SUCCM_IN_AA(zero(X1), one(X1)) -> U15_AA(X1, binaryZH_in_a(X1)) SUCCM_IN_AA(zero(X1), one(X1)) -> BINARYZH_IN_A(X1) SUCCM_IN_AA(one(X1), zero(X2)) -> U16_AA(X1, X2, succM_in_aa(X1, X2)) SUCCM_IN_AA(one(X1), zero(X2)) -> SUCCM_IN_AA(X1, X2) ADDCL_IN_AAA(b, X1, X2) -> U28_AAA(X1, X2, succZN_in_aa(X1, X2)) ADDCL_IN_AAA(b, X1, X2) -> SUCCZN_IN_AA(X1, X2) ADDCL_IN_AAA(X1, X2, X3) -> U29_AAA(X1, X2, X3, addCO_in_aaa(X1, X2, X3)) ADDCL_IN_AAA(X1, X2, X3) -> ADDCO_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(zero(X1), zero(X2), one(X3)) -> U19_AAA(X1, X2, X3, addzJ_in_aaa(X1, X2, X3)) ADDCO_IN_AAA(zero(X1), zero(X2), one(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(zero(zero(X1)), one(b), zero(one(X1))) -> U20_AAA(X1, binaryZH_in_a(X1)) ADDCO_IN_AAA(zero(zero(X1)), one(b), zero(one(X1))) -> BINARYZH_IN_A(X1) ADDCO_IN_AAA(zero(one(X1)), one(b), zero(zero(X2))) -> U21_AAA(X1, X2, succM_in_aa(X1, X2)) ADDCO_IN_AAA(zero(one(X1)), one(b), zero(zero(X2))) -> SUCCM_IN_AA(X1, X2) ADDCO_IN_AAA(zero(X1), one(X2), zero(X3)) -> U22_AAA(X1, X2, X3, addCO_in_aaa(X1, X2, X3)) ADDCO_IN_AAA(zero(X1), one(X2), zero(X3)) -> ADDCO_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(one(b), zero(zero(X1)), zero(one(X1))) -> U23_AAA(X1, binaryZH_in_a(X1)) ADDCO_IN_AAA(one(b), zero(zero(X1)), zero(one(X1))) -> BINARYZH_IN_A(X1) ADDCO_IN_AAA(one(b), zero(one(X1)), zero(zero(X2))) -> U24_AAA(X1, X2, succM_in_aa(X1, X2)) ADDCO_IN_AAA(one(b), zero(one(X1)), zero(zero(X2))) -> SUCCM_IN_AA(X1, X2) ADDCO_IN_AAA(one(X1), zero(X2), zero(X3)) -> U25_AAA(X1, X2, X3, addCO_in_aaa(X1, X2, X3)) ADDCO_IN_AAA(one(X1), zero(X2), zero(X3)) -> ADDCO_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(one(X1), one(X2), one(X3)) -> U26_AAA(X1, X2, X3, addcL_in_aaa(X1, X2, X3)) ADDCO_IN_AAA(one(X1), one(X2), one(X3)) -> ADDCL_IN_AAA(X1, X2, X3) ADDG_IN_AAA(one(X1), X2, one(X3)) -> U66_AAA(X1, X2, X3, addyK_in_aaa(X1, X2, X3)) ADDG_IN_AAA(one(X1), X2, one(X3)) -> ADDYK_IN_AAA(X1, X2, X3) PB_IN_AAAG(X1, X2, X3, X4) -> U59_AAAG(X1, X2, X3, X4, timescF_in_aaa(X1, X2, X3)) U59_AAAG(X1, X2, X3, X4, timescF_out_aaa(X1, X2, X3)) -> U60_AAAG(X1, X2, X3, X4, addC1_in_aag(X2, X3, X4)) U59_AAAG(X1, X2, X3, X4, timescF_out_aaa(X1, X2, X3)) -> ADDC1_IN_AAG(X2, X3, X4) ADDC1_IN_AAG(b, X1, zero(X1)) -> U61_AAG(X1, binaryZP_in_g(X1)) ADDC1_IN_AAG(b, X1, zero(X1)) -> BINARYZP_IN_G(X1) BINARYZP_IN_G(X1) -> U33_G(X1, binaryZH_in_g(X1)) BINARYZP_IN_G(X1) -> BINARYZH_IN_G(X1) BINARYZH_IN_G(zero(X1)) -> U5_G(X1, binaryZH_in_g(X1)) BINARYZH_IN_G(zero(X1)) -> BINARYZH_IN_G(X1) BINARYZH_IN_G(one(X1)) -> U6_G(X1, binaryI_in_g(X1)) BINARYZH_IN_G(one(X1)) -> BINARYI_IN_G(X1) BINARYI_IN_G(zero(X1)) -> U7_G(X1, binaryZH_in_g(X1)) BINARYI_IN_G(zero(X1)) -> BINARYZH_IN_G(X1) BINARYI_IN_G(one(X1)) -> U8_G(X1, binaryI_in_g(X1)) BINARYI_IN_G(one(X1)) -> BINARYI_IN_G(X1) ADDC1_IN_AAG(zero(X1), X2, zero(X3)) -> U62_AAG(X1, X2, X3, addzQ_in_aag(X1, X2, X3)) ADDC1_IN_AAG(zero(X1), X2, zero(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(zero(X1), zero(X2), zero(X3)) -> U34_AAG(X1, X2, X3, addzQ_in_aag(X1, X2, X3)) ADDZQ_IN_AAG(zero(X1), zero(X2), zero(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(zero(one(X1)), one(b), one(one(X1))) -> U35_AAG(X1, binaryI_in_g(X1)) ADDZQ_IN_AAG(zero(one(X1)), one(b), one(one(X1))) -> BINARYI_IN_G(X1) ADDZQ_IN_AAG(zero(zero(X1)), one(b), one(zero(X1))) -> U36_AAG(X1, binaryZH_in_g(X1)) ADDZQ_IN_AAG(zero(zero(X1)), one(b), one(zero(X1))) -> BINARYZH_IN_G(X1) ADDZQ_IN_AAG(zero(X1), one(X2), one(X3)) -> U37_AAG(X1, X2, X3, addzQ_in_aag(X1, X2, X3)) ADDZQ_IN_AAG(zero(X1), one(X2), one(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(one(X1), zero(X2), one(X3)) -> U38_AAG(X1, X2, X3, addyR_in_aag(X1, X2, X3)) ADDZQ_IN_AAG(one(X1), zero(X2), one(X3)) -> ADDYR_IN_AAG(X1, X2, X3) ADDYR_IN_AAG(b, one(X1), one(X1)) -> U55_AAG(X1, binaryI_in_g(X1)) ADDYR_IN_AAG(b, one(X1), one(X1)) -> BINARYI_IN_G(X1) ADDYR_IN_AAG(b, zero(X1), zero(X1)) -> U56_AAG(X1, binaryZH_in_g(X1)) ADDYR_IN_AAG(b, zero(X1), zero(X1)) -> BINARYZH_IN_G(X1) ADDYR_IN_AAG(X1, X2, X3) -> U57_AAG(X1, X2, X3, addzQ_in_aag(X1, X2, X3)) ADDYR_IN_AAG(X1, X2, X3) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(one(X1), one(X2), zero(X3)) -> U39_AAG(X1, X2, X3, addcS_in_aag(X1, X2, X3)) ADDZQ_IN_AAG(one(X1), one(X2), zero(X3)) -> ADDCS_IN_AAG(X1, X2, X3) ADDCS_IN_AAG(X1, b, X2) -> U52_AAG(X1, X2, succZU_in_ag(X1, X2)) ADDCS_IN_AAG(X1, b, X2) -> SUCCZU_IN_AG(X1, X2) SUCCZU_IN_AG(zero(X1), one(X1)) -> U42_AG(X1, binaryZH_in_g(X1)) SUCCZU_IN_AG(zero(X1), one(X1)) -> BINARYZH_IN_G(X1) SUCCZU_IN_AG(one(X1), zero(X2)) -> U43_AG(X1, X2, succT_in_ag(X1, X2)) SUCCZU_IN_AG(one(X1), zero(X2)) -> SUCCT_IN_AG(X1, X2) SUCCT_IN_AG(zero(X1), one(X1)) -> U40_AG(X1, binaryZH_in_g(X1)) SUCCT_IN_AG(zero(X1), one(X1)) -> BINARYZH_IN_G(X1) SUCCT_IN_AG(one(X1), zero(X2)) -> U41_AG(X1, X2, succT_in_ag(X1, X2)) SUCCT_IN_AG(one(X1), zero(X2)) -> SUCCT_IN_AG(X1, X2) ADDCS_IN_AAG(b, X1, X2) -> U53_AAG(X1, X2, succZU_in_ag(X1, X2)) ADDCS_IN_AAG(b, X1, X2) -> SUCCZU_IN_AG(X1, X2) ADDCS_IN_AAG(X1, X2, X3) -> U54_AAG(X1, X2, X3, addCV_in_aag(X1, X2, X3)) ADDCS_IN_AAG(X1, X2, X3) -> ADDCV_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(zero(X1), zero(X2), one(X3)) -> U44_AAG(X1, X2, X3, addzQ_in_aag(X1, X2, X3)) ADDCV_IN_AAG(zero(X1), zero(X2), one(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(zero(zero(X1)), one(b), zero(one(X1))) -> U45_AAG(X1, binaryZH_in_g(X1)) ADDCV_IN_AAG(zero(zero(X1)), one(b), zero(one(X1))) -> BINARYZH_IN_G(X1) ADDCV_IN_AAG(zero(one(X1)), one(b), zero(zero(X2))) -> U46_AAG(X1, X2, succT_in_ag(X1, X2)) ADDCV_IN_AAG(zero(one(X1)), one(b), zero(zero(X2))) -> SUCCT_IN_AG(X1, X2) ADDCV_IN_AAG(zero(X1), one(X2), zero(X3)) -> U47_AAG(X1, X2, X3, addCV_in_aag(X1, X2, X3)) ADDCV_IN_AAG(zero(X1), one(X2), zero(X3)) -> ADDCV_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(one(b), zero(zero(X1)), zero(one(X1))) -> U48_AAG(X1, binaryZH_in_g(X1)) ADDCV_IN_AAG(one(b), zero(zero(X1)), zero(one(X1))) -> BINARYZH_IN_G(X1) ADDCV_IN_AAG(one(b), zero(one(X1)), zero(zero(X2))) -> U49_AAG(X1, X2, succT_in_ag(X1, X2)) ADDCV_IN_AAG(one(b), zero(one(X1)), zero(zero(X2))) -> SUCCT_IN_AG(X1, X2) ADDCV_IN_AAG(one(X1), zero(X2), zero(X3)) -> U50_AAG(X1, X2, X3, addCV_in_aag(X1, X2, X3)) ADDCV_IN_AAG(one(X1), zero(X2), zero(X3)) -> ADDCV_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(one(X1), one(X2), one(X3)) -> U51_AAG(X1, X2, X3, addcS_in_aag(X1, X2, X3)) ADDCV_IN_AAG(one(X1), one(X2), one(X3)) -> ADDCS_IN_AAG(X1, X2, X3) ADDC1_IN_AAG(one(X1), X2, one(X3)) -> U63_AAG(X1, X2, X3, addyR_in_aag(X1, X2, X3)) ADDC1_IN_AAG(one(X1), X2, one(X3)) -> ADDYR_IN_AAG(X1, X2, X3) TIMESA_IN_AAG(one(X1), X2, zero(X3)) -> U77_AAG(X1, X2, X3, pB_in_aaag(X1, X2, X4, zero(X3))) TIMESA_IN_AAG(one(X1), X2, zero(X3)) -> PB_IN_AAAG(X1, X2, X4, zero(X3)) TIMESA_IN_AAG(one(one(b)), X1, X2) -> U78_AAG(X1, X2, addC1_in_aag(X1, X1, X2)) TIMESA_IN_AAG(one(one(b)), X1, X2) -> ADDC1_IN_AAG(X1, X1, X2) TIMESA_IN_AAG(one(zero(X1)), X2, X3) -> U79_AAG(X1, X2, X3, pD_in_aaag(X1, X2, X4, X3)) TIMESA_IN_AAG(one(zero(X1)), X2, X3) -> PD_IN_AAAG(X1, X2, X4, X3) PD_IN_AAAG(X1, X2, X3, X4) -> U67_AAAG(X1, X2, X3, X4, timesF_in_aaa(X1, X2, X3)) PD_IN_AAAG(X1, X2, X3, X4) -> TIMESF_IN_AAA(X1, X2, X3) PD_IN_AAAG(X1, X2, X3, X4) -> U68_AAAG(X1, X2, X3, X4, timescF_in_aaa(X1, X2, X3)) U68_AAAG(X1, X2, X3, X4, timescF_out_aaa(X1, X2, X3)) -> U69_AAAG(X1, X2, X3, X4, addC1_in_aag(X2, zero(X3), X4)) U68_AAAG(X1, X2, X3, X4, timescF_out_aaa(X1, X2, X3)) -> ADDC1_IN_AAG(X2, zero(X3), X4) TIMESA_IN_AAG(one(one(X1)), X2, X3) -> U80_AAG(X1, X2, X3, pE_in_aaaag(X1, X2, X4, X5, X3)) TIMESA_IN_AAG(one(one(X1)), X2, X3) -> PE_IN_AAAAG(X1, X2, X4, X5, X3) PE_IN_AAAAG(X1, X2, X3, X4, X5) -> U70_AAAAG(X1, X2, X3, X4, X5, timesF_in_aaa(X1, X2, X3)) PE_IN_AAAAG(X1, X2, X3, X4, X5) -> TIMESF_IN_AAA(X1, X2, X3) PE_IN_AAAAG(X1, X2, X3, X4, X5) -> U71_AAAAG(X1, X2, X3, X4, X5, timescF_in_aaa(X1, X2, X3)) U71_AAAAG(X1, X2, X3, X4, X5, timescF_out_aaa(X1, X2, X3)) -> U72_AAAAG(X1, X2, X3, X4, X5, addG_in_aaa(X2, X3, X4)) U71_AAAAG(X1, X2, X3, X4, X5, timescF_out_aaa(X1, X2, X3)) -> ADDG_IN_AAA(X2, X3, X4) U71_AAAAG(X1, X2, X3, X4, X5, timescF_out_aaa(X1, X2, X3)) -> U73_AAAAG(X1, X2, X3, X4, X5, addcG_in_aaa(X2, X3, X4)) U73_AAAAG(X1, X2, X3, X4, X5, addcG_out_aaa(X2, X3, X4)) -> U74_AAAAG(X1, X2, X3, X4, X5, addC1_in_ggg(X2, X4, X5)) U73_AAAAG(X1, X2, X3, X4, X5, addcG_out_aaa(X2, X3, X4)) -> ADDC1_IN_GGG(X2, X4, X5) ADDC1_IN_GGG(b, X1, zero(X1)) -> U61_GGG(X1, binaryZP_in_g(X1)) ADDC1_IN_GGG(b, X1, zero(X1)) -> BINARYZP_IN_G(X1) ADDC1_IN_GGG(zero(X1), X2, zero(X3)) -> U62_GGG(X1, X2, X3, addzQ_in_ggg(X1, X2, X3)) ADDC1_IN_GGG(zero(X1), X2, zero(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(zero(X1), zero(X2), zero(X3)) -> U34_GGG(X1, X2, X3, addzQ_in_ggg(X1, X2, X3)) ADDZQ_IN_GGG(zero(X1), zero(X2), zero(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(zero(one(X1)), one(b), one(one(X1))) -> U35_GGG(X1, binaryI_in_g(X1)) ADDZQ_IN_GGG(zero(one(X1)), one(b), one(one(X1))) -> BINARYI_IN_G(X1) ADDZQ_IN_GGG(zero(zero(X1)), one(b), one(zero(X1))) -> U36_GGG(X1, binaryZH_in_g(X1)) ADDZQ_IN_GGG(zero(zero(X1)), one(b), one(zero(X1))) -> BINARYZH_IN_G(X1) ADDZQ_IN_GGG(zero(X1), one(X2), one(X3)) -> U37_GGG(X1, X2, X3, addzQ_in_ggg(X1, X2, X3)) ADDZQ_IN_GGG(zero(X1), one(X2), one(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(one(X1), zero(X2), one(X3)) -> U38_GGG(X1, X2, X3, addyR_in_ggg(X1, X2, X3)) ADDZQ_IN_GGG(one(X1), zero(X2), one(X3)) -> ADDYR_IN_GGG(X1, X2, X3) ADDYR_IN_GGG(b, one(X1), one(X1)) -> U55_GGG(X1, binaryI_in_g(X1)) ADDYR_IN_GGG(b, one(X1), one(X1)) -> BINARYI_IN_G(X1) ADDYR_IN_GGG(b, zero(X1), zero(X1)) -> U56_GGG(X1, binaryZH_in_g(X1)) ADDYR_IN_GGG(b, zero(X1), zero(X1)) -> BINARYZH_IN_G(X1) ADDYR_IN_GGG(X1, X2, X3) -> U57_GGG(X1, X2, X3, addzQ_in_ggg(X1, X2, X3)) ADDYR_IN_GGG(X1, X2, X3) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(one(X1), one(X2), zero(X3)) -> U39_GGG(X1, X2, X3, addcS_in_ggg(X1, X2, X3)) ADDZQ_IN_GGG(one(X1), one(X2), zero(X3)) -> ADDCS_IN_GGG(X1, X2, X3) ADDCS_IN_GGG(X1, b, X2) -> U52_GGG(X1, X2, succZU_in_gg(X1, X2)) ADDCS_IN_GGG(X1, b, X2) -> SUCCZU_IN_GG(X1, X2) SUCCZU_IN_GG(zero(X1), one(X1)) -> U42_GG(X1, binaryZH_in_g(X1)) SUCCZU_IN_GG(zero(X1), one(X1)) -> BINARYZH_IN_G(X1) SUCCZU_IN_GG(one(X1), zero(X2)) -> U43_GG(X1, X2, succT_in_gg(X1, X2)) SUCCZU_IN_GG(one(X1), zero(X2)) -> SUCCT_IN_GG(X1, X2) SUCCT_IN_GG(zero(X1), one(X1)) -> U40_GG(X1, binaryZH_in_g(X1)) SUCCT_IN_GG(zero(X1), one(X1)) -> BINARYZH_IN_G(X1) SUCCT_IN_GG(one(X1), zero(X2)) -> U41_GG(X1, X2, succT_in_gg(X1, X2)) SUCCT_IN_GG(one(X1), zero(X2)) -> SUCCT_IN_GG(X1, X2) ADDCS_IN_GGG(b, X1, X2) -> U53_GGG(X1, X2, succZU_in_gg(X1, X2)) ADDCS_IN_GGG(b, X1, X2) -> SUCCZU_IN_GG(X1, X2) ADDCS_IN_GGG(X1, X2, X3) -> U54_GGG(X1, X2, X3, addCV_in_ggg(X1, X2, X3)) ADDCS_IN_GGG(X1, X2, X3) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(zero(X1), zero(X2), one(X3)) -> U44_GGG(X1, X2, X3, addzQ_in_ggg(X1, X2, X3)) ADDCV_IN_GGG(zero(X1), zero(X2), one(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(zero(zero(X1)), one(b), zero(one(X1))) -> U45_GGG(X1, binaryZH_in_g(X1)) ADDCV_IN_GGG(zero(zero(X1)), one(b), zero(one(X1))) -> BINARYZH_IN_G(X1) ADDCV_IN_GGG(zero(one(X1)), one(b), zero(zero(X2))) -> U46_GGG(X1, X2, succT_in_gg(X1, X2)) ADDCV_IN_GGG(zero(one(X1)), one(b), zero(zero(X2))) -> SUCCT_IN_GG(X1, X2) ADDCV_IN_GGG(zero(X1), one(X2), zero(X3)) -> U47_GGG(X1, X2, X3, addCV_in_ggg(X1, X2, X3)) ADDCV_IN_GGG(zero(X1), one(X2), zero(X3)) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(one(b), zero(zero(X1)), zero(one(X1))) -> U48_GGG(X1, binaryZH_in_g(X1)) ADDCV_IN_GGG(one(b), zero(zero(X1)), zero(one(X1))) -> BINARYZH_IN_G(X1) ADDCV_IN_GGG(one(b), zero(one(X1)), zero(zero(X2))) -> U49_GGG(X1, X2, succT_in_gg(X1, X2)) ADDCV_IN_GGG(one(b), zero(one(X1)), zero(zero(X2))) -> SUCCT_IN_GG(X1, X2) ADDCV_IN_GGG(one(X1), zero(X2), zero(X3)) -> U50_GGG(X1, X2, X3, addCV_in_ggg(X1, X2, X3)) ADDCV_IN_GGG(one(X1), zero(X2), zero(X3)) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(one(X1), one(X2), one(X3)) -> U51_GGG(X1, X2, X3, addcS_in_ggg(X1, X2, X3)) ADDCV_IN_GGG(one(X1), one(X2), one(X3)) -> ADDCS_IN_GGG(X1, X2, X3) ADDC1_IN_GGG(one(X1), X2, one(X3)) -> U63_GGG(X1, X2, X3, addyR_in_ggg(X1, X2, X3)) ADDC1_IN_GGG(one(X1), X2, one(X3)) -> ADDYR_IN_GGG(X1, X2, X3) The TRS R consists of the following rules: timescF_in_aaa(one(b), X1, X1) -> timescF_out_aaa(one(b), X1, X1) timescF_in_aaa(zero(X1), X2, zero(X3)) -> U88_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X3)) timescF_in_aaa(one(X1), X2, X3) -> U89_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U89_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U90_aaa(X1, X2, X3, addcG_in_aaa(X2, X4, X3)) addcG_in_aaa(b, X1, zero(X1)) -> U149_aaa(X1, binaryZcP_in_a(X1)) binaryZcP_in_a(X1) -> U119_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(zero(X1)) -> U91_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(one(X1)) -> U92_a(X1, binarycI_in_a(X1)) binarycI_in_a(b) -> binarycI_out_a(b) binarycI_in_a(zero(X1)) -> U93_a(X1, binaryZcH_in_a(X1)) U93_a(X1, binaryZcH_out_a(X1)) -> binarycI_out_a(zero(X1)) binarycI_in_a(one(X1)) -> U94_a(X1, binarycI_in_a(X1)) U94_a(X1, binarycI_out_a(X1)) -> binarycI_out_a(one(X1)) U92_a(X1, binarycI_out_a(X1)) -> binaryZcH_out_a(one(X1)) U91_a(X1, binaryZcH_out_a(X1)) -> binaryZcH_out_a(zero(X1)) U119_a(X1, binaryZcH_out_a(X1)) -> binaryZcP_out_a(X1) U149_aaa(X1, binaryZcP_out_a(X1)) -> addcG_out_aaa(b, X1, zero(X1)) addcG_in_aaa(zero(X1), X2, zero(X3)) -> U150_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(X1), zero(X2), zero(X3)) -> U95_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(one(X1)), one(b), one(one(X1))) -> U96_aaa(X1, binarycI_in_a(X1)) U96_aaa(X1, binarycI_out_a(X1)) -> addzcJ_out_aaa(zero(one(X1)), one(b), one(one(X1))) addzcJ_in_aaa(zero(zero(X1)), one(b), one(zero(X1))) -> U97_aaa(X1, binaryZcH_in_a(X1)) U97_aaa(X1, binaryZcH_out_a(X1)) -> addzcJ_out_aaa(zero(zero(X1)), one(b), one(zero(X1))) addzcJ_in_aaa(zero(X1), one(X2), one(X3)) -> U98_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), zero(X2), one(X3)) -> U99_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) addycK_in_aaa(b, one(X1), one(X1)) -> U116_aaa(X1, binarycI_in_a(X1)) U116_aaa(X1, binarycI_out_a(X1)) -> addycK_out_aaa(b, one(X1), one(X1)) addycK_in_aaa(b, zero(X1), zero(X1)) -> U117_aaa(X1, binaryZcH_in_a(X1)) U117_aaa(X1, binaryZcH_out_a(X1)) -> addycK_out_aaa(b, zero(X1), zero(X1)) addycK_in_aaa(X1, X2, X3) -> U118_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), one(X2), zero(X3)) -> U100_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) addccL_in_aaa(b, b, one(b)) -> addccL_out_aaa(b, b, one(b)) addccL_in_aaa(X1, b, X2) -> U113_aaa(X1, X2, succZcN_in_aa(X1, X2)) succZcN_in_aa(zero(X1), one(X1)) -> U103_aa(X1, binaryZcH_in_a(X1)) U103_aa(X1, binaryZcH_out_a(X1)) -> succZcN_out_aa(zero(X1), one(X1)) succZcN_in_aa(one(X1), zero(X2)) -> U104_aa(X1, X2, succcM_in_aa(X1, X2)) succcM_in_aa(b, one(b)) -> succcM_out_aa(b, one(b)) succcM_in_aa(zero(X1), one(X1)) -> U101_aa(X1, binaryZcH_in_a(X1)) U101_aa(X1, binaryZcH_out_a(X1)) -> succcM_out_aa(zero(X1), one(X1)) succcM_in_aa(one(X1), zero(X2)) -> U102_aa(X1, X2, succcM_in_aa(X1, X2)) U102_aa(X1, X2, succcM_out_aa(X1, X2)) -> succcM_out_aa(one(X1), zero(X2)) U104_aa(X1, X2, succcM_out_aa(X1, X2)) -> succZcN_out_aa(one(X1), zero(X2)) U113_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(X1, b, X2) addccL_in_aaa(b, X1, X2) -> U114_aaa(X1, X2, succZcN_in_aa(X1, X2)) U114_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(b, X1, X2) addccL_in_aaa(X1, X2, X3) -> U115_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(zero(X1), zero(X2), one(X3)) -> U105_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) U105_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), zero(X2), one(X3)) addCcO_in_aaa(zero(zero(X1)), one(b), zero(one(X1))) -> U106_aaa(X1, binaryZcH_in_a(X1)) U106_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(zero(zero(X1)), one(b), zero(one(X1))) addCcO_in_aaa(zero(one(X1)), one(b), zero(zero(X2))) -> U107_aaa(X1, X2, succcM_in_aa(X1, X2)) U107_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(zero(one(X1)), one(b), zero(zero(X2))) addCcO_in_aaa(zero(X1), one(X2), zero(X3)) -> U108_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(b), zero(zero(X1)), zero(one(X1))) -> U109_aaa(X1, binaryZcH_in_a(X1)) U109_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(one(b), zero(zero(X1)), zero(one(X1))) addCcO_in_aaa(one(b), zero(one(X1)), zero(zero(X2))) -> U110_aaa(X1, X2, succcM_in_aa(X1, X2)) U110_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(one(b), zero(one(X1)), zero(zero(X2))) addCcO_in_aaa(one(X1), zero(X2), zero(X3)) -> U111_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(X1), one(X2), one(X3)) -> U112_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) U112_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), one(X2), one(X3)) U111_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), zero(X2), zero(X3)) U108_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), one(X2), zero(X3)) U115_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addccL_out_aaa(X1, X2, X3) U100_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), one(X2), zero(X3)) U118_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addycK_out_aaa(X1, X2, X3) U99_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), zero(X2), one(X3)) U98_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), one(X2), one(X3)) U95_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), zero(X2), zero(X3)) U150_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addcG_out_aaa(zero(X1), X2, zero(X3)) addcG_in_aaa(one(X1), X2, one(X3)) -> U151_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) U151_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addcG_out_aaa(one(X1), X2, one(X3)) U90_aaa(X1, X2, X3, addcG_out_aaa(X2, X4, X3)) -> timescF_out_aaa(one(X1), X2, X3) U88_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X3)) -> timescF_out_aaa(zero(X1), X2, zero(X3)) The argument filtering Pi contains the following mapping: timesA_in_aag(x1, x2, x3) = timesA_in_aag(x3) zero(x1) = zero(x1) pB_in_aaag(x1, x2, x3, x4) = pB_in_aaag(x4) timesF_in_aaa(x1, x2, x3) = timesF_in_aaa timescF_in_aaa(x1, x2, x3) = timescF_in_aaa timescF_out_aaa(x1, x2, x3) = timescF_out_aaa(x1) U88_aaa(x1, x2, x3, x4) = U88_aaa(x4) U89_aaa(x1, x2, x3, x4) = U89_aaa(x4) U90_aaa(x1, x2, x3, x4) = U90_aaa(x1, x4) addcG_in_aaa(x1, x2, x3) = addcG_in_aaa U149_aaa(x1, x2) = U149_aaa(x2) binaryZcP_in_a(x1) = binaryZcP_in_a U119_a(x1, x2) = U119_a(x2) binaryZcH_in_a(x1) = binaryZcH_in_a U91_a(x1, x2) = U91_a(x2) U92_a(x1, x2) = U92_a(x2) binarycI_in_a(x1) = binarycI_in_a binarycI_out_a(x1) = binarycI_out_a(x1) U93_a(x1, x2) = U93_a(x2) binaryZcH_out_a(x1) = binaryZcH_out_a(x1) U94_a(x1, x2) = U94_a(x2) binaryZcP_out_a(x1) = binaryZcP_out_a(x1) addcG_out_aaa(x1, x2, x3) = addcG_out_aaa(x1, x2, x3) U150_aaa(x1, x2, x3, x4) = U150_aaa(x4) addzcJ_in_aaa(x1, x2, x3) = addzcJ_in_aaa U95_aaa(x1, x2, x3, x4) = U95_aaa(x4) U96_aaa(x1, x2) = U96_aaa(x2) addzcJ_out_aaa(x1, x2, x3) = addzcJ_out_aaa(x1, x2, x3) U97_aaa(x1, x2) = U97_aaa(x2) U98_aaa(x1, x2, x3, x4) = U98_aaa(x4) U99_aaa(x1, x2, x3, x4) = U99_aaa(x4) addycK_in_aaa(x1, x2, x3) = addycK_in_aaa U116_aaa(x1, x2) = U116_aaa(x2) addycK_out_aaa(x1, x2, x3) = addycK_out_aaa(x1, x2, x3) U117_aaa(x1, x2) = U117_aaa(x2) U118_aaa(x1, x2, x3, x4) = U118_aaa(x4) U100_aaa(x1, x2, x3, x4) = U100_aaa(x4) addccL_in_aaa(x1, x2, x3) = addccL_in_aaa addccL_out_aaa(x1, x2, x3) = addccL_out_aaa(x1, x2, x3) U113_aaa(x1, x2, x3) = U113_aaa(x3) succZcN_in_aa(x1, x2) = succZcN_in_aa U103_aa(x1, x2) = U103_aa(x2) succZcN_out_aa(x1, x2) = succZcN_out_aa(x1, x2) U104_aa(x1, x2, x3) = U104_aa(x3) succcM_in_aa(x1, x2) = succcM_in_aa succcM_out_aa(x1, x2) = succcM_out_aa(x1, x2) U101_aa(x1, x2) = U101_aa(x2) U102_aa(x1, x2, x3) = U102_aa(x3) U114_aaa(x1, x2, x3) = U114_aaa(x3) U115_aaa(x1, x2, x3, x4) = U115_aaa(x4) addCcO_in_aaa(x1, x2, x3) = addCcO_in_aaa U105_aaa(x1, x2, x3, x4) = U105_aaa(x4) addCcO_out_aaa(x1, x2, x3) = addCcO_out_aaa(x1, x2, x3) U106_aaa(x1, x2) = U106_aaa(x2) U107_aaa(x1, x2, x3) = U107_aaa(x3) U108_aaa(x1, x2, x3, x4) = U108_aaa(x4) U109_aaa(x1, x2) = U109_aaa(x2) U110_aaa(x1, x2, x3) = U110_aaa(x3) U111_aaa(x1, x2, x3, x4) = U111_aaa(x4) U112_aaa(x1, x2, x3, x4) = U112_aaa(x4) U151_aaa(x1, x2, x3, x4) = U151_aaa(x4) addG_in_aaa(x1, x2, x3) = addG_in_aaa binaryZP_in_a(x1) = binaryZP_in_a binaryZH_in_a(x1) = binaryZH_in_a binaryI_in_a(x1) = binaryI_in_a addzJ_in_aaa(x1, x2, x3) = addzJ_in_aaa addyK_in_aaa(x1, x2, x3) = addyK_in_aaa addcL_in_aaa(x1, x2, x3) = addcL_in_aaa succZN_in_aa(x1, x2) = succZN_in_aa succM_in_aa(x1, x2) = succM_in_aa addCO_in_aaa(x1, x2, x3) = addCO_in_aaa addC1_in_aag(x1, x2, x3) = addC1_in_aag(x3) binaryZP_in_g(x1) = binaryZP_in_g(x1) binaryZH_in_g(x1) = binaryZH_in_g(x1) one(x1) = one(x1) binaryI_in_g(x1) = binaryI_in_g(x1) addzQ_in_aag(x1, x2, x3) = addzQ_in_aag(x3) addyR_in_aag(x1, x2, x3) = addyR_in_aag(x3) addcS_in_aag(x1, x2, x3) = addcS_in_aag(x3) succZU_in_ag(x1, x2) = succZU_in_ag(x2) succT_in_ag(x1, x2) = succT_in_ag(x2) addCV_in_aag(x1, x2, x3) = addCV_in_aag(x3) pD_in_aaag(x1, x2, x3, x4) = pD_in_aaag(x4) pE_in_aaaag(x1, x2, x3, x4, x5) = pE_in_aaaag(x5) addC1_in_ggg(x1, x2, x3) = addC1_in_ggg(x1, x2, x3) b = b addzQ_in_ggg(x1, x2, x3) = addzQ_in_ggg(x1, x2, x3) addyR_in_ggg(x1, x2, x3) = addyR_in_ggg(x1, x2, x3) addcS_in_ggg(x1, x2, x3) = addcS_in_ggg(x1, x2, x3) succZU_in_gg(x1, x2) = succZU_in_gg(x1, x2) succT_in_gg(x1, x2) = succT_in_gg(x1, x2) addCV_in_ggg(x1, x2, x3) = addCV_in_ggg(x1, x2, x3) TIMESA_IN_AAG(x1, x2, x3) = TIMESA_IN_AAG(x3) U75_AAG(x1, x2, x3, x4) = U75_AAG(x3, x4) U76_AAG(x1, x2, x3, x4) = U76_AAG(x3, x4) PB_IN_AAAG(x1, x2, x3, x4) = PB_IN_AAAG(x4) U58_AAAG(x1, x2, x3, x4, x5) = U58_AAAG(x4, x5) TIMESF_IN_AAA(x1, x2, x3) = TIMESF_IN_AAA U1_AAA(x1, x2, x3, x4) = U1_AAA(x4) U2_AAA(x1, x2, x3, x4) = U2_AAA(x4) U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) U4_AAA(x1, x2, x3, x4) = U4_AAA(x1, x4) ADDG_IN_AAA(x1, x2, x3) = ADDG_IN_AAA U64_AAA(x1, x2) = U64_AAA(x2) BINARYZP_IN_A(x1) = BINARYZP_IN_A U33_A(x1, x2) = U33_A(x2) BINARYZH_IN_A(x1) = BINARYZH_IN_A U5_A(x1, x2) = U5_A(x2) U6_A(x1, x2) = U6_A(x2) BINARYI_IN_A(x1) = BINARYI_IN_A U7_A(x1, x2) = U7_A(x2) U8_A(x1, x2) = U8_A(x2) U65_AAA(x1, x2, x3, x4) = U65_AAA(x4) ADDZJ_IN_AAA(x1, x2, x3) = ADDZJ_IN_AAA U9_AAA(x1, x2, x3, x4) = U9_AAA(x4) U10_AAA(x1, x2) = U10_AAA(x2) U11_AAA(x1, x2) = U11_AAA(x2) U12_AAA(x1, x2, x3, x4) = U12_AAA(x4) U13_AAA(x1, x2, x3, x4) = U13_AAA(x4) ADDYK_IN_AAA(x1, x2, x3) = ADDYK_IN_AAA U30_AAA(x1, x2) = U30_AAA(x2) U31_AAA(x1, x2) = U31_AAA(x2) U32_AAA(x1, x2, x3, x4) = U32_AAA(x4) U14_AAA(x1, x2, x3, x4) = U14_AAA(x4) ADDCL_IN_AAA(x1, x2, x3) = ADDCL_IN_AAA U27_AAA(x1, x2, x3) = U27_AAA(x3) SUCCZN_IN_AA(x1, x2) = SUCCZN_IN_AA U17_AA(x1, x2) = U17_AA(x2) U18_AA(x1, x2, x3) = U18_AA(x3) SUCCM_IN_AA(x1, x2) = SUCCM_IN_AA U15_AA(x1, x2) = U15_AA(x2) U16_AA(x1, x2, x3) = U16_AA(x3) U28_AAA(x1, x2, x3) = U28_AAA(x3) U29_AAA(x1, x2, x3, x4) = U29_AAA(x4) ADDCO_IN_AAA(x1, x2, x3) = ADDCO_IN_AAA U19_AAA(x1, x2, x3, x4) = U19_AAA(x4) U20_AAA(x1, x2) = U20_AAA(x2) U21_AAA(x1, x2, x3) = U21_AAA(x3) U22_AAA(x1, x2, x3, x4) = U22_AAA(x4) U23_AAA(x1, x2) = U23_AAA(x2) U24_AAA(x1, x2, x3) = U24_AAA(x3) U25_AAA(x1, x2, x3, x4) = U25_AAA(x4) U26_AAA(x1, x2, x3, x4) = U26_AAA(x4) U66_AAA(x1, x2, x3, x4) = U66_AAA(x4) U59_AAAG(x1, x2, x3, x4, x5) = U59_AAAG(x4, x5) U60_AAAG(x1, x2, x3, x4, x5) = U60_AAAG(x1, x4, x5) ADDC1_IN_AAG(x1, x2, x3) = ADDC1_IN_AAG(x3) U61_AAG(x1, x2) = U61_AAG(x1, x2) BINARYZP_IN_G(x1) = BINARYZP_IN_G(x1) U33_G(x1, x2) = U33_G(x1, x2) BINARYZH_IN_G(x1) = BINARYZH_IN_G(x1) U5_G(x1, x2) = U5_G(x1, x2) U6_G(x1, x2) = U6_G(x1, x2) BINARYI_IN_G(x1) = BINARYI_IN_G(x1) U7_G(x1, x2) = U7_G(x1, x2) U8_G(x1, x2) = U8_G(x1, x2) U62_AAG(x1, x2, x3, x4) = U62_AAG(x3, x4) ADDZQ_IN_AAG(x1, x2, x3) = ADDZQ_IN_AAG(x3) U34_AAG(x1, x2, x3, x4) = U34_AAG(x3, x4) U35_AAG(x1, x2) = U35_AAG(x1, x2) U36_AAG(x1, x2) = U36_AAG(x1, x2) U37_AAG(x1, x2, x3, x4) = U37_AAG(x3, x4) U38_AAG(x1, x2, x3, x4) = U38_AAG(x3, x4) ADDYR_IN_AAG(x1, x2, x3) = ADDYR_IN_AAG(x3) U55_AAG(x1, x2) = U55_AAG(x1, x2) U56_AAG(x1, x2) = U56_AAG(x1, x2) U57_AAG(x1, x2, x3, x4) = U57_AAG(x3, x4) U39_AAG(x1, x2, x3, x4) = U39_AAG(x3, x4) ADDCS_IN_AAG(x1, x2, x3) = ADDCS_IN_AAG(x3) U52_AAG(x1, x2, x3) = U52_AAG(x2, x3) SUCCZU_IN_AG(x1, x2) = SUCCZU_IN_AG(x2) U42_AG(x1, x2) = U42_AG(x1, x2) U43_AG(x1, x2, x3) = U43_AG(x2, x3) SUCCT_IN_AG(x1, x2) = SUCCT_IN_AG(x2) U40_AG(x1, x2) = U40_AG(x1, x2) U41_AG(x1, x2, x3) = U41_AG(x2, x3) U53_AAG(x1, x2, x3) = U53_AAG(x2, x3) U54_AAG(x1, x2, x3, x4) = U54_AAG(x3, x4) ADDCV_IN_AAG(x1, x2, x3) = ADDCV_IN_AAG(x3) U44_AAG(x1, x2, x3, x4) = U44_AAG(x3, x4) U45_AAG(x1, x2) = U45_AAG(x1, x2) U46_AAG(x1, x2, x3) = U46_AAG(x2, x3) U47_AAG(x1, x2, x3, x4) = U47_AAG(x3, x4) U48_AAG(x1, x2) = U48_AAG(x1, x2) U49_AAG(x1, x2, x3) = U49_AAG(x2, x3) U50_AAG(x1, x2, x3, x4) = U50_AAG(x3, x4) U51_AAG(x1, x2, x3, x4) = U51_AAG(x3, x4) U63_AAG(x1, x2, x3, x4) = U63_AAG(x3, x4) U77_AAG(x1, x2, x3, x4) = U77_AAG(x3, x4) U78_AAG(x1, x2, x3) = U78_AAG(x2, x3) U79_AAG(x1, x2, x3, x4) = U79_AAG(x3, x4) PD_IN_AAAG(x1, x2, x3, x4) = PD_IN_AAAG(x4) U67_AAAG(x1, x2, x3, x4, x5) = U67_AAAG(x4, x5) U68_AAAG(x1, x2, x3, x4, x5) = U68_AAAG(x4, x5) U69_AAAG(x1, x2, x3, x4, x5) = U69_AAAG(x1, x4, x5) U80_AAG(x1, x2, x3, x4) = U80_AAG(x3, x4) PE_IN_AAAAG(x1, x2, x3, x4, x5) = PE_IN_AAAAG(x5) U70_AAAAG(x1, x2, x3, x4, x5, x6) = U70_AAAAG(x5, x6) U71_AAAAG(x1, x2, x3, x4, x5, x6) = U71_AAAAG(x5, x6) U72_AAAAG(x1, x2, x3, x4, x5, x6) = U72_AAAAG(x1, x5, x6) U73_AAAAG(x1, x2, x3, x4, x5, x6) = U73_AAAAG(x1, x5, x6) U74_AAAAG(x1, x2, x3, x4, x5, x6) = U74_AAAAG(x1, x2, x5, x6) ADDC1_IN_GGG(x1, x2, x3) = ADDC1_IN_GGG(x1, x2, x3) U61_GGG(x1, x2) = U61_GGG(x1, x2) U62_GGG(x1, x2, x3, x4) = U62_GGG(x1, x2, x3, x4) ADDZQ_IN_GGG(x1, x2, x3) = ADDZQ_IN_GGG(x1, x2, x3) U34_GGG(x1, x2, x3, x4) = U34_GGG(x1, x2, x3, x4) U35_GGG(x1, x2) = U35_GGG(x1, x2) U36_GGG(x1, x2) = U36_GGG(x1, x2) U37_GGG(x1, x2, x3, x4) = U37_GGG(x1, x2, x3, x4) U38_GGG(x1, x2, x3, x4) = U38_GGG(x1, x2, x3, x4) ADDYR_IN_GGG(x1, x2, x3) = ADDYR_IN_GGG(x1, x2, x3) U55_GGG(x1, x2) = U55_GGG(x1, x2) U56_GGG(x1, x2) = U56_GGG(x1, x2) U57_GGG(x1, x2, x3, x4) = U57_GGG(x1, x2, x3, x4) U39_GGG(x1, x2, x3, x4) = U39_GGG(x1, x2, x3, x4) ADDCS_IN_GGG(x1, x2, x3) = ADDCS_IN_GGG(x1, x2, x3) U52_GGG(x1, x2, x3) = U52_GGG(x1, x2, x3) SUCCZU_IN_GG(x1, x2) = SUCCZU_IN_GG(x1, x2) U42_GG(x1, x2) = U42_GG(x1, x2) U43_GG(x1, x2, x3) = U43_GG(x1, x2, x3) SUCCT_IN_GG(x1, x2) = SUCCT_IN_GG(x1, x2) U40_GG(x1, x2) = U40_GG(x1, x2) U41_GG(x1, x2, x3) = U41_GG(x1, x2, x3) U53_GGG(x1, x2, x3) = U53_GGG(x1, x2, x3) U54_GGG(x1, x2, x3, x4) = U54_GGG(x1, x2, x3, x4) ADDCV_IN_GGG(x1, x2, x3) = ADDCV_IN_GGG(x1, x2, x3) U44_GGG(x1, x2, x3, x4) = U44_GGG(x1, x2, x3, x4) U45_GGG(x1, x2) = U45_GGG(x1, x2) U46_GGG(x1, x2, x3) = U46_GGG(x1, x2, x3) U47_GGG(x1, x2, x3, x4) = U47_GGG(x1, x2, x3, x4) U48_GGG(x1, x2) = U48_GGG(x1, x2) U49_GGG(x1, x2, x3) = U49_GGG(x1, x2, x3) U50_GGG(x1, x2, x3, x4) = U50_GGG(x1, x2, x3, x4) U51_GGG(x1, x2, x3, x4) = U51_GGG(x1, x2, x3, x4) U63_GGG(x1, x2, x3, x4) = U63_GGG(x1, x2, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (101) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 10 SCCs with 175 less nodes. ---------------------------------------- (102) Complex Obligation (AND) ---------------------------------------- (103) Obligation: Pi DP problem: The TRS P consists of the following rules: BINARYZH_IN_G(one(X1)) -> BINARYI_IN_G(X1) BINARYI_IN_G(zero(X1)) -> BINARYZH_IN_G(X1) BINARYZH_IN_G(zero(X1)) -> BINARYZH_IN_G(X1) BINARYI_IN_G(one(X1)) -> BINARYI_IN_G(X1) The TRS R consists of the following rules: timescF_in_aaa(one(b), X1, X1) -> timescF_out_aaa(one(b), X1, X1) timescF_in_aaa(zero(X1), X2, zero(X3)) -> U88_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X3)) timescF_in_aaa(one(X1), X2, X3) -> U89_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U89_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U90_aaa(X1, X2, X3, addcG_in_aaa(X2, X4, X3)) addcG_in_aaa(b, X1, zero(X1)) -> U149_aaa(X1, binaryZcP_in_a(X1)) binaryZcP_in_a(X1) -> U119_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(zero(X1)) -> U91_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(one(X1)) -> U92_a(X1, binarycI_in_a(X1)) binarycI_in_a(b) -> binarycI_out_a(b) binarycI_in_a(zero(X1)) -> U93_a(X1, binaryZcH_in_a(X1)) U93_a(X1, binaryZcH_out_a(X1)) -> binarycI_out_a(zero(X1)) binarycI_in_a(one(X1)) -> U94_a(X1, binarycI_in_a(X1)) U94_a(X1, binarycI_out_a(X1)) -> binarycI_out_a(one(X1)) U92_a(X1, binarycI_out_a(X1)) -> binaryZcH_out_a(one(X1)) U91_a(X1, binaryZcH_out_a(X1)) -> binaryZcH_out_a(zero(X1)) U119_a(X1, binaryZcH_out_a(X1)) -> binaryZcP_out_a(X1) U149_aaa(X1, binaryZcP_out_a(X1)) -> addcG_out_aaa(b, X1, zero(X1)) addcG_in_aaa(zero(X1), X2, zero(X3)) -> U150_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(X1), zero(X2), zero(X3)) -> U95_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(one(X1)), one(b), one(one(X1))) -> U96_aaa(X1, binarycI_in_a(X1)) U96_aaa(X1, binarycI_out_a(X1)) -> addzcJ_out_aaa(zero(one(X1)), one(b), one(one(X1))) addzcJ_in_aaa(zero(zero(X1)), one(b), one(zero(X1))) -> U97_aaa(X1, binaryZcH_in_a(X1)) U97_aaa(X1, binaryZcH_out_a(X1)) -> addzcJ_out_aaa(zero(zero(X1)), one(b), one(zero(X1))) addzcJ_in_aaa(zero(X1), one(X2), one(X3)) -> U98_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), zero(X2), one(X3)) -> U99_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) addycK_in_aaa(b, one(X1), one(X1)) -> U116_aaa(X1, binarycI_in_a(X1)) U116_aaa(X1, binarycI_out_a(X1)) -> addycK_out_aaa(b, one(X1), one(X1)) addycK_in_aaa(b, zero(X1), zero(X1)) -> U117_aaa(X1, binaryZcH_in_a(X1)) U117_aaa(X1, binaryZcH_out_a(X1)) -> addycK_out_aaa(b, zero(X1), zero(X1)) addycK_in_aaa(X1, X2, X3) -> U118_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), one(X2), zero(X3)) -> U100_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) addccL_in_aaa(b, b, one(b)) -> addccL_out_aaa(b, b, one(b)) addccL_in_aaa(X1, b, X2) -> U113_aaa(X1, X2, succZcN_in_aa(X1, X2)) succZcN_in_aa(zero(X1), one(X1)) -> U103_aa(X1, binaryZcH_in_a(X1)) U103_aa(X1, binaryZcH_out_a(X1)) -> succZcN_out_aa(zero(X1), one(X1)) succZcN_in_aa(one(X1), zero(X2)) -> U104_aa(X1, X2, succcM_in_aa(X1, X2)) succcM_in_aa(b, one(b)) -> succcM_out_aa(b, one(b)) succcM_in_aa(zero(X1), one(X1)) -> U101_aa(X1, binaryZcH_in_a(X1)) U101_aa(X1, binaryZcH_out_a(X1)) -> succcM_out_aa(zero(X1), one(X1)) succcM_in_aa(one(X1), zero(X2)) -> U102_aa(X1, X2, succcM_in_aa(X1, X2)) U102_aa(X1, X2, succcM_out_aa(X1, X2)) -> succcM_out_aa(one(X1), zero(X2)) U104_aa(X1, X2, succcM_out_aa(X1, X2)) -> succZcN_out_aa(one(X1), zero(X2)) U113_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(X1, b, X2) addccL_in_aaa(b, X1, X2) -> U114_aaa(X1, X2, succZcN_in_aa(X1, X2)) U114_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(b, X1, X2) addccL_in_aaa(X1, X2, X3) -> U115_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(zero(X1), zero(X2), one(X3)) -> U105_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) U105_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), zero(X2), one(X3)) addCcO_in_aaa(zero(zero(X1)), one(b), zero(one(X1))) -> U106_aaa(X1, binaryZcH_in_a(X1)) U106_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(zero(zero(X1)), one(b), zero(one(X1))) addCcO_in_aaa(zero(one(X1)), one(b), zero(zero(X2))) -> U107_aaa(X1, X2, succcM_in_aa(X1, X2)) U107_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(zero(one(X1)), one(b), zero(zero(X2))) addCcO_in_aaa(zero(X1), one(X2), zero(X3)) -> U108_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(b), zero(zero(X1)), zero(one(X1))) -> U109_aaa(X1, binaryZcH_in_a(X1)) U109_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(one(b), zero(zero(X1)), zero(one(X1))) addCcO_in_aaa(one(b), zero(one(X1)), zero(zero(X2))) -> U110_aaa(X1, X2, succcM_in_aa(X1, X2)) U110_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(one(b), zero(one(X1)), zero(zero(X2))) addCcO_in_aaa(one(X1), zero(X2), zero(X3)) -> U111_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(X1), one(X2), one(X3)) -> U112_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) U112_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), one(X2), one(X3)) U111_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), zero(X2), zero(X3)) U108_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), one(X2), zero(X3)) U115_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addccL_out_aaa(X1, X2, X3) U100_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), one(X2), zero(X3)) U118_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addycK_out_aaa(X1, X2, X3) U99_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), zero(X2), one(X3)) U98_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), one(X2), one(X3)) U95_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), zero(X2), zero(X3)) U150_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addcG_out_aaa(zero(X1), X2, zero(X3)) addcG_in_aaa(one(X1), X2, one(X3)) -> U151_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) U151_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addcG_out_aaa(one(X1), X2, one(X3)) U90_aaa(X1, X2, X3, addcG_out_aaa(X2, X4, X3)) -> timescF_out_aaa(one(X1), X2, X3) U88_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X3)) -> timescF_out_aaa(zero(X1), X2, zero(X3)) The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) timescF_in_aaa(x1, x2, x3) = timescF_in_aaa timescF_out_aaa(x1, x2, x3) = timescF_out_aaa(x1) U88_aaa(x1, x2, x3, x4) = U88_aaa(x4) U89_aaa(x1, x2, x3, x4) = U89_aaa(x4) U90_aaa(x1, x2, x3, x4) = U90_aaa(x1, x4) addcG_in_aaa(x1, x2, x3) = addcG_in_aaa U149_aaa(x1, x2) = U149_aaa(x2) binaryZcP_in_a(x1) = binaryZcP_in_a U119_a(x1, x2) = U119_a(x2) binaryZcH_in_a(x1) = binaryZcH_in_a U91_a(x1, x2) = U91_a(x2) U92_a(x1, x2) = U92_a(x2) binarycI_in_a(x1) = binarycI_in_a binarycI_out_a(x1) = binarycI_out_a(x1) U93_a(x1, x2) = U93_a(x2) binaryZcH_out_a(x1) = binaryZcH_out_a(x1) U94_a(x1, x2) = U94_a(x2) binaryZcP_out_a(x1) = binaryZcP_out_a(x1) addcG_out_aaa(x1, x2, x3) = addcG_out_aaa(x1, x2, x3) U150_aaa(x1, x2, x3, x4) = U150_aaa(x4) addzcJ_in_aaa(x1, x2, x3) = addzcJ_in_aaa U95_aaa(x1, x2, x3, x4) = U95_aaa(x4) U96_aaa(x1, x2) = U96_aaa(x2) addzcJ_out_aaa(x1, x2, x3) = addzcJ_out_aaa(x1, x2, x3) U97_aaa(x1, x2) = U97_aaa(x2) U98_aaa(x1, x2, x3, x4) = U98_aaa(x4) U99_aaa(x1, x2, x3, x4) = U99_aaa(x4) addycK_in_aaa(x1, x2, x3) = addycK_in_aaa U116_aaa(x1, x2) = U116_aaa(x2) addycK_out_aaa(x1, x2, x3) = addycK_out_aaa(x1, x2, x3) U117_aaa(x1, x2) = U117_aaa(x2) U118_aaa(x1, x2, x3, x4) = U118_aaa(x4) U100_aaa(x1, x2, x3, x4) = U100_aaa(x4) addccL_in_aaa(x1, x2, x3) = addccL_in_aaa addccL_out_aaa(x1, x2, x3) = addccL_out_aaa(x1, x2, x3) U113_aaa(x1, x2, x3) = U113_aaa(x3) succZcN_in_aa(x1, x2) = succZcN_in_aa U103_aa(x1, x2) = U103_aa(x2) succZcN_out_aa(x1, x2) = succZcN_out_aa(x1, x2) U104_aa(x1, x2, x3) = U104_aa(x3) succcM_in_aa(x1, x2) = succcM_in_aa succcM_out_aa(x1, x2) = succcM_out_aa(x1, x2) U101_aa(x1, x2) = U101_aa(x2) U102_aa(x1, x2, x3) = U102_aa(x3) U114_aaa(x1, x2, x3) = U114_aaa(x3) U115_aaa(x1, x2, x3, x4) = U115_aaa(x4) addCcO_in_aaa(x1, x2, x3) = addCcO_in_aaa U105_aaa(x1, x2, x3, x4) = U105_aaa(x4) addCcO_out_aaa(x1, x2, x3) = addCcO_out_aaa(x1, x2, x3) U106_aaa(x1, x2) = U106_aaa(x2) U107_aaa(x1, x2, x3) = U107_aaa(x3) U108_aaa(x1, x2, x3, x4) = U108_aaa(x4) U109_aaa(x1, x2) = U109_aaa(x2) U110_aaa(x1, x2, x3) = U110_aaa(x3) U111_aaa(x1, x2, x3, x4) = U111_aaa(x4) U112_aaa(x1, x2, x3, x4) = U112_aaa(x4) U151_aaa(x1, x2, x3, x4) = U151_aaa(x4) one(x1) = one(x1) b = b BINARYZH_IN_G(x1) = BINARYZH_IN_G(x1) BINARYI_IN_G(x1) = BINARYI_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (104) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (105) Obligation: Pi DP problem: The TRS P consists of the following rules: BINARYZH_IN_G(one(X1)) -> BINARYI_IN_G(X1) BINARYI_IN_G(zero(X1)) -> BINARYZH_IN_G(X1) BINARYZH_IN_G(zero(X1)) -> BINARYZH_IN_G(X1) BINARYI_IN_G(one(X1)) -> BINARYI_IN_G(X1) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (106) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (107) Obligation: Q DP problem: The TRS P consists of the following rules: BINARYZH_IN_G(one(X1)) -> BINARYI_IN_G(X1) BINARYI_IN_G(zero(X1)) -> BINARYZH_IN_G(X1) BINARYZH_IN_G(zero(X1)) -> BINARYZH_IN_G(X1) BINARYI_IN_G(one(X1)) -> BINARYI_IN_G(X1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (108) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *BINARYI_IN_G(zero(X1)) -> BINARYZH_IN_G(X1) The graph contains the following edges 1 > 1 *BINARYI_IN_G(one(X1)) -> BINARYI_IN_G(X1) The graph contains the following edges 1 > 1 *BINARYZH_IN_G(zero(X1)) -> BINARYZH_IN_G(X1) The graph contains the following edges 1 > 1 *BINARYZH_IN_G(one(X1)) -> BINARYI_IN_G(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (109) YES ---------------------------------------- (110) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCCT_IN_GG(one(X1), zero(X2)) -> SUCCT_IN_GG(X1, X2) The TRS R consists of the following rules: timescF_in_aaa(one(b), X1, X1) -> timescF_out_aaa(one(b), X1, X1) timescF_in_aaa(zero(X1), X2, zero(X3)) -> U88_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X3)) timescF_in_aaa(one(X1), X2, X3) -> U89_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U89_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U90_aaa(X1, X2, X3, addcG_in_aaa(X2, X4, X3)) addcG_in_aaa(b, X1, zero(X1)) -> U149_aaa(X1, binaryZcP_in_a(X1)) binaryZcP_in_a(X1) -> U119_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(zero(X1)) -> U91_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(one(X1)) -> U92_a(X1, binarycI_in_a(X1)) binarycI_in_a(b) -> binarycI_out_a(b) binarycI_in_a(zero(X1)) -> U93_a(X1, binaryZcH_in_a(X1)) U93_a(X1, binaryZcH_out_a(X1)) -> binarycI_out_a(zero(X1)) binarycI_in_a(one(X1)) -> U94_a(X1, binarycI_in_a(X1)) U94_a(X1, binarycI_out_a(X1)) -> binarycI_out_a(one(X1)) U92_a(X1, binarycI_out_a(X1)) -> binaryZcH_out_a(one(X1)) U91_a(X1, binaryZcH_out_a(X1)) -> binaryZcH_out_a(zero(X1)) U119_a(X1, binaryZcH_out_a(X1)) -> binaryZcP_out_a(X1) U149_aaa(X1, binaryZcP_out_a(X1)) -> addcG_out_aaa(b, X1, zero(X1)) addcG_in_aaa(zero(X1), X2, zero(X3)) -> U150_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(X1), zero(X2), zero(X3)) -> U95_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(one(X1)), one(b), one(one(X1))) -> U96_aaa(X1, binarycI_in_a(X1)) U96_aaa(X1, binarycI_out_a(X1)) -> addzcJ_out_aaa(zero(one(X1)), one(b), one(one(X1))) addzcJ_in_aaa(zero(zero(X1)), one(b), one(zero(X1))) -> U97_aaa(X1, binaryZcH_in_a(X1)) U97_aaa(X1, binaryZcH_out_a(X1)) -> addzcJ_out_aaa(zero(zero(X1)), one(b), one(zero(X1))) addzcJ_in_aaa(zero(X1), one(X2), one(X3)) -> U98_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), zero(X2), one(X3)) -> U99_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) addycK_in_aaa(b, one(X1), one(X1)) -> U116_aaa(X1, binarycI_in_a(X1)) U116_aaa(X1, binarycI_out_a(X1)) -> addycK_out_aaa(b, one(X1), one(X1)) addycK_in_aaa(b, zero(X1), zero(X1)) -> U117_aaa(X1, binaryZcH_in_a(X1)) U117_aaa(X1, binaryZcH_out_a(X1)) -> addycK_out_aaa(b, zero(X1), zero(X1)) addycK_in_aaa(X1, X2, X3) -> U118_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), one(X2), zero(X3)) -> U100_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) addccL_in_aaa(b, b, one(b)) -> addccL_out_aaa(b, b, one(b)) addccL_in_aaa(X1, b, X2) -> U113_aaa(X1, X2, succZcN_in_aa(X1, X2)) succZcN_in_aa(zero(X1), one(X1)) -> U103_aa(X1, binaryZcH_in_a(X1)) U103_aa(X1, binaryZcH_out_a(X1)) -> succZcN_out_aa(zero(X1), one(X1)) succZcN_in_aa(one(X1), zero(X2)) -> U104_aa(X1, X2, succcM_in_aa(X1, X2)) succcM_in_aa(b, one(b)) -> succcM_out_aa(b, one(b)) succcM_in_aa(zero(X1), one(X1)) -> U101_aa(X1, binaryZcH_in_a(X1)) U101_aa(X1, binaryZcH_out_a(X1)) -> succcM_out_aa(zero(X1), one(X1)) succcM_in_aa(one(X1), zero(X2)) -> U102_aa(X1, X2, succcM_in_aa(X1, X2)) U102_aa(X1, X2, succcM_out_aa(X1, X2)) -> succcM_out_aa(one(X1), zero(X2)) U104_aa(X1, X2, succcM_out_aa(X1, X2)) -> succZcN_out_aa(one(X1), zero(X2)) U113_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(X1, b, X2) addccL_in_aaa(b, X1, X2) -> U114_aaa(X1, X2, succZcN_in_aa(X1, X2)) U114_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(b, X1, X2) addccL_in_aaa(X1, X2, X3) -> U115_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(zero(X1), zero(X2), one(X3)) -> U105_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) U105_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), zero(X2), one(X3)) addCcO_in_aaa(zero(zero(X1)), one(b), zero(one(X1))) -> U106_aaa(X1, binaryZcH_in_a(X1)) U106_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(zero(zero(X1)), one(b), zero(one(X1))) addCcO_in_aaa(zero(one(X1)), one(b), zero(zero(X2))) -> U107_aaa(X1, X2, succcM_in_aa(X1, X2)) U107_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(zero(one(X1)), one(b), zero(zero(X2))) addCcO_in_aaa(zero(X1), one(X2), zero(X3)) -> U108_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(b), zero(zero(X1)), zero(one(X1))) -> U109_aaa(X1, binaryZcH_in_a(X1)) U109_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(one(b), zero(zero(X1)), zero(one(X1))) addCcO_in_aaa(one(b), zero(one(X1)), zero(zero(X2))) -> U110_aaa(X1, X2, succcM_in_aa(X1, X2)) U110_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(one(b), zero(one(X1)), zero(zero(X2))) addCcO_in_aaa(one(X1), zero(X2), zero(X3)) -> U111_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(X1), one(X2), one(X3)) -> U112_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) U112_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), one(X2), one(X3)) U111_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), zero(X2), zero(X3)) U108_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), one(X2), zero(X3)) U115_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addccL_out_aaa(X1, X2, X3) U100_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), one(X2), zero(X3)) U118_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addycK_out_aaa(X1, X2, X3) U99_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), zero(X2), one(X3)) U98_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), one(X2), one(X3)) U95_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), zero(X2), zero(X3)) U150_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addcG_out_aaa(zero(X1), X2, zero(X3)) addcG_in_aaa(one(X1), X2, one(X3)) -> U151_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) U151_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addcG_out_aaa(one(X1), X2, one(X3)) U90_aaa(X1, X2, X3, addcG_out_aaa(X2, X4, X3)) -> timescF_out_aaa(one(X1), X2, X3) U88_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X3)) -> timescF_out_aaa(zero(X1), X2, zero(X3)) The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) timescF_in_aaa(x1, x2, x3) = timescF_in_aaa timescF_out_aaa(x1, x2, x3) = timescF_out_aaa(x1) U88_aaa(x1, x2, x3, x4) = U88_aaa(x4) U89_aaa(x1, x2, x3, x4) = U89_aaa(x4) U90_aaa(x1, x2, x3, x4) = U90_aaa(x1, x4) addcG_in_aaa(x1, x2, x3) = addcG_in_aaa U149_aaa(x1, x2) = U149_aaa(x2) binaryZcP_in_a(x1) = binaryZcP_in_a U119_a(x1, x2) = U119_a(x2) binaryZcH_in_a(x1) = binaryZcH_in_a U91_a(x1, x2) = U91_a(x2) U92_a(x1, x2) = U92_a(x2) binarycI_in_a(x1) = binarycI_in_a binarycI_out_a(x1) = binarycI_out_a(x1) U93_a(x1, x2) = U93_a(x2) binaryZcH_out_a(x1) = binaryZcH_out_a(x1) U94_a(x1, x2) = U94_a(x2) binaryZcP_out_a(x1) = binaryZcP_out_a(x1) addcG_out_aaa(x1, x2, x3) = addcG_out_aaa(x1, x2, x3) U150_aaa(x1, x2, x3, x4) = U150_aaa(x4) addzcJ_in_aaa(x1, x2, x3) = addzcJ_in_aaa U95_aaa(x1, x2, x3, x4) = U95_aaa(x4) U96_aaa(x1, x2) = U96_aaa(x2) addzcJ_out_aaa(x1, x2, x3) = addzcJ_out_aaa(x1, x2, x3) U97_aaa(x1, x2) = U97_aaa(x2) U98_aaa(x1, x2, x3, x4) = U98_aaa(x4) U99_aaa(x1, x2, x3, x4) = U99_aaa(x4) addycK_in_aaa(x1, x2, x3) = addycK_in_aaa U116_aaa(x1, x2) = U116_aaa(x2) addycK_out_aaa(x1, x2, x3) = addycK_out_aaa(x1, x2, x3) U117_aaa(x1, x2) = U117_aaa(x2) U118_aaa(x1, x2, x3, x4) = U118_aaa(x4) U100_aaa(x1, x2, x3, x4) = U100_aaa(x4) addccL_in_aaa(x1, x2, x3) = addccL_in_aaa