/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern times(g,g,a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) CutEliminatorProof [SOUND, 0 ms] (2) Prolog (3) PrologToPiTRSProof [SOUND, 0 ms] (4) PiTRS (5) DependencyPairsProof [EQUIVALENT, 10 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) PiDP (9) UsableRulesProof [EQUIVALENT, 0 ms] (10) PiDP (11) PiDPToQDPProof [SOUND, 0 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 16 ms] (14) QDP (15) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (16) QDP (17) TransformationProof [SOUND, 0 ms] (18) QDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) QDP (21) QReductionProof [EQUIVALENT, 0 ms] (22) QDP (23) TransformationProof [EQUIVALENT, 0 ms] (24) QDP (25) NonTerminationLoopProof [COMPLETE, 0 ms] (26) NO (27) PrologToPiTRSProof [SOUND, 0 ms] (28) PiTRS (29) DependencyPairsProof [EQUIVALENT, 2 ms] (30) PiDP (31) DependencyGraphProof [EQUIVALENT, 0 ms] (32) PiDP (33) UsableRulesProof [EQUIVALENT, 0 ms] (34) PiDP (35) PiDPToQDPProof [SOUND, 0 ms] (36) QDP (37) TransformationProof [SOUND, 0 ms] (38) QDP (39) UsableRulesProof [EQUIVALENT, 0 ms] (40) QDP (41) QReductionProof [EQUIVALENT, 0 ms] (42) QDP (43) TransformationProof [EQUIVALENT, 0 ms] (44) QDP (45) QDPOrderProof [EQUIVALENT, 28 ms] (46) QDP (47) DependencyGraphProof [EQUIVALENT, 0 ms] (48) AND (49) QDP (50) NonTerminationLoopProof [COMPLETE, 0 ms] (51) NO (52) QDP (53) QDPSizeChangeProof [EQUIVALENT, 0 ms] (54) YES (55) PrologToTRSTransformerProof [SOUND, 197 ms] (56) QTRS (57) DependencyPairsProof [EQUIVALENT, 0 ms] (58) QDP (59) DependencyGraphProof [EQUIVALENT, 0 ms] (60) QDP (61) QDPOrderProof [EQUIVALENT, 49 ms] (62) QDP (63) DependencyGraphProof [EQUIVALENT, 0 ms] (64) AND (65) QDP (66) UsableRulesProof [EQUIVALENT, 0 ms] (67) QDP (68) NonTerminationLoopProof [COMPLETE, 0 ms] (69) NO (70) QDP (71) UsableRulesProof [EQUIVALENT, 0 ms] (72) QDP (73) QDPSizeChangeProof [EQUIVALENT, 0 ms] (74) YES (75) PrologToIRSwTTransformerProof [SOUND, 172 ms] (76) IRSwT (77) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 38 ms] (78) IRSwT (79) IntTRSCompressionProof [EQUIVALENT, 173 ms] (80) IRSwT (81) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (82) IRSwT (83) PrologToDTProblemTransformerProof [SOUND, 220 ms] (84) TRIPLES (85) TriplesToPiDPProof [SOUND, 97 ms] (86) PiDP (87) DependencyGraphProof [EQUIVALENT, 0 ms] (88) PiDP (89) PiDPToQDPProof [SOUND, 0 ms] (90) QDP (91) QDPOrderProof [EQUIVALENT, 65 ms] (92) QDP (93) DependencyGraphProof [EQUIVALENT, 0 ms] (94) AND (95) QDP (96) NonTerminationLoopProof [COMPLETE, 0 ms] (97) NO (98) QDP (99) QDPSizeChangeProof [EQUIVALENT, 0 ms] (100) YES ---------------------------------------- (0) Obligation: Clauses: times(X, Y, Z) :- mult(X, Y, 0, Z). mult(0, Y, 0, Z) :- ','(!, eq(Z, 0)). mult(s(X), Y, 0, Z) :- ','(!, mult(X, Y, Y, Z)). mult(X, Y, W, s(Z)) :- ','(p(W, P), mult(X, Y, P, Z)). p(0, 0). p(s(X), X). eq(X, X). Query: times(g,g,a) ---------------------------------------- (1) CutEliminatorProof (SOUND) Eliminated all cuts by simply ignoring them[PROLOG]. ---------------------------------------- (2) Obligation: Clauses: times(X, Y, Z) :- mult(X, Y, 0, Z). mult(0, Y, 0, Z) :- eq(Z, 0). mult(s(X), Y, 0, Z) :- mult(X, Y, Y, Z). mult(X, Y, W, s(Z)) :- ','(p(W, P), mult(X, Y, P, Z)). p(0, 0). p(s(X), X). eq(X, X). Query: times(g,g,a) ---------------------------------------- (3) 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: (b,b,f) mult_in_4: (b,b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, Z) -> U2_ggga(Y, Z, eq_in_ag(Z, 0)) eq_in_ag(X, X) -> eq_out_ag(X, X) U2_ggga(Y, Z, eq_out_ag(Z, 