addccL_out_aaa(x1, x2, x3) = addccL_out_aaa(x1, x2, x3) U113_aaa(x1, x2, x3) = U113_aaa(x3) succZcN_in_aa(x1, x2) = succZcN_in_aa U103_aa(x1, x2) = U103_aa(x2) succZcN_out_aa(x1, x2) = succZcN_out_aa(x1, x2) U104_aa(x1, x2, x3) = U104_aa(x3) succcM_in_aa(x1, x2) = succcM_in_aa succcM_out_aa(x1, x2) = succcM_out_aa(x1, x2) U101_aa(x1, x2) = U101_aa(x2) U102_aa(x1, x2, x3) = U102_aa(x3) U114_aaa(x1, x2, x3) = U114_aaa(x3) U115_aaa(x1, x2, x3, x4) = U115_aaa(x4) addCcO_in_aaa(x1, x2, x3) = addCcO_in_aaa U105_aaa(x1, x2, x3, x4) = U105_aaa(x4) addCcO_out_aaa(x1, x2, x3) = addCcO_out_aaa(x1, x2, x3) U106_aaa(x1, x2) = U106_aaa(x2) U107_aaa(x1, x2, x3) = U107_aaa(x3) U108_aaa(x1, x2, x3, x4) = U108_aaa(x4) U109_aaa(x1, x2) = U109_aaa(x2) U110_aaa(x1, x2, x3) = U110_aaa(x3) U111_aaa(x1, x2, x3, x4) = U111_aaa(x4) U112_aaa(x1, x2, x3, x4) = U112_aaa(x4) U151_aaa(x1, x2, x3, x4) = U151_aaa(x4) one(x1) = one(x1) b = b SUCCT_IN_GG(x1, x2) = SUCCT_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (111) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (112) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCCT_IN_GG(one(X1), zero(X2)) -> SUCCT_IN_GG(X1, X2) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (113) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: SUCCT_IN_GG(one(X1), zero(X2)) -> SUCCT_IN_GG(X1, X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (115) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *SUCCT_IN_GG(one(X1), zero(X2)) -> SUCCT_IN_GG(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (116) YES ---------------------------------------- (117) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDZQ_IN_GGG(zero(X1), one(X2), one(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(zero(X1), zero(X2), zero(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(one(X1), zero(X2), one(X3)) -> ADDYR_IN_GGG(X1, X2, X3) ADDYR_IN_GGG(X1, X2, X3) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(one(X1), one(X2), zero(X3)) -> ADDCS_IN_GGG(X1, X2, X3) ADDCS_IN_GGG(X1, X2, X3) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(zero(X1), zero(X2), one(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(zero(X1), one(X2), zero(X3)) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(one(X1), zero(X2), zero(X3)) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(one(X1), one(X2), one(X3)) -> ADDCS_IN_GGG(X1, X2, X3) The TRS R consists of the following rules: timescF_in_aaa(one(b), X1, X1) -> timescF_out_aaa(one(b), X1, X1) timescF_in_aaa(zero(X1), X2, zero(X3)) -> U88_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X3)) timescF_in_aaa(one(X1), X2, X3) -> U89_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U89_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U90_aaa(X1, X2, X3, addcG_in_aaa(X2, X4, X3)) addcG_in_aaa(b, X1, zero(X1)) -> U149_aaa(X1, binaryZcP_in_a(X1)) binaryZcP_in_a(X1) -> U119_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(zero(X1)) -> U91_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(one(X1)) -> U92_a(X1, binarycI_in_a(X1)) binarycI_in_a(b) -> binarycI_out_a(b) binarycI_in_a(zero(X1)) -> U93_a(X1, binaryZcH_in_a(X1)) U93_a(X1, binaryZcH_out_a(X1)) -> binarycI_out_a(zero(X1)) binarycI_in_a(one(X1)) -> U94_a(X1, binarycI_in_a(X1)) U94_a(X1, binarycI_out_a(X1)) -> binarycI_out_a(one(X1)) U92_a(X1, binarycI_out_a(X1)) -> binaryZcH_out_a(one(X1)) U91_a(X1, binaryZcH_out_a(X1)) -> binaryZcH_out_a(zero(X1)) U119_a(X1, binaryZcH_out_a(X1)) -> binaryZcP_out_a(X1) U149_aaa(X1, binaryZcP_out_a(X1)) -> addcG_out_aaa(b, X1, zero(X1)) addcG_in_aaa(zero(X1), X2, zero(X3)) -> U150_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(X1), zero(X2), zero(X3)) -> U95_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(one(X1)), one(b), one(one(X1))) -> U96_aaa(X1, binarycI_in_a(X1)) U96_aaa(X1, binarycI_out_a(X1)) -> addzcJ_out_aaa(zero(one(X1)), one(b), one(one(X1))) addzcJ_in_aaa(zero(zero(X1)), one(b), one(zero(X1))) -> U97_aaa(X1, binaryZcH_in_a(X1)) U97_aaa(X1, binaryZcH_out_a(X1)) -> addzcJ_out_aaa(zero(zero(X1)), one(b), one(zero(X1))) addzcJ_in_aaa(zero(X1), one(X2), one(X3)) -> U98_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), zero(X2), one(X3)) -> U99_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) addycK_in_aaa(b, one(X1), one(X1)) -> U116_aaa(X1, binarycI_in_a(X1)) U116_aaa(X1, binarycI_out_a(X1)) -> addycK_out_aaa(b, one(X1), one(X1)) addycK_in_aaa(b, zero(X1), zero(X1)) -> U117_aaa(X1, binaryZcH_in_a(X1)) U117_aaa(X1, binaryZcH_out_a(X1)) -> addycK_out_aaa(b, zero(X1), zero(X1)) addycK_in_aaa(X1, X2, X3) -> U118_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), one(X2), zero(X3)) -> U100_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) addccL_in_aaa(b, b, one(b)) -> addccL_out_aaa(b, b, one(b)) addccL_in_aaa(X1, b, X2) -> U113_aaa(X1, X2, succZcN_in_aa(X1, X2)) succZcN_in_aa(zero(X1), one(X1)) -> U103_aa(X1, binaryZcH_in_a(X1)) U103_aa(X1, binaryZcH_out_a(X1)) -> succZcN_out_aa(zero(X1), one(X1)) succZcN_in_aa(one(X1), zero(X2)) -> U104_aa(X1, X2, succcM_in_aa(X1, X2)) succcM_in_aa(b, one(b)) -> succcM_out_aa(b, one(b)) succcM_in_aa(zero(X1), one(X1)) -> U101_aa(X1, binaryZcH_in_a(X1)) U101_aa(X1, binaryZcH_out_a(X1)) -> succcM_out_aa(zero(X1), one(X1)) succcM_in_aa(one(X1), zero(X2)) -> U102_aa(X1, X2, succcM_in_aa(X1, X2)) U102_aa(X1, X2, succcM_out_aa(X1, X2)) -> succcM_out_aa(one(X1), zero(X2)) U104_aa(X1, X2, succcM_out_aa(X1, X2)) -> succZcN_out_aa(one(X1), zero(X2)) U113_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(X1, b, X2) addccL_in_aaa(b, X1, X2) -> U114_aaa(X1, X2, succZcN_in_aa(X1, X2)) U114_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(b, X1, X2) addccL_in_aaa(X1, X2, X3) -> U115_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(zero(X1), zero(X2), one(X3)) -> U105_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) U105_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), zero(X2), one(X3)) addCcO_in_aaa(zero(zero(X1)), one(b), zero(one(X1))) -> U106_aaa(X1, binaryZcH_in_a(X1)) U106_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(zero(zero(X1)), one(b), zero(one(X1))) addCcO_in_aaa(zero(one(X1)), one(b), zero(zero(X2))) -> U107_aaa(X1, X2, succcM_in_aa(X1, X2)) U107_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(zero(one(X1)), one(b), zero(zero(X2))) addCcO_in_aaa(zero(X1), one(X2), zero(X3)) -> U108_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(b), zero(zero(X1)), zero(one(X1))) -> U109_aaa(X1, binaryZcH_in_a(X1)) U109_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(one(b), zero(zero(X1)), zero(one(X1))) addCcO_in_aaa(one(b), zero(one(X1)), zero(zero(X2))) -> U110_aaa(X1, X2, succcM_in_aa(X1, X2)) U110_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(one(b), zero(one(X1)), zero(zero(X2))) addCcO_in_aaa(one(X1), zero(X2), zero(X3)) -> U111_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(X1), one(X2), one(X3)) -> U112_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) U112_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), one(X2), one(X3)) U111_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), zero(X2), zero(X3)) U108_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), one(X2), zero(X3)) U115_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addccL_out_aaa(X1, X2, X3) U100_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), one(X2), zero(X3)) U118_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addycK_out_aaa(X1, X2, X3) U99_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), zero(X2), one(X3)) U98_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), one(X2), one(X3)) U95_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), zero(X2), zero(X3)) U150_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addcG_out_aaa(zero(X1), X2, zero(X3)) addcG_in_aaa(one(X1), X2, one(X3)) -> U151_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) U151_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addcG_out_aaa(one(X1), X2, one(X3)) U90_aaa(X1, X2, X3, addcG_out_aaa(X2, X4, X3)) -> timescF_out_aaa(one(X1), X2, X3) U88_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X3)) -> timescF_out_aaa(zero(X1), X2, zero(X3)) The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) timescF_in_aaa(x1, x2, x3) = timescF_in_aaa timescF_out_aaa(x1, x2, x3) = timescF_out_aaa(x1) U88_aaa(x1, x2, x3, x4) = U88_aaa(x4) U89_aaa(x1, x2, x3, x4) = U89_aaa(x4) U90_aaa(x1, x2, x3, x4) = U90_aaa(x1, x4) addcG_in_aaa(x1, x2, x3) = addcG_in_aaa U149_aaa(x1, x2) = U149_aaa(x2) binaryZcP_in_a(x1) = binaryZcP_in_a U119_a(x1, x2) = U119_a(x2) binaryZcH_in_a(x1) = binaryZcH_in_a U91_a(x1, x2) = U91_a(x2) U92_a(x1, x2) = U92_a(x2) binarycI_in_a(x1) = binarycI_in_a binarycI_out_a(x1) = binarycI_out_a(x1) U93_a(x1, x2) = U93_a(x2) binaryZcH_out_a(x1) = binaryZcH_out_a(x1) U94_a(x1, x2) = U94_a(x2) binaryZcP_out_a(x1) = binaryZcP_out_a(x1) addcG_out_aaa(x1, x2, x3) = addcG_out_aaa(x1, x2, x3) U150_aaa(x1, x2, x3, x4) = U150_aaa(x4) addzcJ_in_aaa(x1, x2, x3) = addzcJ_in_aaa U95_aaa(x1, x2, x3, x4) = U95_aaa(x4) U96_aaa(x1, x2) = U96_aaa(x2) addzcJ_out_aaa(x1, x2, x3) = addzcJ_out_aaa(x1, x2, x3) U97_aaa(x1, x2) = U97_aaa(x2) U98_aaa(x1, x2, x3, x4) = U98_aaa(x4) U99_aaa(x1, x2, x3, x4) = U99_aaa(x4) addycK_in_aaa(x1, x2, x3) = addycK_in_aaa U116_aaa(x1, x2) = U116_aaa(x2) addycK_out_aaa(x1, x2, x3) = addycK_out_aaa(x1, x2, x3) U117_aaa(x1, x2) = U117_aaa(x2) U118_aaa(x1, x2, x3, x4) = U118_aaa(x4) U100_aaa(x1, x2, x3, x4) = U100_aaa(x4) addccL_in_aaa(x1, x2, x3) = addccL_in_aaa addccL_out_aaa(x1, x2, x3) = addccL_out_aaa(x1, x2, x3) U113_aaa(x1, x2, x3) = U113_aaa(x3) succZcN_in_aa(x1, x2) = succZcN_in_aa U103_aa(x1, x2) = U103_aa(x2) succZcN_out_aa(x1, x2) = succZcN_out_aa(x1, x2) U104_aa(x1, x2, x3) = U104_aa(x3) succcM_in_aa(x1, x2) = succcM_in_aa succcM_out_aa(x1, x2) = succcM_out_aa(x1, x2) U101_aa(x1, x2) = U101_aa(x2) U102_aa(x1, x2, x3) = U102_aa(x3) U114_aaa(x1, x2, x3) = U114_aaa(x3) U115_aaa(x1, x2, x3, x4) = U115_aaa(x4) addCcO_in_aaa(x1, x2, x3) = addCcO_in_aaa U105_aaa(x1, x2, x3, x4) = U105_aaa(x4) addCcO_out_aaa(x1, x2, x3) = addCcO_out_aaa(x1, x2, x3) U106_aaa(x1, x2) = U106_aaa(x2) U107_aaa(x1, x2, x3) = U107_aaa(x3) U108_aaa(x1, x2, x3, x4) = U108_aaa(x4) U109_aaa(x1, x2) = U109_aaa(x2) U110_aaa(x1, x2, x3) = U110_aaa(x3) U111_aaa(x1, x2, x3, x4) = U111_aaa(x4) U112_aaa(x1, x2, x3, x4) = U112_aaa(x4) U151_aaa(x1, x2, x3, x4) = U151_aaa(x4) one(x1) = one(x1) b = b ADDZQ_IN_GGG(x1, x2, x3) = ADDZQ_IN_GGG(x1, x2, x3) ADDYR_IN_GGG(x1, x2, x3) = ADDYR_IN_GGG(x1, x2, x3) ADDCS_IN_GGG(x1, x2, x3) = ADDCS_IN_GGG(x1, x2, x3) ADDCV_IN_GGG(x1, x2, x3) = ADDCV_IN_GGG(x1, x2, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (118) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (119) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDZQ_IN_GGG(zero(X1), one(X2), one(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(zero(X1), zero(X2), zero(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(one(X1), zero(X2), one(X3)) -> ADDYR_IN_GGG(X1, X2, X3) ADDYR_IN_GGG(X1, X2, X3) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(one(X1), one(X2), zero(X3)) -> ADDCS_IN_GGG(X1, X2, X3) ADDCS_IN_GGG(X1, X2, X3) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(zero(X1), zero(X2), one(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(zero(X1), one(X2), zero(X3)) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(one(X1), zero(X2), zero(X3)) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(one(X1), one(X2), one(X3)) -> ADDCS_IN_GGG(X1, X2, X3) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (120) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (121) Obligation: Q DP problem: The TRS P consists of the following rules: ADDZQ_IN_GGG(zero(X1), one(X2), one(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(zero(X1), zero(X2), zero(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(one(X1), zero(X2), one(X3)) -> ADDYR_IN_GGG(X1, X2, X3) ADDYR_IN_GGG(X1, X2, X3) -> ADDZQ_IN_GGG(X1, X2, X3) ADDZQ_IN_GGG(one(X1), one(X2), zero(X3)) -> ADDCS_IN_GGG(X1, X2, X3) ADDCS_IN_GGG(X1, X2, X3) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(zero(X1), zero(X2), one(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(zero(X1), one(X2), zero(X3)) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(one(X1), zero(X2), zero(X3)) -> ADDCV_IN_GGG(X1, X2, X3) ADDCV_IN_GGG(one(X1), one(X2), one(X3)) -> ADDCS_IN_GGG(X1, X2, X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (122) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *ADDYR_IN_GGG(X1, X2, X3) -> ADDZQ_IN_GGG(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3 *ADDCV_IN_GGG(zero(X1), zero(X2), one(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 > 3 *ADDZQ_IN_GGG(one(X1), zero(X2), one(X3)) -> ADDYR_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 > 3 *ADDZQ_IN_GGG(one(X1), one(X2), zero(X3)) -> ADDCS_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 > 3 *ADDCS_IN_GGG(X1, X2, X3) -> ADDCV_IN_GGG(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3 *ADDCV_IN_GGG(one(X1), one(X2), one(X3)) -> ADDCS_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 > 3 *ADDZQ_IN_GGG(zero(X1), one(X2), one(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 > 3 *ADDZQ_IN_GGG(zero(X1), zero(X2), zero(X3)) -> ADDZQ_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 > 3 *ADDCV_IN_GGG(zero(X1), one(X2), zero(X3)) -> ADDCV_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 > 3 *ADDCV_IN_GGG(one(X1), zero(X2), zero(X3)) -> ADDCV_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 > 3 ---------------------------------------- (123) YES ---------------------------------------- (124) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCCT_IN_AG(one(X1), zero(X2)) -> SUCCT_IN_AG(X1, X2) The TRS R consists of the following rules: timescF_in_aaa(one(b), X1, X1) -> timescF_out_aaa(one(b), X1, X1) timescF_in_aaa(zero(X1), X2, zero(X3)) -> U88_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X3)) timescF_in_aaa(one(X1), X2, X3) -> U89_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U89_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U90_aaa(X1, X2, X3, addcG_in_aaa(X2, X4, X3)) addcG_in_aaa(b, X1, zero(X1)) -> U149_aaa(X1, binaryZcP_in_a(X1)) binaryZcP_in_a(X1) -> U119_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(zero(X1)) -> U91_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(one(X1)) -> U92_a(X1, binarycI_in_a(X1)) binarycI_in_a(b) -> binarycI_out_a(b) binarycI_in_a(zero(X1)) -> U93_a(X1, binaryZcH_in_a(X1)) U93_a(X1, binaryZcH_out_a(X1)) -> binarycI_out_a(zero(X1)) binarycI_in_a(one(X1)) -> U94_a(X1, binarycI_in_a(X1)) U94_a(X1, binarycI_out_a(X1)) -> binarycI_out_a(one(X1)) U92_a(X1, binarycI_out_a(X1)) -> binaryZcH_out_a(one(X1)) U91_a(X1, binaryZcH_out_a(X1)) -> binaryZcH_out_a(zero(X1)) U119_a(X1, binaryZcH_out_a(X1)) -> binaryZcP_out_a(X1) U149_aaa(X1, binaryZcP_out_a(X1)) -> addcG_out_aaa(b, X1, zero(X1)) addcG_in_aaa(zero(X1), X2, zero(X3)) -> U150_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(X1), zero(X2), zero(X3)) -> U95_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(one(X1)), one(b), one(one(X1))) -> U96_aaa(X1, binarycI_in_a(X1)) U96_aaa(X1, binarycI_out_a(X1)) -> addzcJ_out_aaa(zero(one(X1)), one(b), one(one(X1))) addzcJ_in_aaa(zero(zero(X1)), one(b), one(zero(X1))) -> U97_aaa(X1, binaryZcH_in_a(X1)) U97_aaa(X1, binaryZcH_out_a(X1)) -> addzcJ_out_aaa(zero(zero(X1)), one(b), one(zero(X1))) addzcJ_in_aaa(zero(X1), one(X2), one(X3)) -> U98_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), zero(X2), one(X3)) -> U99_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) addycK_in_aaa(b, one(X1), one(X1)) -> U116_aaa(X1, binarycI_in_a(X1)) U116_aaa(X1, binarycI_out_a(X1)) -> addycK_out_aaa(b, one(X1), one(X1)) addycK_in_aaa(b, zero(X1), zero(X1)) -> U117_aaa(X1, binaryZcH_in_a(X1)) U117_aaa(X1, binaryZcH_out_a(X1)) -> addycK_out_aaa(b, zero(X1), zero(X1)) addycK_in_aaa(X1, X2, X3) -> U118_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), one(X2), zero(X3)) -> U100_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) addccL_in_aaa(b, b, one(b)) -> addccL_out_aaa(b, b, one(b)) addccL_in_aaa(X1, b, X2) -> U113_aaa(X1, X2, succZcN_in_aa(X1, X2)) succZcN_in_aa(zero(X1), one(X1)) -> U103_aa(X1, binaryZcH_in_a(X1)) U103_aa(X1, binaryZcH_out_a(X1)) -> succZcN_out_aa(zero(X1), one(X1)) succZcN_in_aa(one(X1), zero(X2)) -> U104_aa(X1, X2, succcM_in_aa(X1, X2)) succcM_in_aa(b, one(b)) -> succcM_out_aa(b, one(b)) succcM_in_aa(zero(X1), one(X1)) -> U101_aa(X1, binaryZcH_in_a(X1)) U101_aa(X1, binaryZcH_out_a(X1)) -> succcM_out_aa(zero(X1), one(X1)) succcM_in_aa(one(X1), zero(X2)) -> U102_aa(X1, X2, succcM_in_aa(X1, X2)) U102_aa(X1, X2, succcM_out_aa(X1, X2)) -> succcM_out_aa(one(X1), zero(X2)) U104_aa(X1, X2, succcM_out_aa(X1, X2)) -> succZcN_out_aa(one(X1), zero(X2)) U113_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(X1, b, X2) addccL_in_aaa(b, X1, X2) -> U114_aaa(X1, X2, succZcN_in_aa(X1, X2)) U114_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(b, X1, X2) addccL_in_aaa(X1, X2, X3) -> U115_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(zero(X1), zero(X2), one(X3)) -> U105_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) U105_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), zero(X2), one(X3)) addCcO_in_aaa(zero(zero(X1)), one(b), zero(one(X1))) -> U106_aaa(X1, binaryZcH_in_a(X1)) U106_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(zero(zero(X1)), one(b), zero(one(X1))) addCcO_in_aaa(zero(one(X1)), one(b), zero(zero(X2))) -> U107_aaa(X1, X2, succcM_in_aa(X1, X2)) U107_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(zero(one(X1)), one(b), zero(zero(X2))) addCcO_in_aaa(zero(X1), one(X2), zero(X3)) -> U108_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(b), zero(zero(X1)), zero(one(X1))) -> U109_aaa(X1, binaryZcH_in_a(X1)) U109_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(one(b), zero(zero(X1)), zero(one(X1))) addCcO_in_aaa(one(b), zero(one(X1)), zero(zero(X2))) -> U110_aaa(X1, X2, succcM_in_aa(X1, X2)) U110_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(one(b), zero(one(X1)), zero(zero(X2))) addCcO_in_aaa(one(X1), zero(X2), zero(X3)) -> U111_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(X1), one(X2), one(X3)) -> U112_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) U112_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), one(X2), one(X3)) U111_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), zero(X2), zero(X3)) U108_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), one(X2), zero(X3)) U115_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addccL_out_aaa(X1, X2, X3) U100_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), one(X2), zero(X3)) U118_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addycK_out_aaa(X1, X2, X3) U99_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), zero(X2), one(X3)) U98_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), one(X2), one(X3)) U95_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), zero(X2), zero(X3)) U150_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addcG_out_aaa(zero(X1), X2, zero(X3)) addcG_in_aaa(one(X1), X2, one(X3)) -> U151_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) U151_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addcG_out_aaa(one(X1), X2, one(X3)) U90_aaa(X1, X2, X3, addcG_out_aaa(X2, X4, X3)) -> timescF_out_aaa(one(X1), X2, X3) U88_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X3)) -> timescF_out_aaa(zero(X1), X2, zero(X3)) The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) timescF_in_aaa(x1, x2, x3) = timescF_in_aaa timescF_out_aaa(x1, x2, x3) = timescF_out_aaa(x1) U88_aaa(x1, x2, x3, x4) = U88_aaa(x4) U89_aaa(x1, x2, x3, x4) = U89_aaa(x4) U90_aaa(x1, x2, x3, x4) = U90_aaa(x1, x4) addcG_in_aaa(x1, x2, x3) = addcG_in_aaa U149_aaa(x1, x2) = U149_aaa(x2) binaryZcP_in_a(x1) = binaryZcP_in_a U119_a(x1, x2) = U119_a(x2) binaryZcH_in_a(x1) = binaryZcH_in_a U91_a(x1, x2) = U91_a(x2) U92_a(x1, x2) = U92_a(x2) binarycI_in_a(x1) = binarycI_in_a binarycI_out_a(x1) = binarycI_out_a(x1) U93_a(x1, x2) = U93_a(x2) binaryZcH_out_a(x1) = binaryZcH_out_a(x1) U94_a(x1, x2) = U94_a(x2) binaryZcP_out_a(x1) = binaryZcP_out_a(x1) addcG_out_aaa(x1, x2, x3) = addcG_out_aaa(x1, x2, x3) U150_aaa(x1, x2, x3, x4) = U150_aaa(x4) addzcJ_in_aaa(x1, x2, x3) = addzcJ_in_aaa U95_aaa(x1, x2, x3, x4) = U95_aaa(x4) U96_aaa(x1, x2) = U96_aaa(x2) addzcJ_out_aaa(x1, x2, x3) = addzcJ_out_aaa(x1, x2, x3) U97_aaa(x1, x2) = U97_aaa(x2) U98_aaa(x1, x2, x3, x4) = U98_aaa(x4) U99_aaa(x1, x2, x3, x4) = U99_aaa(x4) addycK_in_aaa(x1, x2, x3) = addycK_in_aaa U116_aaa(x1, x2) = U116_aaa(x2) addycK_out_aaa(x1, x2, x3) = addycK_out_aaa(x1, x2, x3) U117_aaa(x1, x2) = U117_aaa(x2) U118_aaa(x1, x2, x3, x4) = U118_aaa(x4) U100_aaa(x1, x2, x3, x4) = U100_aaa(x4) addccL_in_aaa(x1, x2, x3) = addccL_in_aaa addccL_out_aaa(x1, x2, x3) = addccL_out_aaa(x1, x2, x3) U113_aaa(x1, x2, x3) = U113_aaa(x3) succZcN_in_aa(x1, x2) = succZcN_in_aa U103_aa(x1, x2) = U103_aa(x2) succZcN_out_aa(x1, x2) = succZcN_out_aa(x1, x2) U104_aa(x1, x2, x3) = U104_aa(x3) succcM_in_aa(x1, x2) = succcM_in_aa succcM_out_aa(x1, x2) = succcM_out_aa(x1, x2) U101_aa(x1, x2) = U101_aa(x2) U102_aa(x1, x2, x3) = U102_aa(x3) U114_aaa(x1, x2, x3) = U114_aaa(x3) U115_aaa(x1, x2, x3, x4) = U115_aaa(x4) addCcO_in_aaa(x1, x2, x3) = addCcO_in_aaa U105_aaa(x1, x2, x3, x4) = U105_aaa(x4) addCcO_out_aaa(x1, x2, x3) = addCcO_out_aaa(x1, x2, x3) U106_aaa(x1, x2) = U106_aaa(x2) U107_aaa(x1, x2, x3) = U107_aaa(x3) U108_aaa(x1, x2, x3, x4) = U108_aaa(x4) U109_aaa(x1, x2) = U109_aaa(x2) U110_aaa(x1, x2, x3) = U110_aaa(x3) U111_aaa(x1, x2, x3, x4) = U111_aaa(x4) U112_aaa(x1, x2, x3, x4) = U112_aaa(x4) U151_aaa(x1, x2, x3, x4) = U151_aaa(x4) one(x1) = one(x1) b = b SUCCT_IN_AG(x1, x2) = SUCCT_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (125) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (126) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCCT_IN_AG(one(X1), zero(X2)) -> SUCCT_IN_AG(X1, X2) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) SUCCT_IN_AG(x1, x2) = SUCCT_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (127) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (128) Obligation: Q DP problem: The TRS P consists of the following rules: SUCCT_IN_AG(zero(X2)) -> SUCCT_IN_AG(X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (129) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *SUCCT_IN_AG(zero(X2)) -> SUCCT_IN_AG(X2) The graph contains the following edges 1 > 1 ---------------------------------------- (130) YES ---------------------------------------- (131) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDZQ_IN_AAG(zero(X1), one(X2), one(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(zero(X1), zero(X2), zero(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(one(X1), zero(X2), one(X3)) -> ADDYR_IN_AAG(X1, X2, X3) ADDYR_IN_AAG(X1, X2, X3) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(one(X1), one(X2), zero(X3)) -> ADDCS_IN_AAG(X1, X2, X3) ADDCS_IN_AAG(X1, X2, X3) -> ADDCV_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(zero(X1), zero(X2), one(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(zero(X1), one(X2), zero(X3)) -> ADDCV_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(one(X1), zero(X2), zero(X3)) -> ADDCV_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(one(X1), one(X2), one(X3)) -> ADDCS_IN_AAG(X1, X2, X3) The TRS R consists of the following rules: timescF_in_aaa(one(b), X1, X1) -> timescF_out_aaa(one(b), X1, X1) timescF_in_aaa(zero(X1), X2, zero(X3)) -> U88_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X3)) timescF_in_aaa(one(X1), X2, X3) -> U89_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U89_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U90_aaa(X1, X2, X3, addcG_in_aaa(X2, X4, X3)) addcG_in_aaa(b, X1, zero(X1)) -> U149_aaa(X1, binaryZcP_in_a(X1)) binaryZcP_in_a(X1) -> U119_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(zero(X1)) -> U91_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(one(X1)) -> U92_a(X1, binarycI_in_a(X1)) binarycI_in_a(b) -> binarycI_out_a(b) binarycI_in_a(zero(X1)) -> U93_a(X1, binaryZcH_in_a(X1)) U93_a(X1, binaryZcH_out_a(X1)) -> binarycI_out_a(zero(X1)) binarycI_in_a(one(X1)) -> U94_a(X1, binarycI_in_a(X1)) U94_a(X1, binarycI_out_a(X1)) -> binarycI_out_a(one(X1)) U92_a(X1, binarycI_out_a(X1)) -> binaryZcH_out_a(one(X1)) U91_a(X1, binaryZcH_out_a(X1)) -> binaryZcH_out_a(zero(X1)) U119_a(X1, binaryZcH_out_a(X1)) -> binaryZcP_out_a(X1) U149_aaa(X1, binaryZcP_out_a(X1)) -> addcG_out_aaa(b, X1, zero(X1)) addcG_in_aaa(zero(X1), X2, zero(X3)) -> U150_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(X1), zero(X2), zero(X3)) -> U95_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(one(X1)), one(b), one(one(X1))) -> U96_aaa(X1, binarycI_in_a(X1)) U96_aaa(X1, binarycI_out_a(X1)) -> addzcJ_out_aaa(zero(one(X1)), one(b), one(one(X1))) addzcJ_in_aaa(zero(zero(X1)), one(b), one(zero(X1))) -> U97_aaa(X1, binaryZcH_in_a(X1)) U97_aaa(X1, binaryZcH_out_a(X1)) -> addzcJ_out_aaa(zero(zero(X1)), one(b), one(zero(X1))) addzcJ_in_aaa(zero(X1), one(X2), one(X3)) -> U98_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), zero(X2), one(X3)) -> U99_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) addycK_in_aaa(b, one(X1), one(X1)) -> U116_aaa(X1, binarycI_in_a(X1)) U116_aaa(X1, binarycI_out_a(X1)) -> addycK_out_aaa(b, one(X1), one(X1)) addycK_in_aaa(b, zero(X1), zero(X1)) -> U117_aaa(X1, binaryZcH_in_a(X1)) U117_aaa(X1, binaryZcH_out_a(X1)) -> addycK_out_aaa(b, zero(X1), zero(X1)) addycK_in_aaa(X1, X2, X3) -> U118_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), one(X2), zero(X3)) -> U100_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) addccL_in_aaa(b, b, one(b)) -> addccL_out_aaa(b, b, one(b)) addccL_in_aaa(X1, b, X2) -> U113_aaa(X1, X2, succZcN_in_aa(X1, X2)) succZcN_in_aa(zero(X1), one(X1)) -> U103_aa(X1, binaryZcH_in_a(X1)) U103_aa(X1, binaryZcH_out_a(X1)) -> succZcN_out_aa(zero(X1), one(X1)) succZcN_in_aa(one(X1), zero(X2)) -> U104_aa(X1, X2, succcM_in_aa(X1, X2)) succcM_in_aa(b, one(b)) -> succcM_out_aa(b, one(b)) succcM_in_aa(zero(X1), one(X1)) -> U101_aa(X1, binaryZcH_in_a(X1)) U101_aa(X1, binaryZcH_out_a(X1)) -> succcM_out_aa(zero(X1), one(X1)) succcM_in_aa(one(X1), zero(X2)) -> U102_aa(X1, X2, succcM_in_aa(X1, X2)) U102_aa(X1, X2, succcM_out_aa(X1, X2)) -> succcM_out_aa(one(X1), zero(X2)) U104_aa(X1, X2, succcM_out_aa(X1, X2)) -> succZcN_out_aa(one(X1), zero(X2)) U113_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(X1, b, X2) addccL_in_aaa(b, X1, X2) -> U114_aaa(X1, X2, succZcN_in_aa(X1, X2)) U114_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(b, X1, X2) addccL_in_aaa(X1, X2, X3) -> U115_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(zero(X1), zero(X2), one(X3)) -> U105_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) U105_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), zero(X2), one(X3)) addCcO_in_aaa(zero(zero(X1)), one(b), zero(one(X1))) -> U106_aaa(X1, binaryZcH_in_a(X1)) U106_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(zero(zero(X1)), one(b), zero(one(X1))) addCcO_in_aaa(zero(one(X1)), one(b), zero(zero(X2))) -> U107_aaa(X1, X2, succcM_in_aa(X1, X2)) U107_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(zero(one(X1)), one(b), zero(zero(X2))) addCcO_in_aaa(zero(X1), one(X2), zero(X3)) -> U108_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(b), zero(zero(X1)), zero(one(X1))) -> U109_aaa(X1, binaryZcH_in_a(X1)) U109_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(one(b), zero(zero(X1)), zero(one(X1))) addCcO_in_aaa(one(b), zero(one(X1)), zero(zero(X2))) -> U110_aaa(X1, X2, succcM_in_aa(X1, X2)) U110_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(one(b), zero(one(X1)), zero(zero(X2))) addCcO_in_aaa(one(X1), zero(X2), zero(X3)) -> U111_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(X1), one(X2), one(X3)) -> U112_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) U112_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), one(X2), one(X3)) U111_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), zero(X2), zero(X3)) U108_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), one(X2), zero(X3)) U115_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addccL_out_aaa(X1, X2, X3) U100_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), one(X2), zero(X3)) U118_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addycK_out_aaa(X1, X2, X3) U99_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), zero(X2), one(X3)) U98_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), one(X2), one(X3)) U95_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), zero(X2), zero(X3)) U150_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addcG_out_aaa(zero(X1), X2, zero(X3)) addcG_in_aaa(one(X1), X2, one(X3)) -> U151_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) U151_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addcG_out_aaa(one(X1), X2, one(X3)) U90_aaa(X1, X2, X3, addcG_out_aaa(X2, X4, X3)) -> timescF_out_aaa(one(X1), X2, X3) U88_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X3)) -> timescF_out_aaa(zero(X1), X2, zero(X3)) The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) timescF_in_aaa(x1, x2, x3) = timescF_in_aaa timescF_out_aaa(x1, x2, x3) = timescF_out_aaa(x1) U88_aaa(x1, x2, x3, x4) = U88_aaa(x4) U89_aaa(x1, x2, x3, x4) = U89_aaa(x4) U90_aaa(x1, x2, x3, x4) = U90_aaa(x1, x4) addcG_in_aaa(x1, x2, x3) = addcG_in_aaa U149_aaa(x1, x2) = U149_aaa(x2) binaryZcP_in_a(x1) = binaryZcP_in_a U119_a(x1, x2) = U119_a(x2) binaryZcH_in_a(x1) = binaryZcH_in_a U91_a(x1, x2) = U91_a(x2) U92_a(x1, x2) = U92_a(x2) binarycI_in_a(x1) = binarycI_in_a binarycI_out_a(x1) = binarycI_out_a(x1) U93_a(x1, x2) = U93_a(x2) binaryZcH_out_a(x1) = binaryZcH_out_a(x1) U94_a(x1, x2) = U94_a(x2) binaryZcP_out_a(x1) = binaryZcP_out_a(x1) addcG_out_aaa(x1, x2, x3) = addcG_out_aaa(x1, x2, x3) U150_aaa(x1, x2, x3, x4) = U150_aaa(x4) addzcJ_in_aaa(x1, x2, x3) = addzcJ_in_aaa U95_aaa(x1, x2, x3, x4) = U95_aaa(x4) U96_aaa(x1, x2) = U96_aaa(x2) addzcJ_out_aaa(x1, x2, x3) = addzcJ_out_aaa(x1, x2, x3) U97_aaa(x1, x2) = U97_aaa(x2) U98_aaa(x1, x2, x3, x4) = U98_aaa(x4) U99_aaa(x1, x2, x3, x4) = U99_aaa(x4) addycK_in_aaa(x1, x2, x3) = addycK_in_aaa U116_aaa(x1, x2) = U116_aaa(x2) addycK_out_aaa(x1, x2, x3) = addycK_out_aaa(x1, x2, x3) U117_aaa(x1, x2) = U117_aaa(x2) U118_aaa(x1, x2, x3, x4) = U118_aaa(x4) U100_aaa(x1, x2, x3, x4) = U100_aaa(x4) addccL_in_aaa(x1, x2, x3) = addccL_in_aaa addccL_out_aaa(x1, x2, x3) = addccL_out_aaa(x1, x2, x3) U113_aaa(x1, x2, x3) = U113_aaa(x3) succZcN_in_aa(x1, x2) = succZcN_in_aa U103_aa(x1, x2) = U103_aa(x2) succZcN_out_aa(x1, x2) = succZcN_out_aa(x1, x2) U104_aa(x1, x2, x3) = U104_aa(x3) succcM_in_aa(x1, x2) = succcM_in_aa succcM_out_aa(x1, x2) = succcM_out_aa(x1, x2) U101_aa(x1, x2) = U101_aa(x2) U102_aa(x1, x2, x3) = U102_aa(x3) U114_aaa(x1, x2, x3) = U114_aaa(x3) U115_aaa(x1, x2, x3, x4) = U115_aaa(x4) addCcO_in_aaa(x1, x2, x3) = addCcO_in_aaa U105_aaa(x1, x2, x3, x4) = U105_aaa(x4) addCcO_out_aaa(x1, x2, x3) = addCcO_out_aaa(x1, x2, x3) U106_aaa(x1, x2) = U106_aaa(x2) U107_aaa(x1, x2, x3) = U107_aaa(x3) U108_aaa(x1, x2, x3, x4) = U108_aaa(x4) U109_aaa(x1, x2) = U109_aaa(x2) U110_aaa(x1, x2, x3) = U110_aaa(x3) U111_aaa(x1, x2, x3, x4) = U111_aaa(x4) U112_aaa(x1, x2, x3, x4) = U112_aaa(x4) U151_aaa(x1, x2, x3, x4) = U151_aaa(x4) one(x1) = one(x1) b = b ADDZQ_IN_AAG(x1, x2, x3) = ADDZQ_IN_AAG(x3) ADDYR_IN_AAG(x1, x2, x3) = ADDYR_IN_AAG(x3) ADDCS_IN_AAG(x1, x2, x3) = ADDCS_IN_AAG(x3) ADDCV_IN_AAG(x1, x2, x3) = ADDCV_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (132) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (133) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDZQ_IN_AAG(zero(X1), one(X2), one(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(zero(X1), zero(X2), zero(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(one(X1), zero(X2), one(X3)) -> ADDYR_IN_AAG(X1, X2, X3) ADDYR_IN_AAG(X1, X2, X3) -> ADDZQ_IN_AAG(X1, X2, X3) ADDZQ_IN_AAG(one(X1), one(X2), zero(X3)) -> ADDCS_IN_AAG(X1, X2, X3) ADDCS_IN_AAG(X1, X2, X3) -> ADDCV_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(zero(X1), zero(X2), one(X3)) -> ADDZQ_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(zero(X1), one(X2), zero(X3)) -> ADDCV_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(one(X1), zero(X2), zero(X3)) -> ADDCV_IN_AAG(X1, X2, X3) ADDCV_IN_AAG(one(X1), one(X2), one(X3)) -> ADDCS_IN_AAG(X1, X2, X3) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) ADDZQ_IN_AAG(x1, x2, x3) = ADDZQ_IN_AAG(x3) ADDYR_IN_AAG(x1, x2, x3) = ADDYR_IN_AAG(x3) ADDCS_IN_AAG(x1, x2, x3) = ADDCS_IN_AAG(x3) ADDCV_IN_AAG(x1, x2, x3) = ADDCV_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (134) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (135) Obligation: Q DP problem: The TRS P consists of the following rules: ADDZQ_IN_AAG(one(X3)) -> ADDZQ_IN_AAG(X3) ADDZQ_IN_AAG(zero(X3)) -> ADDZQ_IN_AAG(X3) ADDZQ_IN_AAG(one(X3)) -> ADDYR_IN_AAG(X3) ADDYR_IN_AAG(X3) -> ADDZQ_IN_AAG(X3) ADDZQ_IN_AAG(zero(X3)) -> ADDCS_IN_AAG(X3) ADDCS_IN_AAG(X3) -> ADDCV_IN_AAG(X3) ADDCV_IN_AAG(one(X3)) -> ADDZQ_IN_AAG(X3) ADDCV_IN_AAG(zero(X3)) -> ADDCV_IN_AAG(X3) ADDCV_IN_AAG(one(X3)) -> ADDCS_IN_AAG(X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (136) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *ADDYR_IN_AAG(X3) -> ADDZQ_IN_AAG(X3) The graph contains the following edges 1 >= 1 *ADDCV_IN_AAG(one(X3)) -> ADDZQ_IN_AAG(X3) The graph contains the following edges 1 > 1 *ADDZQ_IN_AAG(one(X3)) -> ADDYR_IN_AAG(X3) The graph contains the following edges 1 > 1 *ADDZQ_IN_AAG(zero(X3)) -> ADDCS_IN_AAG(X3) The graph contains the following edges 1 > 1 *ADDCS_IN_AAG(X3) -> ADDCV_IN_AAG(X3) The graph contains the following edges 1 >= 1 *ADDCV_IN_AAG(one(X3)) -> ADDCS_IN_AAG(X3) The graph contains the following edges 1 > 1 *ADDCV_IN_AAG(zero(X3)) -> ADDCV_IN_AAG(X3) The graph contains the following edges 1 > 1 *ADDZQ_IN_AAG(one(X3)) -> ADDZQ_IN_AAG(X3) The graph contains the following edges 1 > 1 *ADDZQ_IN_AAG(zero(X3)) -> ADDZQ_IN_AAG(X3) The graph contains the following edges 1 > 1 ---------------------------------------- (137) YES ---------------------------------------- (138) Obligation: Pi DP problem: The TRS P consists of the following rules: BINARYZH_IN_A(one(X1)) -> BINARYI_IN_A(X1) BINARYI_IN_A(zero(X1)) -> BINARYZH_IN_A(X1) BINARYZH_IN_A(zero(X1)) -> BINARYZH_IN_A(X1) BINARYI_IN_A(one(X1)) -> BINARYI_IN_A(X1) The TRS R consists of the following rules: timescF_in_aaa(one(b), X1, X1) -> timescF_out_aaa(one(b), X1, X1) timescF_in_aaa(zero(X1), X2, zero(X3)) -> U88_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X3)) timescF_in_aaa(one(X1), X2, X3) -> U89_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U89_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U90_aaa(X1, X2, X3, addcG_in_aaa(X2, X4, X3)) addcG_in_aaa(b, X1, zero(X1)) -> U149_aaa(X1, binaryZcP_in_a(X1)) binaryZcP_in_a(X1) -> U119_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(zero(X1)) -> U91_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(one(X1)) -> U92_a(X1, binarycI_in_a(X1)) binarycI_in_a(b) -> binarycI_out_a(b) binarycI_in_a(zero(X1)) -> U93_a(X1, binaryZcH_in_a(X1)) U93_a(X1, binaryZcH_out_a(X1)) -> binarycI_out_a(zero(X1)) binarycI_in_a(one(X1)) -> U94_a(X1, binarycI_in_a(X1)) U94_a(X1, binarycI_out_a(X1)) -> binarycI_out_a(one(X1)) U92_a(X1, binarycI_out_a(X1)) -> binaryZcH_out_a(one(X1)) U91_a(X1, binaryZcH_out_a(X1)) -> binaryZcH_out_a(zero(X1)) U119_a(X1, binaryZcH_out_a(X1)) -> binaryZcP_out_a(X1) U149_aaa(X1, binaryZcP_out_a(X1)) -> addcG_out_aaa(b, X1, zero(X1)) addcG_in_aaa(zero(X1), X2, zero(X3)) -> U150_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(X1), zero(X2), zero(X3)) -> U95_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(one(X1)), one(b), one(one(X1))) -> U96_aaa(X1, binarycI_in_a(X1)) U96_aaa(X1, binarycI_out_a(X1)) -> addzcJ_out_aaa(zero(one(X1)), one(b), one(one(X1))) addzcJ_in_aaa(zero(zero(X1)), one(b), one(zero(X1))) -> U97_aaa(X1, binaryZcH_in_a(X1)) U97_aaa(X1, binaryZcH_out_a(X1)) -> addzcJ_out_aaa(zero(zero(X1)), one(b), one(zero(X1))) addzcJ_in_aaa(zero(X1), one(X2), one(X3)) -> U98_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), zero(X2), one(X3)) -> U99_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) addycK_in_aaa(b, one(X1), one(X1)) -> U116_aaa(X1, binarycI_in_a(X1)) U116_aaa(X1, binarycI_out_a(X1)) -> addycK_out_aaa(b, one(X1), one(X1)) addycK_in_aaa(b, zero(X1), zero(X1)) -> U117_aaa(X1, binaryZcH_in_a(X1)) U117_aaa(X1, binaryZcH_out_a(X1)) -> addycK_out_aaa(b, zero(X1), zero(X1)) addycK_in_aaa(X1, X2, X3) -> U118_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), one(X2), zero(X3)) -> U100_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) addccL_in_aaa(b, b, one(b)) -> addccL_out_aaa(b, b, one(b)) addccL_in_aaa(X1, b, X2) -> U113_aaa(X1, X2, succZcN_in_aa(X1, X2)) succZcN_in_aa(zero(X1), one(X1)) -> U103_aa(X1, binaryZcH_in_a(X1)) U103_aa(X1, binaryZcH_out_a(X1)) -> succZcN_out_aa(zero(X1), one(X1)) succZcN_in_aa(one(X1), zero(X2)) -> U104_aa(X1, X2, succcM_in_aa(X1, X2)) succcM_in_aa(b, one(b)) -> succcM_out_aa(b, one(b)) succcM_in_aa(zero(X1), one(X1)) -> U101_aa(X1, binaryZcH_in_a(X1)) U101_aa(X1, binaryZcH_out_a(X1)) -> succcM_out_aa(zero(X1), one(X1)) succcM_in_aa(one(X1), zero(X2)) -> U102_aa(X1, X2, succcM_in_aa(X1, X2)) U102_aa(X1, X2, succcM_out_aa(X1, X2)) -> succcM_out_aa(one(X1), zero(X2)) U104_aa(X1, X2, succcM_out_aa(X1, X2)) -> succZcN_out_aa(one(X1), zero(X2)) U113_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(X1, b, X2) addccL_in_aaa(b, X1, X2) -> U114_aaa(X1, X2, succZcN_in_aa(X1, X2)) U114_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(b, X1, X2) addccL_in_aaa(X1, X2, X3) -> U115_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(zero(X1), zero(X2), one(X3)) -> U105_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) U105_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), zero(X2), one(X3)) addCcO_in_aaa(zero(zero(X1)), one(b), zero(one(X1))) -> U106_aaa(X1, binaryZcH_in_a(X1)) U106_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(zero(zero(X1)), one(b), zero(one(X1))) addCcO_in_aaa(zero(one(X1)), one(b), zero(zero(X2))) -> U107_aaa(X1, X2, succcM_in_aa(X1, X2)) U107_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(zero(one(X1)), one(b), zero(zero(X2))) addCcO_in_aaa(zero(X1), one(X2), zero(X3)) -> U108_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(b), zero(zero(X1)), zero(one(X1))) -> U109_aaa(X1, binaryZcH_in_a(X1)) U109_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(one(b), zero(zero(X1)), zero(one(X1))) addCcO_in_aaa(one(b), zero(one(X1)), zero(zero(X2))) -> U110_aaa(X1, X2, succcM_in_aa(X1, X2)) U110_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(one(b), zero(one(X1)), zero(zero(X2))) addCcO_in_aaa(one(X1), zero(X2), zero(X3)) -> U111_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(X1), one(X2), one(X3)) -> U112_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) U112_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), one(X2), one(X3)) U111_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), zero(X2), zero(X3)) U108_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), one(X2), zero(X3)) U115_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addccL_out_aaa(X1, X2, X3) U100_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), one(X2), zero(X3)) U118_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addycK_out_aaa(X1, X2, X3) U99_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), zero(X2), one(X3)) U98_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), one(X2), one(X3)) U95_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), zero(X2), zero(X3)) U150_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addcG_out_aaa(zero(X1), X2, zero(X3)) addcG_in_aaa(one(X1), X2, one(X3)) -> U151_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) U151_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addcG_out_aaa(one(X1), X2, one(X3)) U90_aaa(X1, X2, X3, addcG_out_aaa(X2, X4, X3)) -> timescF_out_aaa(one(X1), X2, X3) U88_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X3)) -> timescF_out_aaa(zero(X1), X2, zero(X3)) The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) timescF_in_aaa(x1, x2, x3) = timescF_in_aaa timescF_out_aaa(x1, x2, x3) = timescF_out_aaa(x1) U88_aaa(x1, x2, x3, x4) = U88_aaa(x4) U89_aaa(x1, x2, x3, x4) = U89_aaa(x4) U90_aaa(x1, x2, x3, x4) = U90_aaa(x1, x4) addcG_in_aaa(x1, x2, x3) = addcG_in_aaa U149_aaa(x1, x2) = U149_aaa(x2) binaryZcP_in_a(x1) = binaryZcP_in_a U119_a(x1, x2) = U119_a(x2) binaryZcH_in_a(x1) = binaryZcH_in_a U91_a(x1, x2) = U91_a(x2) U92_a(x1, x2) = U92_a(x2) binarycI_in_a(x1) = binarycI_in_a binarycI_out_a(x1) = binarycI_out_a(x1) U93_a(x1, x2) = U93_a(x2) binaryZcH_out_a(x1) = binaryZcH_out_a(x1) U94_a(x1, x2) = U94_a(x2) binaryZcP_out_a(x1) = binaryZcP_out_a(x1) addcG_out_aaa(x1, x2, x3) = addcG_out_aaa(x1, x2, x3) U150_aaa(x1, x2, x3, x4) = U150_aaa(x4) addzcJ_in_aaa(x1, x2, x3) = addzcJ_in_aaa U95_aaa(x1, x2, x3, x4) = U95_aaa(x4) U96_aaa(x1, x2) = U96_aaa(x2) addzcJ_out_aaa(x1, x2, x3) = addzcJ_out_aaa(x1, x2, x3) U97_aaa(x1, x2) = U97_aaa(x2) U98_aaa(x1, x2, x3, x4) = U98_aaa(x4) U99_aaa(x1, x2, x3, x4) = U99_aaa(x4) addycK_in_aaa(x1, x2, x3) = addycK_in_aaa U116_aaa(x1, x2) = U116_aaa(x2) addycK_out_aaa(x1, x2, x3) = addycK_out_aaa(x1, x2, x3) U117_aaa(x1, x2) = U117_aaa(x2) U118_aaa(x1, x2, x3, x4) = U118_aaa(x4) U100_aaa(x1, x2, x3, x4) = U100_aaa(x4) addccL_in_aaa(x1, x2, x3) = addccL_in_aaa addccL_out_aaa(x1, x2, x3) = addccL_out_aaa(x1, x2, x3) U113_aaa(x1, x2, x3) = U113_aaa(x3) succZcN_in_aa(x1, x2) = succZcN_in_aa U103_aa(x1, x2) = U103_aa(x2) succZcN_out_aa(x1, x2) = succZcN_out_aa(x1, x2) U104_aa(x1, x2, x3) = U104_aa(x3) succcM_in_aa(x1, x2) = succcM_in_aa succcM_out_aa(x1, x2) = succcM_out_aa(x1, x2) U101_aa(x1, x2) = U101_aa(x2) U102_aa(x1, x2, x3) = U102_aa(x3) U114_aaa(x1, x2, x3) = U114_aaa(x3) U115_aaa(x1, x2, x3, x4) = U115_aaa(x4) addCcO_in_aaa(x1, x2, x3) = addCcO_in_aaa U105_aaa(x1, x2, x3, x4) = U105_aaa(x4) addCcO_out_aaa(x1, x2, x3) = addCcO_out_aaa(x1, x2, x3) U106_aaa(x1, x2) = U106_aaa(x2) U107_aaa(x1, x2, x3) = U107_aaa(x3) U108_aaa(x1, x2, x3, x4) = U108_aaa(x4) U109_aaa(x1, x2) = U109_aaa(x2) U110_aaa(x1, x2, x3) = U110_aaa(x3) U111_aaa(x1, x2, x3, x4) = U111_aaa(x4) U112_aaa(x1, x2, x3, x4) = U112_aaa(x4) U151_aaa(x1, x2, x3, x4) = U151_aaa(x4) one(x1) = one(x1) b = b BINARYZH_IN_A(x1) = BINARYZH_IN_A BINARYI_IN_A(x1) = BINARYI_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (139) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (140) Obligation: Pi DP problem: The TRS P consists of the following rules: BINARYZH_IN_A(one(X1)) -> BINARYI_IN_A(X1) BINARYI_IN_A(zero(X1)) -> BINARYZH_IN_A(X1) BINARYZH_IN_A(zero(X1)) -> BINARYZH_IN_A(X1) BINARYI_IN_A(one(X1)) -> BINARYI_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) BINARYZH_IN_A(x1) = BINARYZH_IN_A BINARYI_IN_A(x1) = BINARYI_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (141) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (142) Obligation: Q DP problem: The TRS P consists of the following rules: BINARYZH_IN_A -> BINARYI_IN_A BINARYI_IN_A -> BINARYZH_IN_A BINARYZH_IN_A -> BINARYZH_IN_A BINARYI_IN_A -> BINARYI_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (143) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCCM_IN_AA(one(X1), zero(X2)) -> SUCCM_IN_AA(X1, X2) The TRS R consists of the following rules: timescF_in_aaa(one(b), X1, X1) -> timescF_out_aaa(one(b), X1, X1) timescF_in_aaa(zero(X1), X2, zero(X3)) -> U88_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X3)) timescF_in_aaa(one(X1), X2, X3) -> U89_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U89_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U90_aaa(X1, X2, X3, addcG_in_aaa(X2, X4, X3)) addcG_in_aaa(b, X1, zero(X1)) -> U149_aaa(X1, binaryZcP_in_a(X1)) binaryZcP_in_a(X1) -> U119_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(zero(X1)) -> U91_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(one(X1)) -> U92_a(X1, binarycI_in_a(X1)) binarycI_in_a(b) -> binarycI_out_a(b) binarycI_in_a(zero(X1)) -> U93_a(X1, binaryZcH_in_a(X1)) U93_a(X1, binaryZcH_out_a(X1)) -> binarycI_out_a(zero(X1)) binarycI_in_a(one(X1)) -> U94_a(X1, binarycI_in_a(X1)) U94_a(X1, binarycI_out_a(X1)) -> binarycI_out_a(one(X1)) U92_a(X1, binarycI_out_a(X1)) -> binaryZcH_out_a(one(X1)) U91_a(X1, binaryZcH_out_a(X1)) -> binaryZcH_out_a(zero(X1)) U119_a(X1, binaryZcH_out_a(X1)) -> binaryZcP_out_a(X1) U149_aaa(X1, binaryZcP_out_a(X1)) -> addcG_out_aaa(b, X1, zero(X1)) addcG_in_aaa(zero(X1), X2, zero(X3)) -> U150_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(X1), zero(X2), zero(X3)) -> U95_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(one(X1)), one(b), one(one(X1))) -> U96_aaa(X1, binarycI_in_a(X1)) U96_aaa(X1, binarycI_out_a(X1)) -> addzcJ_out_aaa(zero(one(X1)), one(b), one(one(X1))) addzcJ_in_aaa(zero(zero(X1)), one(b), one(zero(X1))) -> U97_aaa(X1, binaryZcH_in_a(X1)) U97_aaa(X1, binaryZcH_out_a(X1)) -> addzcJ_out_aaa(zero(zero(X1)), one(b), one(zero(X1))) addzcJ_in_aaa(zero(X1), one(X2), one(X3)) -> U98_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), zero(X2), one(X3)) -> U99_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) addycK_in_aaa(b, one(X1), one(X1)) -> U116_aaa(X1, binarycI_in_a(X1)) U116_aaa(X1, binarycI_out_a(X1)) -> addycK_out_aaa(b, one(X1), one(X1)) addycK_in_aaa(b, zero(X1), zero(X1)) -> U117_aaa(X1, binaryZcH_in_a(X1)) U117_aaa(X1, binaryZcH_out_a(X1)) -> addycK_out_aaa(b, zero(X1), zero(X1)) addycK_in_aaa(X1, X2, X3) -> U118_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), one(X2), zero(X3)) -> U100_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) addccL_in_aaa(b, b, one(b)) -> addccL_out_aaa(b, b, one(b)) addccL_in_aaa(X1, b, X2) -> U113_aaa(X1, X2, succZcN_in_aa(X1, X2)) succZcN_in_aa(zero(X1), one(X1)) -> U103_aa(X1, binaryZcH_in_a(X1)) U103_aa(X1, binaryZcH_out_a(X1)) -> succZcN_out_aa(zero(X1), one(X1)) succZcN_in_aa(one(X1), zero(X2)) -> U104_aa(X1, X2, succcM_in_aa(X1, X2)) succcM_in_aa(b, one(b)) -> succcM_out_aa(b, one(b)) succcM_in_aa(zero(X1), one(X1)) -> U101_aa(X1, binaryZcH_in_a(X1)) U101_aa(X1, binaryZcH_out_a(X1)) -> succcM_out_aa(zero(X1), one(X1)) succcM_in_aa(one(X1), zero(X2)) -> U102_aa(X1, X2, succcM_in_aa(X1, X2)) U102_aa(X1, X2, succcM_out_aa(X1, X2)) -> succcM_out_aa(one(X1), zero(X2)) U104_aa(X1, X2, succcM_out_aa(X1, X2)) -> succZcN_out_aa(one(X1), zero(X2)) U113_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(X1, b, X2) addccL_in_aaa(b, X1, X2) -> U114_aaa(X1, X2, succZcN_in_aa(X1, X2)) U114_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(b, X1, X2) addccL_in_aaa(X1, X2, X3) -> U115_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(zero(X1), zero(X2), one(X3)) -> U105_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) U105_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), zero(X2), one(X3)) addCcO_in_aaa(zero(zero(X1)), one(b), zero(one(X1))) -> U106_aaa(X1, binaryZcH_in_a(X1)) U106_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(zero(zero(X1)), one(b), zero(one(X1))) addCcO_in_aaa(zero(one(X1)), one(b), zero(zero(X2))) -> U107_aaa(X1, X2, succcM_in_aa(X1, X2)) U107_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(zero(one(X1)), one(b), zero(zero(X2))) addCcO_in_aaa(zero(X1), one(X2), zero(X3)) -> U108_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(b), zero(zero(X1)), zero(one(X1))) -> U109_aaa(X1, binaryZcH_in_a(X1)) U109_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(one(b), zero(zero(X1)), zero(one(X1))) addCcO_in_aaa(one(b), zero(one(X1)), zero(zero(X2))) -> U110_aaa(X1, X2, succcM_in_aa(X1, X2)) U110_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(one(b), zero(one(X1)), zero(zero(X2))) addCcO_in_aaa(one(X1), zero(X2), zero(X3)) -> U111_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(X1), one(X2), one(X3)) -> U112_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) U112_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), one(X2), one(X3)) U111_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), zero(X2), zero(X3)) U108_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), one(X2), zero(X3)) U115_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addccL_out_aaa(X1, X2, X3) U100_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), one(X2), zero(X3)) U118_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addycK_out_aaa(X1, X2, X3) U99_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), zero(X2), one(X3)) U98_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), one(X2), one(X3)) U95_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), zero(X2), zero(X3)) U150_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addcG_out_aaa(zero(X1), X2, zero(X3)) addcG_in_aaa(one(X1), X2, one(X3)) -> U151_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) U151_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addcG_out_aaa(one(X1), X2, one(X3)) U90_aaa(X1, X2, X3, addcG_out_aaa(X2, X4, X3)) -> timescF_out_aaa(one(X1), X2, X3) U88_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X3)) -> timescF_out_aaa(zero(X1), X2, zero(X3)) The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) timescF_in_aaa(x1, x2, x3) = timescF_in_aaa timescF_out_aaa(x1, x2, x3) = timescF_out_aaa(x1) U88_aaa(x1, x2, x3, x4) = U88_aaa(x4) U89_aaa(x1, x2, x3, x4) = U89_aaa(x4) U90_aaa(x1, x2, x3, x4) = U90_aaa(x1, x4) addcG_in_aaa(x1, x2, x3) = addcG_in_aaa U149_aaa(x1, x2) = U149_aaa(x2) binaryZcP_in_a(x1) = binaryZcP_in_a U119_a(x1, x2) = U119_a(x2) binaryZcH_in_a(x1) = binaryZcH_in_a U91_a(x1, x2) = U91_a(x2) U92_a(x1, x2) = U92_a(x2) binarycI_in_a(x1) = binarycI_in_a binarycI_out_a(x1) = binarycI_out_a(x1) U93_a(x1, x2) = U93_a(x2) binaryZcH_out_a(x1) = binaryZcH_out_a(x1) U94_a(x1, x2) = U94_a(x2) binaryZcP_out_a(x1) = binaryZcP_out_a(x1) addcG_out_aaa(x1, x2, x3) = addcG_out_aaa(x1, x2, x3) U150_aaa(x1, x2, x3, x4) = U150_aaa(x4) addzcJ_in_aaa(x1, x2, x3) = addzcJ_in_aaa U95_aaa(x1, x2, x3, x4) = U95_aaa(x4) U96_aaa(x1, x2) = U96_aaa(x2) addzcJ_out_aaa(x1, x2, x3) = addzcJ_out_aaa(x1, x2, x3) U97_aaa(x1, x2) = U97_aaa(x2) U98_aaa(x1, x2, x3, x4) = U98_aaa(x4) U99_aaa(x1, x2, x3, x4) = U99_aaa(x4) addycK_in_aaa(x1, x2, x3) = addycK_in_aaa U116_aaa(x1, x2) = U116_aaa(x2) addycK_out_aaa(x1, x2, x3) = addycK_out_aaa(x1, x2, x3) U117_aaa(x1, x2) = U117_aaa(x2) U118_aaa(x1, x2, x3, x4) = U118_aaa(x4) U100_aaa(x1, x2, x3, x4) = U100_aaa(x4) addccL_in_aaa(x1, x2, x3) = addccL_in_aaa addccL_out_aaa(x1, x2, x3) = addccL_out_aaa(x1, x2, x3) U113_aaa(x1, x2, x3) = U113_aaa(x3) succZcN_in_aa(x1, x2) = succZcN_in_aa U103_aa(x1, x2) = U103_aa(x2) succZcN_out_aa(x1, x2) = succZcN_out_aa(x1, x2) U104_aa(x1, x2, x3) = U104_aa(x3) succcM_in_aa(x1, x2) = succcM_in_aa succcM_out_aa(x1, x2) = succcM_out_aa(x1, x2) U101_aa(x1, x2) = U101_aa(x2) U102_aa(x1, x2, x3) = U102_aa(x3) U114_aaa(x1, x2, x3) = U114_aaa(x3) U115_aaa(x1, x2, x3, x4) = U115_aaa(x4) addCcO_in_aaa(x1, x2, x3) = addCcO_in_aaa U105_aaa(x1, x2, x3, x4) = U105_aaa(x4) addCcO_out_aaa(x1, x2, x3) = addCcO_out_aaa(x1, x2, x3) U106_aaa(x1, x2) = U106_aaa(x2) U107_aaa(x1, x2, x3) = U107_aaa(x3) U108_aaa(x1, x2, x3, x4) = U108_aaa(x4) U109_aaa(x1, x2) = U109_aaa(x2) U110_aaa(x1, x2, x3) = U110_aaa(x3) U111_aaa(x1, x2, x3, x4) = U111_aaa(x4) U112_aaa(x1, x2, x3, x4) = U112_aaa(x4) U151_aaa(x1, x2, x3, x4) = U151_aaa(x4) one(x1) = one(x1) b = b SUCCM_IN_AA(x1, x2) = SUCCM_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (144) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (145) Obligation: Pi DP problem: The TRS P consists of the following rules: SUCCM_IN_AA(one(X1), zero(X2)) -> SUCCM_IN_AA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) SUCCM_IN_AA(x1, x2) = SUCCM_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (146) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDZJ_IN_AAA(zero(X1), one(X2), one(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(zero(X1), zero(X2), zero(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(one(X1), zero(X2), one(X3)) -> ADDYK_IN_AAA(X1, X2, X3) ADDYK_IN_AAA(X1, X2, X3) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(one(X1), one(X2), zero(X3)) -> ADDCL_IN_AAA(X1, X2, X3) ADDCL_IN_AAA(X1, X2, X3) -> ADDCO_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(zero(X1), zero(X2), one(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(zero(X1), one(X2), zero(X3)) -> ADDCO_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(one(X1), zero(X2), zero(X3)) -> ADDCO_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(one(X1), one(X2), one(X3)) -> ADDCL_IN_AAA(X1, X2, X3) The TRS R consists of the following rules: timescF_in_aaa(one(b), X1, X1) -> timescF_out_aaa(one(b), X1, X1) timescF_in_aaa(zero(X1), X2, zero(X3)) -> U88_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X3)) timescF_in_aaa(one(X1), X2, X3) -> U89_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U89_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U90_aaa(X1, X2, X3, addcG_in_aaa(X2, X4, X3)) addcG_in_aaa(b, X1, zero(X1)) -> U149_aaa(X1, binaryZcP_in_a(X1)) binaryZcP_in_a(X1) -> U119_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(zero(X1)) -> U91_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(one(X1)) -> U92_a(X1, binarycI_in_a(X1)) binarycI_in_a(b) -> binarycI_out_a(b) binarycI_in_a(zero(X1)) -> U93_a(X1, binaryZcH_in_a(X1)) U93_a(X1, binaryZcH_out_a(X1)) -> binarycI_out_a(zero(X1)) binarycI_in_a(one(X1)) -> U94_a(X1, binarycI_in_a(X1)) U94_a(X1, binarycI_out_a(X1)) -> binarycI_out_a(one(X1)) U92_a(X1, binarycI_out_a(X1)) -> binaryZcH_out_a(one(X1)) U91_a(X1, binaryZcH_out_a(X1)) -> binaryZcH_out_a(zero(X1)) U119_a(X1, binaryZcH_out_a(X1)) -> binaryZcP_out_a(X1) U149_aaa(X1, binaryZcP_out_a(X1)) -> addcG_out_aaa(b, X1, zero(X1)) addcG_in_aaa(zero(X1), X2, zero(X3)) -> U150_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(X1), zero(X2), zero(X3)) -> U95_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(one(X1)), one(b), one(one(X1))) -> U96_aaa(X1, binarycI_in_a(X1)) U96_aaa(X1, binarycI_out_a(X1)) -> addzcJ_out_aaa(zero(one(X1)), one(b), one(one(X1))) addzcJ_in_aaa(zero(zero(X1)), one(b), one(zero(X1))) -> U97_aaa(X1, binaryZcH_in_a(X1)) U97_aaa(X1, binaryZcH_out_a(X1)) -> addzcJ_out_aaa(zero(zero(X1)), one(b), one(zero(X1))) addzcJ_in_aaa(zero(X1), one(X2), one(X3)) -> U98_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), zero(X2), one(X3)) -> U99_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) addycK_in_aaa(b, one(X1), one(X1)) -> U116_aaa(X1, binarycI_in_a(X1)) U116_aaa(X1, binarycI_out_a(X1)) -> addycK_out_aaa(b, one(X1), one(X1)) addycK_in_aaa(b, zero(X1), zero(X1)) -> U117_aaa(X1, binaryZcH_in_a(X1)) U117_aaa(X1, binaryZcH_out_a(X1)) -> addycK_out_aaa(b, zero(X1), zero(X1)) addycK_in_aaa(X1, X2, X3) -> U118_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), one(X2), zero(X3)) -> U100_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) addccL_in_aaa(b, b, one(b)) -> addccL_out_aaa(b, b, one(b)) addccL_in_aaa(X1, b, X2) -> U113_aaa(X1, X2, succZcN_in_aa(X1, X2)) succZcN_in_aa(zero(X1), one(X1)) -> U103_aa(X1, binaryZcH_in_a(X1)) U103_aa(X1, binaryZcH_out_a(X1)) -> succZcN_out_aa(zero(X1), one(X1)) succZcN_in_aa(one(X1), zero(X2)) -> U104_aa(X1, X2, succcM_in_aa(X1, X2)) succcM_in_aa(b, one(b)) -> succcM_out_aa(b, one(b)) succcM_in_aa(zero(X1), one(X1)) -> U101_aa(X1, binaryZcH_in_a(X1)) U101_aa(X1, binaryZcH_out_a(X1)) -> succcM_out_aa(zero(X1), one(X1)) succcM_in_aa(one(X1), zero(X2)) -> U102_aa(X1, X2, succcM_in_aa(X1, X2)) U102_aa(X1, X2, succcM_out_aa(X1, X2)) -> succcM_out_aa(one(X1), zero(X2)) U104_aa(X1, X2, succcM_out_aa(X1, X2)) -> succZcN_out_aa(one(X1), zero(X2)) U113_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(X1, b, X2) addccL_in_aaa(b, X1, X2) -> U114_aaa(X1, X2, succZcN_in_aa(X1, X2)) U114_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(b, X1, X2) addccL_in_aaa(X1, X2, X3) -> U115_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(zero(X1), zero(X2), one(X3)) -> U105_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) U105_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), zero(X2), one(X3)) addCcO_in_aaa(zero(zero(X1)), one(b), zero(one(X1))) -> U106_aaa(X1, binaryZcH_in_a(X1)) U106_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(zero(zero(X1)), one(b), zero(one(X1))) addCcO_in_aaa(zero(one(X1)), one(b), zero(zero(X2))) -> U107_aaa(X1, X2, succcM_in_aa(X1, X2)) U107_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(zero(one(X1)), one(b), zero(zero(X2))) addCcO_in_aaa(zero(X1), one(X2), zero(X3)) -> U108_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(b), zero(zero(X1)), zero(one(X1))) -> U109_aaa(X1, binaryZcH_in_a(X1)) U109_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(one(b), zero(zero(X1)), zero(one(X1))) addCcO_in_aaa(one(b), zero(one(X1)), zero(zero(X2))) -> U110_aaa(X1, X2, succcM_in_aa(X1, X2)) U110_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(one(b), zero(one(X1)), zero(zero(X2))) addCcO_in_aaa(one(X1), zero(X2), zero(X3)) -> U111_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(X1), one(X2), one(X3)) -> U112_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) U112_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), one(X2), one(X3)) U111_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), zero(X2), zero(X3)) U108_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), one(X2), zero(X3)) U115_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addccL_out_aaa(X1, X2, X3) U100_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), one(X2), zero(X3)) U118_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addycK_out_aaa(X1, X2, X3) U99_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), zero(X2), one(X3)) U98_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), one(X2), one(X3)) U95_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), zero(X2), zero(X3)) U150_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addcG_out_aaa(zero(X1), X2, zero(X3)) addcG_in_aaa(one(X1), X2, one(X3)) -> U151_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) U151_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addcG_out_aaa(one(X1), X2, one(X3)) U90_aaa(X1, X2, X3, addcG_out_aaa(X2, X4, X3)) -> timescF_out_aaa(one(X1), X2, X3) U88_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X3)) -> timescF_out_aaa(zero(X1), X2, zero(X3)) The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) timescF_in_aaa(x1, x2, x3) = timescF_in_aaa timescF_out_aaa(x1, x2, x3) = timescF_out_aaa(x1) U88_aaa(x1, x2, x3, x4) = U88_aaa(x4) U89_aaa(x1, x2, x3, x4) = U89_aaa(x4) U90_aaa(x1, x2, x3, x4) = U90_aaa(x1, x4) addcG_in_aaa(x1, x2, x3) = addcG_in_aaa U149_aaa(x1, x2) = U149_aaa(x2) binaryZcP_in_a(x1) = binaryZcP_in_a U119_a(x1, x2) = U119_a(x2) binaryZcH_in_a(x1) = binaryZcH_in_a U91_a(x1, x2) = U91_a(x2) U92_a(x1, x2) = U92_a(x2) binarycI_in_a(x1) = binarycI_in_a binarycI_out_a(x1) = binarycI_out_a(x1) U93_a(x1, x2) = U93_a(x2) binaryZcH_out_a(x1) = binaryZcH_out_a(x1) U94_a(x1, x2) = U94_a(x2) binaryZcP_out_a(x1) = binaryZcP_out_a(x1) addcG_out_aaa(x1, x2, x3) = addcG_out_aaa(x1, x2, x3) U150_aaa(x1, x2, x3, x4) = U150_aaa(x4) addzcJ_in_aaa(x1, x2, x3) = addzcJ_in_aaa U95_aaa(x1, x2, x3, x4) = U95_aaa(x4) U96_aaa(x1, x2) = U96_aaa(x2) addzcJ_out_aaa(x1, x2, x3) = addzcJ_out_aaa(x1, x2, x3) U97_aaa(x1, x2) = U97_aaa(x2) U98_aaa(x1, x2, x3, x4) = U98_aaa(x4) U99_aaa(x1, x2, x3, x4) = U99_aaa(x4) addycK_in_aaa(x1, x2, x3) = addycK_in_aaa U116_aaa(x1, x2) = U116_aaa(x2) addycK_out_aaa(x1, x2, x3) = addycK_out_aaa(x1, x2, x3) U117_aaa(x1, x2) = U117_aaa(x2) U118_aaa(x1, x2, x3, x4) = U118_aaa(x4) U100_aaa(x1, x2, x3, x4) = U100_aaa(x4) addccL_in_aaa(x1, x2, x3) = addccL_in_aaa addccL_out_aaa(x1, x2, x3) = addccL_out_aaa(x1, x2, x3) U113_aaa(x1, x2, x3) = U113_aaa(x3) succZcN_in_aa(x1, x2) = succZcN_in_aa U103_aa(x1, x2) = U103_aa(x2) succZcN_out_aa(x1, x2) = succZcN_out_aa(x1, x2) U104_aa(x1, x2, x3) = U104_aa(x3) succcM_in_aa(x1, x2) = succcM_in_aa succcM_out_aa(x1, x2) = succcM_out_aa(x1, x2) U101_aa(x1, x2) = U101_aa(x2) U102_aa(x1, x2, x3) = U102_aa(x3) U114_aaa(x1, x2, x3) = U114_aaa(x3) U115_aaa(x1, x2, x3, x4) = U115_aaa(x4) addCcO_in_aaa(x1, x2, x3) = addCcO_in_aaa U105_aaa(x1, x2, x3, x4) = U105_aaa(x4) addCcO_out_aaa(x1, x2, x3) = addCcO_out_aaa(x1, x2, x3) U106_aaa(x1, x2) = U106_aaa(x2) U107_aaa(x1, x2, x3) = U107_aaa(x3) U108_aaa(x1, x2, x3, x4) = U108_aaa(x4) U109_aaa(x1, x2) = U109_aaa(x2) U110_aaa(x1, x2, x3) = U110_aaa(x3) U111_aaa(x1, x2, x3, x4) = U111_aaa(x4) U112_aaa(x1, x2, x3, x4) = U112_aaa(x4) U151_aaa(x1, x2, x3, x4) = U151_aaa(x4) one(x1) = one(x1) b = b ADDZJ_IN_AAA(x1, x2, x3) = ADDZJ_IN_AAA ADDYK_IN_AAA(x1, x2, x3) = ADDYK_IN_AAA ADDCL_IN_AAA(x1, x2, x3) = ADDCL_IN_AAA ADDCO_IN_AAA(x1, x2, x3) = ADDCO_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (147) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (148) Obligation: Pi DP problem: The TRS P consists of the following rules: ADDZJ_IN_AAA(zero(X1), one(X2), one(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(zero(X1), zero(X2), zero(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(one(X1), zero(X2), one(X3)) -> ADDYK_IN_AAA(X1, X2, X3) ADDYK_IN_AAA(X1, X2, X3) -> ADDZJ_IN_AAA(X1, X2, X3) ADDZJ_IN_AAA(one(X1), one(X2), zero(X3)) -> ADDCL_IN_AAA(X1, X2, X3) ADDCL_IN_AAA(X1, X2, X3) -> ADDCO_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(zero(X1), zero(X2), one(X3)) -> ADDZJ_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(zero(X1), one(X2), zero(X3)) -> ADDCO_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(one(X1), zero(X2), zero(X3)) -> ADDCO_IN_AAA(X1, X2, X3) ADDCO_IN_AAA(one(X1), one(X2), one(X3)) -> ADDCL_IN_AAA(X1, X2, X3) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) ADDZJ_IN_AAA(x1, x2, x3) = ADDZJ_IN_AAA ADDYK_IN_AAA(x1, x2, x3) = ADDYK_IN_AAA ADDCL_IN_AAA(x1, x2, x3) = ADDCL_IN_AAA ADDCO_IN_AAA(x1, x2, x3) = ADDCO_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (149) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMESF_IN_AAA(one(X1), X2, X3) -> TIMESF_IN_AAA(X1, X2, X4) TIMESF_IN_AAA(zero(X1), X2, zero(X3)) -> TIMESF_IN_AAA(X1, X2, X3) The TRS R consists of the following rules: timescF_in_aaa(one(b), X1, X1) -> timescF_out_aaa(one(b), X1, X1) timescF_in_aaa(zero(X1), X2, zero(X3)) -> U88_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X3)) timescF_in_aaa(one(X1), X2, X3) -> U89_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U89_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U90_aaa(X1, X2, X3, addcG_in_aaa(X2, X4, X3)) addcG_in_aaa(b, X1, zero(X1)) -> U149_aaa(X1, binaryZcP_in_a(X1)) binaryZcP_in_a(X1) -> U119_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(zero(X1)) -> U91_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(one(X1)) -> U92_a(X1, binarycI_in_a(X1)) binarycI_in_a(b) -> binarycI_out_a(b) binarycI_in_a(zero(X1)) -> U93_a(X1, binaryZcH_in_a(X1)) U93_a(X1, binaryZcH_out_a(X1)) -> binarycI_out_a(zero(X1)) binarycI_in_a(one(X1)) -> U94_a(X1, binarycI_in_a(X1)) U94_a(X1, binarycI_out_a(X1)) -> binarycI_out_a(one(X1)) U92_a(X1, binarycI_out_a(X1)) -> binaryZcH_out_a(one(X1)) U91_a(X1, binaryZcH_out_a(X1)) -> binaryZcH_out_a(zero(X1)) U119_a(X1, binaryZcH_out_a(X1)) -> binaryZcP_out_a(X1) U149_aaa(X1, binaryZcP_out_a(X1)) -> addcG_out_aaa(b, X1, zero(X1)) addcG_in_aaa(zero(X1), X2, zero(X3)) -> U150_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(X1), zero(X2), zero(X3)) -> U95_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(one(X1)), one(b), one(one(X1))) -> U96_aaa(X1, binarycI_in_a(X1)) U96_aaa(X1, binarycI_out_a(X1)) -> addzcJ_out_aaa(zero(one(X1)), one(b), one(one(X1))) addzcJ_in_aaa(zero(zero(X1)), one(b), one(zero(X1))) -> U97_aaa(X1, binaryZcH_in_a(X1)) U97_aaa(X1, binaryZcH_out_a(X1)) -> addzcJ_out_aaa(zero(zero(X1)), one(b), one(zero(X1))) addzcJ_in_aaa(zero(X1), one(X2), one(X3)) -> U98_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), zero(X2), one(X3)) -> U99_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) addycK_in_aaa(b, one(X1), one(X1)) -> U116_aaa(X1, binarycI_in_a(X1)) U116_aaa(X1, binarycI_out_a(X1)) -> addycK_out_aaa(b, one(X1), one(X1)) addycK_in_aaa(b, zero(X1), zero(X1)) -> U117_aaa(X1, binaryZcH_in_a(X1)) U117_aaa(X1, binaryZcH_out_a(X1)) -> addycK_out_aaa(b, zero(X1), zero(X1)) addycK_in_aaa(X1, X2, X3) -> U118_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), one(X2), zero(X3)) -> U100_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) addccL_in_aaa(b, b, one(b)) -> addccL_out_aaa(b, b, one(b)) addccL_in_aaa(X1, b, X2) -> U113_aaa(X1, X2, succZcN_in_aa(X1, X2)) succZcN_in_aa(zero(X1), one(X1)) -> U103_aa(X1, binaryZcH_in_a(X1)) U103_aa(X1, binaryZcH_out_a(X1)) -> succZcN_out_aa(zero(X1), one(X1)) succZcN_in_aa(one(X1), zero(X2)) -> U104_aa(X1, X2, succcM_in_aa(X1, X2)) succcM_in_aa(b, one(b)) -> succcM_out_aa(b, one(b)) succcM_in_aa(zero(X1), one(X1)) -> U101_aa(X1, binaryZcH_in_a(X1)) U101_aa(X1, binaryZcH_out_a(X1)) -> succcM_out_aa(zero(X1), one(X1)) succcM_in_aa(one(X1), zero(X2)) -> U102_aa(X1, X2, succcM_in_aa(X1, X2)) U102_aa(X1, X2, succcM_out_aa(X1, X2)) -> succcM_out_aa(one(X1), zero(X2)) U104_aa(X1, X2, succcM_out_aa(X1, X2)) -> succZcN_out_aa(one(X1), zero(X2)) U113_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(X1, b, X2) addccL_in_aaa(b, X1, X2) -> U114_aaa(X1, X2, succZcN_in_aa(X1, X2)) U114_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(b, X1, X2) addccL_in_aaa(X1, X2, X3) -> U115_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(zero(X1), zero(X2), one(X3)) -> U105_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) U105_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), zero(X2), one(X3)) addCcO_in_aaa(zero(zero(X1)), one(b), zero(one(X1))) -> U106_aaa(X1, binaryZcH_in_a(X1)) U106_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(zero(zero(X1)), one(b), zero(one(X1))) addCcO_in_aaa(zero(one(X1)), one(b), zero(zero(X2))) -> U107_aaa(X1, X2, succcM_in_aa(X1, X2)) U107_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(zero(one(X1)), one(b), zero(zero(X2))) addCcO_in_aaa(zero(X1), one(X2), zero(X3)) -> U108_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(b), zero(zero(X1)), zero(one(X1))) -> U109_aaa(X1, binaryZcH_in_a(X1)) U109_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(one(b), zero(zero(X1)), zero(one(X1))) addCcO_in_aaa(one(b), zero(one(X1)), zero(zero(X2))) -> U110_aaa(X1, X2, succcM_in_aa(X1, X2)) U110_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(one(b), zero(one(X1)), zero(zero(X2))) addCcO_in_aaa(one(X1), zero(X2), zero(X3)) -> U111_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(X1), one(X2), one(X3)) -> U112_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) U112_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), one(X2), one(X3)) U111_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), zero(X2), zero(X3)) U108_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), one(X2), zero(X3)) U115_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addccL_out_aaa(X1, X2, X3) U100_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), one(X2), zero(X3)) U118_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addycK_out_aaa(X1, X2, X3) U99_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), zero(X2), one(X3)) U98_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), one(X2), one(X3)) U95_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), zero(X2), zero(X3)) U150_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addcG_out_aaa(zero(X1), X2, zero(X3)) addcG_in_aaa(one(X1), X2, one(X3)) -> U151_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) U151_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addcG_out_aaa(one(X1), X2, one(X3)) U90_aaa(X1, X2, X3, addcG_out_aaa(X2, X4, X3)) -> timescF_out_aaa(one(X1), X2, X3) U88_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X3)) -> timescF_out_aaa(zero(X1), X2, zero(X3)) The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) timescF_in_aaa(x1, x2, x3) = timescF_in_aaa timescF_out_aaa(x1, x2, x3) = timescF_out_aaa(x1) U88_aaa(x1, x2, x3, x4) = U88_aaa(x4) U89_aaa(x1, x2, x3, x4) = U89_aaa(x4) U90_aaa(x1, x2, x3, x4) = U90_aaa(x1, x4) addcG_in_aaa(x1, x2, x3) = addcG_in_aaa U149_aaa(x1, x2) = U149_aaa(x2) binaryZcP_in_a(x1) = binaryZcP_in_a U119_a(x1, x2) = U119_a(x2) binaryZcH_in_a(x1) = binaryZcH_in_a U91_a(x1, x2) = U91_a(x2) U92_a(x1, x2) = U92_a(x2) binarycI_in_a(x1) = binarycI_in_a binarycI_out_a(x1) = binarycI_out_a(x1) U93_a(x1, x2) = U93_a(x2) binaryZcH_out_a(x1) = binaryZcH_out_a(x1) U94_a(x1, x2) = U94_a(x2) binaryZcP_out_a(x1) = binaryZcP_out_a(x1) addcG_out_aaa(x1, x2, x3) = addcG_out_aaa(x1, x2, x3) U150_aaa(x1, x2, x3, x4) = U150_aaa(x4) addzcJ_in_aaa(x1, x2, x3) = addzcJ_in_aaa U95_aaa(x1, x2, x3, x4) = U95_aaa(x4) U96_aaa(x1, x2) = U96_aaa(x2) addzcJ_out_aaa(x1, x2, x3) = addzcJ_out_aaa(x1, x2, x3) U97_aaa(x1, x2) = U97_aaa(x2) U98_aaa(x1, x2, x3, x4) = U98_aaa(x4) U99_aaa(x1, x2, x3, x4) = U99_aaa(x4) addycK_in_aaa(x1, x2, x3) = addycK_in_aaa U116_aaa(x1, x2) = U116_aaa(x2) addycK_out_aaa(x1, x2, x3) = addycK_out_aaa(x1, x2, x3) U117_aaa(x1, x2) = U117_aaa(x2) U118_aaa(x1, x2, x3, x4) = U118_aaa(x4) U100_aaa(x1, x2, x3, x4) = U100_aaa(x4) addccL_in_aaa(x1, x2, x3) = addccL_in_aaa addccL_out_aaa(x1, x2, x3) = addccL_out_aaa(x1, x2, x3) U113_aaa(x1, x2, x3) = U113_aaa(x3) succZcN_in_aa(x1, x2) = succZcN_in_aa U103_aa(x1, x2) = U103_aa(x2) succZcN_out_aa(x1, x2) = succZcN_out_aa(x1, x2) U104_aa(x1, x2, x3) = U104_aa(x3) succcM_in_aa(x1, x2) = succcM_in_aa succcM_out_aa(x1, x2) = succcM_out_aa(x1, x2) U101_aa(x1, x2) = U101_aa(x2) U102_aa(x1, x2, x3) = U102_aa(x3) U114_aaa(x1, x2, x3) = U114_aaa(x3) U115_aaa(x1, x2, x3, x4) = U115_aaa(x4) addCcO_in_aaa(x1, x2, x3) = addCcO_in_aaa U105_aaa(x1, x2, x3, x4) = U105_aaa(x4) addCcO_out_aaa(x1, x2, x3) = addCcO_out_aaa(x1, x2, x3) U106_aaa(x1, x2) = U106_aaa(x2) U107_aaa(x1, x2, x3) = U107_aaa(x3) U108_aaa(x1, x2, x3, x4) = U108_aaa(x4) U109_aaa(x1, x2) = U109_aaa(x2) U110_aaa(x1, x2, x3) = U110_aaa(x3) U111_aaa(x1, x2, x3, x4) = U111_aaa(x4) U112_aaa(x1, x2, x3, x4) = U112_aaa(x4) U151_aaa(x1, x2, x3, x4) = U151_aaa(x4) one(x1) = one(x1) b = b TIMESF_IN_AAA(x1, x2, x3) = TIMESF_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (150) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (151) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMESF_IN_AAA(one(X1), X2, X3) -> TIMESF_IN_AAA(X1, X2, X4) TIMESF_IN_AAA(zero(X1), X2, zero(X3)) -> TIMESF_IN_AAA(X1, X2, X3) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) one(x1) = one(x1) TIMESF_IN_AAA(x1, x2, x3) = TIMESF_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (152) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMESA_IN_AAG(zero(zero(X1)), X2, zero(zero(X3))) -> TIMESA_IN_AAG(X1, X2, X3) The TRS R consists of the following rules: timescF_in_aaa(one(b), X1, X1) -> timescF_out_aaa(one(b), X1, X1) timescF_in_aaa(zero(X1), X2, zero(X3)) -> U88_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X3)) timescF_in_aaa(one(X1), X2, X3) -> U89_aaa(X1, X2, X3, timescF_in_aaa(X1, X2, X4)) U89_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X4)) -> U90_aaa(X1, X2, X3, addcG_in_aaa(X2, X4, X3)) addcG_in_aaa(b, X1, zero(X1)) -> U149_aaa(X1, binaryZcP_in_a(X1)) binaryZcP_in_a(X1) -> U119_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(zero(X1)) -> U91_a(X1, binaryZcH_in_a(X1)) binaryZcH_in_a(one(X1)) -> U92_a(X1, binarycI_in_a(X1)) binarycI_in_a(b) -> binarycI_out_a(b) binarycI_in_a(zero(X1)) -> U93_a(X1, binaryZcH_in_a(X1)) U93_a(X1, binaryZcH_out_a(X1)) -> binarycI_out_a(zero(X1)) binarycI_in_a(one(X1)) -> U94_a(X1, binarycI_in_a(X1)) U94_a(X1, binarycI_out_a(X1)) -> binarycI_out_a(one(X1)) U92_a(X1, binarycI_out_a(X1)) -> binaryZcH_out_a(one(X1)) U91_a(X1, binaryZcH_out_a(X1)) -> binaryZcH_out_a(zero(X1)) U119_a(X1, binaryZcH_out_a(X1)) -> binaryZcP_out_a(X1) U149_aaa(X1, binaryZcP_out_a(X1)) -> addcG_out_aaa(b, X1, zero(X1)) addcG_in_aaa(zero(X1), X2, zero(X3)) -> U150_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(X1), zero(X2), zero(X3)) -> U95_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(zero(one(X1)), one(b), one(one(X1))) -> U96_aaa(X1, binarycI_in_a(X1)) U96_aaa(X1, binarycI_out_a(X1)) -> addzcJ_out_aaa(zero(one(X1)), one(b), one(one(X1))) addzcJ_in_aaa(zero(zero(X1)), one(b), one(zero(X1))) -> U97_aaa(X1, binaryZcH_in_a(X1)) U97_aaa(X1, binaryZcH_out_a(X1)) -> addzcJ_out_aaa(zero(zero(X1)), one(b), one(zero(X1))) addzcJ_in_aaa(zero(X1), one(X2), one(X3)) -> U98_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), zero(X2), one(X3)) -> U99_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) addycK_in_aaa(b, one(X1), one(X1)) -> U116_aaa(X1, binarycI_in_a(X1)) U116_aaa(X1, binarycI_out_a(X1)) -> addycK_out_aaa(b, one(X1), one(X1)) addycK_in_aaa(b, zero(X1), zero(X1)) -> U117_aaa(X1, binaryZcH_in_a(X1)) U117_aaa(X1, binaryZcH_out_a(X1)) -> addycK_out_aaa(b, zero(X1), zero(X1)) addycK_in_aaa(X1, X2, X3) -> U118_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) addzcJ_in_aaa(one(X1), one(X2), zero(X3)) -> U100_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) addccL_in_aaa(b, b, one(b)) -> addccL_out_aaa(b, b, one(b)) addccL_in_aaa(X1, b, X2) -> U113_aaa(X1, X2, succZcN_in_aa(X1, X2)) succZcN_in_aa(zero(X1), one(X1)) -> U103_aa(X1, binaryZcH_in_a(X1)) U103_aa(X1, binaryZcH_out_a(X1)) -> succZcN_out_aa(zero(X1), one(X1)) succZcN_in_aa(one(X1), zero(X2)) -> U104_aa(X1, X2, succcM_in_aa(X1, X2)) succcM_in_aa(b, one(b)) -> succcM_out_aa(b, one(b)) succcM_in_aa(zero(X1), one(X1)) -> U101_aa(X1, binaryZcH_in_a(X1)) U101_aa(X1, binaryZcH_out_a(X1)) -> succcM_out_aa(zero(X1), one(X1)) succcM_in_aa(one(X1), zero(X2)) -> U102_aa(X1, X2, succcM_in_aa(X1, X2)) U102_aa(X1, X2, succcM_out_aa(X1, X2)) -> succcM_out_aa(one(X1), zero(X2)) U104_aa(X1, X2, succcM_out_aa(X1, X2)) -> succZcN_out_aa(one(X1), zero(X2)) U113_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(X1, b, X2) addccL_in_aaa(b, X1, X2) -> U114_aaa(X1, X2, succZcN_in_aa(X1, X2)) U114_aaa(X1, X2, succZcN_out_aa(X1, X2)) -> addccL_out_aaa(b, X1, X2) addccL_in_aaa(X1, X2, X3) -> U115_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(zero(X1), zero(X2), one(X3)) -> U105_aaa(X1, X2, X3, addzcJ_in_aaa(X1, X2, X3)) U105_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), zero(X2), one(X3)) addCcO_in_aaa(zero(zero(X1)), one(b), zero(one(X1))) -> U106_aaa(X1, binaryZcH_in_a(X1)) U106_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(zero(zero(X1)), one(b), zero(one(X1))) addCcO_in_aaa(zero(one(X1)), one(b), zero(zero(X2))) -> U107_aaa(X1, X2, succcM_in_aa(X1, X2)) U107_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(zero(one(X1)), one(b), zero(zero(X2))) addCcO_in_aaa(zero(X1), one(X2), zero(X3)) -> U108_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(b), zero(zero(X1)), zero(one(X1))) -> U109_aaa(X1, binaryZcH_in_a(X1)) U109_aaa(X1, binaryZcH_out_a(X1)) -> addCcO_out_aaa(one(b), zero(zero(X1)), zero(one(X1))) addCcO_in_aaa(one(b), zero(one(X1)), zero(zero(X2))) -> U110_aaa(X1, X2, succcM_in_aa(X1, X2)) U110_aaa(X1, X2, succcM_out_aa(X1, X2)) -> addCcO_out_aaa(one(b), zero(one(X1)), zero(zero(X2))) addCcO_in_aaa(one(X1), zero(X2), zero(X3)) -> U111_aaa(X1, X2, X3, addCcO_in_aaa(X1, X2, X3)) addCcO_in_aaa(one(X1), one(X2), one(X3)) -> U112_aaa(X1, X2, X3, addccL_in_aaa(X1, X2, X3)) U112_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), one(X2), one(X3)) U111_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(one(X1), zero(X2), zero(X3)) U108_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addCcO_out_aaa(zero(X1), one(X2), zero(X3)) U115_aaa(X1, X2, X3, addCcO_out_aaa(X1, X2, X3)) -> addccL_out_aaa(X1, X2, X3) U100_aaa(X1, X2, X3, addccL_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), one(X2), zero(X3)) U118_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addycK_out_aaa(X1, X2, X3) U99_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(one(X1), zero(X2), one(X3)) U98_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), one(X2), one(X3)) U95_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addzcJ_out_aaa(zero(X1), zero(X2), zero(X3)) U150_aaa(X1, X2, X3, addzcJ_out_aaa(X1, X2, X3)) -> addcG_out_aaa(zero(X1), X2, zero(X3)) addcG_in_aaa(one(X1), X2, one(X3)) -> U151_aaa(X1, X2, X3, addycK_in_aaa(X1, X2, X3)) U151_aaa(X1, X2, X3, addycK_out_aaa(X1, X2, X3)) -> addcG_out_aaa(one(X1), X2, one(X3)) U90_aaa(X1, X2, X3, addcG_out_aaa(X2, X4, X3)) -> timescF_out_aaa(one(X1), X2, X3) U88_aaa(X1, X2, X3, timescF_out_aaa(X1, X2, X3)) -> timescF_out_aaa(zero(X1), X2, zero(X3)) The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) timescF_in_aaa(x1, x2, x3) = timescF_in_aaa timescF_out_aaa(x1, x2, x3) = timescF_out_aaa(x1) U88_aaa(x1, x2, x3, x4) = U88_aaa(x4) U89_aaa(x1, x2, x3, x4) = U89_aaa(x4) U90_aaa(x1, x2, x3, x4) = U90_aaa(x1, x4) addcG_in_aaa(x1, x2, x3) = addcG_in_aaa U149_aaa(x1, x2) = U149_aaa(x2) binaryZcP_in_a(x1) = binaryZcP_in_a U119_a(x1, x2) = U119_a(x2) binaryZcH_in_a(x1) = binaryZcH_in_a U91_a(x1, x2) = U91_a(x2) U92_a(x1, x2) = U92_a(x2) binarycI_in_a(x1) = binarycI_in_a binarycI_out_a(x1) = binarycI_out_a(x1) U93_a(x1, x2) = U93_a(x2) binaryZcH_out_a(x1) = binaryZcH_out_a(x1) U94_a(x1, x2) = U94_a(x2) binaryZcP_out_a(x1) = binaryZcP_out_a(x1) addcG_out_aaa(x1, x2, x3) = addcG_out_aaa(x1, x2, x3) U150_aaa(x1, x2, x3, x4) = U150_aaa(x4) addzcJ_in_aaa(x1, x2, x3) = addzcJ_in_aaa U95_aaa(x1, x2, x3, x4) = U95_aaa(x4) U96_aaa(x1, x2) = U96_aaa(x2) addzcJ_out_aaa(x1, x2, x3) = addzcJ_out_aaa(x1, x2, x3) U97_aaa(x1, x2) = U97_aaa(x2) U98_aaa(x1, x2, x3, x4) = U98_aaa(x4) U99_aaa(x1, x2, x3, x4) = U99_aaa(x4) addycK_in_aaa(x1, x2, x3) = addycK_in_aaa U116_aaa(x1, x2) = U116_aaa(x2) addycK_out_aaa(x1, x2, x3) = addycK_out_aaa(x1, x2, x3) U117_aaa(x1, x2) = U117_aaa(x2) U118_aaa(x1, x2, x3, x4) = U118_aaa(x4) U100_aaa(x1, x2, x3, x4) = U100_aaa(x4) addccL_in_aaa(x1, x2, x3) = addccL_in_aaa addccL_out_aaa(x1, x2, x3) = addccL_out_aaa(x1, x2, x3) U113_aaa(x1, x2, x3) = U113_aaa(x3) succZcN_in_aa(x1, x2) = succZcN_in_aa U103_aa(x1, x2) = U103_aa(x2) succZcN_out_aa(x1, x2) = succZcN_out_aa(x1, x2) U104_aa(x1, x2, x3) = U104_aa(x3) succcM_in_aa(x1, x2) = succcM_in_aa succcM_out_aa(x1, x2) = succcM_out_aa(x1, x2) U101_aa(x1, x2) = U101_aa(x2) U102_aa(x1, x2, x3) = U102_aa(x3) U114_aaa(x1, x2, x3) = U114_aaa(x3) U115_aaa(x1, x2, x3, x4) = U115_aaa(x4) addCcO_in_aaa(x1, x2, x3) = addCcO_in_aaa U105_aaa(x1, x2, x3, x4) = U105_aaa(x4) addCcO_out_aaa(x1, x2, x3) = addCcO_out_aaa(x1, x2, x3) U106_aaa(x1, x2) = U106_aaa(x2) U107_aaa(x1, x2, x3) = U107_aaa(x3) U108_aaa(x1, x2, x3, x4) = U108_aaa(x4) U109_aaa(x1, x2) = U109_aaa(x2) U110_aaa(x1, x2, x3) = U110_aaa(x3) U111_aaa(x1, x2, x3, x4) = U111_aaa(x4) U112_aaa(x1, x2, x3, x4) = U112_aaa(x4) U151_aaa(x1, x2, x3, x4) = U151_aaa(x4) one(x1) = one(x1) b = b TIMESA_IN_AAG(x1, x2, x3) = TIMESA_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (153) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (154) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMESA_IN_AAG(zero(zero(X1)), X2, zero(zero(X3))) -> TIMESA_IN_AAG(X1, X2, X3) R is empty. The argument filtering Pi contains the following mapping: zero(x1) = zero(x1) TIMESA_IN_AAG(x1, x2, x3) = TIMESA_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (155) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 2, "program": { "directives": [], "clauses": [ [ "(add (b) (b) (b))", null ], [ "(add X (b) X)", "(binaryZ X)" ], [ "(add (b) Y Y)", "(binaryZ Y)" ], [ "(add X Y Z)", "(addz X Y Z)" ], [ "(addx (one X) (b) (one X))", "(binary X)" ], [ "(addx (zero X) (b) (zero X))", "(binaryZ X)" ], [ "(addx X Y Z)", "(addz X Y Z)" ], [ "(addy (b) (one Y) (one Y))", "(binary Y)" ], [ "(addy (b) (zero Y) (zero Y))", "(binaryZ Y)" ], [ "(addy X Y Z)", "(addz X Y Z)" ], [ "(addz (zero X) (zero Y) (zero Z))", "(addz X Y Z)" ], [ "(addz (zero X) (one Y) (one Z))", "(addx X Y Z)" ], [ "(addz (one X) (zero Y) (one Z))", "(addy X Y Z)" ], [ "(addz (one X) (one Y) (zero Z))", "(addc X Y Z)" ], [ "(addc (b) (b) (one (b)))", null ], [ "(addc X (b) Z)", "(succZ X Z)" ], [ "(addc (b) Y Z)", "(succZ Y Z)" ], [ "(addc X Y Z)", "(addC X Y Z)" ], [ "(addX (zero X) (b) (one X))", "(binaryZ X)" ], [ "(addX (one X) (b) (zero Z))", "(succ X Z)" ], [ "(addX X Y Z)", "(addC X Y Z)" ], [ "(addY (b) (zero Y) (one Y))", "(binaryZ Y)" ], [ "(addY (b) (one Y) (zero Z))", "(succ Y Z)" ], [ "(addY X Y Z)", "(addC X Y Z)" ], [ "(addC (zero X) (zero Y) (one Z))", "(addz X Y Z)" ], [ "(addC (zero X) (one Y) (zero Z))", "(addX X Y Z)" ], [ "(addC (one X) (zero Y) (zero Z))", "(addY X Y Z)" ], [ "(addC (one X) (one Y) (one Z))", "(addc X Y Z)" ], [ "(binary (b))", null ], [ "(binary (zero X))", "(binaryZ X)" ], [ "(binary (one X))", "(binary X)" ], [ "(binaryZ (zero X))", "(binaryZ X)" ], [ "(binaryZ (one X))", "(binary X)" ], [ "(succ (b) (one (b)))", null ], [ "(succ (zero X) (one X))", "(binaryZ X)" ], [ "(succ (one X) (zero Z))", "(succ X Z)" ], [ "(succZ (zero X) (one X))", "(binaryZ X)" ], [ "(succZ (one X) (zero Z))", 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}, "2411": { "goal": [{ "clause": -1, "scope": -1, "term": "(addz T551 T552 T550)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T550"], "free": [], "exprvars": [] } }, "1684": { "goal": [ { "clause": 34, "scope": 13, "term": "(succ T266 X331)" }, { "clause": 35, "scope": 13, "term": "(succ T266 X331)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X331"], "exprvars": [] } }, "1683": { "goal": [{ "clause": 33, "scope": 13, "term": "(succ T266 X331)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X331"], "exprvars": [] } }, "2418": { "goal": [{ "clause": 13, "scope": 22, "term": "(addz T483 T484 T482)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T482"], "free": [], "exprvars": [] } }, "2539": { "goal": [ { "clause": 25, "scope": 28, "term": "(addC T679 T680 T678)" }, { "clause": 26, "scope": 28, "term": "(addC T679 T680 T678)" }, { "clause": 27, "scope": 28, "term": "(addC T679 T680 T678)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T678"], "free": [], "exprvars": [] } }, "2417": { "goal": [{ "clause": 12, "scope": 22, "term": "(addz T483 T484 T482)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T482"], "free": [], "exprvars": [] } }, "2538": { "goal": [{ "clause": 24, "scope": 28, "term": "(addC T679 T680 T678)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T678"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 4, "label": "CASE" }, { "from": 4, "to": 5, "label": "PARALLEL" }, { "from": 4, "to": 6, "label": "PARALLEL" }, { "from": 5, "to": 8, "label": "EVAL with clause\ntimes(one(b), X5, X5).\nand substitutionT1 -> one(b),\nT2 -> T8,\nX5 -> T8,\nT3 -> T8" }, { "from": 5, "to": 9, "label": "EVAL-BACKTRACK" }, { "from": 6, "to": 11, "label": "PARALLEL" }, { "from": 6, "to": 12, "label": "PARALLEL" }, { "from": 8, "to": 10, "label": "SUCCESS" }, { "from": 11, "to": 16, "label": "EVAL with clause\ntimes(zero(X18), X19, zero(X20)) :- times(X18, X19, X20).\nand substitutionX18 -> T24,\nT1 -> zero(T24),\nT2 -> T25,\nX19 -> T25,\nX20 -> T23,\nT3 -> zero(T23),\nT21 -> T24,\nT22 -> T25" }, { "from": 11, "to": 18, "label": "EVAL-BACKTRACK" }, { "from": 12, "to": 212, "label": "EVAL with clause\ntimes(one(X30), X31, X32) :- ','(times(X30, X31, X33), add(X31, zero(X33), X32)).\nand substitutionX30 -> T37,\nT1 -> one(T37),\nT2 -> T38,\nX31 -> T38,\nT3 -> T36,\nX32 -> T36,\nT34 -> T37,\nT35 -> T38" }, { "from": 12, "to": 213, "label": "EVAL-BACKTRACK" }, { "from": 16, "to": 2, "label": "INSTANCE with matching:\nT1 -> T24\nT2 -> T25\nT3 -> T23" }, { "from": 212, "to": 254, "label": "SPLIT 1" }, { "from": 212, "to": 258, "label": "SPLIT 2\nnew knowledge:\nT37 is ground\nreplacements:X33 -> T41,\nT38 -> T42" }, { "from": 254, "to": 297, "label": "CASE" }, { "from": 258, "to": 2225, "label": "CASE" }, { "from": 297, "to": 311, "label": "PARALLEL" }, { "from": 297, "to": 313, "label": "PARALLEL" }, { "from": 311, "to": 334, "label": "EVAL with clause\ntimes(one(b), X42, X42).\nand substitutionT37 -> one(b),\nT38 -> T49,\nX42 -> T49,\nX33 -> T49" }, { "from": 311, "to": 340, "label": "EVAL-BACKTRACK" }, { "from": 313, "to": 377, "label": "PARALLEL" }, { "from": 313, "to": 380, "label": "PARALLEL" }, { "from": 334, "to": 345, "label": "SUCCESS" }, { "from": 377, "to": 433, "label": "EVAL with clause\ntimes(zero(X59), X60, zero(X61)) :- times(X59, X60, X61).\nand substitutionX59 -> T60,\nT37 -> zero(T60),\nT38 -> T61,\nX60 -> T61,\nX61 -> X62,\nX33 -> zero(X62),\nT58 -> T60,\nT59 -> T61" }, { "from": 377, "to": 443, "label": "EVAL-BACKTRACK" }, { "from": 380, "to": 522, "label": "EVAL with clause\ntimes(one(X74), X75, X76) :- ','(times(X74, X75, X77), add(X75, zero(X77), X76)).\nand substitutionX74 -> T70,\nT37 -> one(T70),\nT38 -> T71,\nX75 -> T71,\nX33 -> X78,\nX76 -> X78,\nT68 -> T70,\nT69 -> T71" }, { "from": 380, "to": 526, "label": "EVAL-BACKTRACK" }, { "from": 433, "to": 254, "label": "INSTANCE with matching:\nT37 -> T60\nT38 -> T61\nX33 -> X62" }, { "from": 522, "to": 606, "label": "SPLIT 1" }, { "from": 522, "to": 607, "label": "SPLIT 2\nnew knowledge:\nT70 is ground\nreplacements:X77 -> T74,\nT71 -> T75" }, { "from": 606, "to": 254, "label": "INSTANCE with matching:\nT37 -> T70\nT38 -> T71\nX33 -> X77" }, { "from": 607, "to": 608, "label": "CASE" }, { "from": 608, "to": 609, "label": "BACKTRACK\nfor clause: add(b, b, b)because of non-unification" }, { "from": 609, "to": 611, "label": "BACKTRACK\nfor clause: add(X, b, X) :- binaryZ(X)because of non-unification" }, { "from": 611, "to": 614, "label": "PARALLEL" }, { "from": 611, "to": 615, "label": "PARALLEL" }, { "from": 614, "to": 621, "label": "EVAL with clause\nadd(b, X88, X88) :- binaryZ(X88).\nand substitutionT75 -> b,\nT74 -> T84,\nX88 -> zero(T84),\nX78 -> zero(T84),\nT83 -> T84" }, { "from": 614, "to": 624, "label": "EVAL-BACKTRACK" }, { "from": 615, "to": 1060, "label": "ONLY EVAL with clause\nadd(X129, X130, X131) :- addz(X129, X130, X131).\nand substitutionT75 -> T129,\nX129 -> T129,\nT74 -> T130,\nX130 -> zero(T130),\nX78 -> X132,\nX131 -> X132,\nT127 -> T129,\nT128 -> T130" }, { "from": 621, "to": 626, "label": "CASE" }, { "from": 626, "to": 627, "label": "PARALLEL" }, { "from": 626, "to": 628, "label": "PARALLEL" }, { "from": 627, "to": 629, "label": "ONLY EVAL with clause\nbinaryZ(zero(X96)) :- binaryZ(X96).\nand substitutionT84 -> T95,\nX96 -> T95,\nT94 -> T95" }, { "from": 628, "to": 681, "label": "BACKTRACK\nfor clause: binaryZ(one(X)) :- binary(X)because of non-unification" }, { "from": 629, "to": 632, "label": "CASE" }, { "from": 632, "to": 633, "label": "PARALLEL" }, { "from": 632, "to": 634, "label": "PARALLEL" }, { "from": 633, "to": 635, "label": "EVAL with clause\nbinaryZ(zero(X102)) :- binaryZ(X102).\nand substitutionX102 -> T102,\nT95 -> zero(T102),\nT101 -> T102" }, { "from": 633, "to": 636, "label": "EVAL-BACKTRACK" }, { "from": 634, "to": 639, "label": "EVAL with clause\nbinaryZ(one(X106)) :- binary(X106).\nand substitutionX106 -> T107,\nT95 -> one(T107),\nT106 -> T107" }, { "from": 634, "to": 640, "label": "EVAL-BACKTRACK" }, { "from": 635, "to": 629, "label": "INSTANCE with matching:\nT95 -> T102" }, { "from": 639, "to": 644, "label": "CASE" }, { "from": 644, "to": 648, "label": "PARALLEL" }, { "from": 644, "to": 649, "label": "PARALLEL" }, { "from": 648, "to": 653, "label": "EVAL with clause\nbinary(b).\nand substitutionT107 -> b" }, { "from": 648, "to": 655, "label": "EVAL-BACKTRACK" }, { "from": 649, "to": 665, "label": "PARALLEL" }, { "from": 649, "to": 666, "label": "PARALLEL" }, { "from": 653, "to": 659, "label": "SUCCESS" }, { "from": 665, "to": 675, "label": "EVAL with clause\nbinary(zero(X111)) :- binaryZ(X111).\nand substitutionX111 -> T113,\nT107 -> zero(T113),\nT112 -> T113" }, { "from": 665, "to": 677, "label": "EVAL-BACKTRACK" }, { "from": 666, "to": 679, "label": "EVAL with clause\nbinary(one(X115)) :- binary(X115).\nand substitutionX115 -> T118,\nT107 -> one(T118),\nT117 -> T118" }, { "from": 666, "to": 680, "label": "EVAL-BACKTRACK" }, { "from": 675, "to": 629, "label": "INSTANCE with matching:\nT95 -> T113" }, { "from": 679, "to": 639, "label": "INSTANCE with matching:\nT107 -> T118" }, { "from": 1060, "to": 1080, "label": "CASE" }, { "from": 1080, "to": 1094, "label": "PARALLEL" }, { "from": 1080, "to": 1095, "label": "PARALLEL" }, { "from": 1094, "to": 1120, "label": "EVAL with clause\naddz(zero(X153), zero(X154), zero(X155)) :- addz(X153, X154, X155).