0)) -> mult_out_ggga(0, Y, 0, Z) mult_in_ggga(s(X), Y, 0, Z) -> U3_ggga(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) mult_in_ggga(X, Y, W, s(Z)) -> U4_ggga(X, Y, W, Z, p_in_ga(W, P)) p_in_ga(0, 0) -> p_out_ga(0, 0) p_in_ga(s(X), X) -> p_out_ga(s(X), X) U4_ggga(X, Y, W, Z, p_out_ga(W, P)) -> U5_ggga(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U5_ggga(X, Y, W, Z, mult_out_ggga(X, Y, P, Z)) -> mult_out_ggga(X, Y, W, s(Z)) U3_ggga(X, Y, Z, mult_out_ggga(X, Y, Y, Z)) -> mult_out_ggga(s(X), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 U2_ggga(x1, x2, x3) = U2_ggga(x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x4) s(x1) = s(x1) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x5) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x5) times_out_gga(x1, x2, x3) = times_out_gga(x3) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (4) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, Z) -> U2_ggga(Y, Z, eq_in_ag(Z, 0)) eq_in_ag(X, X) -> eq_out_ag(X, X) U2_ggga(Y, Z, eq_out_ag(Z, 0)) -> mult_out_ggga(0, Y, 0, Z) mult_in_ggga(s(X), Y, 0, Z) -> U3_ggga(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) mult_in_ggga(X, Y, W, s(Z)) -> U4_ggga(X, Y, W, Z, p_in_ga(W, P)) p_in_ga(0, 0) -> p_out_ga(0, 0) p_in_ga(s(X), X) -> p_out_ga(s(X), X) U4_ggga(X, Y, W, Z, p_out_ga(W, P)) -> U5_ggga(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U5_ggga(X, Y, W, Z, mult_out_ggga(X, Y, P, Z)) -> mult_out_ggga(X, Y, W, s(Z)) U3_ggga(X, Y, Z, mult_out_ggga(X, Y, Y, Z)) -> mult_out_ggga(s(X), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 U2_ggga(x1, x2, x3) = U2_ggga(x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x4) s(x1) = s(x1) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x5) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x5) times_out_gga(x1, x2, x3) = times_out_gga(x3) ---------------------------------------- (5) 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_GGA(X, Y, Z) -> U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) TIMES_IN_GGA(X, Y, Z) -> MULT_IN_GGGA(X, Y, 0, Z) MULT_IN_GGGA(0, Y, 0, Z) -> U2_GGGA(Y, Z, eq_in_ag(Z, 0)) MULT_IN_GGGA(0, Y, 0, Z) -> EQ_IN_AG(Z, 0) MULT_IN_GGGA(s(X), Y, 0, Z) -> U3_GGGA(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) MULT_IN_GGGA(s(X), Y, 0, Z) -> MULT_IN_GGGA(X, Y, Y, Z) MULT_IN_GGGA(X, Y, W, s(Z)) -> U4_GGGA(X, Y, W, Z, p_in_ga(W, P)) MULT_IN_GGGA(X, Y, W, s(Z)) -> P_IN_GA(W, P) U4_GGGA(X, Y, W, Z, p_out_ga(W, P)) -> U5_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U4_GGGA(X, Y, W, Z, p_out_ga(W, P)) -> MULT_IN_GGGA(X, Y, P, Z) The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, Z) -> U2_ggga(Y, Z, eq_in_ag(Z, 0)) eq_in_ag(X, X) -> eq_out_ag(X, X) U2_ggga(Y, Z, eq_out_ag(Z, 0)) -> mult_out_ggga(0, Y, 0, Z) mult_in_ggga(s(X), Y, 0, Z) -> U3_ggga(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) mult_in_ggga(X, Y, W, s(Z)) -> U4_ggga(X, Y, W, Z, p_in_ga(W, P)) p_in_ga(0, 0) -> p_out_ga(0, 0) p_in_ga(s(X), X) -> p_out_ga(s(X), X) U4_ggga(X, Y, W, Z, p_out_ga(W, P)) -> U5_ggga(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U5_ggga(X, Y, W, Z, mult_out_ggga(X, Y, P, Z)) -> mult_out_ggga(X, Y, W, s(Z)) U3_ggga(X, Y, Z, mult_out_ggga(X, Y, Y, Z)) -> mult_out_ggga(s(X), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 