\nand substitutionX153 -> T143,\nT129 -> zero(T143),\nT130 -> T144,\nX154 -> T144,\nX155 -> X156,\nX132 -> zero(X156),\nT141 -> T143,\nT142 -> T144" }, { "from": 1094, "to": 1124, "label": "EVAL-BACKTRACK" }, { "from": 1095, "to": 2205, "label": "BACKTRACK\nfor clause: addz(zero(X), one(Y), one(Z)) :- addx(X, Y, Z)because of non-unification" }, { "from": 1120, "to": 1140, "label": "CASE" }, { "from": 1140, "to": 1155, "label": "PARALLEL" }, { "from": 1140, "to": 1157, "label": "PARALLEL" }, { "from": 1155, "to": 1185, "label": "EVAL with clause\naddz(zero(X177), zero(X178), zero(X179)) :- addz(X177, X178, X179).\nand substitutionX177 -> T157,\nT143 -> zero(T157),\nX178 -> T158,\nT144 -> zero(T158),\nX179 -> X180,\nX156 -> zero(X180),\nT155 -> T157,\nT156 -> T158" }, { "from": 1155, "to": 1188, "label": "EVAL-BACKTRACK" }, { "from": 1157, "to": 1220, "label": "PARALLEL" }, { "from": 1157, "to": 1221, "label": "PARALLEL" }, { "from": 1185, "to": 1120, "label": "INSTANCE with matching:\nT143 -> T157\nT144 -> T158\nX156 -> X180" }, { "from": 1220, "to": 1318, "label": "EVAL with clause\naddz(zero(X201), one(X202), one(X203)) :- addx(X201, X202, X203).\nand substitutionX201 -> T171,\nT143 -> zero(T171),\nX202 -> T172,\nT144 -> one(T172),\nX203 -> X204,\nX156 -> one(X204),\nT169 -> T171,\nT170 -> T172" }, { "from": 1220, "to": 1322, "label": "EVAL-BACKTRACK" }, { "from": 1221, "to": 1494, "label": "PARALLEL" }, { "from": 1221, "to": 1496, "label": "PARALLEL" }, { "from": 1318, "to": 1338, "label": "CASE" }, { "from": 1338, "to": 1350, "label": "PARALLEL" }, { "from": 1338, "to": 1351, "label": "PARALLEL" }, { "from": 1350, "to": 1368, "label": "EVAL with clause\naddx(one(X210), b, one(X210)) :- binary(X210).\nand substitutionX210 -> T179,\nT171 -> one(T179),\nT172 -> b,\nX204 -> one(T179),\nT178 -> T179" }, { "from": 1350, "to": 1369, "label": "EVAL-BACKTRACK" }, { "from": 1351, "to": 1387, "label": "PARALLEL" }, { "from": 1351, "to": 1389, "label": "PARALLEL" }, { "from": 1368, "to": 639, "label": "INSTANCE with matching:\nT107 -> T179" }, { "from": 1387, "to": 1401, "label": "EVAL with clause\naddx(zero(X215), b, zero(X215)) :- binaryZ(X215).\nand substitutionX215 -> T185,\nT171 -> zero(T185),\nT172 -> b,\nX204 -> zero(T185),\nT184 -> T185" }, { "from": 1387, "to": 1404, "label": "EVAL-BACKTRACK" }, { "from": 1389, "to": 1458, "label": "ONLY EVAL with clause\naddx(X229, X230, X231) :- addz(X229, X230, X231).\nand substitutionT171 -> T197,\nX229 -> T197,\nT172 -> T198,\nX230 -> T198,\nX204 -> X232,\nX231 -> X232,\nT195 -> T197,\nT196 -> T198" }, { "from": 1401, "to": 629, "label": "INSTANCE with matching:\nT95 -> T185" }, { "from": 1458, "to": 1120, "label": "INSTANCE with matching:\nT143 -> T197\nT144 -> T198\nX156 -> X232" }, { "from": 1494, "to": 1610, "label": "EVAL with clause\naddz(one(X253), zero(X254), one(X255)) :- addy(X253, X254, X255).\nand substitutionX253 -> T211,\nT143 -> one(T211),\nX254 -> T212,\nT144 -> zero(T212),\nX255 -> X256,\nX156 -> one(X256),\nT209 -> T211,\nT210 -> T212" }, { "from": 1494, "to": 1611, "label": "EVAL-BACKTRACK" }, { "from": 1496, "to": 1638, "label": "EVAL with clause\naddz(one(X297), one(X298), zero(X299)) :- addc(X297, X298, X299).\nand substitutionX297 -> T247,\nT143 -> one(T247),\nX298 -> T248,\nT144 -> one(T248),\nX299 -> X300,\nX156 -> zero(X300),\nT245 -> T247,\nT246 -> T248" }, { "from": 1496, "to": 1639, "label": "EVAL-BACKTRACK" }, { "from": 1610, "to": 1612, "label": "CASE" }, { "from": 1612, "to": 1615, "label": "PARALLEL" }, { "from": 1612, "to": 1616, "label": "PARALLEL" }, { "from": 1615, "to": 1617, "label": "EVAL with clause\naddy(b, one(X262), one(X262)) :- binary(X262).\nand substitutionT211 -> b,\nX262 -> T219,\nT212 -> one(T219),\nX256 -> one(T219),\nT218 -> T219" }, { "from": 1615, "to": 1618, "label": "EVAL-BACKTRACK" }, { "from": 1616, "to": 1621, "label": "PARALLEL" }, { "from": 1616, "to": 1622, "label": "PARALLEL" }, { "from": 1617, "to": 639, "label": "INSTANCE with matching:\nT107 -> T219" }, { "from": 1621, "to": 1623, "label": "EVAL with clause\naddy(b, zero(X267), zero(X267)) :- binaryZ(X267).\nand substitutionT211 -> b,\nX267 -> T225,\nT212 -> zero(T225),\nX256 -> zero(T225),\nT224 -> T225" }, { "from": 1621, "to": 1625, "label": "EVAL-BACKTRACK" }, { "from": 1622, "to": 1637, "label": "ONLY EVAL with clause\naddy(X281, X282, X283) :- addz(X281, X282, X283).\nand substitutionT211 -> T237,\nX281 -> T237,\nT212 -> T238,\nX282 -> T238,\nX256 -> X284,\nX283 -> X284,\nT235 -> T237,\nT236 -> T238" }, { "from": 1623, "to": 629, "label": "INSTANCE with matching:\nT95 -> T225" }, { "from": 1637, "to": 1120, "label": "INSTANCE with matching:\nT143 -> T237\nT144 -> T238\nX156 -> X284" }, { "from": 1638, "to": 1640, "label": "CASE" }, { "from": 1640, "to": 1641, "label": "PARALLEL" }, { "from": 1640, "to": 1642, "label": "PARALLEL" }, { "from": 1641, "to": 1643, "label": "EVAL with clause\naddc(b, b, one(b)).\nand substitutionT247 -> b,\nT248 -> b,\nX300 -> one(b)" }, { "from": 1641, "to": 1644, "label": "EVAL-BACKTRACK" }, { "from": 1642, "to": 1646, "label": "PARALLEL" }, { "from": 1642, "to": 1647, "label": "PARALLEL" }, { "from": 1643, "to": 1645, "label": "SUCCESS" }, { "from": 1646, "to": 1648, "label": "EVAL with clause\naddc(X313, b, X314) :- succZ(X313, X314).\nand substitutionT247 -> T254,\nX313 -> T254,\nT248 -> b,\nX300 -> X315,\nX314 -> X315,\nT253 -> T254" }, { "from": 1646, "to": 1649, "label": "EVAL-BACKTRACK" }, { "from": 1647, "to": 1701, "label": "PARALLEL" }, { "from": 1647, "to": 1702, "label": "PARALLEL" }, { "from": 1648, "to": 1650, "label": "CASE" }, { "from": 1650, "to": 1651, "label": "PARALLEL" }, { "from": 1650, "to": 1652, "label": "PARALLEL" }, { "from": 1651, "to": 1653, "label": "EVAL with clause\nsuccZ(zero(X321), one(X321)) :- binaryZ(X321).\nand substitutionX321 -> T261,\nT254 -> zero(T261),\nX315 -> one(T261),\nT260 -> T261" }, { "from": 1651, "to": 1654, "label": "EVAL-BACKTRACK" }, { "from": 1652, "to": 1680, "label": "EVAL with clause\nsuccZ(one(X329), zero(X330)) :- succ(X329, X330).\nand substitutionX329 -> T266,\nT254 -> one(T266),\nX330 -> X331,\nX315 -> zero(X331),\nT265 -> T266" }, { "from": 1652, "to": 1681, "label": "EVAL-BACKTRACK" }, { "from": 1653, "to": 629, "label": "INSTANCE with matching:\nT95 -> T261" }, { "from": 1680, "to": 1682, "label": "CASE" }, { "from": 1682, "to": 1683, "label": "PARALLEL" }, { "from": 1682, "to": 1684, "label": "PARALLEL" }, { "from": 1683, "to": 1685, "label": "EVAL with clause\nsucc(b, one(b)).\nand substitutionT266 -> b,\nX331 -> one(b)" }, { "from": 1683, "to": 1686, "label": "EVAL-BACKTRACK" }, { "from": 1684, "to": 1688, "label": "PARALLEL" }, { "from": 1684, "to": 1689, "label": "PARALLEL" }, { "from": 1685, "to": 1687, "label": "SUCCESS" }, { "from": 1688, "to": 1692, "label": "EVAL with clause\nsucc(zero(X336), one(X336)) :- binaryZ(X336).\nand substitutionX336 -> T272,\nT266 -> zero(T272),\nX331 -> one(T272),\nT271 -> T272" }, { "from": 1688, "to": 1693, "label": "EVAL-BACKTRACK" }, { "from": 1689, "to": 1699, "label": "EVAL with clause\nsucc(one(X344), zero(X345)) :- succ(X344, X345).\nand substitutionX344 -> T277,\nT266 -> one(T277),\nX345 -> X346,\nX331 -> zero(X346),\nT276 -> T277" }, { "from": 1689, "to": 1700, "label": "EVAL-BACKTRACK" }, { "from": 1692, "to": 629, "label": "INSTANCE with matching:\nT95 -> T272" }, { "from": 1699, "to": 1680, "label": "INSTANCE with matching:\nT266 -> T277\nX331 -> X346" }, { "from": 1701, "to": 1703, "label": "EVAL with clause\naddc(b, X359, X360) :- succZ(X359, X360).\nand substitutionT247 -> b,\nT248 -> T283,\nX359 -> T283,\nX300 -> X361,\nX360 -> X361,\nT282 -> T283" }, { "from": 1701, "to": 1704, "label": "EVAL-BACKTRACK" }, { "from": 1702, "to": 2062, "label": "ONLY EVAL with clause\naddc(X375, X376, X377) :- addC(X375, X376, X377).\nand substitutionT247 -> T295,\nX375 -> T295,\nT248 -> T296,\nX376 -> T296,\nX300 -> X378,\nX377 -> X378,\nT293 -> T295,\nT294 -> T296" }, { "from": 1703, "to": 1648, "label": "INSTANCE with matching:\nT254 -> T283\nX315 -> X361" }, { "from": 2062, "to": 2066, "label": "CASE" }, { "from": 2066, "to": 2069, "label": "PARALLEL" }, { "from": 2066, "to": 2070, "label": "PARALLEL" }, { "from": 2069, "to": 2071, "label": "EVAL with clause\naddC(zero(X399), zero(X400), one(X401)) :- addz(X399, X400, X401).\nand substitutionX399 -> T309,\nT295 -> zero(T309),\nX400 -> T310,\nT296 -> zero(T310),\nX401 -> X402,\nX378 -> one(X402),\nT307 -> T309,\nT308 -> T310" }, { "from": 2069, "to": 2072, "label": "EVAL-BACKTRACK" }, { "from": 2070, "to": 2073, "label": "PARALLEL" }, { "from": 2070, "to": 2074, "label": "PARALLEL" }, { "from": 2071, "to": 1120, "label": "INSTANCE with matching:\nT143 -> T309\nT144 -> T310\nX156 -> X402" }, { "from": 2073, "to": 2083, "label": "EVAL with clause\naddC(zero(X423), one(X424), zero(X425)) :- addX(X423, X424, X425).\nand substitutionX423 -> T323,\nT295 -> zero(T323),\nX424 -> T324,\nT296 -> one(T324),\nX425 -> X426,\nX378 -> zero(X426),\nT321 -> T323,\nT322 -> T324" }, { "from": 2073, "to": 2084, "label": "EVAL-BACKTRACK" }, { "from": 2074, "to": 2145, "label": "PARALLEL" }, { "from": 2074, "to": 2146, "label": "PARALLEL" }, { "from": 2083, "to": 2085, "label": "CASE" }, { "from": 2085, "to": 2086, "label": "PARALLEL" }, { "from": 2085, "to": 2087, "label": "PARALLEL" }, { "from": 2086, "to": 2088, "label": "EVAL with clause\naddX(zero(X432), b, one(X432)) :- binaryZ(X432).\nand substitutionX432 -> T331,\nT323 -> zero(T331),\nT324 -> b,\nX426 -> one(T331),\nT330 -> T331" }, { "from": 2086, "to": 2089, "label": "EVAL-BACKTRACK" }, { "from": 2087, "to": 2090, "label": "PARALLEL" }, { "from": 2087, "to": 2091, "label": "PARALLEL" }, { "from": 2088, "to": 629, "label": "INSTANCE with matching:\nT95 -> T331" }, { "from": 2090, "to": 2097, "label": "EVAL with clause\naddX(one(X446), b, zero(X447)) :- succ(X446, X447).\nand substitutionX446 -> T338,\nT323 -> one(T338),\nT324 -> b,\nX447 -> X448,\nX426 -> zero(X448),\nT337 -> T338" }, { "from": 2090, "to": 2098, "label": "EVAL-BACKTRACK" }, { "from": 2091, "to": 2111, "label": "ONLY EVAL with clause\naddX(X461, X462, X463) :- addC(X461, X462, X463).\nand substitutionT323 -> T349,\nX461 -> T349,\nT324 -> T350,\nX462 -> T350,\nX426 -> X464,\nX463 -> X464,\nT347 -> T349,\nT348 -> T350" }, { "from": 2097, "to": 1680, "label": "INSTANCE with matching:\nT266 -> T338\nX331 -> X448" }, { "from": 2111, "to": 2062, "label": "INSTANCE with matching:\nT295 -> T349\nT296 -> T350\nX378 -> X464" }, { "from": 2145, "to": 2149, "label": "EVAL with clause\naddC(one(X485), zero(X486), zero(X487)) :- addY(X485, X486, X487).\nand substitutionX485 -> T363,\nT295 -> one(T363),\nX486 -> T364,\nT296 -> zero(T364),\nX487 -> X488,\nX378 -> zero(X488),\nT361 -> T363,\nT362 -> T364" }, { "from": 2145, "to": 2150, "label": "EVAL-BACKTRACK" }, { "from": 2146, "to": 2189, "label": "EVAL with clause\naddC(one(X539), one(X540), one(X541)) :- addc(X539, X540, X541).\nand substitutionX539 -> T399,\nT295 -> one(T399),\nX540 -> T400,\nT296 -> one(T400),\nX541 -> X542,\nX378 -> one(X542),\nT397 -> T399,\nT398 -> T400" }, { "from": 2146, "to": 2190, "label": "EVAL-BACKTRACK" }, { "from": 2149, "to": 2153, "label": "CASE" }, { "from": 2153, "to": 2154, "label": "PARALLEL" }, { "from": 2153, "to": 2155, "label": "PARALLEL" }, { "from": 2154, "to": 2158, "label": "EVAL with clause\naddY(b, zero(X494), one(X494)) :- binaryZ(X494).\nand substitutionT363 -> b,\nX494 -> T371,\nT364 -> zero(T371),\nX488 -> one(T371),\nT370 -> T371" }, { "from": 2154, "to": 2159, "label": "EVAL-BACKTRACK" }, { "from": 2155, "to": 2173, "label": "PARALLEL" }, { "from": 2155, "to": 2174, "label": "PARALLEL" }, { "from": 2158, "to": 629, "label": "INSTANCE with matching:\nT95 -> T371" }, { "from": 2173, "to": 2176, "label": "EVAL with clause\naddY(b, one(X508), zero(X509)) :- succ(X508, X509).\nand substitutionT363 -> b,\nX508 -> T378,\nT364 -> one(T378),\nX509 -> X510,\nX488 -> zero(X510),\nT377 -> T378" }, { "from": 2173, "to": 2177, "label": "EVAL-BACKTRACK" }, { "from": 2174, "to": 2182, "label": "ONLY EVAL with clause\naddY(X523, X524, X525) :- addC(X523, X524, X525).\nand substitutionT363 -> T389,\nX523 -> T389,\nT364 -> T390,\nX524 -> T390,\nX488 -> X526,\nX525 -> X526,\nT387 -> T389,\nT388 -> T390" }, { "from": 2176, "to": 1680, "label": "INSTANCE with matching:\nT266 -> T378\nX331 -> X510" }, { "from": 2182, "to": 2062, "label": "INSTANCE with matching:\nT295 -> T389\nT296 -> T390\nX378 -> X526" }, { "from": 2189, "to": 1638, "label": "INSTANCE with matching:\nT247 -> T399\nT248 -> T400\nX300 -> X542" }, { "from": 2205, "to": 2208, "label": "PARALLEL" }, { "from": 2205, "to": 2209, "label": "PARALLEL" }, { "from": 2208, "to": 2220, "label": "EVAL with clause\naddz(one(X562), zero(X563), one(X564)) :- addy(X562, X563, X564).\nand substitutionX562 -> T412,\nT129 -> one(T412),\nT130 -> T413,\nX563 -> T413,\nX564 -> X565,\nX132 -> one(X565),\nT410 -> T412,\nT411 -> T413" }, { "from": 2208, "to": 2221, "label": "EVAL-BACKTRACK" }, { "from": 2209, "to": 2224, "label": "BACKTRACK\nfor clause: addz(one(X), one(Y), zero(Z)) :- addc(X, Y, Z)because of non-unification" }, { "from": 2220, "to": 1610, "label": "INSTANCE with matching:\nT211 -> T412\nT212 -> T413\nX256 -> X565" }, { "from": 2225, "to": 2226, "label": "BACKTRACK\nfor clause: add(b, b, b)because of non-unification" }, { "from": 2226, "to": 2227, "label": "BACKTRACK\nfor clause: add(X, b, X) :- binaryZ(X)because of non-unification" }, { "from": 2227, "to": 2230, "label": "PARALLEL" }, { "from": 2227, "to": 2231, "label": "PARALLEL" }, { "from": 2230, "to": 2232, "label": "EVAL with clause\nadd(b, X575, X575) :- binaryZ(X575).\nand substitutionT42 -> b,\nT41 -> T421,\nX575 -> zero(T421),\nT36 -> zero(T421)" }, { "from": 2230, "to": 2233, "label": "EVAL-BACKTRACK" }, { "from": 2231, "to": 2318, "label": "ONLY EVAL with clause\nadd(X613, X614, X615) :- addz(X613, X614, X615).\nand substitutionT42 -> T463,\nX613 -> T463,\nT41 -> T464,\nX614 -> zero(T464),\nT36 -> T462,\nX615 -> T462,\nT460 -> T463,\nT461 -> T464" }, { "from": 2232, "to": 2234, "label": "CASE" }, { "from": 2234, "to": 2237, "label": "PARALLEL" }, { "from": 2234, "to": 2238, "label": "PARALLEL" }, { "from": 2237, "to": 2240, "label": "ONLY EVAL with clause\nbinaryZ(zero(X583)) :- binaryZ(X583).\nand substitutionT421 -> T429,\nX583 -> T429" }, { "from": 2238, "to": 2271, "label": "BACKTRACK\nfor clause: binaryZ(one(X)) :- binary(X)because of non-unification" }, { "from": 2240, "to": 2243, "label": "CASE" }, { "from": 2243, "to": 2246, "label": "PARALLEL" }, { "from": 2243, "to": 2247, "label": "PARALLEL" }, { "from": 2246, "to": 2248, "label": "EVAL with clause\nbinaryZ(zero(X589)) :- binaryZ(X589).\nand substitutionX589 -> T435,\nT429 -> zero(T435)" }, { "from": 2246, "to": 2249, "label": "EVAL-BACKTRACK" }, { "from": 2247, "to": 2254, "label": "EVAL with clause\nbinaryZ(one(X593)) :- binary(X593).\nand substitutionX593 -> T439,\nT429 -> one(T439)" }, { "from": 2247, "to": 2255, "label": "EVAL-BACKTRACK" }, { "from": 2248, "to": 2240, "label": "INSTANCE with matching:\nT429 -> T435" }, { "from": 2254, "to": 2256, "label": "CASE" }, { "from": 2256, "to": 2257, "label": "PARALLEL" }, { "from": 2256, "to": 2258, "label": "PARALLEL" }, { "from": 2257, "to": 2259, "label": "EVAL with clause\nbinary(b).\nand substitutionT439 -> b" }, { "from": 2257, "to": 2260, "label": "EVAL-BACKTRACK" }, { "from": 2258, "to": 2264, "label": "PARALLEL" }, { "from": 2258, "to": 2265, "label": "PARALLEL" }, { "from": 2259, "to": 2261, "label": "SUCCESS" }, { "from": 2264, "to": 2267, "label": "EVAL with clause\nbinary(zero(X598)) :- binaryZ(X598).\nand substitutionX598 -> T444,\nT439 -> zero(T444)" }, { "from": 2264, "to": 2268, "label": "EVAL-BACKTRACK" }, { "from": 2265, "to": 2269, "label": "EVAL with clause\nbinary(one(X602)) :- binary(X602).\nand substitutionX602 -> T448,\nT439 -> one(T448)" }, { "from": 2265, "to": 2270, "label": "EVAL-BACKTRACK" }, { "from": 2267, "to": 2240, "label": "INSTANCE with matching:\nT429 -> T444" }, { "from": 2269, "to": 2254, "label": "INSTANCE with matching:\nT439 -> T448" }, { "from": 2318, "to": 2321, "label": "CASE" }, { "from": 2321, "to": 2327, "label": "PARALLEL" }, { "from": 2321, "to": 2328, "label": "PARALLEL" }, { "from": 2327, "to": 2333, "label": "EVAL with clause\naddz(zero(X631), zero(X632), zero(X633)) :- addz(X631, X632, X633).\nand substitutionX631 -> T483,\nT463 -> zero(T483),\nT464 -> T484,\nX632 -> T484,\nX633 -> T482,\nT462 -> zero(T482),\nT480 -> T483,\nT481 -> T484" }, { "from": 2327, "to": 2334, "label": "EVAL-BACKTRACK" }, { "from": 2328, "to": 2641, "label": "BACKTRACK\nfor clause: addz(zero(X), one(Y), one(Z)) :- addx(X, Y, Z)because of non-unification" }, { "from": 2333, "to": 2335, "label": "CASE" }, { "from": 2335, "to": 2343, "label": "PARALLEL" }, { "from": 2335, "to": 2345, "label": "PARALLEL" }, { "from": 2343, "to": 2357, "label": "EVAL with clause\naddz(zero(X649), zero(X650), zero(X651)) :- addz(X649, X650, X651).\nand substitutionX649 -> T503,\nT483 -> zero(T503),\nX650 -> T504,\nT484 -> zero(T504),\nX651 -> T502,\nT482 -> zero(T502),\nT500 -> T503,\nT501 -> T504" }, { "from": 2343, "to": 2358, "label": "EVAL-BACKTRACK" }, { "from": 2345, "to": 2369, "label": "PARALLEL" }, { "from": 2345, "to": 2370, "label": "PARALLEL" }, { "from": 2357, "to": 2333, "label": "INSTANCE with matching:\nT483 -> T503\nT484 -> T504\nT482 -> T502" }, { "from": 2369, "to": 2371, "label": "EVAL with clause\naddz(zero(X667), one(X668), one(X669)) :- addx(X667, X668, X669).\nand substitutionX667 -> T523,\nT483 -> zero(T523),\nX668 -> T524,\nT484 -> one(T524),\nX669 -> T522,\nT482 -> one(T522),\nT520 -> T523,\nT521 -> T524" }, { "from": 2369, "to": 2372, "label": "EVAL-BACKTRACK" }, { "from": 2370, "to": 2417, "label": "PARALLEL" }, { "from": 2370, "to": 2418, "label": "PARALLEL" }, { "from": 2371, "to": 2373, "label": "CASE" }, { "from": 2373, "to": 2375, "label": "PARALLEL" }, { "from": 2373, "to": 2376, "label": "PARALLEL" }, { "from": 2375, "to": 2380, "label": "EVAL with clause\naddx(one(X675), b, one(X675)) :- binary(X675).\nand substitutionX675 -> T530,\nT523 -> one(T530),\nT524 -> b,\nT522 -> one(T530)" }, { "from": 2375, "to": 2393, "label": "EVAL-BACKTRACK" }, { "from": 2376, "to": 2398, "label": "PARALLEL" }, { "from": 2376, "to": 2399, "label": "PARALLEL" }, { "from": 2380, "to": 2254, "label": "INSTANCE with matching:\nT439 -> T530" }, { "from": 2398, "to": 2402, "label": "EVAL with clause\naddx(zero(X680), b, zero(X680)) :- binaryZ(X680).\nand substitutionX680 -> T535,\nT523 -> zero(T535),\nT524 -> b,\nT522 -> zero(T535)" }, { "from": 2398, "to": 2403, "label": "EVAL-BACKTRACK" }, { "from": 2399, "to": 2411, "label": "ONLY EVAL with clause\naddx(X691, X692, X693) :- addz(X691, X692, X693).\nand substitutionT523 -> T551,\nX691 -> T551,\nT524 -> T552,\nX692 -> T552,\nT522 -> T550,\nX693 -> T550,\nT548 -> T551,\nT549 -> T552" }, { "from": 2402, "to": 2240, "label": "INSTANCE with matching:\nT429 -> T535" }, { "from": 2411, "to": 2333, "label": "INSTANCE with matching:\nT483 -> T551\nT484 -> T552\nT482 -> T550" }, { "from": 2417, "to": 2421, "label": "EVAL with clause\naddz(one(X709), zero(X710), one(X711)) :- addy(X709, X710, X711).\nand substitutionX709 -> T571,\nT483 -> one(T571),\nX710 -> T572,\nT484 -> zero(T572),\nX711 -> T570,\nT482 -> one(T570),\nT568 -> T571,\nT569 -> T572" }, { "from": 2417, "to": 2422, "label": "EVAL-BACKTRACK" }, { "from": 2418, "to": 2461, "label": "EVAL with clause\naddz(one(X745), one(X746), zero(X747)) :- addc(X745, X746, X747).\nand substitutionX745 -> T613,\nT483 -> one(T613),\nX746 -> T614,\nT484 -> one(T614),\nX747 -> T612,\nT482 -> zero(T612),\nT610 -> T613,\nT611 -> T614" }, { "from": 2418, "to": 2463, "label": "EVAL-BACKTRACK" }, { "from": 2421, "to": 2425, "label": "CASE" }, { "from": 2425, "to": 2426, "label": "PARALLEL" }, { "from": 2425, "to": 2427, "label": "PARALLEL" }, { "from": 2426, "to": 2432, "label": "EVAL with clause\naddy(b, one(X717), one(X717)) :- binary(X717).\nand substitutionT571 -> b,\nX717 -> T578,\nT572 -> one(T578),\nT570 -> one(T578)" }, { "from": 2426, "to": 2433, "label": "EVAL-BACKTRACK" }, { "from": 2427, "to": 2436, "label": "PARALLEL" }, { "from": 2427, "to": 2437, "label": "PARALLEL" }, { "from": 2432, "to": 2254, "label": "INSTANCE with matching:\nT439 -> T578" }, { "from": 2436, "to": 2438, "label": "EVAL with clause\naddy(b, zero(X722), zero(X722)) :- binaryZ(X722).\nand substitutionT571 -> b,\nX722 -> T583,\nT572 -> zero(T583),\nT570 -> zero(T583)" }, { "from": 2436, "to": 2439, "label": "EVAL-BACKTRACK" }, { "from": 2437, "to": 2454, "label": "ONLY EVAL with clause\naddy(X733, X734, X735) :- addz(X733, X734, X735).\nand substitutionT571 -> T599,\nX733 -> T599,\nT572 -> T600,\nX734 -> T600,\nT570 -> T598,\nX735 -> T598,\nT596 -> T599,\nT597 -> T600" }, { "from": 2438, "to": 2240, "label": "INSTANCE with matching:\nT429 -> T583" }, { "from": 2454, "to": 2333, "label": "INSTANCE with matching:\nT483 -> T599\nT484 -> T600\nT482 -> T598" }, { "from": 2461, "to": 2465, "label": "CASE" }, { "from": 2465, "to": 2466, "label": "PARALLEL" }, { "from": 2465, "to": 2467, "label": "PARALLEL" }, { "from": 2466, "to": 2468, "label": "EVAL with clause\naddc(b, b, one(b)).\nand substitutionT613 -> b,\nT614 -> b,\nT612 -> one(b)" }, { "from": 2466, "to": 2469, "label": "EVAL-BACKTRACK" }, { "from": 2467, "to": 2471, "label": "PARALLEL" }, { "from": 2467, "to": 2472, "label": "PARALLEL" }, { "from": 2468, "to": 2470, "label": "SUCCESS" }, { "from": 2471, "to": 2473, "label": "EVAL with clause\naddc(X756, b, X757) :- succZ(X756, X757).\nand substitutionT613 -> T625,\nX756 -> T625,\nT614 -> b,\nT612 -> T624,\nX757 -> T624,\nT623 -> T625" }, { "from": 2471, "to": 2474, "label": "EVAL-BACKTRACK" }, { "from": 2472, "to": 2503, "label": "PARALLEL" }, { "from": 2472, "to": 2504, "label": "PARALLEL" }, { "from": 2473, "to": 2475, "label": "CASE" }, { "from": 2475, "to": 2478, "label": "PARALLEL" }, { "from": 2475, "to": 2479, "label": "PARALLEL" }, { "from": 2478, "to": 2481, "label": "EVAL with clause\nsuccZ(zero(X763), one(X763)) :- binaryZ(X763).\nand substitutionX763 -> T631,\nT625 -> zero(T631),\nT624 -> one(T631)" }, { "from": 2478, "to": 2482, "label": "EVAL-BACKTRACK" }, { "from": 2479, "to": 2485, "label": "EVAL with clause\nsuccZ(one(X769), zero(X770)) :- succ(X769, X770).\nand substitutionX769 -> T639,\nT625 -> one(T639),\nX770 -> T638,\nT624 -> zero(T638),\nT637 -> T639" }, { "from": 2479, "to": 2486, "label": "EVAL-BACKTRACK" }, { "from": 2481, "to": 2240, "label": "INSTANCE with matching:\nT429 -> T631" }, { "from": 2485, "to": 2487, "label": "CASE" }, { "from": 2487, "to": 2488, "label": "PARALLEL" }, { "from": 2487, "to": 2489, "label": "PARALLEL" }, { "from": 2488, "to": 2490, "label": "EVAL with clause\nsucc(b, one(b)).\nand substitutionT639 -> b,\nT638 -> one(b)" }, { "from": 2488, "to": 2491, "label": "EVAL-BACKTRACK" }, { "from": 2489, "to": 2493, "label": "PARALLEL" }, { "from": 2489, "to": 2494, "label": "PARALLEL" }, { "from": 2490, "to": 2492, "label": "SUCCESS" }, { "from": 2493, "to": 2497, "label": "EVAL with clause\nsucc(zero(X775), one(X775)) :- binaryZ(X775).\nand substitutionX775 -> T644,\nT639 -> zero(T644),\nT638 -> one(T644)" }, { "from": 2493, "to": 2498, "label": "EVAL-BACKTRACK" }, { "from": 2494, "to": 2501, "label": "EVAL with clause\nsucc(one(X781), zero(X782)) :- succ(X781, X782).\nand substitutionX781 -> T652,\nT639 -> one(T652),\nX782 -> T651,\nT638 -> zero(T651),\nT650 -> T652" }, { "from": 2494, "to": 2502, "label": "EVAL-BACKTRACK" }, { "from": 2497, "to": 2240, "label": "INSTANCE with matching:\nT429 -> T644" }, { "from": 2501, "to": 2485, "label": "INSTANCE with matching:\nT639 -> T652\nT638 -> T651" }, { "from": 2503, "to": 2515, "label": "EVAL with clause\naddc(b, X791, X792) :- succZ(X791, X792).\nand substitutionT613 -> b,\nT614 -> T663,\nX791 -> T663,\nT612 -> T662,\nX792 -> T662,\nT661 -> T663" }, { "from": 2503, "to": 2516, "label": "EVAL-BACKTRACK" }, { "from": 2504, "to": 2534, "label": "ONLY EVAL with clause\naddc(X803, X804, X805) :- addC(X803, X804, X805).\nand substitutionT613 -> T679,\nX803 -> T679,\nT614 -> T680,\nX804 -> T680,\nT612 -> T678,\nX805 -> T678,\nT676 -> T679,\nT677 -> T680" }, { "from": 2515, "to": 2473, "label": "INSTANCE with matching:\nT625 -> T663\nT624 -> T662" }, { "from": 2534, "to": 2536, "label": "CASE" }, { "from": 2536, "to": 2538, "label": "PARALLEL" }, { "from": 2536, "to": 2539, "label": "PARALLEL" }, { "from": 2538, "to": 2542, "label": "EVAL with clause\naddC(zero(X821), zero(X822), one(X823)) :- addz(X821, X822, X823).\nand substitutionX821 -> T699,\nT679 -> zero(T699),\nX822 -> T700,\nT680 -> zero(T700),\nX823 -> T698,\nT678 -> one(T698),\nT696 -> T699,\nT697 -> T700" }, { "from": 2538, "to": 2543, "label": "EVAL-BACKTRACK" }, { "from": 2539, "to": 2544, "label": "PARALLEL" }, { "from": 2539, "to": 2545, "label": "PARALLEL" }, { "from": 2542, "to": 2333, "label": "INSTANCE with matching:\nT483 -> T699\nT484 -> T700\nT482 -> T698" }, { "from": 2544, "to": 2546, "label": "EVAL with clause\naddC(zero(X839), one(X840), zero(X841)) :- addX(X839, X840, X841).\nand substitutionX839 -> T719,\nT679 -> zero(T719),\nX840 -> T720,\nT680 -> one(T720),\nX841 -> T718,\nT678 -> zero(T718),\nT716 -> T719,\nT717 -> T720" }, { "from": 2544, "to": 2547, "label": "EVAL-BACKTRACK" }, { "from": 2545, "to": 2576, "label": "PARALLEL" }, { "from": 2545, "to": 2577, "label": "PARALLEL" }, { "from": 2546, "to": 2548, "label": "CASE" }, { "from": 2548, "to": 2549, "label": "PARALLEL" }, { "from": 2548, "to": 2550, "label": "PARALLEL" }, { "from": 2549, "to": 2551, "label": "EVAL with clause\naddX(zero(X847), b, one(X847)) :- binaryZ(X847).\nand substitutionX847 -> T726,\nT719 -> zero(T726),\nT720 -> b,\nT718 -> one(T726)" }, { "from": 2549, "to": 2552, "label": "EVAL-BACKTRACK" }, { "from": 2550, "to": 2555, "label": "PARALLEL" }, { "from": 2550, "to": 2556, "label": "PARALLEL" }, { "from": 2551, "to": 2240, "label": "INSTANCE with matching:\nT429 -> T726" }, { "from": 2555, "to": 2557, "label": "EVAL with clause\naddX(one(X857), b, zero(X858)) :- succ(X857, X858).\nand substitutionX857 -> T738,\nT719 -> one(T738),\nT720 -> b,\nX858 -> T737,\nT718 -> zero(T737),\nT736 -> T738" }, { "from": 2555, "to": 2558, "label": "EVAL-BACKTRACK" }, { "from": 2556, "to": 2573, "label": "ONLY EVAL with clause\naddX(X868, X869, X870) :- addC(X868, X869, X870).\nand substitutionT719 -> T753,\nX868 -> T753,\nT720 -> T754,\nX869 -> T754,\nT718 -> T752,\nX870 -> T752,\nT750 -> T753,\nT751 -> T754" }, { "from": 2557, "to": 2485, "label": "INSTANCE with matching:\nT639 -> T738\nT638 -> T737" }, { "from": 2573, "to": 2534, "label": "INSTANCE with matching:\nT679 -> T753\nT680 -> T754\nT678 -> T752" }, { "from": 2576, "to": 2583, "label": "EVAL with clause\naddC(one(X886), zero(X887), zero(X888)) :- addY(X886, X887, X888).\nand substitutionX886 -> T773,\nT679 -> one(T773),\nX887 -> T774,\nT680 -> zero(T774),\nX888 -> T772,\nT678 -> zero(T772),\nT770 -> T773,\nT771 -> T774" }, { "from": 2576, "to": 2584, "label": "EVAL-BACKTRACK" }, { "from": 2577, "to": 2638, "label": "EVAL with clause\naddC(one(X927), one(X928), one(X929)) :- addc(X927, X928, X929).\nand substitutionX927 -> T821,\nT679 -> one(T821),\nX928 -> T822,\nT680 -> one(T822),\nX929 -> T820,\nT678 -> one(T820),\nT818 -> T821,\nT819 -> T822" }, { "from": 2577, "to": 2639, "label": "EVAL-BACKTRACK" }, { "from": 2583, "to": 2586, "label": "CASE" }, { "from": 2586, "to": 2588, "label": "PARALLEL" }, { "from": 2586, "to": 2589, "label": "PARALLEL" }, { "from": 2588, "to": 2590, "label": "EVAL with clause\naddY(b, zero(X894), one(X894)) :- binaryZ(X894).\nand substitutionT773 -> b,\nX894 -> T780,\nT774 -> zero(T780),\nT772 -> one(T780)" }, { "from": 2588, "to": 2591, "label": "EVAL-BACKTRACK" }, { "from": 2589, "to": 2594, "label": "PARALLEL" }, { "from": 2589, "to": 2595, "label": "PARALLEL" }, { "from": 2590, "to": 2240, "label": "INSTANCE with matching:\nT429 -> T780" }, { "from": 2594, "to": 2630, "label": "EVAL with clause\naddY(b, one(X904), zero(X905)) :- succ(X904, X905).\nand substitutionT773 -> b,\nX904 -> T792,\nT774 -> one(T792),\nX905 -> T791,\nT772 -> zero(T791),\nT790 -> T792" }, { "from": 2594, "to": 2631, "label": "EVAL-BACKTRACK" }, { "from": 2595, "to": 2632, "label": "ONLY EVAL with clause\naddY(X915, X916, X917) :- addC(X915, X916, X917).\nand substitutionT773 -> T807,\nX915 -> T807,\nT774 -> T808,\nX916 -> T808,\nT772 -> T806,\nX917 -> T806,\nT804 -> T807,\nT805 -> T808" }, { "from": 2630, "to": 2485, "label": "INSTANCE with matching:\nT639 -> T792\nT638 -> T791" }, { "from": 2632, "to": 2534, "label": "INSTANCE with matching:\nT679 -> T807\nT680 -> T808\nT678 -> T806" }, { "from": 2638, "to": 2461, "label": "INSTANCE with matching:\nT613 -> T821\nT614 -> T822\nT612 -> T820" }, { "from": 2641, "to": 2642, "label": "PARALLEL" }, { "from": 2641, "to": 2643, "label": "PARALLEL" }, { "from": 2642, "to": 2646, "label": "EVAL with clause\naddz(one(X945), zero(X946), one(X947)) :- addy(X945, X946, X947).\nand substitutionX945 -> T839,\nT463 -> one(T839),\nT464 -> T840,\nX946 -> T840,\nX947 -> T838,\nT462 -> one(T838),\nT836 -> T839,\nT837 -> T840" }, { "from": 2642, "to": 2647, "label": "EVAL-BACKTRACK" }, { "from": 2643, "to": 2648, "label": "BACKTRACK\nfor clause: addz(one(X), one(Y), zero(Z)) :- addc(X, Y, Z)because of non-unification" }, { "from": 2646, "to": 2421, "label": "INSTANCE with matching:\nT571 -> T839\nT572 -> T840\nT570 -> T838" } ], "type": "Graph" } } ---------------------------------------- (156) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f2_in(T8) -> f2_out1(one(b), T8) f2_in(zero(T23)) -> U1(f2_in(T23), zero(T23)) U1(f2_out1(T24, T25), zero(T23)) -> f2_out1(zero(T24), T25) f2_in(T36) -> U2(f212_in(T36), T36) U2(f212_out1(T37, T38, X33), T36) -> f2_out1(one(T37), T38) f254_in -> f254_out1(one(b)) f254_in -> U3(f254_in) U3(f254_out1(T60)) -> f254_out1(zero(T60)) f254_in -> U4(f522_in) U4(f522_out1(T70, T71, X77, X78)) -> f254_out1(one(T70)) f629_in -> U5(f629_in) U5(f629_out1(T102)) -> f629_out1(zero(T102)) f629_in -> U6(f639_in) U6(f639_out1(T107)) -> f629_out1(one(T107)) f639_in -> f639_out1(b) f639_in -> U7(f629_in) U7(f629_out1(T113)) -> f639_out1(zero(T113)) f639_in -> U8(f639_in) U8(f639_out1(T118)) -> f639_out1(one(T118)) f1120_in -> U9(f1120_in) U9(f1120_out1(T157, T158, X180)) -> f1120_out1(zero(T157), zero(T158), zero(X180)) f1120_in -> U10(f639_in) U10(f639_out1(T179)) -> f1120_out1(zero(one(T179)), one(b), one(one(T179))) f1120_in -> U11(f629_in) U11(f629_out1(T185)) -> f1120_out1(zero(zero(T185)), one(b), one(zero(T185))) f1120_in -> U12(f1120_in) U12(f1120_out1(T197, T198, X232)) -> f1120_out1(zero(T197), one(T198), one(X232)) f1120_in -> U13(f1610_in) U13(f1610_out1(T211, T212, X256)) -> f1120_out1(one(T211), zero(T212), one(X256)) f1120_in -> U14(f1638_in) U14(f1638_out1(T247, T248, X300)) -> f1120_out1(one(T247), one(T248), zero(X300)) f1680_in -> f1680_out1(b, one(b)) f1680_in -> U15(f629_in) U15(f629_out1(T272)) -> f1680_out1(zero(T272), one(T272)) f1680_in -> U16(f1680_in) U16(f1680_out1(T277, X346)) -> f1680_out1(one(T277), zero(X346)) f1648_in -> U17(f629_in) U17(f629_out1(T261)) -> f1648_out1(zero(T261), one(T261)) f1648_in -> U18(f1680_in) U18(f1680_out1(T266, X331)) -> f1648_out1(one(T266), zero(X331)) f2062_in -> U19(f1120_in) U19(f1120_out1(T309, T310, X402)) -> f2062_out1(zero(T309), zero(T310), one(X402)) f2062_in -> U20(f629_in) U20(f629_out1(T331)) -> f2062_out1(zero(zero(T331)), one(b), zero(one(T331))) f2062_in -> U21(f1680_in) U21(f1680_out1(T338, X448)) -> f2062_out1(zero(one(T338)), one(b), zero(zero(X448))) f2062_in -> U22(f2062_in) U22(f2062_out1(T349, T350, X464)) -> f2062_out1(zero(T349), one(T350), zero(X464)) f2062_in -> U23(f629_in) U23(f629_out1(T371)) -> f2062_out1(one(b), zero(zero(T371)), zero(one(T371))) f2062_in -> U24(f1680_in) U24(f1680_out1(T378, X510)) -> f2062_out1(one(b), zero(one(T378)), zero(zero(X510))) f2062_in -> U25(f2062_in) U25(f2062_out1(T389, T390, X526)) -> f2062_out1(one(T389), zero(T390), zero(X526)) f2062_in -> U26(f1638_in) U26(f1638_out1(T399, T400, X542)) -> f2062_out1(one(T399), one(T400), one(X542)) f1638_in -> f1638_out1(b, b, one(b)) f1638_in -> U27(f1648_in) U27(f1648_out1(T254, X315)) -> f1638_out1(T254, b, X315) f1638_in -> U28(f1648_in) U28(f1648_out1(T283, X361)) -> f1638_out1(b, T283, X361) f1638_in -> U29(f2062_in) U29(f2062_out1(T295, T296, X378)) -> f1638_out1(T295, T296, X378) f1610_in -> U30(f639_in) U30(f639_out1(T219)) -> f1610_out1(b, one(T219), one(T219)) f1610_in -> U31(f629_in) U31(f629_out1(T225)) -> f1610_out1(b, zero(T225), zero(T225)) f1610_in -> U32(f1120_in) U32(f1120_out1(T237, T238, X284)) -> f1610_out1(T237, T238, X284) f2240_in(zero(T435)) -> U33(f2240_in(T435), zero(T435)) U33(f2240_out1, zero(T435)) -> f2240_out1 f2240_in(one(T439)) -> U34(f2254_in(T439), one(T439)) U34(f2254_out1, one(T439)) -> f2240_out1 f2254_in(b) -> f2254_out1 f2254_in(zero(T444)) -> U35(f2240_in(T444), zero(T444)) U35(f2240_out1, zero(T444)) -> f2254_out1 f2254_in(one(T448)) -> U36(f2254_in(T448), one(T448)) U36(f2254_out1, one(T448)) -> f2254_out1 f2333_in(zero(T502)) -> U37(f2333_in(T502), zero(T502)) U37(f2333_out1(T503, T504), zero(T502)) -> f2333_out1(zero(T503), zero(T504)) f2333_in(one(one(T530))) -> U38(f2254_in(T530), one(one(T530))) U38(f2254_out1, one(one(T530))) -> f2333_out1(zero(one(T530)), one(b)) f2333_in(one(zero(T535))) -> U39(f2240_in(T535), one(zero(T535))) U39(f2240_out1, one(zero(T535))) -> f2333_out1(zero(zero(T535)), one(b)) f2333_in(one(T550)) -> U40(f2333_in(T550), one(T550)) U40(f2333_out1(T551, T552), one(T550)) -> f2333_out1(zero(T551), one(T552)) f2333_in(one(T570)) -> U41(f2421_in(T570), one(T570)) U41(f2421_out1(T571, T572), one(T570)) -> f2333_out1(one(T571), zero(T572)) f2333_in(zero(T612)) -> U42(f2461_in(T612), zero(T612)) U42(f2461_out1(T613, T614), zero(T612)) -> f2333_out1(one(T613), one(T614)) f2485_in(one(b)) -> f2485_out1(b) f2485_in(one(T644)) -> U43(f2240_in(T644), one(T644)) U43(f2240_out1, one(T644)) -> f2485_out1(zero(T644)) f2485_in(zero(T651)) -> U44(f2485_in(T651), zero(T651)) U44(f2485_out1(T652), zero(T651)) -> f2485_out1(one(T652)) f2473_in(one(T631)) -> U45(f2240_in(T631), one(T631)) U45(f2240_out1, one(T631)) -> f2473_out1(zero(T631)) f2473_in(zero(T638)) -> U46(f2485_in(T638), zero(T638)) U46(f2485_out1(T639), zero(T638)) -> f2473_out1(one(T639)) f2534_in(one(T698)) -> U47(f2333_in(T698), one(T698)) U47(f2333_out1(T699, T700), one(T698)) -> f2534_out1(zero(T699), zero(T700)) f2534_in(zero(one(T726))) -> U48(f2240_in(T726), zero(one(T726))) U48(f2240_out1, zero(one(T726))) -> f2534_out1(zero(zero(T726)), one(b)) f2534_in(zero(zero(T737))) -> U49(f2485_in(T737), zero(zero(T737))) U49(f2485_out1(T738), zero(zero(T737))) -> f2534_out1(zero(one(T738)), one(b)) f2534_in(zero(T752)) -> U50(f2534_in(T752), zero(T752)) U50(f2534_out1(T753, T754), zero(T752)) -> f2534_out1(zero(T753), one(T754)) f2534_in(zero(one(T780))) -> U51(f2240_in(T780), zero(one(T780))) U51(f2240_out1, zero(one(T780))) -> f2534_out1(one(b), zero(zero(T780))) f2534_in(zero(zero(T791))) -> U52(f2485_in(T791), zero(zero(T791))) U52(f2485_out1(T792), zero(zero(T791))) -> f2534_out1(one(b), zero(one(T792))) f2534_in(zero(T806)) -> U53(f2534_in(T806), zero(T806)) U53(f2534_out1(T807, T808), zero(T806)) -> f2534_out1(one(T807), zero(T808)) f2534_in(one(T820)) -> U54(f2461_in(T820), one(T820)) U54(f2461_out1(T821, T822), one(T820)) -> f2534_out1(one(T821), one(T822)) f2461_in(one(b)) -> f2461_out1(b, b) f2461_in(T624) -> U55(f2473_in(T624), T624) U55(f2473_out1(T625), T624) -> f2461_out1(T625, b) f2461_in(T662) -> U56(f2473_in(T662), T662) U56(f2473_out1(T663), T662) -> f2461_out1(b, T663) f2461_in(T678) -> U57(f2534_in(T678), T678) U57(f2534_out1(T679, T680), T678) -> f2461_out1(T679, T680) f2421_in(one(T578)) -> U58(f2254_in(T578), one(T578)) U58(f2254_out1, one(T578)) -> f2421_out1(b, one(T578)) f2421_in(zero(T583)) -> U59(f2240_in(T583), zero(T583)) U59(f2240_out1, zero(T583)) -> f2421_out1(b, zero(T583)) f2421_in(T598) -> U60(f2333_in(T598), T598) U60(f2333_out1(T599, T600), T598) -> f2421_out1(T599, T600) f258_in(zero(T429)) -> U61(f2240_in(T429), zero(T429)) U61(f2240_out1, zero(T429)) -> f258_out1(b, T429) f258_in(zero(T482)) -> U62(f2333_in(T482), zero(T482)) U62(f2333_out1(T483, T484), zero(T482)) -> f258_out1(zero(T483), T484) f258_in(one(T838)) -> U63(f2421_in(T838), one(T838)) U63(f2421_out1(T839, T840), one(T838)) -> f258_out1(one(T839), T840) f607_in -> U64(f629_in) U64(f629_out1(T95)) -> f607_out1(b, T95, zero(T95)) f607_in -> U65(f1120_in) U65(f1120_out1(T143, T144, X156)) -> f607_out1(zero(T143), T144, zero(X156)) f607_in -> U66(f1610_in) U66(f1610_out1(T412, T413, X565)) -> f607_out1(one(T412), T413, one(X565)) f212_in(T36) -> U67(f254_in, T36) U67(f254_out1(T37), T36) -> U68(f258_in(T36), T36, T37) U68(f258_out1(T42, T41), T36, T37) -> f212_out1(T37, T42, T41) f522_in -> U69(f254_in) U69(f254_out1(T70)) -> U70(f607_in, T70) U70(f607_out1(T75, T74, X78), T70) -> f522_out1(T70, T75, T74, X78) Q is empty. ---------------------------------------- (157) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (158) Obligation: Q DP problem: The TRS P consists of the following rules: F2_IN(zero(T23)) -> U1^1(f2_in(T23), zero(T23)) F2_IN(zero(T23)) -> F2_IN(T23) F2_IN(T36) -> U2^1(f212_in(T36), T36) F2_IN(T36) -> F212_IN(T36) F254_IN -> U3^1(f254_in) F254_IN -> F254_IN F254_IN -> U4^1(f522_in) F254_IN -> F522_IN F629_IN -> U5^1(f629_in) F629_IN -> F629_IN F629_IN -> U6^1(f639_in) F629_IN -> F639_IN F639_IN -> U7^1(f629_in) F639_IN -> F629_IN F639_IN -> U8^1(f639_in) F639_IN -> F639_IN F1120_IN -> U9^1(f1120_in) F1120_IN -> F1120_IN F1120_IN -> U10^1(f639_in) F1120_IN -> F639_IN F1120_IN -> U11^1(f629_in) F1120_IN -> F629_IN F1120_IN -> U12^1(f1120_in) F1120_IN -> U13^1(f1610_in) F1120_IN -> F1610_IN F1120_IN -> U14^1(f1638_in) F1120_IN -> F1638_IN F1680_IN -> U15^1(f629_in) F1680_IN -> F629_IN F1680_IN -> U16^1(f1680_in) F1680_IN -> F1680_IN F1648_IN -> U17^1(f629_in) F1648_IN -> F629_IN F1648_IN -> U18^1(f1680_in) F1648_IN -> F1680_IN F2062_IN -> U19^1(f1120_in) F2062_IN -> F1120_IN F2062_IN -> U20^1(f629_in) F2062_IN -> F629_IN F2062_IN -> U21^1(f1680_in) F2062_IN -> F1680_IN F2062_IN -> U22^1(f2062_in) F2062_IN -> F2062_IN F2062_IN -> U23^1(f629_in) F2062_IN -> U24^1(f1680_in) F2062_IN -> U25^1(f2062_in) F2062_IN -> U26^1(f1638_in) F2062_IN -> F1638_IN F1638_IN -> U27^1(f1648_in) F1638_IN -> F1648_IN F1638_IN -> U28^1(f1648_in) F1638_IN -> U29^1(f2062_in) F1638_IN -> F2062_IN F1610_IN -> U30^1(f639_in) F1610_IN -> F639_IN F1610_IN -> U31^1(f629_in) F1610_IN -> F629_IN F1610_IN -> U32^1(f1120_in) F1610_IN -> F1120_IN F2240_IN(zero(T435)) -> U33^1(f2240_in(T435), zero(T435)) F2240_IN(zero(T435)) -> F2240_IN(T435) F2240_IN(one(T439)) -> U34^1(f2254_in(T439), one(T439)) F2240_IN(one(T439)) -> F2254_IN(T439) F2254_IN(zero(T444)) -> U35^1(f2240_in(T444), zero(T444)) F2254_IN(zero(T444)) -> F2240_IN(T444) F2254_IN(one(T448)) -> U36^1(f2254_in(T448), one(T448)) F2254_IN(one(T448)) -> F2254_IN(T448) F2333_IN(zero(T502)) -> U37^1(f2333_in(T502), zero(T502)) F2333_IN(zero(T502)) -> F2333_IN(T502) F2333_IN(one(one(T530))) -> U38^1(f2254_in(T530), one(one(T530))) F2333_IN(one(one(T530))) -> F2254_IN(T530) F2333_IN(one(zero(T535))) -> U39^1(f2240_in(T535), one(zero(T535))) F2333_IN(one(zero(T535))) -> F2240_IN(T535) F2333_IN(one(T550)) -> U40^1(f2333_in(T550), one(T550)) F2333_IN(one(T550)) -> F2333_IN(T550) F2333_IN(one(T570)) -> U41^1(f2421_in(T570), one(T570)) F2333_IN(one(T570)) -> F2421_IN(T570) F2333_IN(zero(T612)) -> U42^1(f2461_in(T612), zero(T612)) F2333_IN(zero(T612)) -> F2461_IN(T612) F2485_IN(one(T644)) -> U43^1(f2240_in(T644), one(T644)) F2485_IN(one(T644)) -> F2240_IN(T644) F2485_IN(zero(T651)) -> U44^1(f2485_in(T651), zero(T651)) F2485_IN(zero(T651)) -> F2485_IN(T651) F2473_IN(one(T631)) -> U45^1(f2240_in(T631), one(T631)) F2473_IN(one(T631)) -> F2240_IN(T631) F2473_IN(zero(T638)) -> U46^1(f2485_in(T638), zero(T638)) F2473_IN(zero(T638)) -> F2485_IN(T638) F2534_IN(one(T698)) -> U47^1(f2333_in(T698), one(T698)) F2534_IN(one(T698)) -> F2333_IN(T698) F2534_IN(zero(one(T726))) -> U48^1(f2240_in(T726), zero(one(T726))) F2534_IN(zero(one(T726))) -> F2240_IN(T726) F2534_IN(zero(zero(T737))) -> U49^1(f2485_in(T737), zero(zero(T737))) F2534_IN(zero(zero(T737))) -> F2485_IN(T737) F2534_IN(zero(T752)) -> U50^1(f2534_in(T752), zero(T752)) F2534_IN(zero(T752)) -> F2534_IN(T752) F2534_IN(zero(one(T780))) -> U51^1(f2240_in(T780), zero(one(T780))) F2534_IN(zero(zero(T791))) -> U52^1(f2485_in(T791), zero(zero(T791))) F2534_IN(zero(T806)) -> U53^1(f2534_in(T806), zero(T806)) F2534_IN(one(T820)) -> U54^1(f2461_in(T820), one(T820)) F2534_IN(one(T820)) -> F2461_IN(T820) F2461_IN(T624) -> U55^1(f2473_in(T624), T624) F2461_IN(T624) -> F2473_IN(T624) F2461_IN(T662) -> U56^1(f2473_in(T662), T662) F2461_IN(T678) -> U57^1(f2534_in(T678), T678) F2461_IN(T678) -> F2534_IN(T678) F2421_IN(one(T578)) -> U58^1(f2254_in(T578), one(T578)) F2421_IN(one(T578)) -> F2254_IN(T578) F2421_IN(zero(T583)) -> U59^1(f2240_in(T583), zero(T583)) F2421_IN(zero(T583)) -> F2240_IN(T583) F2421_IN(T598) -> U60^1(f2333_in(T598), T598) F2421_IN(T598) -> F2333_IN(T598) F258_IN(zero(T429)) -> U61^1(f2240_in(T429), zero(T429)) F258_IN(zero(T429)) -> F2240_IN(T429) F258_IN(zero(T482)) -> U62^1(f2333_in(T482), zero(T482)) F258_IN(zero(T482)) -> F2333_IN(T482) F258_IN(one(T838)) -> U63^1(f2421_in(T838), one(T838)) F258_IN(one(T838)) -> F2421_IN(T838) F607_IN -> U64^1(f629_in) F607_IN -> F629_IN F607_IN -> U65^1(f1120_in) F607_IN -> F1120_IN F607_IN -> U66^1(f1610_in) F607_IN -> F1610_IN F212_IN(T36) -> U67^1(f254_in, T36) F212_IN(T36) -> F254_IN U67^1(f254_out1(T37), T36) -> U68^1(f258_in(T36), T36, T37) U67^1(f254_out1(T37), T36) -> F258_IN(T36) F522_IN -> U69^1(f254_in) F522_IN -> F254_IN U69^1(f254_out1(T70)) -> U70^1(f607_in, T70) U69^1(f254_out1(T70)) -> F607_IN The TRS R consists of the following rules: f2_in(T8) -> f2_out1(one(b), T8) f2_in(zero(T23)) -> U1(f2_in(T23), zero(T23)) U1(f2_out1(T24, T25), zero(T23)) -> f2_out1(zero(T24), T25) f2_in(T36) -> U2(f212_in(T36), T36) U2(f212_out1(T37, T38, X33), T36) -> f2_out1(one(T37), T38) f254_in -> f254_out1(one(b)) f254_in -> U3(f254_in) U3(f254_out1(T60)) -> f254_out1(zero(T60)) f254_in -> U4(f522_in) U4(f522_out1(T70, T71, X77, X78)) -> f254_out1(one(T70)) f629_in -> U5(f629_in) U5(f629_out1(T102)) -> f629_out1(zero(T102)) f629_in -> U6(f639_in) U6(f639_out1(T107)) -> f629_out1(one(T107)) f639_in -> f639_out1(b) f639_in -> U7(f629_in) U7(f629_out1(T113)) -> f639_out1(zero(T113)) f639_in -> U8(f639_in) U8(f639_out1(T118)) -> f639_out1(one(T118)) f1120_in -> U9(f1120_in) U9(f1120_out1(T157, T158, X180)) -> f1120_out1(zero(T157), zero(T158), zero(X180)) f1120_in -> U10(f639_in) U10(f639_out1(T179)) -> f1120_out1(zero(one(T179)), one(b), one(one(T179))) f1120_in -> U11(f629_in) U11(f629_out1(T185)) -> f1120_out1(zero(zero(T185)), one(b), one(zero(T185))) f1120_in -> U12(f1120_in) U12(f1120_out1(T197, T198, X232)) -> f1120_out1(zero(T197), one(T198), one(X232)) f1120_in -> U13(f1610_in) U13(f1610_out1(T211, T212, X256)) -> f1120_out1(one(T211), zero(T212), one(X256)) f1120_in -> U14(f1638_in) U14(f1638_out1(T247, T248, X300)) -> f1120_out1(one(T247), one(T248), zero(X300)) f1680_in -> f1680_out1(b, one(b)) f1680_in -> U15(f629_in) U15(f629_out1(T272)) -> f1680_out1(zero(T272), one(T272)) f1680_in -> U16(f1680_in) U16(f1680_out1(T277, X346)) -> f1680_out1(one(T277), zero(X346)) f1648_in -> U17(f629_in) U17(f629_out1(T261)) -> f1648_out1(zero(T261), one(T261)) f1648_in -> U18(f1680_in) U18(f1680_out1(T266, X331)) -> f1648_out1(one(T266), zero(X331)) f2062_in -> U19(f1120_in) U19(f1120_out1(T309, T310, X402)) -> f2062_out1(zero(T309), zero(T310), one(X402)) f2062_in -> U20(f629_in) U20(f629_out1(T331)) -> f2062_out1(zero(zero(T331)), one(b), zero(one(T331))) f2062_in -> U21(f1680_in) U21(f1680_out1(T338, X448)) -> f2062_out1(zero(one(T338)), one(b), zero(zero(X448))) f2062_in -> U22(f2062_in) U22(f2062_out1(T349, T350, X464)) -> f2062_out1(zero(T349), one(T350), zero(X464)) f2062_in -> U23(f629_in) U23(f629_out1(T371)) -> f2062_out1(one(b), zero(zero(T371)), zero(one(T371))) f2062_in -> U24(f1680_in) U24(f1680_out1(T378, X510)) -> f2062_out1(one(b), zero(one(T378)), zero(zero(X510))) f2062_in -> U25(f2062_in) U25(f2062_out1(T389, T390, X526)) -> f2062_out1(one(T389), zero(T390), zero(X526)) f2062_in -> U26(f1638_in) U26(f1638_out1(T399, T400, X542)) -> f2062_out1(one(T399), one(T400), one(X542)) f1638_in -> f1638_out1(b, b, one(b)) f1638_in -> U27(f1648_in) U27(f1648_out1(T254, X315)) -> f1638_out1(T254, b, X315) f1638_in -> U28(f1648_in) U28(f1648_out1(T283, X361)) -> f1638_out1(b, T283, X361) f1638_in -> U29(f2062_in) U29(f2062_out1(T295, T296, X378)) -> f1638_out1(T295, T296, X378) f1610_in -> U30(f639_in) U30(f639_out1(T219)) -> f1610_out1(b, one(T219), one(T219)) f1610_in -> U31(f629_in) U31(f629_out1(T225)) -> f1610_out1(b, zero(T225), zero(T225)) f1610_in -> U32(f1120_in) U32(f1120_out1(T237, T238, X284)) -> f1610_out1(T237, T238, X284) f2240_in(zero(T435)) -> U33(f2240_in(T435), zero(T435)) U33(f2240_out1, zero(T435)) -> f2240_out1 f2240_in(one(T439)) -> U34(f2254_in(T439), one(T439)) U34(f2254_out1, one(T439)) -> f2240_out1 f2254_in(b) -> f2254_out1 f2254_in(zero(T444)) -> U35(f2240_in(T444), zero(T444)) U35(f2240_out1, zero(T444)) -> f2254_out1 f2254_in(one(T448)) -> U36(f2254_in(T448), one(T448)) U36(f2254_out1, one(T448)) -> f2254_out1 f2333_in(zero(T502)) -> U37(f2333_in(T502), zero(T502)) U37(f2333_out1(T503, T504), zero(T502)) -> f2333_out1(zero(T503), zero(T504)) f2333_in(one(one(T530))) -> U38(f2254_in(T530), one(one(T530))) U38(f2254_out1, one(one(T530))) -> f2333_out1(zero(one(T530)), one(b)) f2333_in(one(zero(T535))) -> U39(f2240_in(T535), one(zero(T535))) U39(f2240_out1, one(zero(T535))) -> f2333_out1(zero(zero(T535)), one(b)) f2333_in(one(T550)) -> U40(f2333_in(T550), one(T550)) U40(f2333_out1(T551, T552), one(T550)) -> f2333_out1(zero(T551), one(T552)) f2333_in(one(T570)) -> U41(f2421_in(T570), one(T570)) U41(f2421_out1(T571, T572), one(T570)) -> f2333_out1(one(T571), zero(T572)) f2333_in(zero(T612)) -> U42(f2461_in(T612), zero(T612)) U42(f2461_out1(T613, T614), zero(T612)) -> f2333_out1(one(T613), one(T614)) f2485_in(one(b)) -> f2485_out1(b) f2485_in(one(T644)) -> U43(f2240_in(T644), one(T644)) U43(f2240_out1, one(T644)) -> f2485_out1(zero(T644)) f2485_in(zero(T651)) -> U44(f2485_in(T651), zero(T651)) U44(f2485_out1(T652), zero(T651)) -> f2485_out1(one(T652)) f2473_in(one(T631)) -> U45(f2240_in(T631), one(T631)) U45(f2240_out1, one(T631)) -> f2473_out1(zero(T631)) f2473_in(zero(T638)) -> U46(f2485_in(T638), zero(T638)) U46(f2485_out1(T639), zero(T638)) -> f2473_out1(one(T639)) f2534_in(one(T698)) -> U47(f2333_in(T698), one(T698)) U47(f2333_out1(T699, T700), one(T698)) -> f2534_out1(zero(T699), zero(T700)) f2534_in(zero(one(T726))) -> U48(f2240_in(T726), zero(one(T726))) U48(f2240_out1, zero(one(T726))) -> f2534_out1(zero(zero(T726)), one(b)) f2534_in(zero(zero(T737))) -> U49(f2485_in(T737), zero(zero(T737))) U49(f2485_out1(T738), zero(zero(T737))) -> f2534_out1(zero(one(T738)), one(b)) f2534_in(zero(T752)) -> U50(f2534_in(T752), zero(T752)) U50(f2534_out1(T753, T754), zero(T752)) -> f2534_out1(zero(T753), one(T754)) f2534_in(zero(one(T780))) -> U51(f2240_in(T780), zero(one(T780))) U51(f2240_out1, zero(one(T780))) -> f2534_out1(one(b), zero(zero(T780))) f2534_in(zero(zero(T791))) -> U52(f2485_in(T791), zero(zero(T791))) U52(f2485_out1(T792), zero(zero(T791))) -> f2534_out1(one(b), zero(one(T792))) f2534_in(zero(T806)) -> U53(f2534_in(T806), zero(T806)) U53(f2534_out1(T807, T808), zero(T806)) -> f2534_out1(one(T807), zero(T808)) f2534_in(one(T820)) -> U54(f2461_in(T820), one(T820)) U54(f2461_out1(T821, T822), one(T820)) -> f2534_out1(one(T821), one(T822)) f2461_in(one(b)) -> f2461_out1(b, b) f2461_in(T624) -> U55(f2473_in(T624), T624) U55(f2473_out1(T625), T624) -> f2461_out1(T625, b) f2461_in(T662) -> U56(f2473_in(T662), T662) U56(f2473_out1(T663), T662) -> f2461_out1(b, T663) f2461_in(T678) -> U57(f2534_in(T678), T678) U57(f2534_out1(T679, T680), T678) -> f2461_out1(T679, T680) f2421_in(one(T578)) -> U58(f2254_in(T578), one(T578)) U58(f2254_out1, one(T578)) -> f2421_out1(b, one(T578)) f2421_in(zero(T583)) -> U59(f2240_in(T583), zero(T583)) U59(f2240_out1, zero(T583)) -> f2421_out1(b, zero(T583)) f2421_in(T598) -> U60(f2333_in(T598), T598) U60(f2333_out1(T599, T600), T598) -> f2421_out1(T599, T600) f258_in(zero(T429)) -> U61(f2240_in(T429), zero(T429)) U61(f2240_out1, zero(T429)) -> f258_out1(b, T429) f258_in(zero(T482)) -> U62(f2333_in(T482), zero(T482)) U62(f2333_out1(T483, T484), zero(T482)) -> f258_out1(zero(T483), T484) f258_in(one(T838)) -> U63(f2421_in(T838), one(T838)) U63(f2421_out1(T839, T840), one(T838)) -> f258_out1(one(T839), T840) f607_in -> U64(f629_in) U64(f629_out1(T95)) -> f607_out1(b, T95, zero(T95)) f607_in -> U65(f1120_in) U65(f1120_out1(T143, T144, X156)) -> f607_out1(zero(T143), T144, zero(X156)) f607_in -> U66(f1610_in) U66(f1610_out1(T412, T413, X565)) -> f607_out1(one(T412), T413, one(X565)) f212_in(T36) -> U67(f254_in, T36) U67(f254_out1(T37), T36) -> U68(f258_in(T36), T36, T37) U68(f258_out1(T42, T41), T36, T37) -> f212_out1(T37, T42, T41) f522_in -> U69(f254_in) U69(f254_out1(T70)) -> U70(f607_in, T70) U70(f607_out1(T75, T74, X78), T70) -> f522_out1(T70, T75, T74, X78) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (159) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 8 SCCs with 100 less nodes. ---------------------------------------- (160) Complex Obligation (AND) ---------------------------------------- (161) Obligation: Q DP problem: The TRS P consists of the following rules: F2240_IN(one(T439)) -> F2254_IN(T439) F2254_IN(zero(T444)) -> F2240_IN(T444) F2240_IN(zero(T435)) -> F2240_IN(T435) F2254_IN(one(T448)) -> F2254_IN(T448) The TRS R consists of the following rules: f2_in(T8) -> f2_out1(one(b), T8) f2_in(zero(T23)) -> U1(f2_in(T23), zero(T23)) U1(f2_out1(T24, T25), zero(T23)) -> f2_out1(zero(T24), T25) f2_in(T36) -> U2(f212_in(T36), T36) U2(f212_out1(T37, T38, X33), T36) -> f2_out1(one(T37), T38) f254_in -> f254_out1(one(b)) f254_in -> U3(f254_in) U3(f254_out1(T60)) -> f254_out1(zero(T60)) f254_in -> U4(f522_in) U4(f522_out1(T70, T71, X77, X78)) -> f254_out1(one(T70)) f629_in -> U5(f629_in) U5(f629_out1(T102)) -> f629_out1(zero(T102)) f629_in -> U6(f639_in) U6(f639_out1(T107)) -> f629_out1(one(T107)) f639_in -> f639_out1(b) f639_in -> U7(f629_in) U7(f629_out1(T113)) -> f639_out1(zero(T113)) f639_in -> U8(f639_in) U8(f639_out1(T118)) -> f639_out1(one(T118)) f1120_in -> U9(f1120_in) U9(f1120_out1(T157, T158, X180)) -> f1120_out1(zero(T157), zero(T158), zero(X180)) f1120_in -> U10(f639_in) U10(f639_out1(T179)) -> f1120_out1(zero(one(T179)), one(b), one(one(T179))) f1120_in -> U11(f629_in) U11(f629_out1(T185)) -> f1120_out1(zero(zero(T185)), one(b), one(zero(T185))) f1120_in -> U12(f1120_in) U12(f1120_out1(T197, T198, X232)) -> f1120_out1(zero(T197), one(T198), one(X232)) f1120_in -> U13(f1610_in) U13(f1610_out1(T211, T212, X256)) -> f1120_out1(one(T211), zero(T212), one(X256)) f1120_in -> U14(f1638_in) U14(f1638_out1(T247, T248, X300)) -> f1120_out1(one(T247), one(T248), zero(X300)) f1680_in -> f1680_out1(b, one(b)) f1680_in -> U15(f629_in) U15(f629_out1(T272)) -> f1680_out1(zero(T272), one(T272)) f1680_in -> U16(f1680_in) U16(f1680_out1(T277, X346)) -> f1680_out1(one(T277), zero(X346)) f1648_in -> U17(f629_in) U17(f629_out1(T261)) -> f1648_out1(zero(T261), one(T261)) f1648_in -> U18(f1680_in) U18(f1680_out1(T266, X331)) -> f1648_out1(one(T266), zero(X331)) f2062_in -> U19(f1120_in) U19(f1120_out1(T309, T310, X402)) -> f2062_out1(zero(T309), zero(T310), one(X402)) f2062_in -> U20(f629_in) U20(f629_out1(T331)) -> f2062_out1(zero(zero(T331)), one(b), zero(one(T331))) f2062_in -> U21(f1680_in) U21(f1680_out1(T338, X448)) -> f2062_out1(zero(one(T338)), one(b), zero(zero(X448))) f2062_in -> U22(f2062_in) U22(f2062_out1(T349, T350, X464)) -> f2062_out1(zero(T349), one(T350), zero(X464)) f2062_in -> U23(f629_in) U23(f629_out1(T371)) -> f2062_out1(one(b), zero(zero(T371)), zero(one(T371))) f2062_in -> U24(f1680_in) U24(f1680_out1(T378, X510)) -> f2062_out1(one(b), zero(one(T378)), zero(zero(X510))) f2062_in -> U25(f2062_in) U25(f2062_out1(T389, T390, X526)) -> f2062_out1(one(T389), zero(T390), zero(X526)) f2062_in -> U26(f1638_in) U26(f1638_out1(T399, T400, X542)) -> f2062_out1(one(T399), one(T400), one(X542)) f1638_in -> f1638_out1(b, b, one(b)) f1638_in -> U27(f1648_in) U27(f1648_out1(T254, X315)) -> f1638_out1(T254, b, X315) f1638_in -> U28(f1648_in) U28(f1648_out1(T283, X361)) -> f1638_out1(b, T283, X361) f1638_in -> U29(f2062_in) U29(f2062_out1(T295, T296, X378)) -> f1638_out1(T295, T296, X378) f1610_in -> U30(f639_in) U30(f639_out1(T219)) -> f1610_out1(b, one(T219), one(T219)) f1610_in -> U31(f629_in) U31(f629_out1(T225)) -> f1610_out1(b, zero(T225), zero(T225)) f1610_in -> U32(f1120_in) U32(f1120_out1(T237, T238, X284)) -> f1610_out1(T237, T238, X284) f2240_in(zero(T435)) -> U33(f2240_in(T435), zero(T435)) U33(f2240_out1, zero(T435)) -> f2240_out1 f2240_in(one(T439)) -> U34(f2254_in(T439), one(T439)) U34(f2254_out1, one(T439)) -> f2240_out1 f2254_in(b) -> f2254_out1 f2254_in(zero(T444)) -> U35(f2240_in(T444), zero(T444)) U35(f2240_out1, zero(T444)) -> f2254_out1 f2254_in(one(T448)) -> U36(f2254_in(T448), one(T448)) U36(f2254_out1, one(T448)) -> f2254_out1 f2333_in(zero(T502)) -> U37(f2333_in(T502), zero(T502)) U37(f2333_out1(T503, T504), zero(T502)) -> f2333_out1(zero(T503), zero(T504)) f2333_in(one(one(T530))) -> U38(f2254_in(T530), one(one(T530))) U38(f2254_out1, one(one(T530))) -> f2333_out1(zero(one(T530)), one(b)) f2333_in(one(zero(T535))) -> U39(f2240_in(T535), one(zero(T535))) U39(f2240_out1, one(zero(T535))) -> f2333_out1(zero(zero(T535)), one(b)) f2333_in(one(T550)) -> U40(f2333_in(T550), one(T550)) U40(f2333_out1(T551, T552), one(T550)) -> f2333_out1(zero(T551), one(T552)) f2333_in(one(T570)) -> U41(f2421_in(T570), one(T570)) U41(f2421_out1(T571, T572), one(T570)) -> f2333_out1(one(T571), zero(T572)) f2333_in(zero(T612)) -> U42(f2461_in(T612), zero(T612)) U42(f2461_out1(T613, T614), zero(T612)) -> f2333_out1(one(T613), one(T614)) f2485_in(one(b)) -> f2485_out1(b) f2485_in(one(T644)) -> U43(f2240_in(T644), one(T644)) U43(f2240_out1, one(T644)) -> f2485_out1(zero(T644)) f2485_in(zero(T651)) -> U44(f2485_in(T651), zero(T651)) U44(f2485_out1(T652), zero(T651)) -> f2485_out1(one(T652)) f2473_in(one(T631)) -> U45(f2240_in(T631), one(T631)) U45(f2240_out1, one(T631)) -> f2473_out1(zero(T631)) f2473_in(zero(T638)) -> U46(f2485_in(T638), zero(T638)) U46(f2485_out1(T639), zero(T638)) -> f2473_out1(one(T639)) f2534_in(one(T698)) -> U47(f2333_in(T698), one(T698)) U47(f2333_out1(T699, T700), one(T698)) -> f2534_out1(zero(T699), zero(T700)) f2534_in(zero(one(T726))) -> U48(f2240_in(T726), zero(one(T726))) U48(f2240_out1, zero(one(T726))) -> f2534_out1(zero(zero(T726)), one(b)) f2534_in(zero(zero(T737))) -> U49(f2485_in(T737), zero(zero(T737))) U49(f2485_out1(T738), zero(zero(T737))) -> f2534_out1(zero(one(T738)), one(b)) f2534_in(zero(T752)) -> U50(f2534_in(T752), zero(T752)) U50(f2534_out1(T753, T754), zero(T752)) -> f2534_out1(zero(T753), one(T754)) f2534_in(zero(one(T780))) -> U51(f2240_in(T780), zero(one(T780))) U51(f2240_out1, zero(one(T780))) -> f2534_out1(one(b), zero(zero(T780))) f2534_in(zero(zero(T791))) -> U52(f2485_in(T791), zero(zero(T791))) U52(f2485_out1(T792), zero(zero(T791))) -> f2534_out1(one(b), zero(one(T792))) f2534_in(zero(T806)) -> U53(f2534_in(T806), zero(T806)) U53(f2534_out1(T807, T808), zero(T806)) -> f2534_out1(one(T807), zero(T808)) f2534_in(one(T820)) -> U54(f2461_in(T820), one(T820)) U54(f2461_out1(T821, T822), one(T820)) -> f2534_out1(one(T821), one(T822)) f2461_in(one(b)) -> f2461_out1(b, b) f2461_in(T624) -> U55(f2473_in(T624), T624) U55(f2473_out1(T625), T624) -> f2461_out1(T625, b) f2461_in(T662) -> U56(f2473_in(T662), T662) U56(f2473_out1(T663), T662) -> f2461_out1(b, T663) f2461_in(T678) -> U57(f2534_in(T678), T678) U57(f2534_out1(T679, T680), T678) -> f2461_out1(T679, T680) f2421_in(one(T578)) -> U58(f2254_in(T578), one(T578)) U58(f2254_out1, one(T578)) -> f2421_out1(b, one(T578)) f2421_in(zero(T583)) -> U59(f2240_in(T583), zero(T583)) U59(f2240_out1, zero(T583)) -> f2421_out1(b, zero(T583)) f2421_in(T598) -> U60(f2333_in(T598), T598) U60(f2333_out1(T599, T600), T598) -> f2421_out1(T599, T600) f258_in(zero(T429)) -> U61(f2240_in(T429), zero(T429)) U61(f2240_out1, zero(T429)) -> f258_out1(b, T429) f258_in(zero(T482)) -> U62(f2333_in(T482), zero(T482)) U62(f2333_out1(T483, T484), zero(T482)) -> f258_out1(zero(T483), T484) f258_in(one(T838)) -> U63(f2421_in(T838), one(T838)) U63(f2421_out1(T839, T840), one(T838)) -> f258_out1(one(T839), T840) f607_in -> U64(f629_in) U64(f629_out1(T95)) -> f607_out1(b, T95, zero(T95)) f607_in -> U65(f1120_in) U65(f1120_out1(T143, T144, X156)) -> f607_out1(zero(T143), T144, zero(X156)) f607_in -> U66(f1610_in) U66(f1610_out1(T412, T413, X565)) -> f607_out1(one(T412), T413, one(X565)) f212_in(T36) -> U67(f254_in, T36) U67(f254_out1(T37), T36) -> U68(f258_in(T36), T36, T37) U68(f258_out1(T42, T41), T36, T37) -> f212_out1(T37, T42, T41) f522_in -> U69(f254_in) U69(f254_out1(T70)) -> U70(f607_in, T70) U70(f607_out1(T75, T74, X78), T70) -> f522_out1(T70, T75, T74, X78) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (162) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (163) Obligation: Q DP problem: The TRS P consists of the following rules: F2240_IN(one(T439)) -> F2254_IN(T439) F2254_IN(zero(T444)) -> F2240_IN(T444) F2240_IN(zero(T435)) -> F2240_IN(T435) F2254_IN(one(T448)) -> F2254_IN(T448) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (164) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *F2254_IN(zero(T444)) -> F2240_IN(T444) The graph contains the following edges 1 > 1 *F2254_IN(one(T448)) -> F2254_IN(T448) The graph contains the following edges 1 > 1 *F2240_IN(zero(T435)) -> F2240_IN(T435) The graph contains the following edges 1 > 1 *F2240_IN(one(T439)) -> F2254_IN(T439) The graph contains the following edges 1 > 1 ---------------------------------------- (165) YES ---------------------------------------- (166) Obligation: Q DP problem: The TRS P consists of the following rules: F2485_IN(zero(T651)) -> F2485_IN(T651) The TRS R consists of the following rules: f2_in(T8) -> f2_out1(one(b), T8) f2_in(zero(T23)) -> U1(f2_in(T23), zero(T23)) U1(f2_out1(T24, T25), zero(T23)) -> f2_out1(zero(T24), T25) f2_in(T36) -> U2(f212_in(T36), T36) U2(f212_out1(T37, T38, X33), T36) -> f2_out1(one(T37), T38) f254_in -> f254_out1(one(b)) f254_in -> U3(f254_in) U3(f254_out1(T60)) -> f254_out1(zero(T60)) f254_in -> U4(f522_in) U4(f522_out1(T70, T71, X77, X78)) -> f254_out1(one(T70)) f629_in -> U5(f629_in) U5(f629_out1(T102)) -> f629_out1(zero(T102)) f629_in -> U6(f639_in) U6(f639_out1(T107)) -> f629_out1(one(T107)) f639_in -> f639_out1(b) f639_in -> U7(f629_in) U7(f629_out1(T113)) -> f639_out1(zero(T113)) f639_in -> U8(f639_in) U8(f639_out1(T118)) -> f639_out1(one(T118)) f1120_in -> U9(f1120_in) U9(f1120_out1(T157, T158, X180)) -> f1120_out1(zero(T157), zero(T158), zero(X180)) f1120_in -> U10(f639_in) U10(f639_out1(T179)) -> f1120_out1(zero(one(T179)), one(b), one(one(T179))) f1120_in -> U11(f629_in) U11(f629_out1(T185)) -> f1120_out1(zero(zero(T185)), one(b), one(zero(T185))) f1120_in -> U12(f1120_in) U12(f1120_out1(T197, T198, X232)) -> f1120_out1(zero(T197), one(T198), one(X232)) f1120_in -> U13(f1610_in) U13(f1610_out1(T211, T212, X256)) -> f1120_out1(one(T211), zero(T212), one(X256)) f1120_in -> U14(f1638_in) U14(f1638_out1(T247, T248, X300)) -> f1120_out1(one(T247), one(T248), zero(X300)) f1680_in -> f1680_out1(b, one(b)) f1680_in -> U15(f629_in) U15(f629_out1(T272)) -> f1680_out1(zero(T272), one(T272)) f1680_in -> U16(f1680_in) U16(f1680_out1(T277, X346)) -> f1680_out1(one(T277), zero(X346)) f1648_in -> U17(f629_in) U17(f629_out1(T261)) -> f1648_out1(zero(T261), one(T261)) f1648_in -> U18(f1680_in) U18(f1680_out1(T266, X331)) -> f1648_out1(one(T266), zero(X331)) f2062_in -> U19(f1120_in) U19(f1120_out1(T309, T310, X402)) -> f2062_out1(zero(T309), zero(T310), one(X402)) f2062_in -> U20(f629_in) U20(f629_out1(T331)) -> f2062_out1(zero(zero(T331)), one(b), zero(one(T331))) f2062_in -> U21(f1680_in) U21(f1680_out1(T338, X448)) -> f2062_out1(zero(one(T338)), one(b), zero(zero(X448))) f2062_in -> U22(f2062_in) U22(f2062_out1(T349, T350, X464)) -> f2062_out1(zero(T349), one(T350), zero(X464)) f2062_in -> U23(f629_in) U23(f629_out1(T371)) -> f2062_out1(one(b), zero(zero(T371)), zero(one(T371))) f2062_in -> U24(f1680_in) U24(f1680_out1(T378, X510)) -> f2062_out1(one(b), zero(one(T378)), zero(zero(X510))) f2062_in -> U25(f2062_in) U25(f2062_out1(T389, T390, X526)) -> f2062_out1(one(T389), zero(T390), zero(X526)) f2062_in -> U26(f1638_in) U26(f1638_out1(T399, T400, X542)) -> f2062_out1(one(T399), one(T400), one(X542)) f1638_in -> f1638_out1(b, b, one(b)) f1638_in -> U27(f1648_in) U27(f1648_out1(T254, X315)) -> f1638_out1(T254, b, X315) f1638_in -> U28(f1648_in) U28(f1648_out1(T283, X361)) -> f1638_out1(b, T283, X361) f1638_in -> U29(f2062_in) U29(f2062_out1(T295, T296, X378)) -> f1638_out1(T295, T296, X378) f1610_in -> U30(f639_in) U30(f639_out1(T219)) -> f1610_out1(b, one(T219), one(T219)) f1610_in -> U31(f629_in) U31(f629_out1(T225)) -> f1610_out1(b, zero(T225), zero(T225)) f1610_in -> U32(f1120_in) U32(f1120_out1(T237, T238, X284)) -> f1610_out1(T237, T238, X284) f2240_in(zero(T435)) -> U33(f2240_in(T435), zero(T435)) U33(f2240_out1, zero(T435)) -> f2240_out1 f2240_in(one(T439)) -> U34(f2254_in(T439), one(T439)) U34(f2254_out1, one(T439)) -> f2240_out1 f2254_in(b) -> f2254_out1 f2254_in(zero(T444)) -> U35(f2240_in(T444), zero(T444)) U35(f2240_out1, zero(T444)) -> f2254_out1 f2254_in(one(T448)) -> U36(f2254_in(T448), one(T448)) U36(f2254_out1, one(T448)) -> f2254_out1 f2333_in(zero(T502)) -> U37(f2333_in(T502), zero(T502)) U37(f2333_out1(T503, T504), zero(T502)) -> f2333_out1(zero(T503), zero(T504)) f2333_in(one(one(T530))) -> U38(f2254_in(T530), one(one(T530))) U38(f2254_out1, one(one(T530))) -> f2333_out1(zero(one(T530)), one(b)) f2333_in(one(zero(T535))) -> U39(f2240_in(T535), one(zero(T535))) U39(f2240_out1, one(zero(T535))) -> f2333_out1(zero(zero(T535)), one(b)) f2333_in(one(T550)) -> U40(f2333_in(T550), one(T550)) U40(f2333_out1(T551, T552), one(T550)) -> f2333_out1(zero(T551), one(T552)) f2333_in(one(T570)) -> U41(f2421_in(T570), one(T570)) U41(f2421_out1(T571, T572), one(T570)) -> f2333_out1(one(T571), zero(T572)) f2333_in(zero(T612)) -> U42(f2461_in(T612), zero(T612)) U42(f2461_out1(T613, T614), zero(T612)) -> f2333_out1(one(T613), one(T614)) f2485_in(one(b)) -> f2485_out1(b) f2485_in(one(T644)) -> U43(f2240_in(T644), one(T644)) U43(f2240_out1, one(T644)) -> f2485_out1(zero(T644)) f2485_in(zero(T651)) -> U44(f2485_in(T651), zero(T651)) U44(f2485_out1(T652), zero(T651)) -> f2485_out1(one(T652)) f2473_in(one(T631)) -> U45(f2240_in(T631), one(T631)) U45(f2240_out1, one(T631)) -> f2473_out1(zero(T631)) f2473_in(zero(T638)) -> U46(f2485_in(T638), zero(T638)) U46(f2485_out1(T639), zero(T638)) -> f2473_out1(one(T639)) f2534_in(one(T698)) -> U47(f2333_in(T698), one(T698)) U47(f2333_out1(T699, T700), one(T698)) -> f2534_out1(zero(T699), zero(T700)) f2534_in(zero(one(T726))) -> U48(f2240_in(T726), zero(one(T726))) U48(f2240_out1, zero(one(T726))) -> f2534_out1(zero(zero(T726)), one(b)) f2534_in(zero(zero(T737))) -> U49(f2485_in(T737), zero(zero(T737))) U49(f2485_out1(T738), zero(zero(T737))) -> f2534_out1(zero(one(T738)), one(b)) f2534_in(zero(T752)) -> U50(f2534_in(T752), zero(T752)) U50(f2534_out1(T753, T754), zero(T752)) -> f2534_out1(zero(T753), one(T754)) f2534_in(zero(one(T780))) -> U51(f2240_in(T780), zero(one(T780))) U51(f2240_out1, zero(one(T780))) -> f2534_out1(one(b), zero(zero(T780))) f2534_in(zero(zero(T791))) -> U52(f2485_in(T791), zero(zero(T791))) U52(f2485_out1(T792), zero(zero(T791))) -> f2534_out1(one(b), zero(one(T792))) f2534_in(zero(T806)) -> U53(f2534_in(T806), zero(T806)) U53(f2534_out1(T807, T808), zero(T806)) -> f2534_out1(one(T807), zero(T808)) f2534_in(one(T820)) -> U54(f2461_in(T820), one(T820)) U54(f2461_out1(T821, T822), one(T820)) -> f2534_out1(one(T821), one(T822)) f2461_in(one(b)) -> f2461_out1(b, b) f2461_in(T624) -> U55(f2473_in(T624), T624) U55(f2473_out1(T625), T624) -> f2461_out1(T625, b) f2461_in(T662) -> U56(f2473_in(T662), T662) U56(f2473_out1(T663), T662) -> f2461_out1(b, T663) f2461_in(T678) -> U57(f2534_in(T678), T678) U57(f2534_out1(T679, T680), T678) -> f2461_out1(T679, T680) f2421_in(one(T578)) -> U58(f2254_in(T578), one(T578)) U58(f2254_out1, one(T578)) -> f2421_out1(b, one(T578)) f2421_in(zero(T583)) -> U59(f2240_in(T583), zero(T583)) U59(f2240_out1, zero(T583)) -> f2421_out1(b, zero(T583)) f2421_in(T598) -> U60(f2333_in(T598), T598) U60(f2333_out1(T599, T600), T598) -> f2421_out1(T599, T600) f258_in(zero(T429)) -> U61(f2240_in(T429), zero(T429)) U61(f2240_out1, zero(T429)) -> f258_out1(b, T429) f258_in(zero(T482)) -> U62(f2333_in(T482), zero(T482)) U62(f2333_out1(T483, T484), zero(T482)) -> f258_out1(zero(T483), T484) f258_in(one(T838)) -> U63(f2421_in(T838), one(T838)) U63(f2421_out1(T839, T840), one(T838)) -> f258_out1(one(T839), T840) f607_in -> U64(f629_in) U64(f629_out1(T95)) -> f607_out1(b, T95, zero(T95)) f607_in -> U65(f1120_in) U65(f1120_out1(T143, T144, X156)) -> f607_out1(zero(T143), T144, zero(X156)) f607_in -> U66(f1610_in) U66(f1610_out1(T412, T413, X565)) -> f607_out1(one(T412), T413, one(X565)) f212_in(T36) -> U67(f254_in, T36) U67(f254_out1(T37), T36) -> U68(f258_in(T36), T36, T37) U68(f258_out1(T42, T41), T36, T37) -> f212_out1(T37, T42, T41) f522_in -> U69(f254_in) U69(f254_out1(T70)) -> U70(f607_in, T70) U70(f607_out1(T75, T74, X78), T70) -> f522_out1(T70, T75, T74, X78) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (167) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (168) Obligation: Q DP problem: The TRS P consists of the following rules: F2485_IN(zero(T651)) -> F2485_IN(T651) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (169) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *F2485_IN(zero(T651)) -> F2485_IN(T651) The graph contains the following edges 1 > 1 ---------------------------------------- (170) YES ---------------------------------------- (171) Obligation: Q DP problem: The TRS P consists of the following rules: F2333_IN(one(T550)) -> F2333_IN(T550) F2333_IN(zero(T502)) -> F2333_IN(T502) F2333_IN(one(T570)) -> F2421_IN(T570) F2421_IN(T598) -> F2333_IN(T598) F2333_IN(zero(T612)) -> F2461_IN(T612) F2461_IN(T678) -> F2534_IN(T678) F2534_IN(one(T698)) -> F2333_IN(T698) F2534_IN(zero(T752)) -> F2534_IN(T752) F2534_IN(one(T820)) -> F2461_IN(T820) The TRS R consists of the following rules: f2_in(T8) -> f2_out1(one(b), T8) f2_in(zero(T23)) -> U1(f2_in(T23), zero(T23)) U1(f2_out1(T24, T25), zero(T23)) -> f2_out1(zero(T24), T25) f2_in(T36) -> U2(f212_in(T36), T36) U2(f212_out1(T37, T38, X33), T36) -> f2_out1(one(T37), T38) f254_in -> f254_out1(one(b)) f254_in -> U3(f254_in) U3(f254_out1(T60)) -> f254_out1(zero(T60)) f254_in -> U4(f522_in) U4(f522_out1(T70, T71, X77, X78)) -> f254_out1(one(T70)) f629_in -> U5(f629_in) U5(f629_out1(T102)) -> f629_out1(zero(T102)) f629_in -> U6(f639_in) U6(f639_out1(T107)) -> f629_out1(one(T107)) f639_in -> f639_out1(b) f639_in -> U7(f629_in) U7(f629_out1(T113)) -> f639_out1(zero(T113)) f639_in -> U8(f639_in) U8(f639_out1(T118)) -> f639_out1(one(T118)) f1120_in -> U9(f1120_in) U9(f1120_out1(T157, T158, X180)) -> f1120_out1(zero(T157), zero(T158), zero(X180)) f1120_in -> U10(f639_in) U10(f639_out1(T179)) -> f1120_out1(zero(one(T179)), one(b), one(one(T179))) f1120_in -> U11(f629_in) U11(f629_out1(T185)) -> f1120_out1(zero(zero(T185)), one(b), one(zero(T185))) f1120_in -> U12(f1120_in) U12(f1120_out1(T197, T198, X232)) -> f1120_out1(zero(T197), one(T198), one(X232)) f1120_in -> U13(f1610_in) U13(f1610_out1(T211, T212, X256)) -> f1120_out1(one(T211), zero(T212), one(X256)) f1120_in -> U14(f1638_in) U14(f1638_out1(T247, T248, X300)) -> f1120_out1(one(T247), one(T248), zero(X300)) f1680_in -> f1680_out1(b, one(b)) f1680_in -> U15(f629_in) U15(f629_out1(T272)) -> f1680_out1(zero(T272), one(T272)) f1680_in -> U16(f1680_in) U16(f1680_out1(T277, X346)) -> f1680_out1(one(T277), zero(X346)) f1648_in -> U17(f629_in) U17(f629_out1(T261)) -> f1648_out1(zero(T261), one(T261)) f1648_in -> U18(f1680_in) U18(f1680_out1(T266, X331)) -> f1648_out1(one(T266), zero(X331)) f2062_in -> U19(f1120_in) U19(f1120_out1(T309, T310, X402)) -> f2062_out1(zero(T309), zero(T310), one(X402)) f2062_in -> U20(f629_in) U20(f629_out1(T331)) -> f2062_out1(zero(zero(T331)), one(b), zero(one(T331))) f2062_in -> U21(f1680_in) U21(f1680_out1(T338, X448)) -> f2062_out1(zero(one(T338)), one(b), zero(zero(X448))) f2062_in -> U22(f2062_in) U22(f2062_out1(T349, T350, X464)) -> f2062_out1(zero(T349), one(T350), zero(X464)) f2062_in -> U23(f629_in) U23(f629_out1(T371)) -> f2062_out1(one(b), zero(zero(T371)), zero(one(T371))) f2062_in -> U24(f1680_in) U24(f1680_out1(T378, X510)) -> f2062_out1(one(b), zero(one(T378)), zero(zero(X510))) f2062_in -> U25(f2062_in) U25(f2062_out1(T389, T390, X526)) -> f2062_out1(one(T389), zero(T390), zero(X526)) f2062_in -> U26(f1638_in) U26(f1638_out1(T399, T400, X542)) -> f2062_out1(one(T399), one(T400), one(X542)) f1638_in -> f1638_out1(b, b, one(b)) f1638_in -> U27(f1648_in) U27(f1648_out1(T254, X315)) -> f1638_out1(T254, b, X315) f1638_in -> U28(f1648_in) U28(f1648_out1(T283, X361)) -> f1638_out1(b, T283, X361) f1638_in -> U29(f2062_in) U29(f2062_out1(T295, T296, X378)) -> f1638_out1(T295, T296, X378) f1610_in -> U30(f639_in) U30(f639_out1(T219)) -> f1610_out1(b, one(T219), one(T219)) f1610_in -> U31(f629_in) U31(f629_out1(T225)) -> f1610_out1(b, zero(T225), zero(T225)) f1610_in -> U32(f1120_in) U32(f1120_out1(T237, T238, X284)) -> f1610_out1(T237, T238, X284) f2240_in(zero(T435)) -> U33(f2240_in(T435), zero(T435)) U33(f2240_out1, zero(T435)) -> f2240_out1 f2240_in(one(T439)) -> U34(f2254_in(T439), one(T439)) U34(f2254_out1, one(T439)) -> f2240_out1 f2254_in(b) -> f2254_out1 f2254_in(zero(T444)) -> U35(f2240_in(T444), zero(T444)) U35(f2240_out1, zero(T444)) -> f2254_out1 f2254_in(one(T448)) -> U36(f2254_in(T448), one(T448)) U36(f2254_out1, one(T448)) -> f2254_out1 f2333_in(zero(T502)) -> U37(f2333_in(T502), zero(T502)) U37(f2333_out1(T503, T504), zero(T502)) -> f2333_out1(zero(T503), zero(T504)) f2333_in(one(one(T530))) -> U38(f2254_in(T530), one(one(T530))) U38(f2254_out1, one(one(T530))) -> f2333_out1(zero(one(T530)), one(b)) f2333_in(one(zero(T535))) -> U39(f2240_in(T535), one(zero(T535))) U39(f2240_out1, one(zero(T535))) -> f2333_out1(zero(zero(T535)), one(b)) f2333_in(one(T550)) -> U40(f2333_in(T550), one(T550)) U40(f2333_out1(T551, T552), one(T550)) -> f2333_out1(zero(T551), one(T552)) f2333_in(one(T570)) -> U41(f2421_in(T570), one(T570)) U41(f2421_out1(T571, T572), one(T570)) -> f2333_out1(one(T571), zero(T572)) f2333_in(zero(T612)) -> U42(f2461_in(T612), zero(T612)) U42(f2461_out1(T613, T614), zero(T612)) -> f2333_out1(one(T613), one(T614)) f2485_in(one(b)) -> f2485_out1(b) f2485_in(one(T644)) -> U43(f2240_in(T644), one(T644)) U43(f2240_out1, one(T644)) -> f2485_out1(zero(T644)) f2485_in(zero(T651)) -> U44(f2485_in(T651), zero(T651)) U44(f2485_out1(T652), zero(T651)) -> f2485_out1(one(T652)) f2473_in(one(T631)) -> U45(f2240_in(T631), one(T631)) U45(f2240_out1, one(T631)) -> f2473_out1(zero(T631)) f2473_in(zero(T638)) -> U46(f2485_in(T638), zero(T638)) U46(f2485_out1(T639), zero(T638)) -> f2473_out1(one(T639)) f2534_in(one(T698)) -> U47(f2333_in(T698), one(T698)) U47(f2333_out1(T699, T700), one(T698)) -> f2534_out1(zero(T699), zero(T700)) f2534_in(zero(one(T726))) -> U48(f2240_in(T726), zero(one(T726))) U48(f2240_out1, zero(one(T726))) -> f2534_out1(zero(zero(T726)), one(b)) f2534_in(zero(zero(T737))) -> U49(f2485_in(T737), zero(zero(T737))) U49(f2485_out1(T738), zero(zero(T737))) -> f2534_out1(zero(one(T738)), one(b)) f2534_in(zero(T752)) -> U50(f2534_in(T752), zero(T752)) U50(f2534_out1(T753, T754), zero(T752)) -> f2534_out1(zero(T753), one(T754)) f2534_in(zero(one(T780))) -> U51(f2240_in(T780), zero(one(T780))) U51(f2240_out1, zero(one(T780))) -> f2534_out1(one(b), zero(zero(T780))) f2534_in(zero(zero(T791))) -> U52(f2485_in(T791), zero(zero(T791))) U52(f2485_out1(T792), zero(zero(T791))) -> f2534_out1(one(b), zero(one(T792))) f2534_in(zero(T806)) -> U53(f2534_in(T806), zero(T806)) U53(f2534_out1(T807, T808), zero(T806)) -> f2534_out1(one(T807), zero(T808)) f2534_in(one(T820)) -> U54(f2461_in(T820), one(T820)) U54(f2461_out1(T821, T822), one(T820)) -> f2534_out1(one(T821), one(T822)) f2461_in(one(b)) -> f2461_out1(b, b) f2461_in(T624) -> U55(f2473_in(T624), T624) U55(f2473_out1(T625), T624) -> f2461_out1(T625, b) f2461_in(T662) -> U56(f2473_in(T662), T662) U56(f2473_out1(T663), T662) -> f2461_out1(b, T663) f2461_in(T678) -> U57(f2534_in(T678), T678) U57(f2534_out1(T679, T680), T678) -> f2461_out1(T679, T680) f2421_in(one(T578)) -> U58(f2254_in(T578), one(T578)) U58(f2254_out1, one(T578)) -> f2421_out1(b, one(T578)) f2421_in(zero(T583)) -> U59(f2240_in(T583), zero(T583)) U59(f2240_out1, zero(T583)) -> f2421_out1(b, zero(T583)) f2421_in(T598) -> U60(f2333_in(T598), T598) U60(f2333_out1(T599, T600), T598) -> f2421_out1(T599, T600) f258_in(zero(T429)) -> U61(f2240_in(T429), zero(T429)) U61(f2240_out1, zero(T429)) -> f258_out1(b, T429) f258_in(zero(T482)) -> U62(f2333_in(T482), zero(T482)) U62(f2333_out1(T483, T484), zero(T482)) -> f258_out1(zero(T483), T484) f258_in(one(T838)) -> U63(f2421_in(T838), one(T838)) U63(f2421_out1(T839, T840), one(T838)) -> f258_out1(one(T839), T840) f607_in -> U64(f629_in) U64(f629_out1(T95)) -> f607_out1(b, T95, zero(T95)) f607_in -> U65(f1120_in) U65(f1120_out1(T143, T144, X156)) -> f607_out1(zero(T143), T144, zero(X156)) f607_in -> U66(f1610_in) U66(f1610_out1(T412, T413, X565)) -> f607_out1(one(T412), T413, one(X565)) f212_in(T36) -> U67(f254_in, T36) U67(f254_out1(T37), T36) -> U68(f258_in(T36), T36, T37) U68(f258_out1(T42, T41), T36, T37) -> f212_out1(T37, T42, T41) f522_in -> U69(f254_in) U69(f254_out1(T70)) -> U70(f607_in, T70) U70(f607_out1(T75, T74, X78), T70) -> f522_out1(T70, T75, T74, X78) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (172) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (173) Obligation: Q DP problem: The TRS P consists of the following rules: F2333_IN(one(T550)) -> F2333_IN(T550) F2333_IN(zero(T502)) -> F2333_IN(T502) F2333_IN(one(T570)) -> F2421_IN(T570) F2421_IN(T598) -> F2333_IN(T598) F2333_IN(zero(T612)) -> F2461_IN(T612) F2461_IN(T678) -> F2534_IN(T678) F2534_IN(one(T698)) -> F2333_IN(T698) F2534_IN(zero(T752)) -> F2534_IN(T752) F2534_IN(one(T820)) -> F2461_IN(T820) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (174) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *F2421_IN(T598) -> F2333_IN(T598) The graph contains the following edges 1 >= 1 *F2534_IN(one(T698)) -> F2333_IN(T698) The graph contains the following edges 1 > 1 *F2333_IN(one(T570)) -> F2421_IN(T570) The graph contains the following edges 1 > 1 *F2333_IN(zero(T612)) -> F2461_IN(T612) The graph contains the following edges 1 > 1 *F2461_IN(T678) -> F2534_IN(T678) The graph contains the following edges 1 >= 1 *F2534_IN(one(T820)) -> F2461_IN(T820) The graph contains the following edges 1 > 1 *F2534_IN(zero(T752)) -> F2534_IN(T752) The graph contains the following edges 1 > 1 *F2333_IN(one(T550)) -> F2333_IN(T550) The graph contains the following edges 1 > 1 *F2333_IN(zero(T502)) -> F2333_IN(T502) The graph contains the following edges 1 > 1 ---------------------------------------- (175) YES ---------------------------------------- (176) Obligation: Q DP problem: The TRS P consists of the following rules: F629_IN -> F639_IN F639_IN -> F629_IN F629_IN -> F629_IN F639_IN -> F639_IN The TRS R consists of the following rules: f2_in(T8) -> f2_out1(one(b), T8) f2_in(zero(T23)) -> U1(f2_in(T23), zero(T23)) U1(f2_out1(T24, T25), zero(T23)) -> f2_out1(zero(T24), T25) f2_in(T36) -> U2(f212_in(T36), T36) U2(f212_out1(T37, T38, X33), T36) -> f2_out1(one(T37), T38) f254_in -> f254_out1(one(b)) f254_in -> U3(f254_in) U3(f254_out1(T60)) -> f254_out1(zero(T60)) f254_in -> U4(f522_in) U4(f522_out1(T70, T71, X77, X78)) -> f254_out1(one(T70)) f629_in -> U5(f629_in) U5(f629_out1(T102)) -> f629_out1(zero(T102)) f629_in -> U6(f639_in) U6(f639_out1(T107)) -> f629_out1(one(T107)) f639_in -> f639_out1(b) f639_in -> U7(f629_in) U7(f629_out1(T113)) -> f639_out1(zero(T113)) f639_in -> U8(f639_in) U8(f639_out1(T118)) -> f639_out1(one(T118)) f1120_in -> U9(f1120_in) U9(f1120_out1(T157, T158, X180)) -> f1120_out1(zero(T157), zero(T158), zero(X180)) f1120_in -> U10(f639_in) U10(f639_out1(T179)) -> f1120_out1(zero(one(T179)), one(b), one(one(T179))) f1120_in -> U11(f629_in) U11(f629_out1(T185)) -> f1120_out1(zero(zero(T185)), one(b), one(zero(T185))) f1120_in -> U12(f1120_in) U12(f1120_out1(T197, T198, X232)) -> f1120_out1(zero(T197), one(T198), one(X232)) f1120_in -> U13(f1610_in) U13(f1610_out1(T211, T212, X256)) -> f1120_out1(one(T211), zero(T212), one(X256)) f1120_in -> U14(f1638_in) U14(f1638_out1(T247, T248, X300)) -> f1120_out1(one(T247), one(T248), zero(X300)) f1680_in -> f1680_out1(b, one(b)) f1680_in -> U15(f629_in) U15(f629_out1(T272)) -> f1680_out1(zero(T272), one(T272)) f1680_in -> U16(f1680_in) U16(f1680_out1(T277, X346)) -> f1680_out1(one(T277), zero(X346)) f1648_in -> U17(f629_in) U17(f629_out1(T261)) -> f1648_out1(zero(T261), one(T261)) f1648_in -> U18(f1680_in) U18(f1680_out1(T266, X331)) -> f1648_out1(one(T266), zero(X331)) f2062_in -> U19(f1120_in) U19(f1120_out1(T309, T310, X402)) -> f2062_out1(zero(T309), zero(T310), one(X402)) f2062_in -> U20(f629_in) U20(f629_out1(T331)) -> f2062_out1(zero(zero(T331)), one(b), zero(one(T331))) f2062_in -> U21(f1680_in) U21(f1680_out1(T338, X448)) -> f2062_out1(zero(one(T338)), one(b), zero(zero(X448))) f2062_in -> U22(f2062_in) U22(f2062_out1(T349, T350, X464)) -> f2062_out1(zero(T349), one(T350), zero(X464)) f2062_in -> U23(f629_in) U23(f629_out1(T371)) -> f2062_out1(one(b), zero(zero(T371)), zero(one(T371))) f2062_in -> U24(f1680_in) U24(f1680_out1(T378, X510)) -> f2062_out1(one(b), zero(one(T378)), zero(zero(X510))) f2062_in -> U25(f2062_in) U25(f2062_out1(T389, T390, X526)) -> f2062_out1(one(T389), zero(T390), zero(X526)) f2062_in -> U26(f1638_in) U26(f1638_out1(T399, T400, X542)) -> f2062_out1(one(T399), one(T400), one(X542)) f1638_in -> f1638_out1(b, b, one(b)) f1638_in -> U27(f1648_in) U27(f1648_out1(T254, X315)) -> f1638_out1(T254, b, X315) f1638_in -> U28(f1648_in) U28(f1648_out1(T283, X361)) -> f1638_out1(b, T283, X361) f1638_in -> U29(f2062_in) U29(f2062_out1(T295, T296, X378)) -> f1638_out1(T295, T296, X378) f1610_in -> U30(f639_in) U30(f639_out1(T219)) -> f1610_out1(b, one(T219), one(T219)) f1610_in -> U31(f629_in) U31(f629_out1(T225)) -> f1610_out1(b, zero(T225), zero(T225)) f1610_in -> U32(f1120_in) U32(f1120_out1(T237, T238, X284)) -> f1610_out1(T237, T238, X284) f2240_in(zero(T435)) -> U33(f2240_in(T435), zero(T435)) U33(f2240_out1, zero(T435)) -> f2240_out1 f2240_in(one(T439)) -> U34(f2254_in(T439), one(T439)) U34(f2254_out1, one(T439)) -> f2240_out1 f2254_in(b) -> f2254_out1 f2254_in(zero(T444)) -> U35(f2240_in(T444), zero(T444)) U35(f2240_out1, zero(T444)) -> f2254_out1 f2254_in(one(T448)) -> U36(f2254_in(T448), one(T448)) U36(f2254_out1, one(T448)) -> f2254_out1 f2333_in(zero(T502)) -> U37(f2333_in(T502), zero(T502)) U37(f2333_out1(T503, T504), zero(T502)) -> f2333_out1(zero(T503), zero(T504)) f2333_in(one(one(T530))) -> U38(f2254_in(T530), one(one(T530))) U38(f2254_out1, one(one(T530))) -> f2333_out1(zero(one(T530)), one(b)) f2333_in(one(zero(T535))) -> U39(f2240_in(T535), one(zero(T535))) U39(f2240_out1, one(zero(T535))) -> f2333_out1(zero(zero(T535)), one(b)) f2333_in(one(T550)) -> U40(f2333_in(T550), one(T550)) U40(f2333_out1(T551, T552), one(T550)) -> f2333_out1(zero(T551), one(T552)) f2333_in(one(T570)) -> U41(f2421_in(T570), one(T570)) U41(f2421_out1(T571, T572), one(T570)) -> f2333_out1(one(T571), zero(T572)) f2333_in(zero(T612)) -> U42(f2461_in(T612), zero(T612)) U42(f2461_out1(T613, T614), zero(T612)) -> f2333_out1(one(T613), one(T614)) f2485_in(one(b)) -> f2485_out1(b) f2485_in(one(T644)) -> U43(f2240_in(T644), one(T644)) U43(f2240_out1, one(T644)) -> f2485_out1(zero(T644)) f2485_in(zero(T651)) -> U44(f2485_in(T651), zero(T651)) U44(f2485_out1(T652), zero(T651)) -> f2485_out1(one(T652)) f2473_in(one(T631)) -> U45(f2240_in(T631), one(T631)) U45(f2240_out1, one(T631)) -> f2473_out1(zero(T631)) f2473_in(zero(T638)) -> U46(f2485_in(T638), zero(T638)) U46(f2485_out1(T639), zero(T638)) -> f2473_out1(one(T639)) f2534_in(one(T698)) -> U47(f2333_in(T698), one(T698)) U47(f2333_out1(T699, T700), one(T698)) -> f2534_out1(zero(T699), zero(T700)) f2534_in(zero(one(T726))) -> U48(f2240_in(T726), zero(one(T726))) U48(f2240_out1, zero(one(T726))) -> f2534_out1(zero(zero(T726)), one(b)) f2534_in(zero(zero(T737))) -> U49(f2485_in(T737), zero(zero(T737))) U49(f2485_out1(T738), zero(zero(T737))) -> f2534_out1(zero(one(T738)), one(b)) f2534_in(zero(T752)) -> U50(f2534_in(T752), zero(T752)) U50(f2534_out1(T753, T754), zero(T752)) -> f2534_out1(zero(T753), one(T754)) f2534_in(zero(one(T780))) -> U51(f2240_in(T780), zero(one(T780))) U51(f2240_out1, zero(one(T780))) -> f2534_out1(one(b), zero(zero(T780))) f2534_in(zero(zero(T791))) -> U52(f2485_in(T791), zero(zero(T791))) U52(f2485_out1(T792), zero(zero(T791))) -> f2534_out1(one(b), zero(one(T792))) f2534_in(zero(T806)) -> U53(f2534_in(T806), zero(T806)) U53(f2534_out1(T807, T808), zero(T806)) -> f2534_out1(one(T807), zero(T808)) f2534_in(one(T820)) -> U54(f2461_in(T820), one(T820)) U54(f2461_out1(T821, T822), one(T820)) -> f2534_out1(one(T821), one(T822)) f2461_in(one(b)) -> f2461_out1(b, b) f2461_in(T624) -> U55(f2473_in(T624), T624) U55(f2473_out1(T625), T624) -> f2461_out1(T625, b) f2461_in(T662) -> U56(f2473_in(T662), T662) U56(f2473_out1(T663), T662) -> f2461_out1(b, T663) f2461_in(T678) -> U57(f2534_in(T678), T678) U57(f2534_out1(T679, T680), T678) -> f2461_out1(T679, T680) f2421_in(one(T578)) -> U58(f2254_in(T578), one(T578)) U58(f2254_out1, one(T578)) -> f2421_out1(b, one(T578)) f2421_in(zero(T583)) -> U59(f2240_in(T583), zero(T583)) U59(f2240_out1, zero(T583)) -> f2421_out1(b, zero(T583)) f2421_in(T598) -> U60(f2333_in(T598), T598) U60(f2333_out1(T599, T600), T598) -> f2421_out1(T599, T600) f258_in(zero(T429)) -> U61(f2240_in(T429), zero(T429)) U61(f2240_out1, zero(T429)) -> f258_out1(b, T429) f258_in(zero(T482)) -> U62(f2333_in(T482), zero(T482)) U62(f2333_out1(T483, T484), zero(T482)) -> f258_out1(zero(T483), T484) f258_in(one(T838)) -> U63(f2421_in(T838), one(T838)) U63(f2421_out1(T839, T840), one(T838)) -> f258_out1(one(T839), T840) f607_in -> U64(f629_in) U64(f629_out1(T95)) -> f607_out1(b, T95, zero(T95)) f607_in -> U65(f1120_in) U65(f1120_out1(T143, T144, X156)) -> f607_out1(zero(T143), T144, zero(X156)) f607_in -> U66(f1610_in) U66(f1610_out1(T412, T413, X565)) -> f607_out1(one(T412), T413, one(X565)) f212_in(T36) -> U67(f254_in, T36) U67(f254_out1(T37), T36) -> U68(f258_in(T36), T36, T37) U68(f258_out1(T42, T41), T36, T37) -> f212_out1(T37, T42, T41) f522_in -> U69(f254_in) U69(f254_out1(T70)) -> U70(f607_in, T70) U70(f607_out1(T75, T74, X78), T70) -> f522_out1(T70, T75, T74, X78) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (177) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (178) Obligation: Q DP problem: The TRS P consists of the following rules: F629_IN -> F639_IN F639_IN -> F629_IN F629_IN -> F629_IN F639_IN -> F639_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (179) Obligation: Q DP problem: The TRS P consists of the following rules: F1680_IN -> F1680_IN The TRS R consists of the following rules: f2_in(T8) -> f2_out1(one(b), T8) f2_in(zero(T23)) -> U1(f2_in(T23), zero(T23)) U1(f2_out1(T24, T25), zero(T23)) -> f2_out1(zero(T24), T25) f2_in(T36) -> U2(f212_in(T36), T36) U2(f212_out1(T37, T38, X33), T36) -> f2_out1(one(T37), T38) f254_in -> f254_out1(one(b)) f254_in -> U3(f254_in) U3(f254_out1(T60)) -> f254_out1(zero(T60)) f254_in -> U4(f522_in) U4(f522_out1(T70, T71, X77, X78)) -> f254_out1(one(T70)) f629_in -> U5(f629_in) U5(f629_out1(T102)) -> f629_out1(zero(T102)) f629_in -> U6(f639_in) U6(f639_out1(T107)) -> f629_out1(one(T107)) f639_in -> f639_out1(b) f639_in -> U7(f629_in) U7(f629_out1(T113)) -> f639_out1(zero(T113)) f639_in -> U8(f639_in) U8(f639_out1(T118)) -> f639_out1(one(T118)) f1120_in -> U9(f1120_in) U9(f1120_out1(T157, T158, X180)) -> f1120_out1(zero(T157), zero(T158), zero(X180)) f1120_in -> U10(f639_in) U10(f639_out1(T179)) -> f1120_out1(zero(one(T179)), one(b), one(one(T179))) f1120_in -> U11(f629_in) U11(f629_out1(T185)) -> f1120_out1(zero(zero(T185)), one(b), one(zero(T185))) f1120_in -> U12(f1120_in) U12(f1120_out1(T197, T198, X232)) -> f1120_out1(zero(T197), one(T198), one(X232)) f1120_in -> U13(f1610_in) U13(f1610_out1(T211, T212, X256)) -> f1120_out1(one(T211), zero(T212), one(X256)) f1120_in -> U14(f1638_in) U14(f1638_out1(T247, T248, X300)) -> f1120_out1(one(T247), one(T248), zero(X300)) f1680_in -> f1680_out1(b, one(b)) f1680_in -> U15(f629_in) U15(f629_out1(T272)) -> f1680_out1(zero(T272), one(T272)) f1680_in -> U16(f1680_in) U16(f1680_out1(T277, X346)) -> f1680_out1(one(T277), zero(X346)) f1648_in -> U17(f629_in) U17(f629_out1(T261)) -> f1648_out1(zero(T261), one(T261)) f1648_in -> U18(f1680_in) U18(f1680_out1(T266, X331)) -> f1648_out1(one(T266), zero(X331)) f2062_in -> U19(f1120_in) U19(f1120_out1(T309, T310, X402)) -> f2062_out1(zero(T309), zero(T310), one(X402)) f2062_in -> U20(f629_in) U20(f629_out1(T331)) -> f2062_out1(zero(zero(T331)), one(b), zero(one(T331))) f2062_in -> U21(f1680_in) U21(f1680_out1(T338, X448)) -> f2062_out1(zero(one(T338)), one(b), zero(zero(X448))) f2062_in -> U22(f2062_in) U22(f2062_out1(T349, T350, X464)) -> f2062_out1(zero(T349), one(T350), zero(X464)) f2062_in -> U23(f629_in) U23(f629_out1(T371)) -> f2062_out1(one(b), zero(zero(T371)), zero(one(T371))) f2062_in -> U24(f1680_in) U24(f1680_out1(T378, X510)) -> f2062_out1(one(b), zero(one(T378)), zero(zero(X510))) f2062_in -> U25(f2062_in) U25(f2062_out1(T389, T390, X526)) -> f2062_out1(one(T389), zero(T390), zero(X526)) f2062_in -> U26(f1638_in) U26(f1638_out1(T399, T400, X542)) -> f2062_out1(one(T399), one(T400), one(X542)) f1638_in -> f1638_out1(b, b, one(b)) f1638_in -> U27(f1648_in) U27(f1648_out1(T254, X315)) -> f1638_out1(T254, b, X315) f1638_in -> U28(f1648_in) U28(f1648_out1(T283, X361)) -> f1638_out1(b, T283, X361) f1638_in -> U29(f2062_in) U29(f2062_out1(T295, T296, X378)) -> f1638_out1(T295, T296, X378) f1610_in -> U30(f639_in) U30(f639_out1(T219)) -> f1610_out1(b, one(T219), one(T219)) f1610_in -> U31(f629_in) U31(f629_out1(T225)) -> f1610_out1(b, zero(T225), zero(T225)) f1610_in -> U32(f1120_in) U32(f1120_out1(T237, T238, X284)) -> f1610_out1(T237, T238, X284) f2240_in(zero(T435)) -> U33(f2240_in(T435), zero(T435)) U33(f2240_out1, zero(T435)) -> f2240_out1 f2240_in(one(T439)) -> U34(f2254_in(T439), one(T439)) U34(f2254_out1, one(T439)) -> f2240_out1 f2254_in(b) -> f2254_out1 f2254_in(zero(T444)) -> U35(f2240_in(T444), zero(T444)) U35(f2240_out1, zero(T444)) -> f2254_out1 f2254_in(one(T448)) -> U36(f2254_in(T448), one(T448)) U36(f2254_out1, one(T448)) -> f2254_out1 f2333_in(zero(T502)) -> U37(f2333_in(T502), zero(T502)) U37(f2333_out1(T503, T504), zero(T502)) -> f2333_out1(zero(T503), zero(T504)) f2333_in(one(one(T530))) -> U38(f2254_in(T530), one(one(T530))) U38(f2254_out1, one(one(T530))) -> f2333_out1(zero(one(T530)), one(b)) f2333_in(one(zero(T535))) -> U39(f2240_in(T535), one(zero(T535))) U39(f2240_out1, one(zero(T535))) -> f2333_out1(zero(zero(T535)), one(b)) f2333_in(one(T550)) -> U40(f2333_in(T550), one(T550)) U40(f2333_out1(T551, T552), one(T550)) -> f2333_out1(zero(T551), one(T552)) f2333_in(one(T570)) -> U41(f2421_in(T570), one(T570)) U41(f2421_out1(T571, T572), one(T570)) -> f2333_out1(one(T571), zero(T572)) f2333_in(zero(T612)) -> U42(f2461_in(T612), zero(T612)) U42(f2461_out1(T613, T614), zero(T612)) -> f2333_out1(one(T613), one(T614)) f2485_in(one(b)) -> f2485_out1(b) f2485_in(one(T644)) -> U43(f2240_in(T644), one(T644)) U43(f2240_out1, one(T644)) -> f2485_out1(zero(T644)) f2485_in(zero(T651)) -> U44(f2485_in(T651), zero(T651)) U44(f2485_out1(T652), zero(T651)) -> f2485_out1(one(T652)) f2473_in(one(T631)) -> U45(f2240_in(T631), one(T631)) U45(f2240_out1, one(T631)) -> f2473_out1(zero(T631)) f2473_in(zero(T638)) -> U46(f2485_in(T638), zero(T638)) U46(f2485_out1(T639), zero(T638)) -> f2473_out1(one(T639)) f2534_in(one(T698)) -> U47(f2333_in(T698), one(T698)) U47(f2333_out1(T699, T700), one(T698)) -> f2534_out1(zero(T699), zero(T700)) f2534_in(zero(one(T726))) -> U48(f2240_in(T726), zero(one(T726))) U48(f2240_out1, zero(one(T726))) -> f2534_out1(zero(zero(T726)), one(b)) f2534_in(zero(zero(T737))) -> U49(f2485_in(T737), zero(zero(T737))) U49(f2485_out1(T738), zero(zero(T737))) -> f2534_out1(zero(one(T738)), one(b)) f2534_in(zero(T752)) -> U50(f2534_in(T752), zero(T752)) U50(f2534_out1(T753, T754), zero(T752)) -> f2534_out1(zero(T753), one(T754)) f2534_in(zero(one(T780))) -> U51(f2240_in(T780), zero(one(T780))) U51(f2240_out1, zero(one(T780))) -> f2534_out1(one(b), zero(zero(T780))) f2534_in(zero(zero(T791))) -> U52(f2485_in(T791), zero(zero(T791))) U52(f2485_out1(T792), zero(zero(T791))) -> f2534_out1(one(b), zero(one(T792))) f2534_in(zero(T806)) -> U53(f2534_in(T806), zero(T806)) U53(f2534_out1(T807, T808), zero(T806)) -> f2534_out1(one(T807), zero(T808)) f2534_in(one(T820)) -> U54(f2461_in(T820), one(T820)) U54(f2461_out1(T821, T822), one(T820)) -> f2534_out1(one(T821), one(T822)) f2461_in(one(b)) -> f2461_out1(b, b) f2461_in(T624) -> U55(f2473_in(T624), T624) U55(f2473_out1(T625), T624) -> f2461_out1(T625, b) f2461_in(T662) -> U56(f2473_in(T662), T662) U56(f2473_out1(T663), T662) -> f2461_out1(b, T663) f2461_in(T678) -> U57(f2534_in(T678), T678) U57(f2534_out1(T679, T680), T678) -> f2461_out1(T679, T680) f2421_in(one(T578)) -> U58(f2254_in(T578), one(T578)) U58(f2254_out1, one(T578)) -> f2421_out1(b, one(T578)) f2421_in(zero(T583)) -> U59(f2240_in(T583), zero(T583)) U59(f2240_out1, zero(T583)) -> f2421_out1(b, zero(T583)) f2421_in(T598) -> U60(f2333_in(T598), T598) U60(f2333_out1(T599, T600), T598) -> f2421_out1(T599, T600) f258_in(zero(T429)) -> U61(f2240_in(T429), zero(T429)) U61(f2240_out1, zero(T429)) -> f258_out1(b, T429) f258_in(zero(T482)) -> U62(f2333_in(T482), zero(T482)) U62(f2333_out1(T483, T484), zero(T482)) -> f258_out1(zero(T483), T484) f258_in(one(T838)) -> U63(f2421_in(T838), one(T838)) U63(f2421_out1(T839, T840), one(T838)) -> f258_out1(one(T839), T840) f607_in -> U64(f629_in) U64(f629_out1(T95)) -> f607_out1(b, T95, zero(T95)) f607_in -> U65(f1120_in) U65(f1120_out1(T143, T144, X156)) -> f607_out1(zero(T143), T144, zero(X156)) f607_in -> U66(f1610_in) U66(f1610_out1(T412, T413, X565)) -> f607_out1(one(T412), T413, one(X565)) f212_in(T36) -> U67(f254_in, T36) U67(f254_out1(T37), T36) -> U68(f258_in(T36), T36, T37) U68(f258_out1(T42, T41), T36, T37) -> f212_out1(T37, T42, T41) f522_in -> U69(f254_in) U69(f254_out1(T70)) -> U70(f607_in, T70) U70(f607_out1(T75, T74, X78), T70) -> f522_out1(T70, T75, T74, X78) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (180) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (181) Obligation: Q DP problem: The TRS P consists of the following rules: F1680_IN -> F1680_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (182) Obligation: Q DP problem: The TRS P consists of the following rules: F1120_IN -> F1610_IN F1610_IN -> F1120_IN F1120_IN -> F1120_IN F1120_IN -> F1638_IN F1638_IN -> F2062_IN F2062_IN -> F1120_IN F2062_IN -> F2062_IN F2062_IN -> F1638_IN The TRS R consists of the following rules: f2_in(T8) -> f2_out1(one(b), T8) f2_in(zero(T23)) -> U1(f2_in(T23), zero(T23)) U1(f2_out1(T24, T25), zero(T23)) -> f2_out1(zero(T24), T25) f2_in(T36) -> U2(f212_in(T36), T36) U2(f212_out1(T37, T38, X33), T36) -> f2_out1(one(T37), T38) f254_in -> f254_out1(one(b)) f254_in -> U3(f254_in) U3(f254_out1(T60)) -> f254_out1(zero(T60)) f254_in -> U4(f522_in) U4(f522_out1(T70, T71, X77, X78)) -> f254_out1(one(T70)) f629_in -> U5(f629_in) U5(f629_out1(T102)) -> f629_out1(zero(T102)) f629_in -> U6(f639_in) U6(f639_out1(T107)) -> f629_out1(one(T107)) f639_in -> f639_out1(b) f639_in -> U7(f629_in) U7(f629_out1(T113)) -> f639_out1(zero(T113)) f639_in -> U8(f639_in) U8(f639_out1(T118)) -> f639_out1(one(T118)) f1120_in -> U9(f1120_in) U9(f1120_out1(T157, T158, X180)) -> f1120_out1(zero(T157), zero(T158), zero(X180)) f1120_in -> U10(f639_in) U10(f639_out1(T179)) -> f1120_out1(zero(one(T179)), one(b), one(one(T179))) f1120_in -> U11(f629_in) U11(f629_out1(T185)) -> f1120_out1(zero(zero(T185)), one(b), one(zero(T185))) f1120_in -> U12(f1120_in) U12(f1120_out1(T197, T198, X232)) -> f1120_out1(zero(T197), one(T198), one(X232)) f1120_in -> U13(f1610_in) U13(f1610_out1(T211, T212, X256)) -> f1120_out1(one(T211), zero(T212), one(X256)) f1120_in -> U14(f1638_in) U14(f1638_out1(T247, T248, X300)) -> f1120_out1(one(T247), one(T248), zero(X300)) f1680_in -> f1680_out1(b, one(b)) f1680_in -> U15(f629_in) U15(f629_out1(T272)) -> f1680_out1(zero(T272), one(T272)) f1680_in -> U16(f1680_in) U16(f1680_out1(T277, X346)) -> f1680_out1(one(T277), zero(X346)) f1648_in -> U17(f629_in) U17(f629_out1(T261)) -> f1648_out1(zero(T261), one(T261)) f1648_in -> U18(f1680_in) U18(f1680_out1(T266, X331)) -> f1648_out1(one(T266), zero(X331)) f2062_in -> U19(f1120_in) U19(f1120_out1(T309, T310, X402)) -> f2062_out1(zero(T309), zero(T310), one(X402)) f2062_in -> U20(f629_in) U20(f629_out1(T331)) -> f2062_out1(zero(zero(T331)), one(b), zero(one(T331))) f2062_in -> U21(f1680_in) U21(f1680_out1(T338, X448)) -> f2062_out1(zero(one(T338)), one(b), zero(zero(X448))) f2062_in -> U22(f2062_in) U22(f2062_out1(T349, T350, X464)) -> f2062_out1(zero(T349), one(T350), zero(X464)) f2062_in -> U23(f629_in) U23(f629_out1(T371)) -> f2062_out1(one(b), zero(zero(T371)), zero(one(T371))) f2062_in -> U24(f1680_in) U24(f1680_out1(T378, X510)) -> f2062_out1(one(b), zero(one(T378)), zero(zero(X510))) f2062_in -> U25(f2062_in) U25(f2062_out1(T389, T390, X526)) -> f2062_out1(one(T389), zero(T390), zero(X526)) f2062_in -> U26(f1638_in) U26(f1638_out1(T399, T400, X542)) -> f2062_out1(one(T399), one(T400), one(X542)) f1638_in -> f1638_out1(b, b, one(b)) f1638_in -> U27(f1648_in) U27(f1648_out1(T254, X315)) -> f1638_out1(T254, b, X315) f1638_in -> U28(f1648_in) U28(f1648_out1(T283, X361)) -> f1638_out1(b, T283, X361) f1638_in -> U29(f2062_in) U29(f2062_out1(T295, T296, X378)) -> f1638_out1(T295, T296, X378) f1610_in -> U30(f639_in) U30(f639_out1(T219)) -> f1610_out1(b, one(T219), one(T219)) f1610_in -> U31(f629_in) U31(f629_out1(T225)) -> f1610_out1(b, zero(T225), zero(T225)) f1610_in -> U32(f1120_in) U32(f1120_out1(T237, T238, X284)) -> f1610_out1(T237, T238, X284) f2240_in(zero(T435)) -> U33(f2240_in(T435), zero(T435)) U33(f2240_out1, zero(T435)) -> f2240_out1 f2240_in(one(T439)) -> U34(f2254_in(T439), one(T439)) U34(f2254_out1, one(T439)) -> f2240_out1 f2254_in(b) -> f2254_out1 f2254_in(zero(T444)) -> U35(f2240_in(T444), zero(T444)) U35(f2240_out1, zero(T444)) -> f2254_out1 f2254_in(one(T448)) -> U36(f2254_in(T448), one(T448)) U36(f2254_out1, one(T448)) -> f2254_out1 f2333_in(zero(T502)) -> U37(f2333_in(T502), zero(T502)) U37(f2333_out1(T503, T504), zero(T502)) -> f2333_out1(zero(T503), zero(T504)) f2333_in(one(one(T530))) -> U38(f2254_in(T530), one(one(T530))) U38(f2254_out1, one(one(T530))) -> f2333_out1(zero(one(T530)), one(b)) f2333_in(one(zero(T535))) -> U39(f2240_in(T535), one(zero(T535))) U39(f2240_out1, one(zero(T535))) -> f2333_out1(zero(zero(T535)), one(b)) f2333_in(one(T550)) -> U40(f2333_in(T550), one(T550)) U40(f2333_out1(T551, T552), one(T550)) -> f2333_out1(zero(T551), one(T552)) f2333_in(one(T570)) -> U41(f2421_in(T570), one(T570)) U41(f2421_out1(T571, T572), one(T570)) -> f2333_out1(one(T571), zero(T572)) f2333_in(zero(T612)) -> U42(f2461_in(T612), zero(T612)) U42(f2461_out1(T613, T614), zero(T612)) -> f2333_out1(one(T613), one(T614)) f2485_in(one(b)) -> f2485_out1(b) f2485_in(one(T644)) -> U43(f2240_in(T644), one(T644)) U43(f2240_out1, one(T644)) -> f2485_out1(zero(T644)) f2485_in(zero(T651)) -> U44(f2485_in(T651), zero(T651)) U44(f2485_out1(T652), zero(T651)) -> f2485_out1(one(T652)) f2473_in(one(T631)) -> U45(f2240_in(T631), one(T631)) U45(f2240_out1, one(T631)) -> f2473_out1(zero(T631)) f2473_in(zero(T638)) -> U46(f2485_in(T638), zero(T638)) U46(f2485_out1(T639), zero(T638)) -> f2473_out1(one(T639)) f2534_in(one(T698)) -> U47(f2333_in(T698), one(T698)) U47(f2333_out1(T699, T700), one(T698)) -> f2534_out1(zero(T699), zero(T700)) f2534_in(zero(one(T726))) -> U48(f2240_in(T726), zero(one(T726))) U48(f2240_out1, zero(one(T726))) -> f2534_out1(zero(zero(T726)), one(b)) f2534_in(zero(zero(T737))) -> U49(f2485_in(T737), zero(zero(T737))) U49(f2485_out1(T738), zero(zero(T737))) -> f2534_out1(zero(one(T738)), one(b)) f2534_in(zero(T752)) -> U50(f2534_in(T752), zero(T752)) U50(f2534_out1(T753, T754), zero(T752)) -> f2534_out1(zero(T753), one(T754)) f2534_in(zero(one(T780))) -> U51(f2240_in(T780), zero(one(T780))) U51(f2240_out1, zero(one(T780))) -> f2534_out1(one(b), zero(zero(T780))) f2534_in(zero(zero(T791))) -> U52(f2485_in(T791), zero(zero(T791))) U52(f2485_out1(T792), zero(zero(T791))) -> f2534_out1(one(b), zero(one(T792))) f2534_in(zero(T806)) -> U53(f2534_in(T806), zero(T806)) U53(f2534_out1(T807, T808), zero(T806)) -> f2534_out1(one(T807), zero(T808)) f2534_in(one(T820)) -> U54(f2461_in(T820), one(T820)) U54(f2461_out1(T821, T822), one(T820)) -> f2534_out1(one(T821), one(T822)) f2461_in(one(b)) -> f2461_out1(b, b) f2461_in(T624) -> U55(f2473_in(T624), T624) U55(f2473_out1(T625), T624) -> f2461_out1(T625, b) f2461_in(T662) -> U56(f2473_in(T662), T662) U56(f2473_out1(T663), T662) -> f2461_out1(b, T663) f2461_in(T678) -> U57(f2534_in(T678), T678) U57(f2534_out1(T679, T680), T678) -> f2461_out1(T679, T680) f2421_in(one(T578)) -> U58(f2254_in(T578), one(T578)) U58(f2254_out1, one(T578)) -> f2421_out1(b, one(T578)) f2421_in(zero(T583)) -> U59(f2240_in(T583), zero(T583)) U59(f2240_out1, zero(T583)) -> f2421_out1(b, zero(T583)) f2421_in(T598) -> U60(f2333_in(T598), T598) U60(f2333_out1(T599, T600), T598) -> f2421_out1(T599, T600) f258_in(zero(T429)) -> U61(f2240_in(T429), zero(T429)) U61(f2240_out1, zero(T429)) -> f258_out1(b, T429) f258_in(zero(T482)) -> U62(f2333_in(T482), zero(T482)) U62(f2333_out1(T483, T484), zero(T482)) -> f258_out1(zero(T483), T484) f258_in(one(T838)) -> U63(f2421_in(T838), one(T838)) U63(f2421_out1(T839, T840), one(T838)) -> f258_out1(one(T839), T840) f607_in -> U64(f629_in) U64(f629_out1(T95)) -> f607_out1(b, T95, zero(T95)) f607_in -> U65(f1120_in) U65(f1120_out1(T143, T144, X156)) -> f607_out1(zero(T143), T144, zero(X156)) f607_in -> U66(f1610_in) U66(f1610_out1(T412, T413, X565)) -> f607_out1(one(T412), T413, one(X565)) f212_in(T36) -> U67(f254_in, T36) U67(f254_out1(T37), T36) -> U68(f258_in(T36), T36, T37) U68(f258_out1(T42, T41), T36, T37) -> f212_out1(T37, T42, T41) f522_in -> U69(f254_in) U69(f254_out1(T70)) -> U70(f607_in, T70) U70(f607_out1(T75, T74, X78), T70) -> f522_out1(T70, T75, T74, X78) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (183) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (184) Obligation: Q DP problem: The TRS P consists of the following rules: F1120_IN -> F1610_IN F1610_IN -> F1120_IN F1120_IN -> F1120_IN F1120_IN -> F1638_IN F1638_IN -> F2062_IN F2062_IN -> F1120_IN F2062_IN -> F2062_IN F2062_IN -> F1638_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (185) Obligation: Q DP problem: The TRS P consists of the following rules: F254_IN -> F522_IN F522_IN -> F254_IN F254_IN -> F254_IN The TRS R consists of the following rules: f2_in(T8) -> f2_out1(one(b), T8) f2_in(zero(T23)) -> U1(f2_in(T23), zero(T23)) U1(f2_out1(T24, T25), zero(T23)) -> f2_out1(zero(T24), T25) f2_in(T36) -> U2(f212_in(T36), T36) U2(f212_out1(T37, T38, X33), T36) -> f2_out1(one(T37), T38) f254_in -> f254_out1(one(b)) f254_in -> U3(f254_in) U3(f254_out1(T60)) -> f254_out1(zero(T60)) f254_in -> U4(f522_in) U4(f522_out1(T70, T71, X77, X78)) -> f254_out1(one(T70)) f629_in -> U5(f629_in) U5(f629_out1(T102)) -> f629_out1(zero(T102)) f629_in -> U6(f639_in) U6(f639_out1(T107)) -> f629_out1(one(T107)) f639_in -> f639_out1(b) f639_in -> U7(f629_in) U7(f629_out1(T113)) -> f639_out1(zero(T113)) f639_in -> U8(f639_in) U8(f639_out1(T118)) -> f639_out1(one(T118)) f1120_in -> U9(f1120_in) U9(f1120_out1(T157, T158, X180)) -> f1120_out1(zero(T157), zero(T158), zero(X180)) f1120_in -> U10(f639_in) U10(f639_out1(T179)) -> f1120_out1(zero(one(T179)), one(b), one(one(T179))) f1120_in -> U11(f629_in) U11(f629_out1(T185)) -> f1120_out1(zero(zero(T185)), one(b), one(zero(T185))) f1120_in -> U12(f1120_in) U12(f1120_out1(T197, T198, X232)) -> f1120_out1(zero(T197), one(T198), one(X232)) f1120_in -> U13(f1610_in) U13(f1610_out1(T211, T212, X256)) -> f1120_out1(one(T211), zero(T212), one(X256)) f1120_in -> U14(f1638_in) U14(f1638_out1(T247, T248, X300)) -> f1120_out1(one(T247), one(T248), zero(X300)) f1680_in -> f1680_out1(b, one(b)) f1680_in -> U15(f629_in) U15(f629_out1(T272)) -> f1680_out1(zero(T272), one(T272)) f1680_in -> U16(f1680_in) U16(f1680_out1(T277, X346)) -> f1680_out1(one(T277), zero(X346)) f1648_in -> U17(f629_in) U17(f629_out1(T261)) -> f1648_out1(zero(T261), one(T261)) f1648_in -> U18(f1680_in) U18(f1680_out1(T266, X331)) -> f1648_out1(one(T266), zero(X331)) f2062_in -> U19(f1120_in) U19(f1120_out1(T309, T310, X402)) -> f2062_out1(zero(T309), zero(T310), one(X402)) f2062_in -> U20(f629_in) U20(f629_out1(T331)) -> f2062_out1(zero(zero(T331)), one(b), zero(one(T331))) f2062_in -> U21(f1680_in) U21(f1680_out1(T338, X448)) -> f2062_out1(zero(one(T338)), one(b), zero(zero(X448))) f2062_in -> U22(f2062_in) U22(f2062_out1(T349, T350, X464)) -> f2062_out1(zero(T349), one(T350), zero(X464)) f2062_in -> U23(f629_in) U23(f629_out1(T371)) -> f2062_out1(one(b), zero(zero(T371)), zero(one(T371))) f2062_in -> U24(f1680_in) U24(f1680_out1(T378, X510)) -> f2062_out1(one(b), zero(one(T378)), zero(zero(X510))) f2062_in -> U25(f2062_in) U25(f2062_out1(T389, T390, X526)) -> f2062_out1(one(T389), zero(T390), zero(X526)) f2062_in -> U26(f1638_in) U26(f1638_out1(T399, T400, X542)) -> f2062_out1(one(T399), one(T400), one(X542)) f1638_in -> f1638_out1(b, b, one(b)) f1638_in -> U27(f1648_in) U27(f1648_out1(T254, X315)) -> f1638_out1(T254, b, X315) f1638_in -> U28(f1648_in) U28(f1648_out1(T283, X361)) -> f1638_out1(b, T283, X361) f1638_in -> U29(f2062_in) U29(f2062_out1(T295, T296, X378)) -> f1638_out1(T295, T296, X378) f1610_in -> U30(f639_in) U30(f639_out1(T219)) -> f1610_out1(b, one(T219), one(T219)) f1610_in -> U31(f629_in) U31(f629_out1(T225)) -> f1610_out1(b, zero(T225), zero(T225)) f1610_in -> U32(f1120_in) U32(f1120_out1(T237, T238, X284)) -> f1610_out1(T237, T238, X284) f2240_in(zero(T435)) -> U33(f2240_in(T435), zero(T435)) U33(f2240_out1, zero(T435)) -> f2240_out1 f2240_in(one(T439)) -> U34(f2254_in(T439), one(T439)) U34(f2254_out1, one(T439)) -> f2240_out1 f2254_in(b) -> f2254_out1 f2254_in(zero(T444)) -> U35(f2240_in(T444), zero(T444)) U35(f2240_out1, zero(T444)) -> f2254_out1 f2254_in(one(T448)) -> U36(f2254_in(T448), one(T448)) U36(f2254_out1, one(T448)) -> f2254_out1 f2333_in(zero(T502)) -> U37(f2333_in(T502), zero(T502)) U37(f2333_out1(T503, T504), zero(T502)) -> f2333_out1(zero(T503), zero(T504)) f2333_in(one(one(T530))) -> U38(f2254_in(T530), one(one(T530))) U38(f2254_out1, one(one(T530))) -> f2333_out1(zero(one(T530)), one(b)) f2333_in(one(zero(T535))) -> U39(f2240_in(T535), one(zero(T535))) U39(f2240_out1, one(zero(T535))) -> f2333_out1(zero(zero(T535)), one(b)) f2333_in(one(T550)) -> U40(f2333_in(T550), one(T550)) U40(f2333_out1(T551, T552), one(T550)) -> f2333_out1(zero(T551), one(T552)) f2333_in(one(T570)) -> U41(f2421_in(T570), one(T570)) U41(f2421_out1(T571, T572), one(T570)) -> f2333_out1(one(T571), zero(T572)) f2333_in(zero(T612)) -> U42(f2461_in(T612), zero(T612)) U42(f2461_out1(T613, T614), zero(T612)) -> f2333_out1(one(T613), one(T614)) f2485_in(one(b)) -> f2485_out1(b) f2485_in(one(T644)) -> U43(f2240_in(T644), one(T644)) U43(f2240_out1, one(T644)) -> f2485_out1(zero(T644)) f2485_in(zero(T651)) -> U44(f2485_in(T651), zero(T651)) U44(f2485_out1(T652), zero(T651)) -> f2485_out1(one(T652)) f2473_in(one(T631)) -> U45(f2240_in(T631), one(T631)) U45(f2240_out1, one(T631)) -> f2473_out1(zero(T631)) f2473_in(zero(T638)) -> U46(f2485_in(T638), zero(T638)) U46(f2485_out1(T639), zero(T638)) -> f2473_out1(one(T639)) f2534_in(one(T698)) -> U47(f2333_in(T698), one(T698)) U47(f2333_out1(T699, T700), one(T698)) -> f2534_out1(zero(T699), zero(T700)) f2534_in(zero(one(T726))) -> U48(f2240_in(T726), zero(one(T726))) U48(f2240_out1, zero(one(T726))) -> f2534_out1(zero(zero(T726)), one(b)) f2534_in(zero(zero(T737))) -> U49(f2485_in(T737), zero(zero(T737))) U49(f2485_out1(T738), zero(zero(T737))) -> f2534_out1(zero(one(T738)), one(b)) f2534_in(zero(T752)) -> U50(f2534_in(T752), zero(T752)) U50(f2534_out1(T753, T754), zero(T752)) -> f2534_out1(zero(T753), one(T754)) f2534_in(zero(one(T780))) -> U51(f2240_in(T780), zero(one(T780))) U51(f2240_out1, zero(one(T780))) -> f2534_out1(one(b), zero(zero(T780))) f2534_in(zero(zero(T791))) -> U52(f2485_in(T791), zero(zero(T791))) U52(f2485_out1(T792), zero(zero(T791))) -> f2534_out1(one(b), zero(one(T792))) f2534_in(zero(T806)) -> U53(f2534_in(T806), zero(T806)) U53(f2534_out1(T807, T808), zero(T806)) -> f2534_out1(one(T807), zero(T808)) f2534_in(one(T820)) -> U54(f2461_in(T820), one(T820)) U54(f2461_out1(T821, T822), one(T820)) -> f2534_out1(one(T821), one(T822)) f2461_in(one(b)) -> f2461_out1(b, b) f2461_in(T624) -> U55(f2473_in(T624), T624) U55(f2473_out1(T625), T624) -> f2461_out1(T625, b) f2461_in(T662) -> U56(f2473_in(T662), T662) U56(f2473_out1(T663), T662) -> f2461_out1(b, T663) f2461_in(T678) -> U57(f2534_in(T678), T678) U57(f2534_out1(T679, T680), T678) -> f2461_out1(T679, T680) f2421_in(one(T578)) -> U58(f2254_in(T578), one(T578)) U58(f2254_out1, one(T578)) -> f2421_out1(b, one(T578)) f2421_in(zero(T583)) -> U59(f2240_in(T583), zero(T583)) U59(f2240_out1, zero(T583)) -> f2421_out1(b, zero(T583)) f2421_in(T598) -> U60(f2333_in(T598), T598) U60(f2333_out1(T599, T600), T598) -> f2421_out1(T599, T600) f258_in(zero(T429)) -> U61(f2240_in(T429), zero(T429)) U61(f2240_out1, zero(T429)) -> f258_out1(b, T429) f258_in(zero(T482)) -> U62(f2333_in(T482), zero(T482)) U62(f2333_out1(T483, T484), zero(T482)) -> f258_out1(zero(T483), T484) f258_in(one(T838)) -> U63(f2421_in(T838), one(T838)) U63(f2421_out1(T839, T840), one(T838)) -> f258_out1(one(T839), T840) f607_in -> U64(f629_in) U64(f629_out1(T95)) -> f607_out1(b, T95, zero(T95)) f607_in -> U65(f1120_in) U65(f1120_out1(T143, T144, X156)) -> f607_out1(zero(T143), T144, zero(X156)) f607_in -> U66(f1610_in) U66(f1610_out1(T412, T413, X565)) -> f607_out1(one(T412), T413, one(X565)) f212_in(T36) -> U67(f254_in, T36) U67(f254_out1(T37), T36) -> U68(f258_in(T36), T36, T37) U68(f258_out1(T42, T41), T36, T37) -> f212_out1(T37, T42, T41) f522_in -> U69(f254_in) U69(f254_out1(T70)) -> U70(f607_in, T70) U70(f607_out1(T75, T74, X78), T70) -> f522_out1(T70, T75, T74, X78) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (186) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (187) Obligation: Q DP problem: The TRS P consists of the following rules: F254_IN -> F522_IN F522_IN -> F254_IN F254_IN -> F254_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (188) Obligation: Q DP problem: The TRS P consists of the following rules: F2_IN(zero(T23)) -> F2_IN(T23) The TRS R consists of the following rules: f2_in(T8) -> f2_out1(one(b), T8) f2_in(zero(T23)) -> U1(f2_in(T23), zero(T23)) U1(f2_out1(T24, T25), zero(T23)) -> f2_out1(zero(T24), T25) f2_in(T36) -> U2(f212_in(T36), T36) U2(f212_out1(T37, T38, X33), T36) -> f2_out1(one(T37), T38) f254_in -> f254_out1(one(b)) f254_in -> U3(f254_in) U3(f254_out1(T60)) -> f254_out1(zero(T60)) f254_in -> U4(f522_in) U4(f522_out1(T70, T71, X77, X78)) -> f254_out1(one(T70)) f629_in -> U5(f629_in) U5(f629_out1(T102)) -> f629_out1(zero(T102)) f629_in -> U6(f639_in) U6(f639_out1(T107)) -> f629_out1(one(T107)) f639_in -> f639_out1(b) f639_in -> U7(f629_in) U7(f629_out1(T113)) -> f639_out1(zero(T113)) f639_in -> U8(f639_in) U8(f639_out1(T118)) -> f639_out1(one(T118)) f1120_in -> U9(f1120_in) U9(f1120_out1(T157, T158, X180)) -> f1120_out1(zero(T157), zero(T158), zero(X180)) f1120_in -> U10(f639_in) U10(f639_out1(T179)) -> f1120_out1(zero(one(T179)), one(b), one(one(T179))) f1120_in -> U11(f629_in) U11(f629_out1(T185)) -> f1120_out1(zero(zero(T185)), one(b), one(zero(T185))) f1120_in -> U12(f1120_in) U12(f1120_out1(T197, T198, X232)) -> f1120_out1(zero(T197), one(T198), one(X232)) f1120_in -> U13(f1610_in) U13(f1610_out1(T211, T212, X256)) -> f1120_out1(one(T211), zero(T212), one(X256)) f1120_in -> U14(f1638_in) U14(f1638_out1(T247, T248, X300)) -> f1120_out1(one(T247), one(T248), zero(X300)) f1680_in -> f1680_out1(b, one(b)) f1680_in -> U15(f629_in) U15(f629_out1(T272)) -> f1680_out1(zero(T272), one(T272)) f1680_in -> U16(f1680_in) U16(f1680_out1(T277, X346)) -> f1680_out1(one(T277), zero(X346)) f1648_in -> U17(f629_in) U17(f629_out1(T261)) -> f1648_out1(zero(T261), one(T261)) f1648_in -> U18(f1680_in) U18(f1680_out1(T266, X331)) -> f1648_out1(one(T266), zero(X331)) f2062_in -> U19(f1120_in) U19(f1120_out1(T309, T310, X402)) -> f2062_out1(zero(T309), zero(T310), one(X402)) f2062_in -> U20(f629_in) U20(f629_out1(T331)) -> f2062_out1(zero(zero(T331)), one(b), zero(one(T331))) f2062_in -> U21(f1680_in) U21(f1680_out1(T338, X448)) -> f2062_out1(zero(one(T338)), one(b), zero(zero(X448))) f2062_in -> U22(f2062_in) U22(f2062_out1(T349, T350, X464)) -> f2062_out1(zero(T349), one(T350), zero(X464)) f2062_in -> U23(f629_in) U23(f629_out1(T371)) -> f2062_out1(one(b), zero(zero(T371)), zero(one(T371))) f2062_in -> U24(f1680_in) U24(f1680_out1(T378, X510)) -> f2062_out1(one(b), zero(one(T378)), zero(zero(X510))) f2062_in -> U25(f2062_in) U25(f2062_out1(T389, T390, X526)) -> f2062_out1(one(T389), zero(T390), zero(X526)) f2062_in -> U26(f1638_in) U26(f1638_out1(T399, T400, X542)) -> f2062_out1(one(T399), one(T400), one(X542)) f1638_in -> f1638_out1(b, b, one(b)) f1638_in -> U27(f1648_in) U27(f1648_out1(T254, X315)) -> f1638_out1(T254, b, X315) f1638_in -> U28(f1648_in) U28(f1648_out1(T283, X361)) -> f1638_out1(b, T283, X361) f1638_in -> U29(f2062_in) U29(f2062_out1(T295, T296, X378)) -> f1638_out1(T295, T296, X378) f1610_in -> U30(f639_in) U30(f639_out1(T219)) -> f1610_out1(b, one(T219), one(T219)) f1610_in -> U31(f629_in) U31(f629_out1(T225)) -> f1610_out1(b, zero(T225), zero(T225)) f1610_in -> U32(f1120_in) U32(f1120_out1(T237, T238, X284)) -> f1610_out1(T237, T238, X284) f2240_in(zero(T435)) -> U33(f2240_in(T435), zero(T435)) U33(f2240_out1, zero(T435)) -> f2240_out1 f2240_in(one(T439)) -> U34(f2254_in(T439), one(T439)) U34(f2254_out1, one(T439)) -> f2240_out1 f2254_in(b) -> f2254_out1 f2254_in(zero(T444)) -> U35(f2240_in(T444), zero(T444)) U35(f2240_out1, zero(T444)) -> f2254_out1 f2254_in(one(T448)) -> U36(f2254_in(T448), one(T448)) U36(f2254_out1, one(T448)) -> f2254_out1 f2333_in(zero(T502)) -> U37(f2333_in(T502), zero(T502)) U37(f2333_out1(T503, T504), zero(T502)) -> f2333_out1(zero(T503), zero(T504)) f2333_in(one(one(T530))) -> U38(f2254_in(T530), one(one(T530))) U38(f2254_out1, one(one(T530))) -> f2333_out1(zero(one(T530)), one(b)) f2333_in(one(zero(T535))) -> U39(f2240_in(T535), one(zero(T535))) U39(f2240_out1, one(zero(T535))) -> f2333_out1(zero(zero(T535)), one(b)) f2333_in(one(T550)) -> U40(f2333_in(T550), one(T550)) U40(f2333_out1(T551, T552), one(T550)) -> f2333_out1(zero(T551), one(T552)) f2333_in(one(T570)) -> U41(f2421_in(T570), one(T570)) U41(f2421_out1(T571, T572), one(T570)) -> f2333_out1(one(T571), zero(T572)) f2333_in(zero(T612)) -> U42(f2461_in(T612), zero(T612)) U42(f2461_out1(T613, T614), zero(T612)) -> f2333_out1(one(T613), one(T614)) f2485_in(one(b)) -> f2485_out1(b) f2485_in(one(T644)) -> U43(f2240_in(T644), one(T644)) U43(f2240_out1, one(T644)) -> f2485_out1(zero(T644)) f2485_in(zero(T651)) -> U44(f2485_in(T651), zero(T651)) U44(f2485_out1(T652), zero(T651)) -> f2485_out1(one(T652)) f2473_in(one(T631)) -> U45(f2240_in(T631), one(T631)) U45(f2240_out1, one(T631)) -> f2473_out1(zero(T631)) f2473_in(zero(T638)) -> U46(f2485_in(T638), zero(T638)) U46(f2485_out1(T639), zero(T638)) -> f2473_out1(one(T639)) f2534_in(one(T698)) -> U47(f2333_in(T698), one(T698)) U47(f2333_out1(T699, T700), one(T698)) -> f2534_out1(zero(T699), zero(T700)) f2534_in(zero(one(T726))) -> U48(f2240_in(T726), zero(one(T726))) U48(f2240_out1, zero(one(T726))) -> f2534_out1(zero(zero(T726)), one(b)) f2534_in(zero(zero(T737))) -> U49(f2485_in(T737), zero(zero(T737))) U49(f2485_out1(T738), zero(zero(T737))) -> f2534_out1(zero(one(T738)), one(b)) f2534_in(zero(T752)) -> U50(f2534_in(T752), zero(T752)) U50(f2534_out1(T753, T754), zero(T752)) -> f2534_out1(zero(T753), one(T754)) f2534_in(zero(one(T780))) -> U51(f2240_in(T780), zero(one(T780))) U51(f2240_out1, zero(one(T780))) -> f2534_out1(one(b), zero(zero(T780))) f2534_in(zero(zero(T791))) -> U52(f2485_in(T791), zero(zero(T791))) U52(f2485_out1(T792), zero(zero(T791))) -> f2534_out1(one(b), zero(one(T792))) f2534_in(zero(T806)) -> U53(f2534_in(T806), zero(T806)) U53(f2534_out1(T807, T808), zero(T806)) -> f2534_out1(one(T807), zero(T808)) f2534_in(one(T820)) -> U54(f2461_in(T820), one(T820)) U54(f2461_out1(T821, T822), one(T820)) -> f2534_out1(one(T821), one(T822)) f2461_in(one(b)) -> f2461_out1(b, b) f2461_in(T624) -> U55(f2473_in(T624), T624) U55(f2473_out1(T625), T624) -> f2461_out1(T625, b) f2461_in(T662) -> U56(f2473_in(T662), T662) U56(f2473_out1(T663), T662) -> f2461_out1(b, T663) f2461_in(T678) -> U57(f2534_in(T678), T678) U57(f2534_out1(T679, T680), T678) -> f2461_out1(T679, T680) f2421_in(one(T578)) -> U58(f2254_in(T578), one(T578)) U58(f2254_out1, one(T578)) -> f2421_out1(b, one(T578)) f2421_in(zero(T583)) -> U59(f2240_in(T583), zero(T583)) U59(f2240_out1, zero(T583)) -> f2421_out1(b, zero(T583)) f2421_in(T598) -> U60(f2333_in(T598), T598) U60(f2333_out1(T599, T600), T598) -> f2421_out1(T599, T600) f258_in(zero(T429)) -> U61(f2240_in(T429), zero(T429)) U61(f2240_out1, zero(T429)) -> f258_out1(b, T429) f258_in(zero(T482)) -> U62(f2333_in(T482), zero(T482)) U62(f2333_out1(T483, T484), zero(T482)) -> f258_out1(zero(T483), T484) f258_in(one(T838)) -> U63(f2421_in(T838), one(T838)) U63(f2421_out1(T839, T840), one(T838)) -> f258_out1(one(T839), T840) f607_in -> U64(f629_in) U64(f629_out1(T95)) -> f607_out1(b, T95, zero(T95)) f607_in -> U65(f1120_in) U65(f1120_out1(T143, T144, X156)) -> f607_out1(zero(T143), T144, zero(X156)) f607_in -> U66(f1610_in) U66(f1610_out1(T412, T413, X565)) -> f607_out1(one(T412), T413, one(X565)) f212_in(T36) -> U67(f254_in, T36) U67(f254_out1(T37), T36) -> U68(f258_in(T36), T36, T37) U68(f258_out1(T42, T41), T36, T37) -> f212_out1(T37, T42, T41) f522_in -> U69(f254_in) U69(f254_out1(T70)) -> U70(f607_in, T70) U70(f607_out1(T75, T74, X78), T70) -> f522_out1(T70, T75, T74, X78) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (189) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (190) Obligation: Q DP problem: The TRS P consists of the following rules: F2_IN(zero(T23)) -> F2_IN(T23) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (191) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *F2_IN(zero(T23)) -> F2_IN(T23) The graph contains the following edges 1 > 1 ---------------------------------------- (192) YES ---------------------------------------- (193) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 3, "program": { "directives": [], "clauses": [ [ "(add (b) (b) (b))", null ], [ "(add X (b) X)", "(binaryZ X)" ], [ "(add (b) Y Y)", "(binaryZ Y)" ], [ "(add X Y Z)", "(addz X Y Z)" ], [ "(addx (one X) (b) (one X))", "(binary X)" ], [ "(addx (zero X) (b) (zero X))", "(binaryZ X)" ], [ "(addx X Y Z)", "(addz X Y Z)" ], [ "(addy (b) (one Y) (one Y))", "(binary Y)" ], [ "(addy (b) (zero Y) (zero Y))", "(binaryZ Y)" ], [ "(addy X Y Z)", "(addz X Y Z)" ], [ "(addz (zero X) (zero Y) (zero Z))", "(addz X Y Z)" ], [ "(addz (zero X) (one Y) (one Z))", "(addx X Y Z)" ], [ "(addz (one X) (zero Y) (one Z))", "(addy X Y Z)" ], [ "(addz (one X) (one Y) (zero Z))", "(addc X Y Z)" ], [ "(addc (b) (b) (one (b)))", null ], [ "(addc X (b) Z)", "(succZ X Z)" ], [ "(addc (b) Y Z)", "(succZ Y Z)" ], [ "(addc X Y Z)", "(addC X Y Z)" ], [ "(addX (zero X) (b) (one X))", "(binaryZ X)" ], [ "(addX (one X) (b) (zero Z))", "(succ X Z)" ], [ "(addX X Y Z)", "(addC X Y Z)" ], [ "(addY (b) (zero Y) (one Y))", "(binaryZ Y)" ], [ "(addY (b) (one Y) (zero Z))", "(succ Y Z)" ], [ "(addY X Y Z)", "(addC X Y Z)" ], [ "(addC (zero X) (zero Y) (one Z))", "(addz X Y Z)" ], [ "(addC (zero X) (one Y) (zero Z))", "(addX X Y Z)" ], [ "(addC (one X) (zero Y) (zero Z))", "(addY X Y Z)" ], [ "(addC (one X) (one Y) (one Z))", "(addc X Y Z)" ], [ "(binary (b))", null ], [ "(binary (zero X))", "(binaryZ X)" ], [ "(binary (one X))", "(binary X)" ], [ "(binaryZ (zero X))", "(binaryZ X)" ], [ "(binaryZ (one X))", "(binary X)" ], [ "(succ (b) (one (b)))", null ], [ "(succ (zero X) (one X))", "(binaryZ X)" ], [ "(succ (one X) (zero Z))", "(succ X Z)" ], [ "(succZ (zero X) (one X))", "(binaryZ X)" ], [ "(succZ (one X) (zero Z))", "(succ X Z)" ], [ "(times (one (b)) X X)", null ], [ "(times (zero R) S (zero RS))", "(times R S RS)" ], [ "(times (one R) S RSS)", "(',' (times R S RS) (add S (zero RS) RSS))" ] ] }, "graph": { "nodes": { "2305": { "goal": [ { "clause": 10, "scope": 22, "term": "(addz T483 T484 T482)" }, { "clause": 11, "scope": 22, "term": "(addz T483 T484 T482)" }, { "clause": 12, "scope": 22, "term": "(addz T483 T484 T482)" }, { "clause": 13, "scope": 22, "term": "(addz T483 T484 T482)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T482"], "free": [], "exprvars": [] } }, "2304": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1698": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2303": { "goal": [{ "clause": -1, "scope": -1, "term": "(addz T483 T484 T482)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T482"], "free": [], "exprvars": [] } }, "1697": { 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["T678"], "free": [], "exprvars": [] } }, "2413": { "goal": [{ "clause": 9, "scope": 24, "term": "(addy T571 T572 T570)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T570"], "free": [], "exprvars": [] } }, "2412": { "goal": [{ "clause": 8, "scope": 24, "term": "(addy T571 T572 T570)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T570"], "free": [], "exprvars": [] } }, "2410": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 13, "label": "CASE" }, { "from": 13, "to": 14, "label": "PARALLEL" }, { "from": 13, "to": 15, "label": "PARALLEL" }, { "from": 14, "to": 17, "label": "EVAL with clause\ntimes(one(b), X5, X5).\nand substitutionT1 -> one(b),\nT2 -> T8,\nX5 -> T8,\nT3 -> T8" }, { "from": 14, "to": 19, "label": "EVAL-BACKTRACK" }, { "from": 15, "to": 21, "label": "PARALLEL" }, { "from": 15, "to": 22, "label": "PARALLEL" }, { "from": 17, "to": 20, "label": "SUCCESS" }, { "from": 21, "to": 23, "label": "EVAL with clause\ntimes(zero(X18), X19, zero(X20)) :- times(X18, X19, X20).\nand substitutionX18 -> T24,\nT1 -> zero(T24),\nT2 -> T25,\nX19 -> T25,\nX20 -> T23,\nT3 -> zero(T23),\nT21 -> T24,\nT22 -> T25" }, { "from": 21, "to": 24, "label": "EVAL-BACKTRACK" }, { "from": 22, "to": 40, "label": "EVAL with clause\ntimes(one(X30), X31, X32) :- ','(times(X30, X31, X33), add(X31, zero(X33), X32)).\nand substitutionX30 -> T37,\nT1 -> one(T37),\nT2 -> T38,\nX31 -> T38,\nT3 -> T36,\nX32 -> T36,\nT34 -> T37,\nT35 -> T38" }, { "from": 22, "to": 41, "label": "EVAL-BACKTRACK" }, { "from": 23, "to": 3, "label": "INSTANCE with matching:\nT1 -> T24\nT2 -> T25\nT3 -> T23" }, { "from": 40, "to": 121, "label": "SPLIT 1" }, { "from": 40, "to": 122, "label": "SPLIT 2\nnew knowledge:\nT37 is ground\nreplacements:X33 -> T41,\nT38 -> T42" }, { "from": 121, "to": 123, "label": "CASE" }, { "from": 122, "to": 2170, "label": "CASE" }, { "from": 123, "to": 124, "label": "PARALLEL" }, { "from": 123, "to": 125, "label": "PARALLEL" }, { "from": 124, "to": 126, "label": "EVAL with clause\ntimes(one(b), X42, X42).\nand substitutionT37 -> one(b),\nT38 -> T49,\nX42 -> T49,\nX33 -> T49" }, { "from": 124, "to": 127, "label": "EVAL-BACKTRACK" }, { "from": 125, "to": 129, "label": "PARALLEL" }, { "from": 125, "to": 131, "label": "PARALLEL" }, { "from": 126, "to": 128, "label": "SUCCESS" }, { "from": 129, "to": 201, "label": "EVAL with clause\ntimes(zero(X59), X60, zero(X61)) :- times(X59, X60, X61).\nand substitutionX59 -> T60,\nT37 -> zero(T60),\nT38 -> T61,\nX60 -> T61,\nX61 -> X62,\nX33 -> zero(X62),\nT58 -> T60,\nT59 -> T61" }, { "from": 129, "to": 208, "label": "EVAL-BACKTRACK" }, { "from": 131, "to": 1055, "label": "EVAL with clause\ntimes(one(X74), X75, X76) :- ','(times(X74, X75, X77), add(X75, zero(X77), X76)).\nand substitutionX74 -> T70,\nT37 -> one(T70),\nT38 -> T71,\nX75 -> T71,\nX33 -> X78,\nX76 -> X78,\nT68 -> T70,\nT69 -> T71" }, { "from": 131, "to": 1056, "label": "EVAL-BACKTRACK" }, { "from": 201, "to": 121, "label": "INSTANCE with matching:\nT37 -> T60\nT38 -> T61\nX33 -> X62" }, { "from": 1055, "to": 1057, "label": "SPLIT 1" }, { "from": 1055, "to": 1058, "label": "SPLIT 2\nnew knowledge:\nT70 is ground\nreplacements:X77 -> T74,\nT71 -> T75" }, { "from": 1057, "to": 121, "label": "INSTANCE with matching:\nT37 -> T70\nT38 -> T71\nX33 -> X77" }, { "from": 1058, "to": 1065, "label": "CASE" }, { "from": 1065, "to": 1067, "label": "BACKTRACK\nfor clause: add(b, b, b)because of non-unification" }, { "from": 1067, "to": 1071, "label": "BACKTRACK\nfor clause: add(X, b, X) :- binaryZ(X)because of non-unification" }, { "from": 1071, "to": 1077, "label": "PARALLEL" }, { "from": 1071, "to": 1079, "label": "PARALLEL" }, { "from": 1077, "to": 1089, "label": "EVAL with clause\nadd(b, X88, X88) :- binaryZ(X88).\nand substitutionT75 -> b,\nT74 -> T84,\nX88 -> zero(T84),\nX78 -> zero(T84),\nT83 -> T84" }, { "from": 1077, "to": 1091, "label": "EVAL-BACKTRACK" }, { "from": 1079, "to": 1357, "label": "ONLY EVAL with clause\nadd(X129, X130, X131) :- addz(X129, X130, X131).\nand substitutionT75 -> T129,\nX129 -> T129,\nT74 -> T130,\nX130 -> zero(T130),\nX78 -> X132,\nX131 -> X132,\nT127 -> T129,\nT128 -> T130" }, { "from": 1089, "to": 1099, "label": "CASE" }, { "from": 1099, "to": 1109, "label": "PARALLEL" }, { "from": 1099, "to": 1110, "label": "PARALLEL" }, { "from": 1109, "to": 1122, "label": "ONLY EVAL with clause\nbinaryZ(zero(X96)) :- binaryZ(X96).\nand substitutionT84 -> T95,\nX96 -> T95,\nT94 -> T95" }, { "from": 1110, "to": 1316, "label": "BACKTRACK\nfor clause: binaryZ(one(X)) :- binary(X)because of non-unification" }, { "from": 1122, "to": 1129, "label": "CASE" }, { "from": 1129, "to": 1133, "label": "PARALLEL" }, { "from": 1129, "to": 1135, "label": "PARALLEL" }, { "from": 1133, "to": 1252, "label": "EVAL with clause\nbinaryZ(zero(X102)) :- binaryZ(X102).\nand substitutionX102 -> T102,\nT95 -> zero(T102),\nT101 -> T102" }, { "from": 1133, "to": 1255, "label": "EVAL-BACKTRACK" }, { "from": 1135, "to": 1266, "label": "EVAL with clause\nbinaryZ(one(X106)) :- binary(X106).\nand substitutionX106 -> T107,\nT95 -> one(T107),\nT106 -> T107" }, { "from": 1135, "to": 1273, "label": "EVAL-BACKTRACK" }, { "from": 1252, "to": 1122, "label": "INSTANCE with matching:\nT95 -> T102" }, { "from": 1266, "to": 1274, "label": "CASE" }, { "from": 1274, "to": 1275, "label": "PARALLEL" }, { "from": 1274, "to": 1276, "label": "PARALLEL" }, { "from": 1275, "to": 1277, "label": "EVAL with clause\nbinary(b).\nand substitutionT107 -> b" }, { "from": 1275, "to": 1278, "label": "EVAL-BACKTRACK" }, { "from": 1276, "to": 1280, "label": "PARALLEL" }, { "from": 1276, "to": 1281, "label": "PARALLEL" }, { "from": 1277, "to": 1279, "label": "SUCCESS" }, { "from": 1280, "to": 1286, "label": "EVAL with clause\nbinary(zero(X111)) :- binaryZ(X111).\nand substitutionX111 -> T113,\nT107 -> zero(T113),\nT112 -> T113" }, { "from": 1280, "to": 1288, "label": "EVAL-BACKTRACK" }, { "from": 1281, "to": 1306, "label": "EVAL with clause\nbinary(one(X115)) :- binary(X115).\nand substitutionX115 -> T118,\nT107 -> one(T118),\nT117 -> T118" }, { "from": 1281, "to": 1308, "label": "EVAL-BACKTRACK" }, { "from": 1286, "to": 1122, "label": "INSTANCE with matching:\nT95 -> T113" }, { "from": 1306, "to": 1266, "label": "INSTANCE with matching:\nT107 -> T118" }, { "from": 1357, "to": 1595, "label": "CASE" }, { "from": 1595, "to": 1598, "label": "PARALLEL" }, { "from": 1595, "to": 1599, "label": "PARALLEL" }, { "from": 1598, "to": 1602, "label": "EVAL with clause\naddz(zero(X153), zero(X154), zero(X155)) :- addz(X153, X154, X155).\nand substitutionX153 -> T143,\nT129 -> zero(T143),\nT130 -> T144,\nX154 -> T144,\nX155 -> X156,\nX132 -> zero(X156),\nT141 -> T143,\nT142 -> T144" }, { "from": 1598, "to": 1603, "label": "EVAL-BACKTRACK" }, { "from": 1599, "to": 2164, "label": "BACKTRACK\nfor clause: addz(zero(X), one(Y), one(Z)) :- addx(X, Y, Z)because of non-unification" }, { "from": 1602, "to": 1604, "label": "CASE" }, { "from": 1604, "to": 1605, "label": "PARALLEL" }, { "from": 1604, "to": 1606, "label": "PARALLEL" }, { "from": 1605, "to": 1608, "label": "EVAL with clause\naddz(zero(X177), zero(X178), zero(X179)) :- addz(X177, X178, X179).\nand substitutionX177 -> T157,\nT143 -> zero(T157),\nX178 -> T158,\nT144 -> zero(T158),\nX179 -> X180,\nX156 -> zero(X180),\nT155 -> T157,\nT156 -> T158" }, { "from": 1605, "to": 1609, "label": "EVAL-BACKTRACK" }, { "from": 1606, "to": 1619, "label": "PARALLEL" }, { "from": 1606, "to": 1620, "label": "PARALLEL" }, { "from": 1608, "to": 1602, "label": "INSTANCE with matching:\nT143 -> T157\nT144 -> T158\nX156 -> X180" }, { "from": 1619, "to": 1624, "label": "EVAL with clause\naddz(zero(X201), one(X202), one(X203)) :- addx(X201, X202, X203).\nand substitutionX201 -> T171,\nT143 -> zero(T171),\nX202 -> T172,\nT144 -> one(T172),\nX203 -> X204,\nX156 -> one(X204),\nT169 -> T171,\nT170 -> T172" }, { "from": 1619, "to": 1626, "label": "EVAL-BACKTRACK" }, { "from": 1620, "to": 1666, "label": "PARALLEL" }, { "from": 1620, "to": 1667, "label": "PARALLEL" }, { "from": 1624, "to": 1627, "label": "CASE" }, { "from": 1627, "to": 1628, "label": "PARALLEL" }, { "from": 1627, "to": 1629, "label": "PARALLEL" }, { "from": 1628, "to": 1632, "label": "EVAL with clause\naddx(one(X210), b, one(X210)) :- binary(X210).\nand substitutionX210 -> T179,\nT171 -> one(T179),\nT172 -> b,\nX204 -> one(T179),\nT178 -> T179" }, { "from": 1628, "to": 1633, "label": "EVAL-BACKTRACK" }, { "from": 1629, "to": 1635, "label": "PARALLEL" }, { "from": 1629, "to": 1636, "label": "PARALLEL" }, { "from": 1632, "to": 1266, "label": "INSTANCE with matching:\nT107 -> T179" }, { "from": 1635, "to": 1657, "label": "EVAL with clause\naddx(zero(X215), b, zero(X215)) :- binaryZ(X215).\nand substitutionX215 -> T185,\nT171 -> zero(T185),\nT172 -> b,\nX204 -> zero(T185),\nT184 -> T185" }, { "from": 1635, "to": 1658, "label": "EVAL-BACKTRACK" }, { "from": 1636, "to": 1665, "label": "ONLY EVAL with clause\naddx(X229, X230, X231) :- addz(X229, X230, X231).\nand substitutionT171 -> T197,\nX229 -> T197,\nT172 -> T198,\nX230 -> T198,\nX204 -> X232,\nX231 -> X232,\nT195 -> T197,\nT196 -> T198" }, { "from": 1657, "to": 1122, "label": "INSTANCE with matching:\nT95 -> T185" }, { "from": 1665, "to": 1602, "label": "INSTANCE with matching:\nT143 -> T197\nT144 -> T198\nX156 -> X232" }, { "from": 1666, "to": 1668, "label": "EVAL with clause\naddz(one(X253), zero(X254), one(X255)) :- addy(X253, X254, X255).\nand substitutionX253 -> T211,\nT143 -> one(T211),\nX254 -> T212,\nT144 -> zero(T212),\nX255 -> X256,\nX156 -> one(X256),\nT209 -> T211,\nT210 -> T212" }, { "from": 1666, "to": 1669, "label": "EVAL-BACKTRACK" }, { "from": 1667, "to": 1690, "label": "EVAL with clause\naddz(one(X297), one(X298), zero(X299)) :- addc(X297, X298, X299).\nand substitutionX297 -> T247,\nT143 -> one(T247),\nX298 -> T248,\nT144 -> one(T248),\nX299 -> X300,\nX156 -> zero(X300),\nT245 -> T247,\nT246 -> T248" }, { "from": 1667, "to": 1691, "label": "EVAL-BACKTRACK" }, { "from": 1668, "to": 1670, "label": "CASE" }, { "from": 1670, "to": 1671, "label": "PARALLEL" }, { "from": 1670, "to": 1672, "label": "PARALLEL" }, { "from": 1671, "to": 1673, "label": "EVAL with clause\naddy(b, one(X262), one(X262)) :- binary(X262).\nand substitutionT211 -> b,\nX262 -> T219,\nT212 -> one(T219),\nX256 -> one(T219),\nT218 -> T219" }, { "from": 1671, "to": 1674, "label": "EVAL-BACKTRACK" }, { "from": 1672, "to": 1675, "label": "PARALLEL" }, { "from": 1672, "to": 1676, "label": "PARALLEL" }, { "from": 1673, "to": 1266, "label": "INSTANCE with matching:\nT107 -> T219" }, { "from": 1675, "to": 1677, "label": "EVAL with clause\naddy(b, zero(X267), zero(X267)) :- binaryZ(X267).\nand substitutionT211 -> b,\nX267 -> T225,\nT212 -> zero(T225),\nX256 -> zero(T225),\nT224 -> T225" }, { "from": 1675, "to": 1678, "label": "EVAL-BACKTRACK" }, { "from": 1676, "to": 1679, "label": "ONLY EVAL with clause\naddy(X281, X282, X283) :- addz(X281, X282, X283).\nand substitutionT211 -> T237,\nX281 -> T237,\nT212 -> T238,\nX282 -> T238,\nX256 -> X284,\nX283 -> X284,\nT235 -> T237,\nT236 -> T238" }, { "from": 1677, "to": 1122, "label": "INSTANCE with matching:\nT95 -> T225" }, { "from": 1679, "to": 1602, "label": "INSTANCE with matching:\nT143 -> T237\nT144 -> T238\nX156 -> X284" }, { "from": 1690, "to": 1694, "label": "CASE" }, { "from": 1694, "to": 1695, "label": "PARALLEL" }, { "from": 1694, "to": 1696, "label": "PARALLEL" }, { "from": 1695, "to": 1697, "label": "EVAL with clause\naddc(b, b, one(b)).\nand substitutionT247 -> b,\nT248 -> b,\nX300 -> one(b)" }, { "from": 1695, "to": 1698, "label": "EVAL-BACKTRACK" }, { "from": 1696, "to": 1854, "label": "PARALLEL" }, { "from": 1696, "to": 1855, "label": "PARALLEL" }, { "from": 1697, "to": 1705, "label": "SUCCESS" }, { "from": 1854, "to": 1857, "label": "EVAL with clause\naddc(X313, b, X314) :- succZ(X313, X314).\nand substitutionT247 -> T254,\nX313 -> T254,\nT248 -> b,\nX300 -> X315,\nX314 -> X315,\nT253 -> T254" }, { "from": 1854, "to": 1858, "label": "EVAL-BACKTRACK" }, { "from": 1855, "to": 2043, "label": "PARALLEL" }, { "from": 1855, "to": 2044, "label": "PARALLEL" }, { "from": 1857, "to": 1859, "label": "CASE" }, { "from": 1859, "to": 1860, "label": "PARALLEL" }, { "from": 1859, "to": 1861, "label": "PARALLEL" }, { "from": 1860, "to": 1862, "label": "EVAL with clause\nsuccZ(zero(X321), one(X321)) :- binaryZ(X321).\nand substitutionX321 -> T261,\nT254 -> zero(T261),\nX315 -> one(T261),\nT260 -> T261" }, { "from": 1860, "to": 1863, "label": "EVAL-BACKTRACK" }, { "from": 1861, "to": 1872, "label": "EVAL with clause\nsuccZ(one(X329), zero(X330)) :- succ(X329, X330).\nand substitutionX329 -> T266,\nT254 -> one(T266),\nX330 -> X331,\nX315 -> zero(X331),\nT265 -> T266" }, { "from": 1861, "to": 1873, "label": "EVAL-BACKTRACK" }, { "from": 1862, "to": 1122, "label": "INSTANCE with matching:\nT95 -> T261" }, { "from": 1872, "to": 1874, "label": "CASE" }, { "from": 1874, "to": 1875, "label": "PARALLEL" }, { "from": 1874, "to": 1876, "label": "PARALLEL" }, { "from": 1875, "to": 1879, "label": "EVAL with clause\nsucc(b, one(b)).\nand substitutionT266 -> b,\nX331 -> one(b)" }, { "from": 1875, "to": 1880, "label": "EVAL-BACKTRACK" }, { "from": 1876, "to": 1882, "label": "PARALLEL" }, { "from": 1876, "to": 1883, "label": "PARALLEL" }, { "from": 1879, "to": 1881, "label": "SUCCESS" }, { "from": 1882, "to": 2034, "label": "EVAL with clause\nsucc(zero(X336), one(X336)) :- binaryZ(X336).\nand substitutionX336 -> T272,\nT266 -> zero(T272),\nX331 -> one(T272),\nT271 -> T272" }, { "from": 1882, "to": 2035, "label": "EVAL-BACKTRACK" }, { "from": 1883, "to": 2041, "label": "EVAL with clause\nsucc(one(X344), zero(X345)) :- succ(X344, X345).\nand substitutionX344 -> T277,\nT266 -> one(T277),\nX345 -> X346,\nX331 -> zero(X346),\nT276 -> T277" }, { "from": 1883, "to": 2042, "label": "EVAL-BACKTRACK" }, { "from": 2034, "to": 1122, "label": "INSTANCE with matching:\nT95 -> T272" }, { "from": 2041, "to": 1872, "label": "INSTANCE with matching:\nT266 -> T277\nX331 -> X346" }, { "from": 2043, "to": 2053, "label": "EVAL with clause\naddc(b, X359, X360) :- succZ(X359, X360).\nand substitutionT247 -> b,\nT248 -> T283,\nX359 -> T283,\nX300 -> X361,\nX360 -> X361,\nT282 -> T283" }, { "from": 2043, "to": 2054, "label": "EVAL-BACKTRACK" }, { "from": 2044, "to": 2061, "label": "ONLY EVAL with clause\naddc(X375, X376, X377) :- addC(X375, X376, X377).\nand substitutionT247 -> T295,\nX375 -> T295,\nT248 -> T296,\nX376 -> T296,\nX300 -> X378,\nX377 -> X378,\nT293 -> T295,\nT294 -> T296" }, { "from": 2053, "to": 1857, "label": "INSTANCE with matching:\nT254 -> T283\nX315 -> X361" }, { "from": 2061, "to": 2063, "label": "CASE" }, { "from": 2063, "to": 2095, "label": "PARALLEL" }, { "from": 2063, "to": 2096, "label": "PARALLEL" }, { "from": 2095, "to": 2099, "label": "EVAL with clause\naddC(zero(X399), zero(X400), one(X401)) :- addz(X399, X400, X401).\nand substitutionX399 -> T309,\nT295 -> zero(T309),\nX400 -> T310,\nT296 -> zero(T310),\nX401 -> X402,\nX378 -> one(X402),\nT307 -> T309,\nT308 -> T310" }, { "from": 2095, "to": 2100, "label": "EVAL-BACKTRACK" }, { "from": 2096, "to": 2109, "label": "PARALLEL" }, { "from": 2096, "to": 2110, "label": "PARALLEL" }, { "from": 2099, "to": 1602, "label": "INSTANCE with matching:\nT143 -> T309\nT144 -> T310\nX156 -> X402" }, { "from": 2109, "to": 2112, "label": "EVAL with clause\naddC(zero(X423), one(X424), zero(X425)) :- addX(X423, X424, X425).