U2_ggga(x1, x2, x3) = U2_ggga(x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x4) s(x1) = s(x1) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x5) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x5) times_out_gga(x1, x2, x3) = times_out_gga(x3) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x4) MULT_IN_GGGA(x1, x2, x3, x4) = MULT_IN_GGGA(x1, x2, x3) U2_GGGA(x1, x2, x3) = U2_GGGA(x3) EQ_IN_AG(x1, x2) = EQ_IN_AG(x2) U3_GGGA(x1, x2, x3, x4) = U3_GGGA(x4) U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x1, x2, x5) P_IN_GA(x1, x2) = P_IN_GA(x1) U5_GGGA(x1, x2, x3, x4, x5) = U5_GGGA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_GGA(X, Y, Z) -> U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) TIMES_IN_GGA(X, Y, Z) -> MULT_IN_GGGA(X, Y, 0, Z) MULT_IN_GGGA(0, Y, 0, Z) -> U2_GGGA(Y, Z, eq_in_ag(Z, 0)) MULT_IN_GGGA(0, Y, 0, Z) -> EQ_IN_AG(Z, 0) MULT_IN_GGGA(s(X), Y, 0, Z) -> U3_GGGA(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) MULT_IN_GGGA(s(X), Y, 0, Z) -> MULT_IN_GGGA(X, Y, Y, Z) MULT_IN_GGGA(X, Y, W, s(Z)) -> U4_GGGA(X, Y, W, Z, p_in_ga(W, P)) MULT_IN_GGGA(X, Y, W, s(Z)) -> P_IN_GA(W, P) U4_GGGA(X, Y, W, Z, p_out_ga(W, P)) -> U5_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U4_GGGA(X, Y, W, Z, p_out_ga(W, P)) -> MULT_IN_GGGA(X, Y, P, Z) The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, Z) -> U2_ggga(Y, Z, eq_in_ag(Z, 0)) eq_in_ag(X, X) -> eq_out_ag(X, X) U2_ggga(Y, Z, eq_out_ag(Z, 0)) -> mult_out_ggga(0, Y, 0, Z) mult_in_ggga(s(X), Y, 0, Z) -> U3_ggga(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) mult_in_ggga(X, Y, W, s(Z)) -> U4_ggga(X, Y, W, Z, p_in_ga(W, P)) p_in_ga(0, 0) -> p_out_ga(0, 0) p_in_ga(s(X), X) -> p_out_ga(s(X), X) U4_ggga(X, Y, W, Z, p_out_ga(W, P)) -> U5_ggga(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U5_ggga(X, Y, W, Z, mult_out_ggga(X, Y, P, Z)) -> mult_out_ggga(X, Y, W, s(Z)) U3_ggga(X, Y, Z, mult_out_ggga(X, Y, Y, Z)) -> mult_out_ggga(s(X), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 U2_ggga(x1, x2, x3) = U2_ggga(x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x4) s(x1) = s(x1) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x5) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x5) times_out_gga(x1, x2, x3) = times_out_gga(x3) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x4) MULT_IN_GGGA(x1, x2, x3, x4) = MULT_IN_GGGA(x1, x2, x3) U2_GGGA(x1, x2, x3) = U2_GGGA(x3) EQ_IN_AG(x1, x2) = EQ_IN_AG(x2) U3_GGGA(x1, x2, x3, x4) = U3_GGGA(x4) U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x1, x2, x5) P_IN_GA(x1, x2) = P_IN_GA(x1) U5_GGGA(x1, x2, x3, x4, x5) = U5_GGGA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: MULT_IN_GGGA(X, Y, W, s(Z)) -> U4_GGGA(X, Y, W, Z, p_in_ga(W, P)) U4_GGGA(X, Y, W, Z, p_out_ga(W, P)) -> MULT_IN_GGGA(X, Y, P, Z) MULT_IN_GGGA(s(X), Y, 0, Z) -> MULT_IN_GGGA(X, Y, Y, Z) The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, Z) -> U2_ggga(Y, Z, eq_in_ag(Z, 0)) eq_in_ag(X, X) -> eq_out_ag(X, X) U2_ggga(Y, Z, eq_out_ag(Z, 0)) -> mult_out_ggga(0, Y, 0, Z) mult_in_ggga(s(X), Y, 0, Z) -> U3_ggga(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) mult_in_ggga(X, Y, W, s(Z)) -> U4_ggga(X, Y, W, Z, p_in_ga(W, P)) p_in_ga(0, 0) -> p_out_ga(0, 0) p_in_ga(s(X), X) -> p_out_ga(s(X), X) U4_ggga(X, Y, W, Z, p_out_ga(W, P)) -> U5_ggga(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U5_ggga(X, Y, W, Z, mult_out_ggga(X, Y, P, Z)) -> mult_out_ggga(X, Y, W, s(Z)) U3_ggga(X, Y, Z, mult_out_ggga(X, Y, Y, Z)) -> mult_out_ggga(s(X), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 U2_ggga(x1, x2, x3) = U2_ggga(x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x4) s(x1) = s(x1) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x5) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x5) times_out_gga(x1, x2, x3) = times_out_gga(x3) MULT_IN_GGGA(x1, x2, x3, x4) = MULT_IN_GGGA(x1, x2, x3) U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x1, x2, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (10) Obligation: Pi DP problem: The TRS P consists of the following rules: MULT_IN_GGGA(X, Y, W, s(Z)) -> U4_GGGA(X, Y, W, Z, p_in_ga(W, P)) U4_GGGA(X, Y, W, Z, p_out_ga(W, P)) -> MULT_IN_GGGA(X, Y, P, Z) MULT_IN_GGGA(s(X), Y, 0, Z) -> MULT_IN_GGGA(X, Y, Y, Z) The TRS R consists of the following rules: p_in_ga(0, 0) -> p_out_ga(0, 0) p_in_ga(s(X), X) -> p_out_ga(s(X), X) The argument filtering Pi contains the following mapping: 0 = 0 s(x1) = s(x1) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) MULT_IN_GGGA(x1, x2, x3, x4) = MULT_IN_GGGA(x1, x2, x3) U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x1, x2, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (11) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: MULT_IN_GGGA(X, Y, W) -> U4_GGGA(X, Y, p_in_ga(W)) U4_GGGA(X, Y, p_out_ga(P)) -> MULT_IN_GGGA(X, Y, P) MULT_IN_GGGA(s(X), Y, 0) -> MULT_IN_GGGA(X, Y, Y) The TRS R consists of the following rules: p_in_ga(0) -> p_out_ga(0) p_in_ga(s(X)) -> p_out_ga(X) The set Q consists of the following terms: p_in_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (13) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MULT_IN_GGGA(s(X), Y, 0) -> MULT_IN_GGGA(X, Y, Y) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. MULT_IN_GGGA(x1, x2, x3) = x1 U4_GGGA(x1, x2, x3) = x1 s(x1) = s(x1) Knuth-Bendix order [KBO] with precedence:trivial and weight map: s_1=1 dummyConstant=1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: none ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: MULT_IN_GGGA(X, Y, W) -> U4_GGGA(X, Y, p_in_ga(W)) U4_GGGA(X, Y, p_out_ga(P)) -> MULT_IN_GGGA(X, Y, P) The TRS R consists of the following rules: p_in_ga(0) -> p_out_ga(0) p_in_ga(s(X)) -> p_out_ga(X) The set Q consists of the following terms: p_in_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (15) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: p_in_ga(s(X)) -> p_out_ga(X) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0) = 0 POL(MULT_IN_GGGA(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U4_GGGA(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(p_in_ga(x_1)) = x_1 POL(p_out_ga(x_1)) = x_1 POL(s(x_1)) = 2*x_1 ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: MULT_IN_GGGA(X, Y, W) -> U4_GGGA(X, Y, p_in_ga(W)) U4_GGGA(X, Y, p_out_ga(P)) -> MULT_IN_GGGA(X, Y, P) The TRS R consists of the following rules: p_in_ga(0) -> p_out_ga(0) The set Q consists of the following terms: p_in_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (17) TransformationProof (SOUND) By narrowing [LPAR04] the rule MULT_IN_GGGA(X, Y, W) -> U4_GGGA(X, Y, p_in_ga(W)) at position [2] we obtained the following new rules [LPAR04]: (MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, p_out_ga(0)),MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, p_out_ga(0))) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GGGA(X, Y, p_out_ga(P)) -> MULT_IN_GGGA(X, Y, P) MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, p_out_ga(0)) The TRS R consists of the following rules: p_in_ga(0) -> p_out_ga(0) The set Q consists of the following terms: p_in_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GGGA(X, Y, p_out_ga(P)) -> MULT_IN_GGGA(X, Y, P) MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, p_out_ga(0)) R is empty. The set Q consists of the following terms: p_in_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. p_in_ga(x0) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GGGA(X, Y, p_out_ga(P)) -> MULT_IN_GGGA(X, Y, P) MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, p_out_ga(0)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U4_GGGA(X, Y, p_out_ga(P)) -> MULT_IN_GGGA(X, Y, P) we