\nand substitutionX423 -> T323,\nT295 -> zero(T323),\nX424 -> T324,\nT296 -> one(T324),\nX425 -> X426,\nX378 -> zero(X426),\nT321 -> T323,\nT322 -> T324" }, { "from": 2109, "to": 2113, "label": "EVAL-BACKTRACK" }, { "from": 2110, "to": 2132, "label": "PARALLEL" }, { "from": 2110, "to": 2133, "label": "PARALLEL" }, { "from": 2112, "to": 2114, "label": "CASE" }, { "from": 2114, "to": 2115, "label": "PARALLEL" }, { "from": 2114, "to": 2116, "label": "PARALLEL" }, { "from": 2115, "to": 2117, "label": "EVAL with clause\naddX(zero(X432), b, one(X432)) :- binaryZ(X432).\nand substitutionX432 -> T331,\nT323 -> zero(T331),\nT324 -> b,\nX426 -> one(T331),\nT330 -> T331" }, { "from": 2115, "to": 2118, "label": "EVAL-BACKTRACK" }, { "from": 2116, "to": 2119, "label": "PARALLEL" }, { "from": 2116, "to": 2120, "label": "PARALLEL" }, { "from": 2117, "to": 1122, "label": "INSTANCE with matching:\nT95 -> T331" }, { "from": 2119, "to": 2121, "label": "EVAL with clause\naddX(one(X446), b, zero(X447)) :- succ(X446, X447).\nand substitutionX446 -> T338,\nT323 -> one(T338),\nT324 -> b,\nX447 -> X448,\nX426 -> zero(X448),\nT337 -> T338" }, { "from": 2119, "to": 2122, "label": "EVAL-BACKTRACK" }, { "from": 2120, "to": 2123, "label": "ONLY EVAL with clause\naddX(X461, X462, X463) :- addC(X461, X462, X463).\nand substitutionT323 -> T349,\nX461 -> T349,\nT324 -> T350,\nX462 -> T350,\nX426 -> X464,\nX463 -> X464,\nT347 -> T349,\nT348 -> T350" }, { "from": 2121, "to": 1872, "label": "INSTANCE with matching:\nT266 -> T338\nX331 -> X448" }, { "from": 2123, "to": 2061, "label": "INSTANCE with matching:\nT295 -> T349\nT296 -> T350\nX378 -> X464" }, { "from": 2132, "to": 2134, "label": "EVAL with clause\naddC(one(X485), zero(X486), zero(X487)) :- addY(X485, X486, X487).\nand substitutionX485 -> T363,\nT295 -> one(T363),\nX486 -> T364,\nT296 -> zero(T364),\nX487 -> X488,\nX378 -> zero(X488),\nT361 -> T363,\nT362 -> T364" }, { "from": 2132, "to": 2135, "label": "EVAL-BACKTRACK" }, { "from": 2133, "to": 2162, "label": "EVAL with clause\naddC(one(X539), one(X540), one(X541)) :- addc(X539, X540, X541).\nand substitutionX539 -> T399,\nT295 -> one(T399),\nX540 -> T400,\nT296 -> one(T400),\nX541 -> X542,\nX378 -> one(X542),\nT397 -> T399,\nT398 -> T400" }, { "from": 2133, "to": 2163, "label": "EVAL-BACKTRACK" }, { "from": 2134, "to": 2136, "label": "CASE" }, { "from": 2136, "to": 2137, "label": "PARALLEL" }, { "from": 2136, "to": 2138, "label": "PARALLEL" }, { "from": 2137, "to": 2139, "label": "EVAL with clause\naddY(b, zero(X494), one(X494)) :- binaryZ(X494).\nand substitutionT363 -> b,\nX494 -> T371,\nT364 -> zero(T371),\nX488 -> one(T371),\nT370 -> T371" }, { "from": 2137, "to": 2140, "label": "EVAL-BACKTRACK" }, { "from": 2138, "to": 2141, "label": "PARALLEL" }, { "from": 2138, "to": 2142, "label": "PARALLEL" }, { "from": 2139, "to": 1122, "label": "INSTANCE with matching:\nT95 -> T371" }, { "from": 2141, "to": 2143, "label": "EVAL with clause\naddY(b, one(X508), zero(X509)) :- succ(X508, X509).\nand substitutionT363 -> b,\nX508 -> T378,\nT364 -> one(T378),\nX509 -> X510,\nX488 -> zero(X510),\nT377 -> T378" }, { "from": 2141, "to": 2144, "label": "EVAL-BACKTRACK" }, { "from": 2142, "to": 2161, "label": "ONLY EVAL with clause\naddY(X523, X524, X525) :- addC(X523, X524, X525).\nand substitutionT363 -> T389,\nX523 -> T389,\nT364 -> T390,\nX524 -> T390,\nX488 -> X526,\nX525 -> X526,\nT387 -> T389,\nT388 -> T390" }, { "from": 2143, "to": 1872, "label": "INSTANCE with matching:\nT266 -> T378\nX331 -> X510" }, { "from": 2161, "to": 2061, "label": "INSTANCE with matching:\nT295 -> T389\nT296 -> T390\nX378 -> X526" }, { "from": 2162, "to": 1690, "label": "INSTANCE with matching:\nT247 -> T399\nT248 -> T400\nX300 -> X542" }, { "from": 2164, "to": 2165, "label": "PARALLEL" }, { "from": 2164, "to": 2166, "label": "PARALLEL" }, { "from": 2165, "to": 2167, "label": "EVAL with clause\naddz(one(X562), zero(X563), one(X564)) :- addy(X562, X563, X564).\nand substitutionX562 -> T412,\nT129 -> one(T412),\nT130 -> T413,\nX563 -> T413,\nX564 -> X565,\nX132 -> one(X565),\nT410 -> T412,\nT411 -> T413" }, { "from": 2165, "to": 2168, "label": "EVAL-BACKTRACK" }, { "from": 2166, "to": 2169, "label": "BACKTRACK\nfor clause: addz(one(X), one(Y), zero(Z)) :- addc(X, Y, Z)because of non-unification" }, { "from": 2167, "to": 1668, "label": "INSTANCE with matching:\nT211 -> T412\nT212 -> T413\nX256 -> X565" }, { "from": 2170, "to": 2191, "label": "BACKTRACK\nfor clause: add(b, b, b)because of non-unification" }, { "from": 2191, "to": 2192, "label": "BACKTRACK\nfor clause: add(X, b, X) :- binaryZ(X)because of non-unification" }, { "from": 2192, "to": 2193, "label": "PARALLEL" }, { "from": 2192, "to": 2194, "label": "PARALLEL" }, { "from": 2193, "to": 2195, "label": "EVAL with clause\nadd(b, X575, X575) :- binaryZ(X575).\nand substitutionT42 -> b,\nT41 -> T421,\nX575 -> zero(T421),\nT36 -> zero(T421)" }, { "from": 2193, "to": 2196, "label": "EVAL-BACKTRACK" }, { "from": 2194, "to": 2296, "label": "ONLY EVAL with clause\nadd(X613, X614, X615) :- addz(X613, X614, X615).\nand substitutionT42 -> T463,\nX613 -> T463,\nT41 -> T464,\nX614 -> zero(T464),\nT36 -> T462,\nX615 -> T462,\nT460 -> T463,\nT461 -> T464" }, { "from": 2195, "to": 2197, "label": "CASE" }, { "from": 2197, "to": 2198, "label": "PARALLEL" }, { "from": 2197, "to": 2199, "label": "PARALLEL" }, { "from": 2198, "to": 2200, "label": "ONLY EVAL with clause\nbinaryZ(zero(X583)) :- binaryZ(X583).\nand substitutionT421 -> T429,\nX583 -> T429" }, { "from": 2199, "to": 2291, "label": "BACKTRACK\nfor clause: binaryZ(one(X)) :- binary(X)because of non-unification" }, { "from": 2200, "to": 2215, "label": "CASE" }, { "from": 2215, "to": 2216, "label": "PARALLEL" }, { "from": 2215, "to": 2217, "label": "PARALLEL" }, { "from": 2216, "to": 2218, "label": "EVAL with clause\nbinaryZ(zero(X589)) :- binaryZ(X589).\nand substitutionX589 -> T435,\nT429 -> zero(T435)" }, { "from": 2216, "to": 2219, "label": "EVAL-BACKTRACK" }, { "from": 2217, "to": 2262, "label": "EVAL with clause\nbinaryZ(one(X593)) :- binary(X593).\nand substitutionX593 -> T439,\nT429 -> one(T439)" }, { "from": 2217, "to": 2263, "label": "EVAL-BACKTRACK" }, { "from": 2218, "to": 2200, "label": "INSTANCE with matching:\nT429 -> T435" }, { "from": 2262, "to": 2274, "label": "CASE" }, { "from": 2274, "to": 2275, "label": "PARALLEL" }, { "from": 2274, "to": 2276, "label": "PARALLEL" }, { "from": 2275, "to": 2277, "label": "EVAL with clause\nbinary(b).\nand substitutionT439 -> b" }, { "from": 2275, "to": 2278, "label": "EVAL-BACKTRACK" }, { "from": 2276, "to": 2280, "label": "PARALLEL" }, { "from": 2276, "to": 2281, "label": "PARALLEL" }, { "from": 2277, "to": 2279, "label": "SUCCESS" }, { "from": 2280, "to": 2282, "label": "EVAL with clause\nbinary(zero(X598)) :- binaryZ(X598).\nand substitutionX598 -> T444,\nT439 -> zero(T444)" }, { "from": 2280, "to": 2283, "label": "EVAL-BACKTRACK" }, { "from": 2281, "to": 2287, "label": "EVAL with clause\nbinary(one(X602)) :- binary(X602).\nand substitutionX602 -> T448,\nT439 -> one(T448)" }, { "from": 2281, "to": 2288, "label": "EVAL-BACKTRACK" }, { "from": 2282, "to": 2200, "label": "INSTANCE with matching:\nT429 -> T444" }, { "from": 2287, "to": 2262, "label": "INSTANCE with matching:\nT439 -> T448" }, { "from": 2296, "to": 2299, "label": "CASE" }, { "from": 2299, "to": 2300, "label": "PARALLEL" }, { "from": 2299, "to": 2301, "label": "PARALLEL" }, { "from": 2300, "to": 2303, "label": "EVAL with clause\naddz(zero(X631), zero(X632), zero(X633)) :- addz(X631, X632, X633).\nand substitutionX631 -> T483,\nT463 -> zero(T483),\nT464 -> T484,\nX632 -> T484,\nX633 -> T482,\nT462 -> zero(T482),\nT480 -> T483,\nT481 -> T484" }, { "from": 2300, "to": 2304, "label": "EVAL-BACKTRACK" }, { "from": 2301, "to": 2623, "label": "BACKTRACK\nfor clause: addz(zero(X), one(Y), one(Z)) :- addx(X, Y, Z)because of non-unification" }, { "from": 2303, "to": 2305, "label": "CASE" }, { "from": 2305, "to": 2316, "label": "PARALLEL" }, { "from": 2305, "to": 2317, "label": "PARALLEL" }, { "from": 2316, "to": 2322, "label": "EVAL with clause\naddz(zero(X649), zero(X650), zero(X651)) :- addz(X649, X650, X651).\nand substitutionX649 -> T503,\nT483 -> zero(T503),\nX650 -> T504,\nT484 -> zero(T504),\nX651 -> T502,\nT482 -> zero(T502),\nT500 -> T503,\nT501 -> T504" }, { "from": 2316, "to": 2323, "label": "EVAL-BACKTRACK" }, { "from": 2317, "to": 2331, "label": "PARALLEL" }, { "from": 2317, "to": 2332, "label": "PARALLEL" }, { "from": 2322, "to": 2303, "label": "INSTANCE with matching:\nT483 -> T503\nT484 -> T504\nT482 -> T502" }, { "from": 2331, "to": 2338, "label": "EVAL with clause\naddz(zero(X667), one(X668), one(X669)) :- addx(X667, X668, X669).\nand substitutionX667 -> T523,\nT483 -> zero(T523),\nX668 -> T524,\nT484 -> one(T524),\nX669 -> T522,\nT482 -> one(T522),\nT520 -> T523,\nT521 -> T524" }, { "from": 2331, "to": 2339, "label": "EVAL-BACKTRACK" }, { "from": 2332, "to": 2394, "label": "PARALLEL" }, { "from": 2332, "to": 2395, "label": "PARALLEL" }, { "from": 2338, "to": 2348, "label": "CASE" }, { "from": 2348, "to": 2351, "label": "PARALLEL" }, { "from": 2348, "to": 2352, "label": "PARALLEL" }, { "from": 2351, "to": 2353, "label": "EVAL with clause\naddx(one(X675), b, one(X675)) :- binary(X675).\nand substitutionX675 -> T530,\nT523 -> one(T530),\nT524 -> b,\nT522 -> one(T530)" }, { "from": 2351, "to": 2356, "label": "EVAL-BACKTRACK" }, { "from": 2352, "to": 2359, "label": "PARALLEL" }, { "from": 2352, "to": 2360, "label": "PARALLEL" }, { "from": 2353, "to": 2262, "label": "INSTANCE with matching:\nT439 -> T530" }, { "from": 2359, "to": 2361, "label": "EVAL with clause\naddx(zero(X680), b, zero(X680)) :- binaryZ(X680).\nand substitutionX680 -> T535,\nT523 -> zero(T535),\nT524 -> b,\nT522 -> zero(T535)" }, { "from": 2359, "to": 2362, "label": "EVAL-BACKTRACK" }, { "from": 2360, "to": 2381, "label": "ONLY EVAL with clause\naddx(X691, X692, X693) :- addz(X691, X692, X693).\nand substitutionT523 -> T551,\nX691 -> T551,\nT524 -> T552,\nX692 -> T552,\nT522 -> T550,\nX693 -> T550,\nT548 -> T551,\nT549 -> T552" }, { "from": 2361, "to": 2200, "label": "INSTANCE with matching:\nT429 -> T535" }, { "from": 2381, "to": 2303, "label": "INSTANCE with matching:\nT483 -> T551\nT484 -> T552\nT482 -> T550" }, { "from": 2394, "to": 2404, "label": "EVAL with clause\naddz(one(X709), zero(X710), one(X711)) :- addy(X709, X710, X711).\nand substitutionX709 -> T571,\nT483 -> one(T571),\nX710 -> T572,\nT484 -> zero(T572),\nX711 -> T570,\nT482 -> one(T570),\nT568 -> T571,\nT569 -> T572" }, { "from": 2394, "to": 2405, "label": "EVAL-BACKTRACK" }, { "from": 2395, "to": 2442, "label": "EVAL with clause\naddz(one(X745), one(X746), zero(X747)) :- addc(X745, X746, X747).\nand substitutionX745 -> T613,\nT483 -> one(T613),\nX746 -> T614,\nT484 -> one(T614),\nX747 -> T612,\nT482 -> zero(T612),\nT610 -> T613,\nT611 -> T614" }, { "from": 2395, "to": 2443, "label": "EVAL-BACKTRACK" }, { "from": 2404, "to": 2406, "label": "CASE" }, { "from": 2406, "to": 2407, "label": "PARALLEL" }, { "from": 2406, "to": 2408, "label": "PARALLEL" }, { "from": 2407, "to": 2409, "label": "EVAL with clause\naddy(b, one(X717), one(X717)) :- binary(X717).\nand substitutionT571 -> b,\nX717 -> T578,\nT572 -> one(T578),\nT570 -> one(T578)" }, { "from": 2407, "to": 2410, "label": "EVAL-BACKTRACK" }, { "from": 2408, "to": 2412, "label": "PARALLEL" }, { "from": 2408, "to": 2413, "label": "PARALLEL" }, { "from": 2409, "to": 2262, "label": "INSTANCE with matching:\nT439 -> T578" }, { "from": 2412, "to": 2415, "label": "EVAL with clause\naddy(b, zero(X722), zero(X722)) :- binaryZ(X722).\nand substitutionT571 -> b,\nX722 -> T583,\nT572 -> zero(T583),\nT570 -> zero(T583)" }, { "from": 2412, "to": 2416, "label": "EVAL-BACKTRACK" }, { "from": 2413, "to": 2429, "label": "ONLY EVAL with clause\naddy(X733, X734, X735) :- addz(X733, X734, X735).\nand substitutionT571 -> T599,\nX733 -> T599,\nT572 -> T600,\nX734 -> T600,\nT570 -> T598,\nX735 -> T598,\nT596 -> T599,\nT597 -> T600" }, { "from": 2415, "to": 2200, "label": "INSTANCE with matching:\nT429 -> T583" }, { "from": 2429, "to": 2303, "label": "INSTANCE with matching:\nT483 -> T599\nT484 -> T600\nT482 -> T598" }, { "from": 2442, "to": 2444, "label": "CASE" }, { "from": 2444, "to": 2445, "label": "PARALLEL" }, { "from": 2444, "to": 2446, "label": "PARALLEL" }, { "from": 2445, "to": 2447, "label": "EVAL with clause\naddc(b, b, one(b)).\nand substitutionT613 -> b,\nT614 -> b,\nT612 -> one(b)" }, { "from": 2445, "to": 2448, "label": "EVAL-BACKTRACK" }, { "from": 2446, "to": 2452, "label": "PARALLEL" }, { "from": 2446, "to": 2453, "label": "PARALLEL" }, { "from": 2447, "to": 2449, "label": "SUCCESS" }, { "from": 2452, "to": 2455, "label": "EVAL with clause\naddc(X756, b, X757) :- succZ(X756, X757).\nand substitutionT613 -> T625,\nX756 -> T625,\nT614 -> b,\nT612 -> T624,\nX757 -> T624,\nT623 -> T625" }, { "from": 2452, "to": 2456, "label": "EVAL-BACKTRACK" }, { "from": 2453, "to": 2523, "label": "PARALLEL" }, { "from": 2453, "to": 2524, "label": "PARALLEL" }, { "from": 2455, "to": 2457, "label": "CASE" }, { "from": 2457, "to": 2458, "label": "PARALLEL" }, { "from": 2457, "to": 2459, "label": "PARALLEL" }, { "from": 2458, "to": 2462, "label": "EVAL with clause\nsuccZ(zero(X763), one(X763)) :- binaryZ(X763).\nand substitutionX763 -> T631,\nT625 -> zero(T631),\nT624 -> one(T631)" }, { "from": 2458, "to": 2464, "label": "EVAL-BACKTRACK" }, { "from": 2459, "to": 2507, "label": "EVAL with clause\nsuccZ(one(X769), zero(X770)) :- succ(X769, X770).\nand substitutionX769 -> T639,\nT625 -> one(T639),\nX770 -> T638,\nT624 -> zero(T638),\nT637 -> T639" }, { "from": 2459, "to": 2508, "label": "EVAL-BACKTRACK" }, { "from": 2462, "to": 2200, "label": "INSTANCE with matching:\nT429 -> T631" }, { "from": 2507, "to": 2509, "label": "CASE" }, { "from": 2509, "to": 2510, "label": "PARALLEL" }, { "from": 2509, "to": 2511, "label": "PARALLEL" }, { "from": 2510, "to": 2512, "label": "EVAL with clause\nsucc(b, one(b)).\nand substitutionT639 -> b,\nT638 -> one(b)" }, { "from": 2510, "to": 2513, "label": "EVAL-BACKTRACK" }, { "from": 2511, "to": 2517, "label": "PARALLEL" }, { "from": 2511, "to": 2518, "label": "PARALLEL" }, { "from": 2512, "to": 2514, "label": "SUCCESS" }, { "from": 2517, "to": 2519, "label": "EVAL with clause\nsucc(zero(X775), one(X775)) :- binaryZ(X775).\nand substitutionX775 -> T644,\nT639 -> zero(T644),\nT638 -> one(T644)" }, { "from": 2517, "to": 2520, "label": "EVAL-BACKTRACK" }, { "from": 2518, "to": 2521, "label": "EVAL with clause\nsucc(one(X781), zero(X782)) :- succ(X781, X782).\nand substitutionX781 -> T652,\nT639 -> one(T652),\nX782 -> T651,\nT638 -> zero(T651),\nT650 -> T652" }, { "from": 2518, "to": 2522, "label": "EVAL-BACKTRACK" }, { "from": 2519, "to": 2200, "label": "INSTANCE with matching:\nT429 -> T644" }, { "from": 2521, "to": 2507, "label": "INSTANCE with matching:\nT639 -> T652\nT638 -> T651" }, { "from": 2523, "to": 2528, "label": "EVAL with clause\naddc(b, X791, X792) :- succZ(X791, X792).\nand substitutionT613 -> b,\nT614 -> T663,\nX791 -> T663,\nT612 -> T662,\nX792 -> T662,\nT661 -> T663" }, { "from": 2523, "to": 2529, "label": "EVAL-BACKTRACK" }, { "from": 2524, "to": 2535, "label": "ONLY EVAL with clause\naddc(X803, X804, X805) :- addC(X803, X804, X805).\nand substitutionT613 -> T679,\nX803 -> T679,\nT614 -> T680,\nX804 -> T680,\nT612 -> T678,\nX805 -> T678,\nT676 -> T679,\nT677 -> T680" }, { "from": 2528, "to": 2455, "label": "INSTANCE with matching:\nT625 -> T663\nT624 -> T662" }, { "from": 2535, "to": 2537, "label": "CASE" }, { "from": 2537, "to": 2540, "label": "PARALLEL" }, { "from": 2537, "to": 2541, "label": "PARALLEL" }, { "from": 2540, "to": 2569, "label": "EVAL with clause\naddC(zero(X821), zero(X822), one(X823)) :- addz(X821, X822, X823).\nand substitutionX821 -> T699,\nT679 -> zero(T699),\nX822 -> T700,\nT680 -> zero(T700),\nX823 -> T698,\nT678 -> one(T698),\nT696 -> T699,\nT697 -> T700" }, { "from": 2540, "to": 2570, "label": "EVAL-BACKTRACK" }, { "from": 2541, "to": 2571, "label": "PARALLEL" }, { "from": 2541, "to": 2572, "label": "PARALLEL" }, { "from": 2569, "to": 2303, "label": "INSTANCE with matching:\nT483 -> T699\nT484 -> T700\nT482 -> T698" }, { "from": 2571, "to": 2574, "label": "EVAL with clause\naddC(zero(X839), one(X840), zero(X841)) :- addX(X839, X840, X841).\nand substitutionX839 -> T719,\nT679 -> zero(T719),\nX840 -> T720,\nT680 -> one(T720),\nX841 -> T718,\nT678 -> zero(T718),\nT716 -> T719,\nT717 -> T720" }, { "from": 2571, "to": 2575, "label": "EVAL-BACKTRACK" }, { "from": 2572, "to": 2597, "label": "PARALLEL" }, { "from": 2572, "to": 2598, "label": "PARALLEL" }, { "from": 2574, "to": 2578, "label": "CASE" }, { "from": 2578, "to": 2579, "label": "PARALLEL" }, { "from": 2578, "to": 2580, "label": "PARALLEL" }, { "from": 2579, "to": 2581, "label": "EVAL with clause\naddX(zero(X847), b, one(X847)) :- binaryZ(X847).\nand substitutionX847 -> T726,\nT719 -> zero(T726),\nT720 -> b,\nT718 -> one(T726)" }, { "from": 2579, "to": 2582, "label": "EVAL-BACKTRACK" }, { "from": 2580, "to": 2585, "label": "PARALLEL" }, { "from": 2580, "to": 2587, "label": "PARALLEL" }, { "from": 2581, "to": 2200, "label": "INSTANCE with matching:\nT429 -> T726" }, { "from": 2585, "to": 2592, "label": "EVAL with clause\naddX(one(X857), b, zero(X858)) :- succ(X857, X858).\nand substitutionX857 -> T738,\nT719 -> one(T738),\nT720 -> b,\nX858 -> T737,\nT718 -> zero(T737),\nT736 -> T738" }, { "from": 2585, "to": 2593, "label": "EVAL-BACKTRACK" }, { "from": 2587, "to": 2596, "label": "ONLY EVAL with clause\naddX(X868, X869, X870) :- addC(X868, X869, X870).\nand substitutionT719 -> T753,\nX868 -> T753,\nT720 -> T754,\nX869 -> T754,\nT718 -> T752,\nX870 -> T752,\nT750 -> T753,\nT751 -> T754" }, { "from": 2592, "to": 2507, "label": "INSTANCE with matching:\nT639 -> T738\nT638 -> T737" }, { "from": 2596, "to": 2535, "label": "INSTANCE with matching:\nT679 -> T753\nT680 -> T754\nT678 -> T752" }, { "from": 2597, "to": 2599, "label": "EVAL with clause\naddC(one(X886), zero(X887), zero(X888)) :- addY(X886, X887, X888).\nand substitutionX886 -> T773,\nT679 -> one(T773),\nX887 -> T774,\nT680 -> zero(T774),\nX888 -> T772,\nT678 -> zero(T772),\nT770 -> T773,\nT771 -> T774" }, { "from": 2597, "to": 2600, "label": "EVAL-BACKTRACK" }, { "from": 2598, "to": 2621, "label": "EVAL with clause\naddC(one(X927), one(X928), one(X929)) :- addc(X927, X928, X929).\nand substitutionX927 -> T821,\nT679 -> one(T821),\nX928 -> T822,\nT680 -> one(T822),\nX929 -> T820,\nT678 -> one(T820),\nT818 -> T821,\nT819 -> T822" }, { "from": 2598, "to": 2622, "label": "EVAL-BACKTRACK" }, { "from": 2599, "to": 2601, "label": "CASE" }, { "from": 2601, "to": 2602, "label": "PARALLEL" }, { "from": 2601, "to": 2603, "label": "PARALLEL" }, { "from": 2602, "to": 2604, "label": "EVAL with clause\naddY(b, zero(X894), one(X894)) :- binaryZ(X894).\nand substitutionT773 -> b,\nX894 -> T780,\nT774 -> zero(T780),\nT772 -> one(T780)" }, { "from": 2602, "to": 2605, "label": "EVAL-BACKTRACK" }, { "from": 2603, "to": 2606, "label": "PARALLEL" }, { "from": 2603, "to": 2607, "label": "PARALLEL" }, { "from": 2604, "to": 2200, "label": "INSTANCE with matching:\nT429 -> T780" }, { "from": 2606, "to": 2608, "label": "EVAL with clause\naddY(b, one(X904), zero(X905)) :- succ(X904, X905).\nand substitutionT773 -> b,\nX904 -> T792,\nT774 -> one(T792),\nX905 -> T791,\nT772 -> zero(T791),\nT790 -> T792" }, { "from": 2606, "to": 2609, "label": "EVAL-BACKTRACK" }, { "from": 2607, "to": 2613, "label": "ONLY EVAL with clause\naddY(X915, X916, X917) :- addC(X915, X916, X917).\nand substitutionT773 -> T807,\nX915 -> T807,\nT774 -> T808,\nX916 -> T808,\nT772 -> T806,\nX917 -> T806,\nT804 -> T807,\nT805 -> T808" }, { "from": 2608, "to": 2507, "label": "INSTANCE with matching:\nT639 -> T792\nT638 -> T791" }, { "from": 2613, "to": 2535, "label": "INSTANCE with matching:\nT679 -> T807\nT680 -> T808\nT678 -> T806" }, { "from": 2621, "to": 2442, "label": "INSTANCE with matching:\nT613 -> T821\nT614 -> T822\nT612 -> T820" }, { "from": 2623, "to": 2624, "label": "PARALLEL" }, { "from": 2623, "to": 2625, "label": "PARALLEL" }, { "from": 2624, "to": 2628, "label": "EVAL with clause\naddz(one(X945), zero(X946), one(X947)) :- addy(X945, X946, X947).\nand substitutionX945 -> T839,\nT463 -> one(T839),\nT464 -> T840,\nX946 -> T840,\nX947 -> T838,\nT462 -> one(T838),\nT836 -> T839,\nT837 -> T840" }, { "from": 2624, "to": 2629, "label": "EVAL-BACKTRACK" }, { "from": 2625, "to": 2633, "label": "BACKTRACK\nfor clause: addz(one(X), one(Y), zero(Z)) :- addc(X, Y, Z)because of non-unification" }, { "from": 2628, "to": 2404, "label": "INSTANCE with matching:\nT571 -> T839\nT572 -> T840\nT570 -> T838" } ], "type": "Graph" } } ---------------------------------------- (194) Complex Obligation (AND) ---------------------------------------- (195) Obligation: Rules: f2274_out(T439) -> f2262_out(T439) :|: TRUE f2262_in(x) -> f2274_in(x) :|: TRUE f2218_in(T435) -> f2200_in(T435) :|: TRUE f2200_out(x1) -> f2218_out(x1) :|: TRUE f2215_in(T429) -> f2217_in(T429) :|: TRUE f2216_out(x2) -> f2215_out(x2) :|: TRUE f2215_in(x3) -> f2216_in(x3) :|: TRUE f2217_out(x4) -> f2215_out(x4) :|: TRUE f2200_in(x5) -> f2215_in(x5) :|: TRUE f2215_out(x6) -> f2200_out(x6) :|: TRUE f2280_out(x7) -> f2276_out(x7) :|: TRUE f2281_out(x8) -> f2276_out(x8) :|: TRUE f2276_in(x9) -> f2280_in(x9) :|: TRUE f2276_in(x10) -> f2281_in(x10) :|: TRUE f2274_in(x11) -> f2275_in(x11) :|: TRUE f2276_out(x12) -> f2274_out(x12) :|: TRUE f2274_in(x13) -> f2276_in(x13) :|: TRUE f2275_out(x14) -> f2274_out(x14) :|: TRUE f2262_out(x15) -> f2217_out(one(x15)) :|: TRUE f2217_in(x16) -> f2263_in :|: TRUE f2263_out -> f2217_out(x17) :|: TRUE f2217_in(one(x18)) -> f2262_in(x18) :|: TRUE f2281_in(x19) -> f2288_in :|: TRUE f2287_out(T448) -> f2281_out(one(T448)) :|: TRUE f2281_in(one(x20)) -> f2287_in(x20) :|: TRUE f2288_out -> f2281_out(x21) :|: TRUE f2216_in(x22) -> f2219_in :|: TRUE f2216_in(zero(x23)) -> f2218_in(x23) :|: TRUE f2218_out(x24) -> f2216_out(zero(x24)) :|: TRUE f2219_out -> f2216_out(x25) :|: TRUE f2287_in(x26) -> f2262_in(x26) :|: TRUE f2262_out(x27) -> f2287_out(x27) :|: TRUE f2283_out -> f2280_out(x28) :|: TRUE f2282_out(T444) -> f2280_out(zero(T444)) :|: TRUE f2280_in(x29) -> f2283_in :|: TRUE f2280_in(zero(x30)) -> f2282_in(x30) :|: TRUE f2200_out(x31) -> f2282_out(x31) :|: TRUE f2282_in(x32) -> f2200_in(x32) :|: TRUE f3_in(T3) -> f13_in(T3) :|: TRUE f13_out(x33) -> f3_out(x33) :|: TRUE f13_in(x34) -> f15_in(x34) :|: TRUE f15_out(x35) -> f13_out(x35) :|: TRUE f13_in(x36) -> f14_in(x36) :|: TRUE f14_out(x37) -> f13_out(x37) :|: TRUE f15_in(x38) -> f22_in(x38) :|: TRUE f15_in(x39) -> f21_in(x39) :|: TRUE f21_out(x40) -> f15_out(x40) :|: TRUE f22_out(x41) -> f15_out(x41) :|: TRUE f40_out(T36) -> f22_out(T36) :|: TRUE f22_in(x42) -> f41_in :|: TRUE f22_in(x43) -> f40_in(x43) :|: TRUE f41_out -> f22_out(x44) :|: TRUE f122_out(x45) -> f40_out(x45) :|: TRUE f40_in(x46) -> f121_in :|: TRUE f121_out -> f122_in(x47) :|: TRUE f2170_out(x48) -> f122_out(x48) :|: TRUE f122_in(x49) -> f2170_in(x49) :|: TRUE f2191_out(x50) -> f2170_out(x50) :|: TRUE f2170_in(x51) -> f2191_in(x51) :|: TRUE f2192_out(x52) -> f2191_out(x52) :|: TRUE f2191_in(x53) -> f2192_in(x53) :|: TRUE f2194_out(x54) -> f2192_out(x54) :|: TRUE f2193_out(x55) -> f2192_out(x55) :|: TRUE f2192_in(x56) -> f2194_in(x56) :|: TRUE f2192_in(x57) -> f2193_in(x57) :|: TRUE f2194_in(T462) -> f2296_in(T462) :|: TRUE f2296_out(x58) -> f2194_out(x58) :|: TRUE f2296_in(x59) -> f2299_in(x59) :|: TRUE f2299_out(x60) -> f2296_out(x60) :|: TRUE f2299_in(x61) -> f2300_in(x61) :|: TRUE f2300_out(x62) -> f2299_out(x62) :|: TRUE f2299_in(x63) -> f2301_in(x63) :|: TRUE f2301_out(x64) -> f2299_out(x64) :|: TRUE f2303_out(T482) -> f2300_out(zero(T482)) :|: TRUE f2304_out -> f2300_out(x65) :|: TRUE f2300_in(zero(x66)) -> f2303_in(x66) :|: TRUE f2300_in(x67) -> f2304_in :|: TRUE f2303_in(x68) -> f2305_in(x68) :|: TRUE f2305_out(x69) -> f2303_out(x69) :|: TRUE f2317_out(x70) -> f2305_out(x70) :|: TRUE f2316_out(x71) -> f2305_out(x71) :|: TRUE f2305_in(x72) -> f2317_in(x72) :|: TRUE f2305_in(x73) -> f2316_in(x73) :|: TRUE f2331_out(x74) -> f2317_out(x74) :|: TRUE f2317_in(x75) -> f2331_in(x75) :|: TRUE f2317_in(x76) -> f2332_in(x76) :|: TRUE f2332_out(x77) -> f2317_out(x77) :|: TRUE f2332_in(x78) -> f2395_in(x78) :|: TRUE f2394_out(x79) -> f2332_out(x79) :|: TRUE f2395_out(x80) -> f2332_out(x80) :|: TRUE f2332_in(x81) -> f2394_in(x81) :|: TRUE f2395_in(zero(T612)) -> f2442_in(T612) :|: TRUE f2395_in(x82) -> f2443_in :|: TRUE f2443_out -> f2395_out(x83) :|: TRUE f2442_out(x84) -> f2395_out(zero(x84)) :|: TRUE f2442_in(x85) -> f2444_in(x85) :|: TRUE f2444_out(x86) -> f2442_out(x86) :|: TRUE f2445_out(x87) -> f2444_out(x87) :|: TRUE f2444_in(x88) -> f2446_in(x88) :|: TRUE f2444_in(x89) -> f2445_in(x89) :|: TRUE f2446_out(x90) -> f2444_out(x90) :|: TRUE f2446_in(x91) -> f2452_in(x91) :|: TRUE f2452_out(x92) -> f2446_out(x92) :|: TRUE f2446_in(x93) -> f2453_in(x93) :|: TRUE f2453_out(x94) -> f2446_out(x94) :|: TRUE f2453_in(x95) -> f2523_in(x95) :|: TRUE f2524_out(x96) -> f2453_out(x96) :|: TRUE f2523_out(x97) -> f2453_out(x97) :|: TRUE f2453_in(x98) -> f2524_in(x98) :|: TRUE f2535_out(T678) -> f2524_out(T678) :|: TRUE f2524_in(x99) -> f2535_in(x99) :|: TRUE f2537_out(x100) -> f2535_out(x100) :|: TRUE f2535_in(x101) -> f2537_in(x101) :|: TRUE f2540_out(x102) -> f2537_out(x102) :|: TRUE f2537_in(x103) -> f2540_in(x103) :|: TRUE f2537_in(x104) -> f2541_in(x104) :|: TRUE f2541_out(x105) -> f2537_out(x105) :|: TRUE f2541_in(x106) -> f2571_in(x106) :|: TRUE f2571_out(x107) -> f2541_out(x107) :|: TRUE f2541_in(x108) -> f2572_in(x108) :|: TRUE f2572_out(x109) -> f2541_out(x109) :|: TRUE f2598_out(x110) -> f2572_out(x110) :|: TRUE f2572_in(x111) -> f2598_in(x111) :|: TRUE f2597_out(x112) -> f2572_out(x112) :|: TRUE f2572_in(x113) -> f2597_in(x113) :|: TRUE f2599_out(T772) -> f2597_out(zero(T772)) :|: TRUE f2597_in(x114) -> f2600_in :|: TRUE f2600_out -> f2597_out(x115) :|: TRUE f2597_in(zero(x116)) -> f2599_in(x116) :|: TRUE f2601_out(x117) -> f2599_out(x117) :|: TRUE f2599_in(x118) -> f2601_in(x118) :|: TRUE f2601_in(x119) -> f2602_in(x119) :|: TRUE f2601_in(x120) -> f2603_in(x120) :|: TRUE f2603_out(x121) -> f2601_out(x121) :|: TRUE f2602_out(x122) -> f2601_out(x122) :|: TRUE f2603_in(x123) -> f2607_in(x123) :|: TRUE f2606_out(x124) -> f2603_out(x124) :|: TRUE f2607_out(x125) -> f2603_out(x125) :|: TRUE f2603_in(x126) -> f2606_in(x126) :|: TRUE f2608_out(T791) -> f2606_out(zero(T791)) :|: TRUE f2606_in(x127) -> f2609_in :|: TRUE f2609_out -> f2606_out(x128) :|: TRUE f2606_in(zero(x129)) -> f2608_in(x129) :|: TRUE f2608_in(x130) -> f2507_in(x130) :|: TRUE f2507_out(x131) -> f2608_out(x131) :|: TRUE f2507_in(T638) -> f2509_in(T638) :|: TRUE f2509_out(x132) -> f2507_out(x132) :|: TRUE f2511_out(x133) -> f2509_out(x133) :|: TRUE f2510_out(x134) -> f2509_out(x134) :|: TRUE f2509_in(x135) -> f2511_in(x135) :|: TRUE f2509_in(x136) -> f2510_in(x136) :|: TRUE f2517_out(x137) -> f2511_out(x137) :|: TRUE f2511_in(x138) -> f2517_in(x138) :|: TRUE f2518_out(x139) -> f2511_out(x139) :|: TRUE f2511_in(x140) -> f2518_in(x140) :|: TRUE f2517_in(one(T644)) -> f2519_in(T644) :|: TRUE f2517_in(x141) -> f2520_in :|: TRUE f2520_out -> f2517_out(x142) :|: TRUE f2519_out(x143) -> f2517_out(one(x143)) :|: TRUE f2519_in(x144) -> f2200_in(x144) :|: TRUE f2200_out(x145) -> f2519_out(x145) :|: TRUE f2331_in(x146) -> f2339_in :|: TRUE f2331_in(one(T522)) -> f2338_in(T522) :|: TRUE f2338_out(x147) -> f2331_out(one(x147)) :|: TRUE f2339_out -> f2331_out(x148) :|: TRUE f2348_out(x149) -> f2338_out(x149) :|: TRUE f2338_in(x150) -> f2348_in(x150) :|: TRUE f2351_out(x151) -> f2348_out(x151) :|: TRUE f2348_in(x152) -> f2351_in(x152) :|: TRUE f2348_in(x153) -> f2352_in(x153) :|: TRUE f2352_out(x154) -> f2348_out(x154) :|: TRUE f2359_out(x155) -> f2352_out(x155) :|: TRUE f2360_out(x156) -> f2352_out(x156) :|: TRUE f2352_in(x157) -> f2360_in(x157) :|: TRUE f2352_in(x158) -> f2359_in(x158) :|: TRUE f2361_out(T535) -> f2359_out(zero(T535)) :|: TRUE f2362_out -> f2359_out(x159) :|: TRUE f2359_in(zero(x160)) -> f2361_in(x160) :|: TRUE f2359_in(x161) -> f2362_in :|: TRUE f2361_in(x162) -> f2200_in(x162) :|: TRUE f2200_out(x163) -> f2361_out(x163) :|: TRUE f2623_out(x164) -> f2301_out(x164) :|: TRUE f2301_in(x165) -> f2623_in(x165) :|: TRUE f2623_in(x166) -> f2625_in(x166) :|: TRUE f2624_out(x167) -> f2623_out(x167) :|: TRUE f2623_in(x168) -> f2624_in(x168) :|: TRUE f2625_out(x169) -> f2623_out(x169) :|: TRUE f2624_in(one(T838)) -> f2628_in(T838) :|: TRUE f2628_out(x170) -> f2624_out(one(x170)) :|: TRUE f2624_in(x171) -> f2629_in :|: TRUE f2629_out -> f2624_out(x172) :|: TRUE f2404_out(x173) -> f2628_out(x173) :|: TRUE f2628_in(x174) -> f2404_in(x174) :|: TRUE f2404_in(T570) -> f2406_in(T570) :|: TRUE f2406_out(x175) -> f2404_out(x175) :|: TRUE f2407_out(x176) -> f2406_out(x176) :|: TRUE f2408_out(x177) -> f2406_out(x177) :|: TRUE f2406_in(x178) -> f2407_in(x178) :|: TRUE f2406_in(x179) -> f2408_in(x179) :|: TRUE f2412_out(x180) -> f2408_out(x180) :|: TRUE f2408_in(x181) -> f2413_in(x181) :|: TRUE f2413_out(x182) -> f2408_out(x182) :|: TRUE f2408_in(x183) -> f2412_in(x183) :|: TRUE f2412_in(x184) -> f2416_in :|: TRUE f2415_out(T583) -> f2412_out(zero(T583)) :|: TRUE f2416_out -> f2412_out(x185) :|: TRUE f2412_in(zero(x186)) -> f2415_in(x186) :|: TRUE f2200_out(x187) -> f2415_out(x187) :|: TRUE f2415_in(x188) -> f2200_in(x188) :|: TRUE f2394_in(x189) -> f2405_in :|: TRUE f2394_in(one(x190)) -> f2404_in(x190) :|: TRUE f2405_out -> f2394_out(x191) :|: TRUE f2404_out(x192) -> f2394_out(one(x192)) :|: TRUE f2429_out(T598) -> f2413_out(T598) :|: TRUE f2413_in(x193) -> f2429_in(x193) :|: TRUE f2429_in(x194) -> f2303_in(x194) :|: TRUE f2303_out(x195) -> f2429_out(x195) :|: TRUE f2455_out(T624) -> f2452_out(T624) :|: TRUE f2456_out -> f2452_out(x196) :|: TRUE f2452_in(x197) -> f2456_in :|: TRUE f2452_in(x198) -> f2455_in(x198) :|: TRUE f2455_in(x199) -> f2457_in(x199) :|: TRUE f2457_out(x200) -> f2455_out(x200) :|: TRUE f2459_out(x201) -> f2457_out(x201) :|: TRUE f2457_in(x202) -> f2458_in(x202) :|: TRUE f2458_out(x203) -> f2457_out(x203) :|: TRUE f2457_in(x204) -> f2459_in(x204) :|: TRUE f2458_in(one(T631)) -> f2462_in(T631) :|: TRUE f2462_out(x205) -> f2458_out(one(x205)) :|: TRUE f2458_in(x206) -> f2464_in :|: TRUE f2464_out -> f2458_out(x207) :|: TRUE f2462_in(x208) -> f2200_in(x208) :|: TRUE f2200_out(x209) -> f2462_out(x209) :|: TRUE f2528_out(T662) -> f2523_out(T662) :|: TRUE f2523_in(x210) -> f2529_in :|: TRUE f2523_in(x211) -> f2528_in(x211) :|: TRUE f2529_out -> f2523_out(x212) :|: TRUE f2455_out(x213) -> f2528_out(x213) :|: TRUE f2528_in(x214) -> f2455_in(x214) :|: TRUE f2508_out -> f2459_out(x215) :|: TRUE f2459_in(zero(x216)) -> f2507_in(x216) :|: TRUE f2507_out(x217) -> f2459_out(zero(x217)) :|: TRUE f2459_in(x218) -> f2508_in :|: TRUE f2602_in(one(T780)) -> f2604_in(T780) :|: TRUE f2602_in(x219) -> f2605_in :|: TRUE f2605_out -> f2602_out(x220) :|: TRUE f2604_out(x221) -> f2602_out(one(x221)) :|: TRUE f2200_out(x222) -> f2604_out(x222) :|: TRUE f2604_in(x223) -> f2200_in(x223) :|: TRUE f2195_out(T421) -> f2193_out(zero(T421)) :|: TRUE f2196_out -> f2193_out(x224) :|: TRUE f2193_in(zero(x225)) -> f2195_in(x225) :|: TRUE f2193_in(x226) -> f2196_in :|: TRUE f2195_in(x227) -> f2197_in(x227) :|: TRUE f2197_out(x228) -> f2195_out(x228) :|: TRUE f2197_in(x229) -> f2198_in(x229) :|: TRUE f2199_out(x230) -> f2197_out(x230) :|: TRUE f2197_in(x231) -> f2199_in(x231) :|: TRUE f2198_out(x232) -> f2197_out(x232) :|: TRUE f2198_in(x233) -> f2200_in(x233) :|: TRUE f2200_out(x234) -> f2198_out(x234) :|: TRUE f2409_out(T578) -> f2407_out(one(T578)) :|: TRUE f2407_in(one(x235)) -> f2409_in(x235) :|: TRUE f2407_in(x236) -> f2410_in :|: TRUE f2410_out -> f2407_out(x237) :|: TRUE f2409_in(x238) -> f2262_in(x238) :|: TRUE f2262_out(x239) -> f2409_out(x239) :|: TRUE f2571_in(x240) -> f2575_in :|: TRUE f2571_in(zero(T718)) -> f2574_in(T718) :|: TRUE f2574_out(x241) -> f2571_out(zero(x241)) :|: TRUE f2575_out -> f2571_out(x242) :|: TRUE f2574_in(x243) -> f2578_in(x243) :|: TRUE f2578_out(x244) -> f2574_out(x244) :|: TRUE f2580_out(x245) -> f2578_out(x245) :|: TRUE f2578_in(x246) -> f2579_in(x246) :|: TRUE f2579_out(x247) -> f2578_out(x247) :|: TRUE f2578_in(x248) -> f2580_in(x248) :|: TRUE f2579_in(x249) -> f2582_in :|: TRUE f2582_out -> f2579_out(x250) :|: TRUE f2579_in(one(T726)) -> f2581_in(T726) :|: TRUE f2581_out(x251) -> f2579_out(one(x251)) :|: TRUE f2581_in(x252) -> f2200_in(x252) :|: TRUE f2200_out(x253) -> f2581_out(x253) :|: TRUE f2351_in(one(T530)) -> f2353_in(T530) :|: TRUE f2353_out(x254) -> f2351_out(one(x254)) :|: TRUE f2356_out -> f2351_out(x255) :|: TRUE f2351_in(x256) -> f2356_in :|: TRUE f2262_out(x257) -> f2353_out(x257) :|: TRUE f2353_in(x258) -> f2262_in(x258) :|: TRUE f2585_out(x259) -> f2580_out(x259) :|: TRUE f2580_in(x260) -> f2587_in(x260) :|: TRUE f2587_out(x261) -> f2580_out(x261) :|: TRUE f2580_in(x262) -> f2585_in(x262) :|: TRUE f2593_out -> f2585_out(x263) :|: TRUE f2585_in(x264) -> f2593_in :|: TRUE f2592_out(T737) -> f2585_out(zero(T737)) :|: TRUE f2585_in(zero(x265)) -> f2592_in(x265) :|: TRUE f2592_in(x266) -> f2507_in(x266) :|: TRUE f2507_out(x267) -> f2592_out(x267) :|: TRUE Start term: f3_in(T3) ---------------------------------------- (196) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (197) TRUE ---------------------------------------- (198) Obligation: Rules: f2507_out(T651) -> f2521_out(T651) :|: TRUE f2521_in(x) -> f2507_in(x) :|: TRUE f2511_out(T638) -> f2509_out(T638) :|: TRUE f2510_out(x1) -> f2509_out(x1) :|: TRUE f2509_in(x2) -> f2511_in(x2) :|: TRUE f2509_in(x3) -> f2510_in(x3) :|: TRUE f2517_out(x4) -> f2511_out(x4) :|: TRUE f2511_in(x5) -> f2517_in(x5) :|: TRUE f2518_out(x6) -> f2511_out(x6) :|: TRUE f2511_in(x7) -> f2518_in(x7) :|: TRUE f2521_out(x8) -> f2518_out(zero(x8)) :|: TRUE f2518_in(x9) -> f2522_in :|: TRUE f2522_out -> f2518_out(x10) :|: TRUE f2518_in(zero(x11)) -> f2521_in(x11) :|: TRUE f2507_in(x12) -> f2509_in(x12) :|: TRUE f2509_out(x13) -> f2507_out(x13) :|: TRUE f3_in(T3) -> f13_in(T3) :|: TRUE f13_out(x14) -> f3_out(x14) :|: TRUE f13_in(x15) -> f15_in(x15) :|: TRUE f15_out(x16) -> f13_out(x16) :|: TRUE f13_in(x17) -> f14_in(x17) :|: TRUE f14_out(x18) -> f13_out(x18) :|: TRUE f15_in(x19) -> f22_in(x19) :|: TRUE f15_in(x20) -> f21_in(x20) :|: TRUE f21_out(x21) -> f15_out(x21) :|: TRUE f22_out(x22) -> f15_out(x22) :|: TRUE f40_out(T36) -> f22_out(T36) :|: TRUE f22_in(x23) -> f41_in :|: TRUE f22_in(x24) -> f40_in(x24) :|: TRUE f41_out -> f22_out(x25) :|: TRUE f122_out(x26) -> f40_out(x26) :|: TRUE f40_in(x27) -> f121_in :|: TRUE f121_out -> f122_in(x28) :|: TRUE f2170_out(x29) -> f122_out(x29) :|: TRUE f122_in(x30) -> f2170_in(x30) :|: TRUE f2191_out(x31) -> f2170_out(x31) :|: TRUE f2170_in(x32) -> f2191_in(x32) :|: TRUE f2192_out(x33) -> f2191_out(x33) :|: TRUE f2191_in(x34) -> f2192_in(x34) :|: TRUE f2194_out(x35) -> f2192_out(x35) :|: TRUE f2193_out(x36) -> f2192_out(x36) :|: TRUE f2192_in(x37) -> f2194_in(x37) :|: TRUE f2192_in(x38) -> f2193_in(x38) :|: TRUE f2194_in(T462) -> f2296_in(T462) :|: TRUE f2296_out(x39) -> f2194_out(x39) :|: TRUE f2296_in(x40) -> f2299_in(x40) :|: TRUE f2299_out(x41) -> f2296_out(x41) :|: TRUE f2299_in(x42) -> f2300_in(x42) :|: TRUE f2300_out(x43) -> f2299_out(x43) :|: TRUE f2299_in(x44) -> f2301_in(x44) :|: TRUE f2301_out(x45) -> f2299_out(x45) :|: TRUE f2303_out(T482) -> f2300_out(zero(T482)) :|: TRUE f2304_out -> f2300_out(x46) :|: TRUE f2300_in(zero(x47)) -> f2303_in(x47) :|: TRUE f2300_in(x48) -> f2304_in :|: TRUE f2303_in(x49) -> f2305_in(x49) :|: TRUE f2305_out(x50) -> f2303_out(x50) :|: TRUE f2317_out(x51) -> f2305_out(x51) :|: TRUE f2316_out(x52) -> f2305_out(x52) :|: TRUE f2305_in(x53) -> f2317_in(x53) :|: TRUE f2305_in(x54) -> f2316_in(x54) :|: TRUE f2331_out(x55) -> f2317_out(x55) :|: TRUE f2317_in(x56) -> f2331_in(x56) :|: TRUE f2317_in(x57) -> f2332_in(x57) :|: TRUE f2332_out(x58) -> f2317_out(x58) :|: TRUE f2332_in(x59) -> f2395_in(x59) :|: TRUE f2394_out(x60) -> f2332_out(x60) :|: TRUE f2395_out(x61) -> f2332_out(x61) :|: TRUE f2332_in(x62) -> f2394_in(x62) :|: TRUE f2395_in(zero(T612)) -> f2442_in(T612) :|: TRUE f2395_in(x63) -> f2443_in :|: TRUE f2443_out -> f2395_out(x64) :|: TRUE f2442_out(x65) -> f2395_out(zero(x65)) :|: TRUE f2442_in(x66) -> f2444_in(x66) :|: TRUE f2444_out(x67) -> f2442_out(x67) :|: TRUE f2445_out(x68) -> f2444_out(x68) :|: TRUE f2444_in(x69) -> f2446_in(x69) :|: TRUE f2444_in(x70) -> f2445_in(x70) :|: TRUE f2446_out(x71) -> f2444_out(x71) :|: TRUE f2446_in(x72) -> f2452_in(x72) :|: TRUE f2452_out(x73) -> f2446_out(x73) :|: TRUE f2446_in(x74) -> f2453_in(x74) :|: TRUE f2453_out(x75) -> f2446_out(x75) :|: TRUE f2453_in(x76) -> f2523_in(x76) :|: TRUE f2524_out(x77) -> f2453_out(x77) :|: TRUE f2523_out(x78) -> f2453_out(x78) :|: TRUE f2453_in(x79) -> f2524_in(x79) :|: TRUE f2535_out(T678) -> f2524_out(T678) :|: TRUE f2524_in(x80) -> f2535_in(x80) :|: TRUE f2537_out(x81) -> f2535_out(x81) :|: TRUE f2535_in(x82) -> f2537_in(x82) :|: TRUE f2540_out(x83) -> f2537_out(x83) :|: TRUE f2537_in(x84) -> f2540_in(x84) :|: TRUE f2537_in(x85) -> f2541_in(x85) :|: TRUE f2541_out(x86) -> f2537_out(x86) :|: TRUE f2541_in(x87) -> f2571_in(x87) :|: TRUE f2571_out(x88) -> f2541_out(x88) :|: TRUE f2541_in(x89) -> f2572_in(x89) :|: TRUE f2572_out(x90) -> f2541_out(x90) :|: TRUE f2571_in(x91) -> f2575_in :|: TRUE f2571_in(zero(T718)) -> f2574_in(T718) :|: TRUE f2574_out(x92) -> f2571_out(zero(x92)) :|: TRUE f2575_out -> f2571_out(x93) :|: TRUE f2574_in(x94) -> f2578_in(x94) :|: TRUE f2578_out(x95) -> f2574_out(x95) :|: TRUE f2580_out(x96) -> f2578_out(x96) :|: TRUE f2578_in(x97) -> f2579_in(x97) :|: TRUE f2579_out(x98) -> f2578_out(x98) :|: TRUE f2578_in(x99) -> f2580_in(x99) :|: TRUE f2585_out(x100) -> f2580_out(x100) :|: TRUE f2580_in(x101) -> f2587_in(x101) :|: TRUE f2587_out(x102) -> f2580_out(x102) :|: TRUE f2580_in(x103) -> f2585_in(x103) :|: TRUE f2593_out -> f2585_out(x104) :|: TRUE f2585_in(x105) -> f2593_in :|: TRUE f2592_out(T737) -> f2585_out(zero(T737)) :|: TRUE f2585_in(zero(x106)) -> f2592_in(x106) :|: TRUE f2592_in(x107) -> f2507_in(x107) :|: TRUE f2507_out(x108) -> f2592_out(x108) :|: TRUE f2623_out(x109) -> f2301_out(x109) :|: TRUE f2301_in(x110) -> f2623_in(x110) :|: TRUE f2623_in(x111) -> f2625_in(x111) :|: TRUE f2624_out(x112) -> f2623_out(x112) :|: TRUE f2623_in(x113) -> f2624_in(x113) :|: TRUE f2625_out(x114) -> f2623_out(x114) :|: TRUE f2624_in(one(T838)) -> f2628_in(T838) :|: TRUE f2628_out(x115) -> f2624_out(one(x115)) :|: TRUE f2624_in(x116) -> f2629_in :|: TRUE f2629_out -> f2624_out(x117) :|: TRUE f2404_out(x118) -> f2628_out(x118) :|: TRUE f2628_in(x119) -> f2404_in(x119) :|: TRUE f2404_in(T570) -> f2406_in(T570) :|: TRUE f2406_out(x120) -> f2404_out(x120) :|: TRUE f2407_out(x121) -> f2406_out(x121) :|: TRUE f2408_out(x122) -> f2406_out(x122) :|: TRUE f2406_in(x123) -> f2407_in(x123) :|: TRUE f2406_in(x124) -> f2408_in(x124) :|: TRUE f2412_out(x125) -> f2408_out(x125) :|: TRUE f2408_in(x126) -> f2413_in(x126) :|: TRUE f2413_out(x127) -> f2408_out(x127) :|: TRUE f2408_in(x128) -> f2412_in(x128) :|: TRUE f2429_out(T598) -> f2413_out(T598) :|: TRUE f2413_in(x129) -> f2429_in(x129) :|: TRUE f2429_in(x130) -> f2303_in(x130) :|: TRUE f2303_out(x131) -> f2429_out(x131) :|: TRUE f2528_out(T662) -> f2523_out(T662) :|: TRUE f2523_in(x132) -> f2529_in :|: TRUE f2523_in(x133) -> f2528_in(x133) :|: TRUE f2529_out -> f2523_out(x134) :|: TRUE f2455_out(x135) -> f2528_out(x135) :|: TRUE f2528_in(x136) -> f2455_in(x136) :|: TRUE f2455_in(T624) -> f2457_in(T624) :|: TRUE f2457_out(x137) -> f2455_out(x137) :|: TRUE f2459_out(x138) -> f2457_out(x138) :|: TRUE f2457_in(x139) -> f2458_in(x139) :|: TRUE f2458_out(x140) -> f2457_out(x140) :|: TRUE f2457_in(x141) -> f2459_in(x141) :|: TRUE f2508_out -> f2459_out(x142) :|: TRUE f2459_in(zero(x143)) -> f2507_in(x143) :|: TRUE f2507_out(x144) -> f2459_out(zero(x144)) :|: TRUE f2459_in(x145) -> f2508_in :|: TRUE f2598_out(x146) -> f2572_out(x146) :|: TRUE f2572_in(x147) -> f2598_in(x147) :|: TRUE f2597_out(x148) -> f2572_out(x148) :|: TRUE f2572_in(x149) -> f2597_in(x149) :|: TRUE f2599_out(T772) -> f2597_out(zero(T772)) :|: TRUE f2597_in(x150) -> f2600_in :|: TRUE f2600_out -> f2597_out(x151) :|: TRUE f2597_in(zero(x152)) -> f2599_in(x152) :|: TRUE f2601_out(x153) -> f2599_out(x153) :|: TRUE f2599_in(x154) -> f2601_in(x154) :|: TRUE f2601_in(x155) -> f2602_in(x155) :|: TRUE f2601_in(x156) -> f2603_in(x156) :|: TRUE f2603_out(x157) -> f2601_out(x157) :|: TRUE f2602_out(x158) -> f2601_out(x158) :|: TRUE f2603_in(x159) -> f2607_in(x159) :|: TRUE f2606_out(x160) -> f2603_out(x160) :|: TRUE f2607_out(x161) -> f2603_out(x161) :|: TRUE f2603_in(x162) -> f2606_in(x162) :|: TRUE f2608_out(T791) -> f2606_out(zero(T791)) :|: TRUE f2606_in(x163) -> f2609_in :|: TRUE f2609_out -> f2606_out(x164) :|: TRUE f2606_in(zero(x165)) -> f2608_in(x165) :|: TRUE f2608_in(x166) -> f2507_in(x166) :|: TRUE f2507_out(x167) -> f2608_out(x167) :|: TRUE f2455_out(x168) -> f2452_out(x168) :|: TRUE f2456_out -> f2452_out(x169) :|: TRUE f2452_in(x170) -> f2456_in :|: TRUE f2452_in(x171) -> f2455_in(x171) :|: TRUE Start term: f3_in(T3) ---------------------------------------- (199) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (200) TRUE ---------------------------------------- (201) Obligation: Rules: f2580_out(T718) -> f2578_out(T718) :|: TRUE f2578_in(x) -> f2579_in(x) :|: TRUE f2579_out(x1) -> f2578_out(x1) :|: TRUE f2578_in(x2) -> f2580_in(x2) :|: TRUE f2587_in(T752) -> f2596_in(T752) :|: TRUE f2596_out(x3) -> f2587_out(x3) :|: TRUE f2537_out(T678) -> f2535_out(T678) :|: TRUE f2535_in(x4) -> f2537_in(x4) :|: TRUE f2541_in(x5) -> f2571_in(x5) :|: TRUE f2571_out(x6) -> f2541_out(x6) :|: TRUE f2541_in(x7) -> f2572_in(x7) :|: TRUE f2572_out(x8) -> f2541_out(x8) :|: TRUE f2603_in(T772) -> f2607_in(T772) :|: TRUE f2606_out(x9) -> f2603_out(x9) :|: TRUE f2607_out(x10) -> f2603_out(x10) :|: TRUE f2603_in(x11) -> f2606_in(x11) :|: TRUE f2429_in(T598) -> f2303_in(T598) :|: TRUE f2303_out(x12) -> f2429_out(x12) :|: TRUE f2574_in(x13) -> f2578_in(x13) :|: TRUE f2578_out(x14) -> f2574_out(x14) :|: TRUE f2359_out(T522) -> f2352_out(T522) :|: TRUE f2360_out(x15) -> f2352_out(x15) :|: TRUE f2352_in(x16) -> f2360_in(x16) :|: TRUE f2352_in(x17) -> f2359_in(x17) :|: TRUE f2598_in(one(T820)) -> f2621_in(T820) :|: TRUE f2621_out(x18) -> f2598_out(one(x18)) :|: TRUE f2598_in(x19) -> f2622_in :|: TRUE f2622_out -> f2598_out(x20) :|: TRUE f2322_out(T502) -> f2316_out(zero(T502)) :|: TRUE f2323_out -> f2316_out(T482) :|: TRUE f2316_in(zero(x21)) -> f2322_in(x21) :|: TRUE f2316_in(x22) -> f2323_in :|: TRUE f2395_in(zero(T612)) -> f2442_in(T612) :|: TRUE f2395_in(x23) -> f2443_in :|: TRUE f2443_out -> f2395_out(x24) :|: TRUE f2442_out(x25) -> f2395_out(zero(x25)) :|: TRUE f2317_out(x26) -> f2305_out(x26) :|: TRUE f2316_out(x27) -> f2305_out(x27) :|: TRUE f2305_in(x28) -> f2317_in(x28) :|: TRUE f2305_in(x29) -> f2316_in(x29) :|: TRUE f2613_out(T806) -> f2607_out(T806) :|: TRUE f2607_in(x30) -> f2613_in(x30) :|: TRUE f2535_out(x31) -> f2524_out(x31) :|: TRUE f2524_in(x32) -> f2535_in(x32) :|: TRUE f2351_out(x33) -> f2348_out(x33) :|: TRUE f2348_in(x34) -> f2351_in(x34) :|: TRUE f2348_in(x35) -> f2352_in(x35) :|: TRUE f2352_out(x36) -> f2348_out(x36) :|: TRUE f2442_in(x37) -> f2444_in(x37) :|: TRUE f2444_out(x38) -> f2442_out(x38) :|: TRUE f2404_in(T570) -> f2406_in(T570) :|: TRUE f2406_out(x39) -> f2404_out(x39) :|: TRUE f2303_out(T698) -> f2569_out(T698) :|: TRUE f2569_in(x40) -> f2303_in(x40) :|: TRUE f2360_in(T550) -> f2381_in(T550) :|: TRUE f2381_out(x41) -> f2360_out(x41) :|: TRUE f2322_in(x42) -> f2303_in(x42) :|: TRUE f2303_out(x43) -> f2322_out(x43) :|: TRUE f2429_out(x44) -> f2413_out(x44) :|: TRUE f2413_in(x45) -> f2429_in(x45) :|: TRUE f2613_in(x46) -> f2535_in(x46) :|: TRUE f2535_out(x47) -> f2613_out(x47) :|: TRUE f2535_out(x48) -> f2596_out(x48) :|: TRUE f2596_in(x49) -> f2535_in(x49) :|: TRUE f2621_in(x50) -> f2442_in(x50) :|: TRUE f2442_out(x51) -> f2621_out(x51) :|: TRUE f2540_out(x52) -> f2537_out(x52) :|: TRUE f2537_in(x53) -> f2540_in(x53) :|: TRUE f2537_in(x54) -> f2541_in(x54) :|: TRUE f2541_out(x55) -> f2537_out(x55) :|: TRUE f2445_out(x56) -> f2444_out(x56) :|: TRUE f2444_in(x57) -> f2446_in(x57) :|: TRUE f2444_in(x58) -> f2445_in(x58) :|: TRUE f2446_out(x59) -> f2444_out(x59) :|: TRUE f2446_in(x60) -> f2452_in(x60) :|: TRUE f2452_out(x61) -> f2446_out(x61) :|: TRUE f2446_in(x62) -> f2453_in(x62) :|: TRUE f2453_out(x63) -> f2446_out(x63) :|: TRUE f2571_in(x64) -> f2575_in :|: TRUE f2571_in(zero(x65)) -> f2574_in(x65) :|: TRUE f2574_out(x66) -> f2571_out(zero(x66)) :|: TRUE f2575_out -> f2571_out(x67) :|: TRUE f2598_out(x68) -> f2572_out(x68) :|: TRUE f2572_in(x69) -> f2598_in(x69) :|: TRUE f2597_out(x70) -> f2572_out(x70) :|: TRUE f2572_in(x71) -> f2597_in(x71) :|: TRUE f2412_out(x72) -> f2408_out(x72) :|: TRUE f2408_in(x73) -> f2413_in(x73) :|: TRUE f2413_out(x74) -> f2408_out(x74) :|: TRUE f2408_in(x75) -> f2412_in(x75) :|: TRUE f2453_in(x76) -> f2523_in(x76) :|: TRUE f2524_out(x77) -> f2453_out(x77) :|: TRUE f2523_out(x78) -> f2453_out(x78) :|: TRUE f2453_in(x79) -> f2524_in(x79) :|: TRUE f2394_in(x80) -> f2405_in :|: TRUE f2394_in(one(x81)) -> f2404_in(x81) :|: TRUE f2405_out -> f2394_out(x82) :|: TRUE f2404_out(x83) -> f2394_out(one(x83)) :|: TRUE f2585_out(x84) -> f2580_out(x84) :|: TRUE f2580_in(x85) -> f2587_in(x85) :|: TRUE f2587_out(x86) -> f2580_out(x86) :|: TRUE f2580_in(x87) -> f2585_in(x87) :|: TRUE f2601_in(x88) -> f2602_in(x88) :|: TRUE f2601_in(x89) -> f2603_in(x89) :|: TRUE f2603_out(x90) -> f2601_out(x90) :|: TRUE f2602_out(x91) -> f2601_out(x91) :|: TRUE f2381_in(x92) -> f2303_in(x92) :|: TRUE f2303_out(x93) -> f2381_out(x93) :|: TRUE f2407_out(x94) -> f2406_out(x94) :|: TRUE f2408_out(x95) -> f2406_out(x95) :|: TRUE f2406_in(x96) -> f2407_in(x96) :|: TRUE f2406_in(x97) -> f2408_in(x97) :|: TRUE f2332_in(x98) -> f2395_in(x98) :|: TRUE f2394_out(x99) -> f2332_out(x99) :|: TRUE f2395_out(x100) -> f2332_out(x100) :|: TRUE f2332_in(x101) -> f2394_in(x101) :|: TRUE f2540_in(one(x102)) -> f2569_in(x102) :|: TRUE f2569_out(x103) -> f2540_out(one(x103)) :|: TRUE f2540_in(x104) -> f2570_in :|: TRUE f2570_out -> f2540_out(x105) :|: TRUE f2599_out(x106) -> f2597_out(zero(x106)) :|: TRUE f2597_in(x107) -> f2600_in :|: TRUE f2600_out -> f2597_out(x108) :|: TRUE f2597_in(zero(x109)) -> f2599_in(x109) :|: TRUE f2348_out(x110) -> f2338_out(x110) :|: TRUE f2338_in(x111) -> f2348_in(x111) :|: TRUE f2601_out(x112) -> f2599_out(x112) :|: TRUE f2599_in(x113) -> f2601_in(x113) :|: TRUE f2331_out(x114) -> f2317_out(x114) :|: TRUE f2317_in(x115) -> f2331_in(x115) :|: TRUE f2317_in(x116) -> f2332_in(x116) :|: TRUE f2332_out(x117) -> f2317_out(x117) :|: TRUE f2303_in(x118) -> f2305_in(x118) :|: TRUE f2305_out(x119) -> f2303_out(x119) :|: TRUE f2331_in(x120) -> f2339_in :|: TRUE f2331_in(one(x121)) -> f2338_in(x121) :|: TRUE f2338_out(x122) -> f2331_out(one(x122)) :|: TRUE f2339_out -> f2331_out(x123) :|: TRUE f3_in(T3) -> f13_in(T3) :|: TRUE f13_out(x124) -> f3_out(x124) :|: TRUE f13_in(x125) -> f15_in(x125) :|: TRUE f15_out(x126) -> f13_out(x126) :|: TRUE f13_in(x127) -> f14_in(x127) :|: TRUE f14_out(x128) -> f13_out(x128) :|: TRUE f15_in(x129) -> f22_in(x129) :|: TRUE f15_in(x130) -> f21_in(x130) :|: TRUE f21_out(x131) -> f15_out(x131) :|: TRUE f22_out(x132) -> f15_out(x132) :|: TRUE f40_out(T36) -> f22_out(T36) :|: TRUE f22_in(x133) -> f41_in :|: TRUE f22_in(x134) -> f40_in(x134) :|: TRUE f41_out -> f22_out(x135) :|: TRUE f122_out(x136) -> f40_out(x136) :|: TRUE f40_in(x137) -> f121_in :|: TRUE f121_out -> f122_in(x138) :|: TRUE f2170_out(x139) -> f122_out(x139) :|: TRUE f122_in(x140) -> f2170_in(x140) :|: TRUE f2191_out(x141) -> f2170_out(x141) :|: TRUE f2170_in(x142) -> f2191_in(x142) :|: TRUE f2192_out(x143) -> f2191_out(x143) :|: TRUE f2191_in(x144) -> f2192_in(x144) :|: TRUE f2194_out(x145) -> f2192_out(x145) :|: TRUE f2193_out(x146) -> f2192_out(x146) :|: TRUE f2192_in(x147) -> f2194_in(x147) :|: TRUE f2192_in(x148) -> f2193_in(x148) :|: TRUE f2194_in(T462) -> f2296_in(T462) :|: TRUE f2296_out(x149) -> f2194_out(x149) :|: TRUE f2296_in(x150) -> f2299_in(x150) :|: TRUE f2299_out(x151) -> f2296_out(x151) :|: TRUE f2299_in(x152) -> f2300_in(x152) :|: TRUE f2300_out(x153) -> f2299_out(x153) :|: TRUE f2299_in(x154) -> f2301_in(x154) :|: TRUE f2301_out(x155) -> f2299_out(x155) :|: TRUE f2303_out(x156) -> f2300_out(zero(x156)) :|: TRUE f2304_out -> f2300_out(x157) :|: TRUE f2300_in(zero(x158)) -> f2303_in(x158) :|: TRUE f2300_in(x159) -> f2304_in :|: TRUE f2623_out(x160) -> f2301_out(x160) :|: TRUE f2301_in(x161) -> f2623_in(x161) :|: TRUE f2623_in(x162) -> f2625_in(x162) :|: TRUE f2624_out(x163) -> f2623_out(x163) :|: TRUE f2623_in(x164) -> f2624_in(x164) :|: TRUE f2625_out(x165) -> f2623_out(x165) :|: TRUE f2624_in(one(T838)) -> f2628_in(T838) :|: TRUE f2628_out(x166) -> f2624_out(one(x166)) :|: TRUE f2624_in(x167) -> f2629_in :|: TRUE f2629_out -> f2624_out(x168) :|: TRUE f2404_out(x169) -> f2628_out(x169) :|: TRUE f2628_in(x170) -> f2404_in(x170) :|: TRUE Start term: f3_in(T3) ---------------------------------------- (202) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (203) TRUE ---------------------------------------- (204) Obligation: Rules: f1274_in -> f1276_in :|: TRUE f1276_out -> f1274_out :|: TRUE f1274_in -> f1275_in :|: TRUE f1275_out -> f1274_out :|: TRUE f1252_out -> f1133_out :|: TRUE f1255_out -> f1133_out :|: TRUE f1133_in -> f1255_in :|: TRUE f1133_in -> f1252_in :|: TRUE f1129_in -> f1135_in :|: TRUE f1133_out -> f1129_out :|: TRUE f1129_in -> f1133_in :|: TRUE f1135_out -> f1129_out :|: TRUE f1266_in -> f1274_in :|: TRUE f1274_out -> f1266_out :|: TRUE f1276_in -> f1281_in :|: TRUE f1276_in -> f1280_in :|: TRUE f1281_out -> f1276_out :|: TRUE f1280_out -> f1276_out :|: TRUE f1122_out -> f1252_out :|: TRUE f1252_in -> f1122_in :|: TRUE f1273_out -> f1135_out :|: TRUE f1135_in -> f1273_in :|: TRUE f1135_in -> f1266_in :|: TRUE f1266_out -> f1135_out :|: TRUE f1281_in -> f1306_in :|: TRUE f1308_out -> f1281_out :|: TRUE f1281_in -> f1308_in :|: TRUE f1306_out -> f1281_out :|: TRUE f1280_in -> f1288_in :|: TRUE f1280_in -> f1286_in :|: TRUE f1286_out -> f1280_out :|: TRUE f1288_out -> f1280_out :|: TRUE f1122_out -> f1286_out :|: TRUE f1286_in -> f1122_in :|: TRUE f1306_in -> f1266_in :|: TRUE f1266_out -> f1306_out :|: TRUE f1129_out -> f1122_out :|: TRUE f1122_in -> f1129_in :|: TRUE f3_in(T3) -> f13_in(T3) :|: TRUE f13_out(x) -> f3_out(x) :|: TRUE f13_in(x1) -> f15_in(x1) :|: TRUE f15_out(x2) -> f13_out(x2) :|: TRUE f13_in(x3) -> f14_in(x3) :|: TRUE f14_out(x4) -> f13_out(x4) :|: TRUE f15_in(x5) -> f22_in(x5) :|: TRUE f15_in(x6) -> f21_in(x6) :|: TRUE f21_out(x7) -> f15_out(x7) :|: TRUE f22_out(x8) -> f15_out(x8) :|: TRUE f40_out(T36) -> f22_out(T36) :|: TRUE f22_in(x9) -> f41_in :|: TRUE f22_in(x10) -> f40_in(x10) :|: TRUE f41_out -> f22_out(x11) :|: TRUE f122_out(x12) -> f40_out(x12) :|: TRUE f40_in(x13) -> f121_in :|: TRUE f121_out -> f122_in(x14) :|: TRUE f123_out -> f121_out :|: TRUE f121_in -> f123_in :|: TRUE f123_in -> f124_in :|: TRUE f124_out -> f123_out :|: TRUE f123_in -> f125_in :|: TRUE f125_out -> f123_out :|: TRUE f125_in -> f129_in :|: TRUE f129_out -> f125_out :|: TRUE f125_in -> f131_in :|: TRUE f131_out -> f125_out :|: TRUE f131_in -> f1055_in :|: TRUE f1056_out -> f131_out :|: TRUE f1055_out -> f131_out :|: TRUE f131_in -> f1056_in :|: TRUE f1057_out -> f1058_in :|: TRUE f1055_in -> f1057_in :|: TRUE f1058_out -> f1055_out :|: TRUE f1065_out -> f1058_out :|: TRUE f1058_in -> f1065_in :|: TRUE f1065_in -> f1067_in :|: TRUE f1067_out -> f1065_out :|: TRUE f1071_out -> f1067_out :|: TRUE f1067_in -> f1071_in :|: TRUE f1071_in -> f1077_in :|: TRUE f1079_out -> f1071_out :|: TRUE f1077_out -> f1071_out :|: TRUE f1071_in -> f1079_in :|: TRUE f1079_in -> f1357_in :|: TRUE f1357_out -> f1079_out :|: TRUE f1357_in -> f1595_in :|: TRUE f1595_out -> f1357_out :|: TRUE f1595_in -> f1598_in :|: TRUE f1595_in -> f1599_in :|: TRUE f1598_out -> f1595_out :|: TRUE f1599_out -> f1595_out :|: TRUE f1602_out -> f1598_out :|: TRUE f1598_in -> f1602_in :|: TRUE f1598_in -> f1603_in :|: TRUE f1603_out -> f1598_out :|: TRUE f1604_out -> f1602_out :|: TRUE f1602_in -> f1604_in :|: TRUE f1604_in -> f1606_in :|: TRUE f1604_in -> f1605_in :|: TRUE f1606_out -> f1604_out :|: TRUE f1605_out -> f1604_out :|: TRUE f1619_out -> f1606_out :|: TRUE f1620_out -> f1606_out :|: TRUE f1606_in -> f1620_in :|: TRUE f1606_in -> f1619_in :|: TRUE f1666_out -> f1620_out :|: TRUE f1620_in -> f1667_in :|: TRUE f1620_in -> f1666_in :|: TRUE f1667_out -> f1620_out :|: TRUE f1667_in -> f1691_in :|: TRUE f1691_out -> f1667_out :|: TRUE f1690_out -> f1667_out :|: TRUE f1667_in -> f1690_in :|: TRUE f1694_out -> f1690_out :|: TRUE f1690_in -> f1694_in :|: TRUE f1696_out -> f1694_out :|: TRUE f1695_out -> f1694_out :|: TRUE f1694_in -> f1695_in :|: TRUE f1694_in -> f1696_in :|: TRUE f1855_out -> f1696_out :|: TRUE f1854_out -> f1696_out :|: TRUE f1696_in -> f1855_in :|: TRUE f1696_in -> f1854_in :|: TRUE f1855_in -> f2044_in :|: TRUE f2044_out -> f1855_out :|: TRUE f2043_out -> f1855_out :|: TRUE f1855_in -> f2043_in :|: TRUE f2044_in -> f2061_in :|: TRUE f2061_out -> f2044_out :|: TRUE f2061_in -> f2063_in :|: TRUE f2063_out -> f2061_out :|: TRUE f2063_in -> f2095_in :|: TRUE f2063_in -> f2096_in :|: TRUE f2095_out -> f2063_out :|: TRUE f2096_out -> f2063_out :|: TRUE f2096_in -> f2109_in :|: TRUE f2110_out -> f2096_out :|: TRUE f2096_in -> f2110_in :|: TRUE f2109_out -> f2096_out :|: TRUE f2133_out -> f2110_out :|: TRUE f2110_in -> f2132_in :|: TRUE f2110_in -> f2133_in :|: TRUE f2132_out -> f2110_out :|: TRUE f2132_in -> f2134_in :|: TRUE f2135_out -> f2132_out :|: TRUE f2132_in -> f2135_in :|: TRUE f2134_out -> f2132_out :|: TRUE f2136_out -> f2134_out :|: TRUE f2134_in -> f2136_in :|: TRUE f2136_in -> f2138_in :|: TRUE f2136_in -> f2137_in :|: TRUE f2137_out -> f2136_out :|: TRUE f2138_out -> f2136_out :|: TRUE f2141_out -> f2138_out :|: TRUE f2142_out -> f2138_out :|: TRUE f2138_in -> f2142_in :|: TRUE f2138_in -> f2141_in :|: TRUE f2141_in -> f2143_in :|: TRUE f2144_out -> f2141_out :|: TRUE f2141_in -> f2144_in :|: TRUE f2143_out -> f2141_out :|: TRUE f1872_out -> f2143_out :|: TRUE f2143_in -> f1872_in :|: TRUE f1874_out -> f1872_out :|: TRUE f1872_in -> f1874_in :|: TRUE f1875_out -> f1874_out :|: TRUE f1874_in -> f1876_in :|: TRUE f1876_out -> f1874_out :|: TRUE f1874_in -> f1875_in :|: TRUE f1876_in -> f1882_in :|: TRUE f1876_in -> f1883_in :|: TRUE f1883_out -> f1876_out :|: TRUE f1882_out -> f1876_out :|: TRUE f1882_in -> f2034_in :|: TRUE f1882_in -> f2035_in :|: TRUE f2035_out -> f1882_out :|: TRUE f2034_out -> f1882_out :|: TRUE f2034_in -> f1122_in :|: TRUE f1122_out -> f2034_out :|: TRUE f1619_in -> f1626_in :|: TRUE f1619_in -> f1624_in :|: TRUE f1624_out -> f1619_out :|: TRUE f1626_out -> f1619_out :|: TRUE f1627_out -> f1624_out :|: TRUE f1624_in -> f1627_in :|: TRUE f1628_out -> f1627_out :|: TRUE f1629_out -> f1627_out :|: TRUE f1627_in -> f1629_in :|: TRUE f1627_in -> f1628_in :|: TRUE f1632_out -> f1628_out :|: TRUE f1628_in -> f1632_in :|: TRUE f1628_in -> f1633_in :|: TRUE f1633_out -> f1628_out :|: TRUE f1632_in -> f1266_in :|: TRUE f1266_out -> f1632_out :|: TRUE f2053_out -> f2043_out :|: TRUE f2043_in -> f2054_in :|: TRUE f2043_in -> f2053_in :|: TRUE f2054_out -> f2043_out :|: TRUE f2053_in -> f1857_in :|: TRUE f1857_out -> f2053_out :|: TRUE f1857_in -> f1859_in :|: TRUE f1859_out -> f1857_out :|: TRUE f1859_in -> f1861_in :|: TRUE f1860_out -> f1859_out :|: TRUE f1859_in -> f1860_in :|: TRUE f1861_out -> f1859_out :|: TRUE f1861_in -> f1873_in :|: TRUE f1872_out -> f1861_out :|: TRUE f1873_out -> f1861_out :|: TRUE f1861_in -> f1872_in :|: TRUE f1669_out -> f1666_out :|: TRUE f1666_in -> f1669_in :|: TRUE f1668_out -> f1666_out :|: TRUE f1666_in -> f1668_in :|: TRUE f1670_out -> f1668_out :|: TRUE f1668_in -> f1670_in :|: TRUE f1671_out -> f1670_out :|: TRUE f1670_in -> f1671_in :|: TRUE f1670_in -> f1672_in :|: TRUE f1672_out -> f1670_out :|: TRUE f1674_out -> f1671_out :|: TRUE f1671_in -> f1674_in :|: TRUE f1671_in -> f1673_in :|: TRUE f1673_out -> f1671_out :|: TRUE f1673_in -> f1266_in :|: TRUE f1266_out -> f1673_out :|: TRUE f1672_in -> f1676_in :|: TRUE f1672_in -> f1675_in :|: TRUE f1675_out -> f1672_out :|: TRUE f1676_out -> f1672_out :|: TRUE f1675_in -> f1678_in :|: TRUE f1677_out -> f1675_out :|: TRUE f1678_out -> f1675_out :|: TRUE f1675_in -> f1677_in :|: TRUE f1122_out -> f1677_out :|: TRUE f1677_in -> f1122_in :|: TRUE f1599_in -> f2164_in :|: TRUE f2164_out -> f1599_out :|: TRUE f2164_in -> f2165_in :|: TRUE f2165_out -> f2164_out :|: TRUE f2164_in -> f2166_in :|: TRUE f2166_out -> f2164_out :|: TRUE f2168_out -> f2165_out :|: TRUE f2165_in -> f2167_in :|: TRUE f2167_out -> f2165_out :|: TRUE f2165_in -> f2168_in :|: TRUE f1668_out -> f2167_out :|: TRUE f2167_in -> f1668_in :|: TRUE f1679_out -> f1676_out :|: TRUE f1676_in -> f1679_in :|: TRUE f1679_in -> f1602_in :|: TRUE f1602_out -> f1679_out :|: TRUE f1858_out -> f1854_out :|: TRUE f1854_in -> f1858_in :|: TRUE f1857_out -> f1854_out :|: TRUE f1854_in -> f1857_in :|: TRUE f1860_in -> f1863_in :|: TRUE f1863_out -> f1860_out :|: TRUE f1860_in -> f1862_in :|: TRUE f1862_out -> f1860_out :|: TRUE f1122_out -> f1862_out :|: TRUE f1862_in -> f1122_in :|: TRUE f1635_out -> f1629_out :|: TRUE f1629_in -> f1636_in :|: TRUE f1636_out -> f1629_out :|: TRUE f1629_in -> f1635_in :|: TRUE f1657_out -> f1635_out :|: TRUE f1635_in -> f1658_in :|: TRUE f1658_out -> f1635_out :|: TRUE f1635_in -> f1657_in :|: TRUE f1122_out -> f1657_out :|: TRUE f1657_in -> f1122_in :|: TRUE f2137_in -> f2139_in :|: TRUE f2139_out -> f2137_out :|: TRUE f2137_in -> f2140_in :|: TRUE f2140_out -> f2137_out :|: TRUE f2139_in -> f1122_in :|: TRUE f1122_out -> f2139_out :|: TRUE f2109_in -> f2113_in :|: TRUE f2112_out -> f2109_out :|: TRUE f2113_out -> f2109_out :|: TRUE f2109_in -> f2112_in :|: TRUE f2114_out -> f2112_out :|: TRUE f2112_in -> f2114_in :|: TRUE f2116_out -> f2114_out :|: TRUE f2115_out -> f2114_out :|: TRUE f2114_in -> f2116_in :|: TRUE f2114_in -> f2115_in :|: TRUE f2120_out -> f2116_out :|: TRUE f2116_in -> f2120_in :|: TRUE f2119_out -> f2116_out :|: TRUE f2116_in -> f2119_in :|: TRUE f2121_out -> f2119_out :|: TRUE f2119_in -> f2121_in :|: TRUE f2122_out -> f2119_out :|: TRUE f2119_in -> f2122_in :|: TRUE f2121_in -> f1872_in :|: TRUE f1872_out -> f2121_out :|: TRUE f1077_in -> f1089_in :|: TRUE f1091_out -> f1077_out :|: TRUE f1077_in -> f1091_in :|: TRUE f1089_out -> f1077_out :|: TRUE f1099_out -> f1089_out :|: TRUE f1089_in -> f1099_in :|: TRUE f1099_in -> f1110_in :|: TRUE f1110_out -> f1099_out :|: TRUE f1109_out -> f1099_out :|: TRUE f1099_in -> f1109_in :|: TRUE f1109_in -> f1122_in :|: TRUE f1122_out -> f1109_out :|: TRUE f2117_out -> f2115_out :|: TRUE f2115_in -> f2118_in :|: TRUE f2118_out -> f2115_out :|: TRUE f2115_in -> f2117_in :|: TRUE f2117_in -> f1122_in :|: TRUE f1122_out -> f2117_out :|: TRUE Start term: f3_in(T3) ---------------------------------------- (205) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (206) TRUE ---------------------------------------- (207) Obligation: Rules: f1875_out -> f1874_out :|: TRUE f1874_in -> f1876_in :|: TRUE f1876_out -> f1874_out :|: TRUE f1874_in -> f1875_in :|: TRUE f1876_in -> f1882_in :|: TRUE f1876_in -> f1883_in :|: TRUE f1883_out -> f1876_out :|: TRUE f1882_out -> f1876_out :|: TRUE f1883_in -> f2042_in :|: TRUE f2042_out -> f1883_out :|: TRUE f2041_out -> f1883_out :|: TRUE f1883_in -> f2041_in :|: TRUE f1874_out -> f1872_out :|: TRUE f1872_in -> f1874_in :|: TRUE f1872_out -> f2041_out :|: TRUE f2041_in -> f1872_in :|: TRUE f3_in(T3) -> f13_in(T3) :|: TRUE f13_out(x) -> f3_out(x) :|: TRUE f13_in(x1) -> f15_in(x1) :|: TRUE f15_out(x2) -> f13_out(x2) :|: TRUE f13_in(x3) -> f14_in(x3) :|: TRUE f14_out(x4) -> f13_out(x4) :|: TRUE f15_in(x5) -> f22_in(x5) :|: TRUE f15_in(x6) -> f21_in(x6) :|: TRUE f21_out(x7) -> f15_out(x7) :|: TRUE f22_out(x8) -> f15_out(x8) :|: TRUE f40_out(T36) -> f22_out(T36) :|: TRUE f22_in(x9) -> f41_in :|: TRUE f22_in(x10) -> f40_in(x10) :|: TRUE f41_out -> f22_out(x11) :|: TRUE f122_out(x12) -> f40_out(x12) :|: TRUE f40_in(x13) -> f121_in :|: TRUE f121_out -> f122_in(x14) :|: TRUE f123_out -> f121_out :|: TRUE f121_in -> f123_in :|: TRUE f123_in -> f124_in :|: TRUE f124_out -> f123_out :|: TRUE f123_in -> f125_in :|: TRUE f125_out -> f123_out :|: TRUE f125_in -> f129_in :|: TRUE f129_out -> f125_out :|: TRUE f125_in -> f131_in :|: TRUE f131_out -> f125_out :|: TRUE f131_in -> f1055_in :|: TRUE f1056_out -> f131_out :|: TRUE f1055_out -> f131_out :|: TRUE f131_in -> f1056_in :|: TRUE f1057_out -> f1058_in :|: TRUE f1055_in -> f1057_in :|: TRUE f1058_out -> f1055_out :|: TRUE f1065_out -> f1058_out :|: TRUE f1058_in -> f1065_in :|: TRUE f1065_in -> f1067_in :|: TRUE f1067_out -> f1065_out :|: TRUE f1071_out -> f1067_out :|: TRUE f1067_in -> f1071_in :|: TRUE f1071_in -> f1077_in :|: TRUE f1079_out -> f1071_out :|: TRUE f1077_out -> f1071_out :|: TRUE f1071_in -> f1079_in :|: TRUE f1079_in -> f1357_in :|: TRUE f1357_out -> f1079_out :|: TRUE f1357_in -> f1595_in :|: TRUE f1595_out -> f1357_out :|: TRUE f1595_in -> f1598_in :|: TRUE f1595_in -> f1599_in :|: TRUE f1598_out -> f1595_out :|: TRUE f1599_out -> f1595_out :|: TRUE f1602_out -> f1598_out :|: TRUE f1598_in -> f1602_in :|: TRUE f1598_in -> f1603_in :|: TRUE f1603_out -> f1598_out :|: TRUE f1604_out -> f1602_out :|: TRUE f1602_in -> f1604_in :|: TRUE f1604_in -> f1606_in :|: TRUE f1604_in -> f1605_in :|: TRUE f1606_out -> f1604_out :|: TRUE f1605_out -> f1604_out :|: TRUE f1619_out -> f1606_out :|: TRUE f1620_out -> f1606_out :|: TRUE f1606_in -> f1620_in :|: TRUE f1606_in -> f1619_in :|: TRUE f1666_out -> f1620_out :|: TRUE f1620_in -> f1667_in :|: TRUE f1620_in -> f1666_in :|: TRUE f1667_out -> f1620_out :|: TRUE f1667_in -> f1691_in :|: TRUE f1691_out -> f1667_out :|: TRUE f1690_out -> f1667_out :|: TRUE f1667_in -> f1690_in :|: TRUE f1694_out -> f1690_out :|: TRUE f1690_in -> f1694_in :|: TRUE f1696_out -> f1694_out :|: TRUE f1695_out -> f1694_out :|: TRUE f1694_in -> f1695_in :|: TRUE f1694_in -> f1696_in :|: TRUE f1855_out -> f1696_out :|: TRUE f1854_out -> f1696_out :|: TRUE f1696_in -> f1855_in :|: TRUE f1696_in -> f1854_in :|: TRUE f1855_in -> f2044_in :|: TRUE f2044_out -> f1855_out :|: TRUE f2043_out -> f1855_out :|: TRUE f1855_in -> f2043_in :|: TRUE f2044_in -> f2061_in :|: TRUE f2061_out -> f2044_out :|: TRUE f2061_in -> f2063_in :|: TRUE f2063_out -> f2061_out :|: TRUE f2063_in -> f2095_in :|: TRUE f2063_in -> f2096_in :|: TRUE f2095_out -> f2063_out :|: TRUE f2096_out -> f2063_out :|: TRUE f2096_in -> f2109_in :|: TRUE f2110_out -> f2096_out :|: TRUE f2096_in -> f2110_in :|: TRUE f2109_out -> f2096_out :|: TRUE f2109_in -> f2113_in :|: TRUE f2112_out -> f2109_out :|: TRUE f2113_out -> f2109_out :|: TRUE f2109_in -> f2112_in :|: TRUE f2114_out -> f2112_out :|: TRUE f2112_in -> f2114_in :|: TRUE f2116_out -> f2114_out :|: TRUE f2115_out -> f2114_out :|: TRUE f2114_in -> f2116_in :|: TRUE f2114_in -> f2115_in :|: TRUE f2120_out -> f2116_out :|: TRUE f2116_in -> f2120_in :|: TRUE f2119_out -> f2116_out :|: TRUE f2116_in -> f2119_in :|: TRUE f2121_out -> f2119_out :|: TRUE f2119_in -> f2121_in :|: TRUE f2122_out -> f2119_out :|: TRUE f2119_in -> f2122_in :|: TRUE f2121_in -> f1872_in :|: TRUE f1872_out -> f2121_out :|: TRUE f2133_out -> f2110_out :|: TRUE f2110_in -> f2132_in :|: TRUE f2110_in -> f2133_in :|: TRUE f2132_out -> f2110_out :|: TRUE f2132_in -> f2134_in :|: TRUE f2135_out -> f2132_out :|: TRUE f2132_in -> f2135_in :|: TRUE f2134_out -> f2132_out :|: TRUE f2136_out -> f2134_out :|: TRUE f2134_in -> f2136_in :|: TRUE f2136_in -> f2138_in :|: TRUE f2136_in -> f2137_in :|: TRUE f2137_out -> f2136_out :|: TRUE f2138_out -> f2136_out :|: TRUE f2141_out -> f2138_out :|: TRUE f2142_out -> f2138_out :|: TRUE f2138_in -> f2142_in :|: TRUE f2138_in -> f2141_in :|: TRUE f2141_in -> f2143_in :|: TRUE f2144_out -> f2141_out :|: TRUE f2141_in -> f2144_in :|: TRUE f2143_out -> f2141_out :|: TRUE f1872_out -> f2143_out :|: TRUE f2143_in -> f1872_in :|: TRUE f1599_in -> f2164_in :|: TRUE f2164_out -> f1599_out :|: TRUE f2164_in -> f2165_in :|: TRUE f2165_out -> f2164_out :|: TRUE f2164_in -> f2166_in :|: TRUE f2166_out -> f2164_out :|: TRUE f2168_out -> f2165_out :|: TRUE f2165_in -> f2167_in :|: TRUE f2167_out -> f2165_out :|: TRUE f2165_in -> f2168_in :|: TRUE f1668_out -> f2167_out :|: TRUE f2167_in -> f1668_in :|: TRUE f1670_out -> f1668_out :|: TRUE f1668_in -> f1670_in :|: TRUE f1671_out -> f1670_out :|: TRUE f1670_in -> f1671_in :|: TRUE f1670_in -> f1672_in :|: TRUE f1672_out -> f1670_out :|: TRUE f1672_in -> f1676_in :|: TRUE f1672_in -> f1675_in :|: TRUE f1675_out -> f1672_out :|: TRUE f1676_out -> f1672_out :|: TRUE f1679_out -> f1676_out :|: TRUE f1676_in -> f1679_in :|: TRUE f1679_in -> f1602_in :|: TRUE f1602_out -> f1679_out :|: TRUE f2053_out -> f2043_out :|: TRUE f2043_in -> f2054_in :|: TRUE f2043_in -> f2053_in :|: TRUE f2054_out -> f2043_out :|: TRUE f2053_in -> f1857_in :|: TRUE f1857_out -> f2053_out :|: TRUE f1857_in -> f1859_in :|: TRUE f1859_out -> f1857_out :|: TRUE f1859_in -> f1861_in :|: TRUE f1860_out -> f1859_out :|: TRUE f1859_in -> f1860_in :|: TRUE f1861_out -> f1859_out :|: TRUE f1861_in -> f1873_in :|: TRUE f1872_out -> f1861_out :|: TRUE f1873_out -> f1861_out :|: TRUE f1861_in -> f1872_in :|: TRUE f1858_out -> f1854_out :|: TRUE f1854_in -> f1858_in :|: TRUE f1857_out -> f1854_out :|: TRUE f1854_in -> f1857_in :|: TRUE Start term: f3_in(T3) ---------------------------------------- (208) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (209) TRUE ---------------------------------------- (210) Obligation: Rules: f1666_out -> f1620_out :|: TRUE f1620_in -> f1667_in :|: TRUE f1620_in -> f1666_in :|: TRUE f1667_out -> f1620_out :|: TRUE f1602_out -> f2099_out :|: TRUE f2099_in -> f1602_in :|: TRUE f1627_out -> f1624_out :|: TRUE f1624_in -> f1627_in :|: TRUE f1855_in -> f2044_in :|: TRUE f2044_out -> f1855_out :|: TRUE f2043_out -> f1855_out :|: TRUE f1855_in -> f2043_in :|: TRUE f2109_in -> f2113_in :|: TRUE f2112_out -> f2109_out :|: TRUE f2113_out -> f2109_out :|: TRUE f2109_in -> f2112_in :|: TRUE f2133_out -> f2110_out :|: TRUE f2110_in -> f2132_in :|: TRUE f2110_in -> f2133_in :|: TRUE f2132_out -> f2110_out :|: TRUE f2096_in -> f2109_in :|: TRUE f2110_out -> f2096_out :|: TRUE f2096_in -> f2110_in :|: TRUE f2109_out -> f2096_out :|: TRUE f1669_out -> f1666_out :|: TRUE f1666_in -> f1669_in :|: TRUE f1668_out -> f1666_out :|: TRUE f1666_in -> f1668_in :|: TRUE f2136_out -> f2134_out :|: TRUE f2134_in -> f2136_in :|: TRUE f2142_in -> f2161_in :|: TRUE f2161_out -> f2142_out :|: TRUE f2063_in -> f2095_in :|: TRUE f2063_in -> f2096_in :|: TRUE f2095_out -> f2063_out :|: TRUE f2096_out -> f2063_out :|: TRUE f2133_in -> f2163_in :|: TRUE f2162_out -> f2133_out :|: TRUE f2133_in -> f2162_in :|: TRUE f2163_out -> f2133_out :|: TRUE f2162_in -> f1690_in :|: TRUE f1690_out -> f2162_out :|: TRUE f1604_in -> f1606_in :|: TRUE f1604_in -> f1605_in :|: TRUE f1606_out -> f1604_out :|: TRUE f1605_out -> f1604_out :|: TRUE f2136_in -> f2138_in :|: TRUE f2136_in -> f2137_in :|: TRUE f2137_out -> f2136_out :|: TRUE f2138_out -> f2136_out :|: TRUE f2120_in -> f2123_in :|: TRUE f2123_out -> f2120_out :|: TRUE f2123_in -> f2061_in :|: TRUE f2061_out -> f2123_out :|: TRUE f2141_out -> f2138_out :|: TRUE f2142_out -> f2138_out :|: TRUE f2138_in -> f2142_in :|: TRUE f2138_in -> f2141_in :|: TRUE f2061_in -> f2063_in :|: TRUE f2063_out -> f2061_out :|: TRUE f1604_out -> f1602_out :|: TRUE f1602_in -> f1604_in :|: TRUE f1619_out -> f1606_out :|: TRUE f1620_out -> f1606_out :|: TRUE f1606_in -> f1620_in :|: TRUE f1606_in -> f1619_in :|: TRUE f1602_out -> f1608_out :|: TRUE f1608_in -> f1602_in :|: TRUE f1636_in -> f1665_in :|: TRUE f1665_out -> f1636_out :|: TRUE f1602_out -> f1665_out :|: TRUE f1665_in -> f1602_in :|: TRUE f1609_out -> f1605_out :|: TRUE f1605_in -> f1608_in :|: TRUE f1605_in -> f1609_in :|: TRUE f1608_out -> f1605_out :|: TRUE f1619_in -> f1626_in :|: TRUE f1619_in -> f1624_in :|: TRUE f1624_out -> f1619_out :|: TRUE f1626_out -> f1619_out :|: TRUE f2044_in -> f2061_in :|: TRUE f2061_out -> f2044_out :|: TRUE f2116_out -> f2114_out :|: TRUE f2115_out -> f2114_out :|: TRUE f2114_in -> f2116_in :|: TRUE f2114_in -> f2115_in :|: TRUE f1671_out -> f1670_out :|: TRUE f1670_in -> f1671_in :|: TRUE f1670_in -> f1672_in :|: TRUE f1672_out -> f1670_out :|: TRUE f2120_out -> f2116_out :|: TRUE f2116_in -> f2120_in :|: TRUE f2119_out -> f2116_out :|: TRUE f2116_in -> f2119_in :|: TRUE f1628_out -> f1627_out :|: TRUE f1629_out -> f1627_out :|: TRUE f1627_in -> f1629_in :|: TRUE f1627_in -> f1628_in :|: TRUE f2100_out -> f2095_out :|: TRUE f2095_in -> f2100_in :|: TRUE f2095_in -> f2099_in :|: TRUE f2099_out -> f2095_out :|: TRUE f1667_in -> f1691_in :|: TRUE f1691_out -> f1667_out :|: TRUE f1690_out -> f1667_out :|: TRUE f1667_in -> f1690_in :|: TRUE f1635_out -> f1629_out :|: TRUE f1629_in -> f1636_in :|: TRUE f1636_out -> f1629_out :|: TRUE f1629_in -> f1635_in :|: TRUE f1672_in -> f1676_in :|: TRUE f1672_in -> f1675_in :|: TRUE f1675_out -> f1672_out :|: TRUE f1676_out -> f1672_out :|: TRUE f1679_in -> f1602_in :|: TRUE f1602_out -> f1679_out :|: TRUE f1855_out -> f1696_out :|: TRUE f1854_out -> f1696_out :|: TRUE f1696_in -> f1855_in :|: TRUE f1696_in -> f1854_in :|: TRUE f1679_out -> f1676_out :|: TRUE f1676_in -> f1679_in :|: TRUE f1696_out -> f1694_out :|: TRUE f1695_out -> f1694_out :|: TRUE f1694_in -> f1695_in :|: TRUE f1694_in -> f1696_in :|: TRUE f1694_out -> f1690_out :|: TRUE f1690_in -> f1694_in :|: TRUE f2161_in -> f2061_in :|: TRUE f2061_out -> f2161_out :|: TRUE f2114_out -> f2112_out :|: TRUE f2112_in -> f2114_in :|: TRUE f2132_in -> f2134_in :|: TRUE f2135_out -> f2132_out :|: TRUE f2132_in -> f2135_in :|: TRUE f2134_out -> f2132_out :|: TRUE f1670_out -> f1668_out :|: TRUE f1668_in -> f1670_in :|: TRUE f3_in(T3) -> f13_in(T3) :|: TRUE f13_out(x) -> f3_out(x) :|: TRUE f13_in(x1) -> f15_in(x1) :|: TRUE f15_out(x2) -> f13_out(x2) :|: TRUE f13_in(x3) -> f14_in(x3) :|: TRUE f14_out(x4) -> f13_out(x4) :|: TRUE f15_in(x5) -> f22_in(x5) :|: TRUE f15_in(x6) -> f21_in(x6) :|: TRUE f21_out(x7) -> f15_out(x7) :|: TRUE f22_out(x8) -> f15_out(x8) :|: TRUE f40_out(T36) -> f22_out(T36) :|: TRUE f22_in(x9) -> f41_in :|: TRUE f22_in(x10) -> f40_in(x10) :|: TRUE f41_out -> f22_out(x11) :|: TRUE f122_out(x12) -> f40_out(x12) :|: TRUE f40_in(x13) -> f121_in :|: TRUE f121_out -> f122_in(x14) :|: TRUE f123_out -> f121_out :|: TRUE f121_in -> f123_in :|: TRUE f123_in -> f124_in :|: TRUE f124_out -> f123_out :|: TRUE f123_in -> f125_in :|: TRUE f125_out -> f123_out :|: TRUE f125_in -> f129_in :|: TRUE f129_out -> f125_out :|: TRUE f125_in -> f131_in :|: TRUE f131_out -> f125_out :|: TRUE f131_in -> f1055_in :|: TRUE f1056_out -> f131_out :|: TRUE f1055_out -> f131_out :|: TRUE f131_in -> f1056_in :|: TRUE f1057_out -> f1058_in :|: TRUE f1055_in -> f1057_in :|: TRUE f1058_out -> f1055_out :|: TRUE f1065_out -> f1058_out :|: TRUE f1058_in -> f1065_in :|: TRUE f1065_in -> f1067_in :|: TRUE f1067_out -> f1065_out :|: TRUE f1071_out -> f1067_out :|: TRUE f1067_in -> f1071_in :|: TRUE f1071_in -> f1077_in :|: TRUE f1079_out -> f1071_out :|: TRUE f1077_out -> f1071_out :|: TRUE f1071_in -> f1079_in :|: TRUE f1079_in -> f1357_in :|: TRUE f1357_out -> f1079_out :|: TRUE f1357_in -> f1595_in :|: TRUE f1595_out -> f1357_out :|: TRUE f1595_in -> f1598_in :|: TRUE f1595_in -> f1599_in :|: TRUE f1598_out -> f1595_out :|: TRUE f1599_out -> f1595_out :|: TRUE f1599_in -> f2164_in :|: TRUE f2164_out -> f1599_out :|: TRUE f2164_in -> f2165_in :|: TRUE f2165_out -> f2164_out :|: TRUE f2164_in -> f2166_in :|: TRUE f2166_out -> f2164_out :|: TRUE f2168_out -> f2165_out :|: TRUE f2165_in -> f2167_in :|: TRUE f2167_out -> f2165_out :|: TRUE f2165_in -> f2168_in :|: TRUE f1668_out -> f2167_out :|: TRUE f2167_in -> f1668_in :|: TRUE f1602_out -> f1598_out :|: TRUE f1598_in -> f1602_in :|: TRUE f1598_in -> f1603_in :|: TRUE f1603_out -> f1598_out :|: TRUE Start term: f3_in(T3) ---------------------------------------- (211) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (212) TRUE ---------------------------------------- (213) Obligation: Rules: f1602_out -> f2099_out :|: TRUE f2099_in -> f1602_in :|: TRUE f1627_out -> f1624_out :|: TRUE f1624_in -> f1627_in :|: TRUE f2137_in -> f2139_in :|: TRUE f2139_out -> f2137_out :|: TRUE f2137_in -> f2140_in :|: TRUE f2140_out -> f2137_out :|: TRUE f2096_in -> f2109_in :|: TRUE f2110_out -> f2096_out :|: TRUE f2096_in -> f2110_in :|: TRUE f2109_out -> f2096_out :|: TRUE f1668_out -> f2167_out :|: TRUE f2167_in -> f1668_in :|: TRUE f1252_out -> f1133_out :|: TRUE f1255_out -> f1133_out :|: TRUE f1133_in -> f1255_in :|: TRUE f1133_in -> f1252_in :|: TRUE f1697_in -> f1697_out :|: TRUE f121_out -> f201_out :|: TRUE f201_in -> f121_in :|: TRUE f2063_in -> f2095_in :|: TRUE f2063_in -> f2096_in :|: TRUE f2095_out -> f2063_out :|: TRUE f2096_out -> f2063_out :|: TRUE f2139_in -> f1122_in :|: TRUE f1122_out -> f2139_out :|: TRUE f1604_in -> f1606_in :|: TRUE f1604_in -> f1605_in :|: TRUE f1606_out -> f1604_out :|: TRUE f1605_out -> f1604_out :|: TRUE f2034_in -> f1122_in :|: TRUE f1122_out -> f2034_out :|: TRUE f2120_in -> f2123_in :|: TRUE f2123_out -> f2120_out :|: TRUE f1632_out -> f1628_out :|: TRUE f1628_in -> f1632_in :|: TRUE f1628_in -> f1633_in :|: TRUE f1633_out -> f1628_out :|: TRUE f1619_out -> f1606_out :|: TRUE f1620_out -> f1606_out :|: TRUE f1606_in -> f1620_in :|: TRUE f1606_in -> f1619_in :|: TRUE f1099_in -> f1110_in :|: TRUE f1110_out -> f1099_out :|: TRUE f1109_out -> f1099_out :|: TRUE f1099_in -> f1109_in :|: TRUE f1602_out -> f1665_out :|: TRUE f1665_in -> f1602_in :|: TRUE f121_out -> f1057_out :|: TRUE f1057_in -> f121_in :|: TRUE f2044_in -> f2061_in :|: TRUE f2061_out -> f2044_out :|: TRUE f2117_in -> f1122_in :|: TRUE f1122_out -> f2117_out :|: TRUE f1632_in -> f1266_in :|: TRUE f1266_out -> f1632_out :|: TRUE f125_in -> f129_in :|: TRUE f129_out -> f125_out :|: TRUE f125_in -> f131_in :|: TRUE f131_out -> f125_out :|: TRUE f2053_in -> f1857_in :|: TRUE f1857_out -> f2053_out :|: TRUE f1860_in -> f1863_in :|: TRUE f1863_out -> f1860_out :|: TRUE f1860_in -> f1862_in :|: TRUE f1862_out -> f1860_out :|: TRUE f1859_in -> f1861_in :|: TRUE f1860_out -> f1859_out :|: TRUE f1859_in -> f1860_in :|: TRUE f1861_out -> f1859_out :|: TRUE f1065_in -> f1067_in :|: TRUE f1067_out -> f1065_out :|: TRUE f2100_out -> f2095_out :|: TRUE f2095_in -> f2100_in :|: TRUE f2095_in -> f2099_in :|: TRUE f2099_out -> f2095_out :|: TRUE f1667_in -> f1691_in :|: TRUE f1691_out -> f1667_out :|: TRUE f1690_out -> f1667_out :|: TRUE f1667_in -> f1690_in :|: TRUE f2169_out -> f2166_out :|: TRUE f2166_in -> f2169_in :|: TRUE f1872_out -> f2143_out :|: TRUE f2143_in -> f1872_in :|: TRUE f1855_out -> f1696_out :|: TRUE f1854_out -> f1696_out :|: TRUE f1696_in -> f1855_in :|: TRUE f1696_in -> f1854_in :|: TRUE f1857_in -> f1859_in :|: TRUE f1859_out -> f1857_out :|: TRUE f1599_in -> f2164_in :|: TRUE f2164_out -> f1599_out :|: TRUE f1129_out -> f1122_out :|: TRUE f1122_in -> f1129_in :|: TRUE f1666_out -> f1620_out :|: TRUE f1620_in -> f1667_in :|: TRUE f1620_in -> f1666_in :|: TRUE f1667_out -> f1620_out :|: TRUE f1855_in -> f2044_in :|: TRUE f2044_out -> f1855_out :|: TRUE f2043_out -> f1855_out :|: TRUE f1855_in -> f2043_in :|: TRUE f2109_in -> f2113_in :|: TRUE f2112_out -> f2109_out :|: TRUE f2113_out -> f2109_out :|: TRUE f2109_in -> f2112_in :|: TRUE f201_out -> f129_out :|: TRUE f129_in -> f201_in :|: TRUE f208_out -> f129_out :|: TRUE f129_in -> f208_in :|: TRUE f2133_in -> f2163_in :|: TRUE f2162_out -> f2133_out :|: TRUE f2133_in -> f2162_in :|: TRUE f2163_out -> f2133_out :|: TRUE f1266_in -> f1274_in :|: TRUE f1274_out -> f1266_out :|: TRUE f1065_out -> f1058_out :|: TRUE f1058_in -> f1065_in :|: TRUE f1277_out -> f1275_out :|: TRUE f1278_out -> f1275_out :|: TRUE f1275_in -> f1278_in :|: TRUE f1275_in -> f1277_in :|: TRUE f1071_in -> f1077_in :|: TRUE f1079_out -> f1071_out :|: TRUE f1077_out -> f1071_out :|: TRUE f1071_in -> f1079_in :|: TRUE f1673_in -> f1266_in :|: TRUE f1266_out -> f1673_out :|: TRUE f1674_out -> f1671_out :|: TRUE f1671_in -> f1674_in :|: TRUE f1671_in -> f1673_in :|: TRUE f1673_out -> f1671_out :|: TRUE f2141_out -> f2138_out :|: TRUE f2142_out -> f2138_out :|: TRUE f2138_in -> f2142_in :|: TRUE f2138_in -> f2141_in :|: TRUE f1604_out -> f1602_out :|: TRUE f1602_in -> f1604_in :|: TRUE f2117_out -> f2115_out :|: TRUE f2115_in -> f2118_in :|: TRUE f2118_out -> f2115_out :|: TRUE f2115_in -> f2117_in :|: TRUE f1872_out -> f2041_out :|: TRUE f2041_in -> f1872_in :|: TRUE f131_in -> f1055_in :|: TRUE f1056_out -> f131_out :|: TRUE f1055_out -> f131_out :|: TRUE f131_in -> f1056_in :|: TRUE f1883_in -> f2042_in :|: TRUE f2042_out -> f1883_out :|: TRUE f2041_out -> f1883_out :|: TRUE f1883_in -> f2041_in :|: TRUE f1628_out -> f1627_out :|: TRUE f1629_out -> f1627_out :|: TRUE f1627_in -> f1629_in :|: TRUE f1627_in -> f1628_in :|: TRUE f1874_out -> f1872_out :|: TRUE f1872_in -> f1874_in :|: TRUE f1635_out -> f1629_out :|: TRUE f1629_in -> f1636_in :|: TRUE f1636_out -> f1629_out :|: TRUE f1629_in -> f1635_in :|: TRUE f1679_in -> f1602_in :|: TRUE f1602_out -> f1679_out :|: TRUE f1122_out -> f1677_out :|: TRUE f1677_in -> f1122_in :|: TRUE f1595_in -> f1598_in :|: TRUE f1595_in -> f1599_in :|: TRUE f1598_out -> f1595_out :|: TRUE f1599_out -> f1595_out :|: TRUE f1079_in -> f1357_in :|: TRUE f1357_out -> f1079_out :|: TRUE f1694_out -> f1690_out :|: TRUE f1690_in -> f1694_in :|: TRUE f1882_in -> f2034_in :|: TRUE f1882_in -> f2035_in :|: TRUE f2035_out -> f1882_out :|: TRUE f2034_out -> f1882_out :|: TRUE f1077_in -> f1089_in :|: TRUE f1091_out -> f1077_out :|: TRUE f1077_in -> f1091_in :|: TRUE f1089_out -> f1077_out :|: TRUE f1670_out -> f1668_out :|: TRUE f1668_in -> f1670_in :|: TRUE f2133_out -> f2110_out :|: TRUE f2110_in -> f2132_in :|: TRUE f2110_in -> f2133_in :|: TRUE f2132_out -> f2110_out :|: TRUE f1675_in -> f1678_in :|: TRUE f1677_out -> f1675_out :|: TRUE f1678_out -> f1675_out :|: TRUE f1675_in -> f1677_in :|: TRUE f1669_out -> f1666_out :|: TRUE f1666_in -> f1669_in :|: TRUE f1668_out -> f1666_out :|: TRUE f1666_in -> f1668_in :|: TRUE f1357_in -> f1595_in :|: TRUE f1595_out -> f1357_out :|: TRUE f2141_in -> f2143_in :|: TRUE f2144_out -> f2141_out :|: TRUE f2141_in -> f2144_in :|: TRUE f2143_out -> f2141_out :|: TRUE f2162_in -> f1690_in :|: TRUE f1690_out -> f2162_out :|: TRUE f1657_out -> f1635_out :|: TRUE f1635_in -> f1658_in :|: TRUE f1658_out -> f1635_out :|: TRUE f1635_in -> f1657_in :|: TRUE f2136_in -> f2138_in :|: TRUE f2136_in -> f2137_in :|: TRUE f2137_out -> f2136_out :|: TRUE f2138_out -> f2136_out :|: TRUE f1280_in -> f1288_in :|: TRUE f1280_in -> f1286_in :|: TRUE f1286_out -> f1280_out :|: TRUE f1288_out -> f1280_out :|: TRUE f1122_out -> f1286_out :|: TRUE f1286_in -> f1122_in :|: TRUE f1602_out -> f1608_out :|: TRUE f1608_in -> f1602_in :|: TRUE f123_in -> f124_in :|: TRUE f124_out -> f123_out :|: TRUE f123_in -> f125_in :|: TRUE f125_out -> f123_out :|: TRUE f2168_out -> f2165_out :|: TRUE f2165_in -> f2167_in :|: TRUE f2167_out -> f2165_out :|: TRUE f2165_in -> f2168_in :|: TRUE f1619_in -> f1626_in :|: TRUE f1619_in -> f1624_in :|: TRUE f1624_out -> f1619_out :|: TRUE f1626_out -> f1619_out :|: TRUE f1876_in -> f1882_in :|: TRUE f1876_in -> f1883_in :|: TRUE f1883_out -> f1876_out :|: TRUE f1882_out -> f1876_out :|: TRUE f1099_out -> f1089_out :|: TRUE f1089_in -> f1099_in :|: TRUE f2116_out -> f2114_out :|: TRUE f2115_out -> f2114_out :|: TRUE f2114_in -> f2116_in :|: TRUE f2114_in -> f2115_in :|: TRUE f1057_out -> f1058_in :|: TRUE f1055_in -> f1057_in :|: TRUE f1058_out -> f1055_out :|: TRUE f1671_out -> f1670_out :|: TRUE f1670_in -> f1671_in :|: TRUE f1670_in -> f1672_in :|: TRUE f1672_out -> f1670_out :|: TRUE f1879_in -> f1879_out :|: TRUE f2120_out -> f2116_out :|: TRUE f2116_in -> f2120_in :|: TRUE f2119_out -> f2116_out :|: TRUE f2116_in -> f2119_in :|: TRUE f1129_in -> f1135_in :|: TRUE f1133_out -> f1129_out :|: TRUE f1129_in -> f1133_in :|: TRUE f1135_out -> f1129_out :|: TRUE f123_out -> f121_out :|: TRUE f121_in -> f123_in :|: TRUE f1672_in -> f1676_in :|: TRUE f1672_in -> f1675_in :|: TRUE f1675_out -> f1672_out :|: TRUE f1676_out -> f1672_out :|: TRUE f2164_in -> f2165_in :|: TRUE f2165_out -> f2164_out :|: TRUE f2164_in -> f2166_in :|: TRUE f2166_out -> f2164_out :|: TRUE f1696_out -> f1694_out :|: TRUE f1695_out -> f1694_out :|: TRUE f1694_in -> f1695_in :|: TRUE f1694_in -> f1696_in :|: TRUE f1281_in -> f1306_in :|: TRUE f1308_out -> f1281_out :|: TRUE f1281_in -> f1308_in :|: TRUE f1306_out -> f1281_out :|: TRUE f1122_out -> f1862_out :|: TRUE f1862_in -> f1122_in :|: TRUE f1109_in -> f1122_in :|: TRUE f1122_out -> f1109_out :|: TRUE f1861_in -> f1873_in :|: TRUE f1872_out -> f1861_out :|: TRUE f1873_out -> f1861_out :|: TRUE f1861_in -> f1872_in :|: TRUE f1697_out -> f1695_out :|: TRUE f1698_out -> f1695_out :|: TRUE f1695_in -> f1697_in :|: TRUE f1695_in -> f1698_in :|: TRUE f2053_out -> f2043_out :|: TRUE f2043_in -> f2054_in :|: TRUE f2043_in -> f2053_in :|: TRUE f2054_out -> f2043_out :|: TRUE f1274_in -> f1276_in :|: TRUE f1276_out -> f1274_out :|: TRUE f1274_in -> f1275_in :|: TRUE f1275_out -> f1274_out :|: TRUE f2136_out -> f2134_out :|: TRUE f2134_in -> f2136_in :|: TRUE f2142_in -> f2161_in :|: TRUE f2161_out -> f2142_out :|: TRUE f1875_in -> f1880_in :|: TRUE f1880_out -> f1875_out :|: TRUE f1879_out -> f1875_out :|: TRUE f1875_in -> f1879_in :|: TRUE f2123_in -> f2061_in :|: TRUE f2061_out -> f2123_out :|: TRUE f2061_in -> f2063_in :|: TRUE f2063_out -> f2061_out :|: TRUE f2121_out -> f2119_out :|: TRUE f2119_in -> f2121_in :|: TRUE f2122_out -> f2119_out :|: TRUE f2119_in -> f2122_in :|: TRUE f1110_in -> f1316_in :|: TRUE f1316_out -> f1110_out :|: TRUE f1071_out -> f1067_out :|: TRUE f1067_in -> f1071_in :|: TRUE f1875_out -> f1874_out :|: TRUE f1874_in -> f1876_in :|: TRUE f1876_out -> f1874_out :|: TRUE f1874_in -> f1875_in :|: TRUE f1636_in -> f1665_in :|: TRUE f1665_out -> f1636_out :|: TRUE f1609_out -> f1605_out :|: TRUE f1605_in -> f1608_in :|: TRUE f1605_in -> f1609_in :|: TRUE f1608_out -> f1605_out :|: TRUE f2121_in -> f1872_in :|: TRUE f1872_out -> f2121_out :|: TRUE f1122_out -> f1657_out :|: TRUE f1657_in -> f1122_in :|: TRUE f1602_out -> f1598_out :|: TRUE f1598_in -> f1602_in :|: TRUE f1598_in -> f1603_in :|: TRUE f1603_out -> f1598_out :|: TRUE f1276_in -> f1281_in :|: TRUE f1276_in -> f1280_in :|: TRUE f1281_out -> f1276_out :|: TRUE f1280_out -> f1276_out :|: TRUE f1273_out -> f1135_out :|: TRUE f1135_in -> f1273_in :|: TRUE f1135_in -> f1266_in :|: TRUE f1266_out -> f1135_out :|: TRUE f1122_out -> f1252_out :|: TRUE f1252_in -> f1122_in :|: TRUE f1679_out -> f1676_out :|: TRUE f1676_in -> f1679_in :|: TRUE f1858_out -> f1854_out :|: TRUE f1854_in -> f1858_in :|: TRUE f1857_out -> f1854_out :|: TRUE f1854_in -> f1857_in :|: TRUE f2161_in -> f2061_in :|: TRUE f2061_out -> f2161_out :|: TRUE f2114_out -> f2112_out :|: TRUE f2112_in -> f2114_in :|: TRUE f1306_in -> f1266_in :|: TRUE f1266_out -> f1306_out :|: TRUE f2132_in -> f2134_in :|: TRUE f2135_out -> f2132_out :|: TRUE f2132_in -> f2135_in :|: TRUE f2134_out -> f2132_out :|: TRUE f1277_in -> f1277_out :|: TRUE f3_in(T3) -> f13_in(T3) :|: TRUE f13_out(x) -> f3_out(x) :|: TRUE f13_in(x1) -> f15_in(x1) :|: TRUE f15_out(x2) -> f13_out(x2) :|: TRUE f13_in(x3) -> f14_in(x3) :|: TRUE f14_out(x4) -> f13_out(x4) :|: TRUE f15_in(x5) -> f22_in(x5) :|: TRUE f15_in(x6) -> f21_in(x6) :|: TRUE f21_out(x7) -> f15_out(x7) :|: TRUE f22_out(x8) -> f15_out(x8) :|: TRUE f40_out(T36) -> f22_out(T36) :|: TRUE f22_in(x9) -> f41_in :|: TRUE f22_in(x10) -> f40_in(x10) :|: TRUE f41_out -> f22_out(x11) :|: TRUE f122_out(x12) -> f40_out(x12) :|: TRUE f40_in(x13) -> f121_in :|: TRUE f121_out -> f122_in(x14) :|: TRUE Start term: f3_in(T3) ---------------------------------------- (214) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f201_in -> f121_in :|: TRUE f1057_in -> f121_in :|: TRUE f125_in -> f129_in :|: TRUE f125_in -> f131_in :|: TRUE f129_in -> f201_in :|: TRUE f131_in -> f1055_in :|: TRUE f123_in -> f125_in :|: TRUE f1055_in -> f1057_in :|: TRUE f121_in -> f123_in :|: TRUE ---------------------------------------- (215) Obligation: Rules: f201_in -> f121_in :|: TRUE f1057_in -> f121_in :|: TRUE f125_in -> f129_in :|: TRUE f125_in -> f131_in :|: TRUE f129_in -> f201_in :|: TRUE f131_in -> f1055_in :|: TRUE f123_in -> f125_in :|: TRUE f1055_in -> f1057_in :|: TRUE f121_in -> f123_in :|: TRUE ---------------------------------------- (216) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (217) Obligation: Rules: f123_in -> f123_in :|: TRUE ---------------------------------------- (218) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (219) Obligation: Rules: f123_in -> f123_in :|: TRUE ---------------------------------------- (220) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f123_in -> f123_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (221) Obligation: Termination digraph: Nodes: (1) f123_in -> f123_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (222) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f123_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (223) Obligation: Rules: f123_in -> f123_in :|: TRUE ---------------------------------------- (224) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Proved unsatisfiability of the following formula, indicating that the system is never left after entering: ((run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T)) Proved satisfiability of the following formula, indicating that the system is entered at least once: (run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) ---------------------------------------- (225) NO ---------------------------------------- (226) Obligation: Rules: f15_in(T3) -> f22_in(T3) :|: TRUE f15_in(x) -> f21_in(x) :|: TRUE f21_out(x1) -> f15_out(x1) :|: TRUE f22_out(x2) -> f15_out(x2) :|: TRUE f24_out -> f21_out(x3) :|: TRUE f23_out(T23) -> f21_out(zero(T23)) :|: TRUE f21_in(x4) -> f24_in :|: TRUE f21_in(zero(x5)) -> f23_in(x5) :|: TRUE f13_in(x6) -> f15_in(x6) :|: TRUE f15_out(x7) -> f13_out(x7) :|: TRUE f13_in(x8) -> f14_in(x8) :|: TRUE f14_out(x9) -> f13_out(x9) :|: TRUE f23_in(x10) -> f3_in(x10) :|: TRUE f3_out(x11) -> f23_out(x11) :|: TRUE f3_in(x12) -> f13_in(x12) :|: TRUE f13_out(x13) -> f3_out(x13) :|: TRUE Start term: f3_in(T3) ---------------------------------------- (227) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f15_in(x) -> f21_in(x) :|: TRUE f21_in(zero(x5)) -> f23_in(x5) :|: TRUE f13_in(x6) -> f15_in(x6) :|: TRUE f23_in(x10) -> f3_in(x10) :|: TRUE f3_in(x12) -> f13_in(x12) :|: TRUE ---------------------------------------- (228) Obligation: Rules: f15_in(x) -> f21_in(x) :|: TRUE f21_in(zero(x5)) -> f23_in(x5) :|: TRUE f13_in(x6) -> f15_in(x6) :|: TRUE f23_in(x10) -> f3_in(x10) :|: TRUE f3_in(x12) -> f13_in(x12) :|: TRUE ---------------------------------------- (229) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (230) Obligation: Rules: f13_in(zero(x5:0)) -> f13_in(x5:0) :|: TRUE ---------------------------------------- (231) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (232) Obligation: Rules: f13_in(zero(x5:0)) -> f13_in(x5:0) :|: TRUE ---------------------------------------- (233) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f13_in(zero(x5:0)) -> f13_in(x5:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (234) Obligation: Termination digraph: Nodes: (1) f13_in(zero(x5:0)) -> f13_in(x5:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (235) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f13_in(VARIABLE) zero(VARIABLE) Removed predefined arithmetic. ---------------------------------------- (236) Obligation: Rules: f13_in(zero(x5:0)) -> f13_in(x5:0) ---------------------------------------- (237) IRSwTToQDPProof (SOUND) Removed the integers and created a QDP-Problem. ---------------------------------------- (238) Obligation: Q DP problem: The TRS P consists of the following rules: f13_in(zero(x5:0)) -> f13_in(x5:0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (239) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *f13_in(zero(x5:0)) -> f13_in(x5:0) The graph contains the following edges 1 > 1 ---------------------------------------- (240) YES