obtained the following new rules [LPAR04]: (U4_GGGA(z0, z1, p_out_ga(0)) -> MULT_IN_GGGA(z0, z1, 0),U4_GGGA(z0, z1, p_out_ga(0)) -> MULT_IN_GGGA(z0, z1, 0)) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, p_out_ga(0)) U4_GGGA(z0, z1, p_out_ga(0)) -> MULT_IN_GGGA(z0, z1, 0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (25) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = U4_GGGA(z0, z1, p_out_ga(0)) evaluates to t =U4_GGGA(z0, z1, p_out_ga(0)) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence U4_GGGA(z0, z1, p_out_ga(0)) -> MULT_IN_GGGA(z0, z1, 0) with rule U4_GGGA(z0', z1', p_out_ga(0)) -> MULT_IN_GGGA(z0', z1', 0) at position [] and matcher [z0' / z0, z1' / z1] MULT_IN_GGGA(z0, z1, 0) -> U4_GGGA(z0, z1, p_out_ga(0)) with rule MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, p_out_ga(0)) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (26) NO ---------------------------------------- (27) 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: (b,b,f) mult_in_4: (b,b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, Z) -> U2_ggga(Y, Z, eq_in_ag(Z, 0)) eq_in_ag(X, X) -> eq_out_ag(X, X) U2_ggga(Y, Z, eq_out_ag(Z, 0)) -> mult_out_ggga(0, Y, 0, Z) mult_in_ggga(s(X), Y, 0, Z) -> U3_ggga(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) mult_in_ggga(X, Y, W, s(Z)) -> U4_ggga(X, Y, W, Z, p_in_ga(W, P)) p_in_ga(0, 0) -> p_out_ga(0, 0) p_in_ga(s(X), X) -> p_out_ga(s(X), X) U4_ggga(X, Y, W, Z, p_out_ga(W, P)) -> U5_ggga(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U5_ggga(X, Y, W, Z, mult_out_ggga(X, Y, P, Z)) -> mult_out_ggga(X, Y, W, s(Z)) U3_ggga(X, Y, Z, mult_out_ggga(X, Y, Y, Z)) -> mult_out_ggga(s(X), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 U2_ggga(x1, x2, x3) = U2_ggga(x1, x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1, x2) mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x1, x2, x3, x4) s(x1) = s(x1) U3_ggga(x1, x2, x3, x4) = U3_ggga(x1, x2, x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x3, x5) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x1, x2, x3, x5) times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (28) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, Z) -> U2_ggga(Y, Z, eq_in_ag(Z, 0)) eq_in_ag(X, X) -> eq_out_ag(X, X) U2_ggga(Y, Z, eq_out_ag(Z, 0)) -> mult_out_ggga(0, Y, 0, Z) mult_in_ggga(s(X), Y, 0, Z) -> U3_ggga(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) mult_in_ggga(X, Y, W, s(Z)) -> U4_ggga(X, Y, W, Z, p_in_ga(W, P)) p_in_ga(0, 0) -> p_out_ga(0, 0) p_in_ga(s(X), X) -> p_out_ga(s(X), X) U4_ggga(X, Y, W, Z, p_out_ga(W, P)) -> U5_ggga(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U5_ggga(X, Y, W, Z, mult_out_ggga(X, Y, P, Z)) -> mult_out_ggga(X, Y, W, s(Z)) U3_ggga(X, Y, Z, mult_out_ggga(X, Y, Y, Z)) -> mult_out_ggga(s(X), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 U2_ggga(x1, x2, x3) = U2_ggga(x1, x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1, x2) mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x1, x2, x3, x4) s(x1) = s(x1) U3_ggga(x1, x2, x3, x4) = U3_ggga(x1, x2, x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x3, x5) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x1, x2, x3, x5) times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) ---------------------------------------- (29) 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_GGA(X, Y, Z) -> U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) TIMES_IN_GGA(X, Y, Z) -> MULT_IN_GGGA(X, Y, 0, Z) MULT_IN_GGGA(0, Y, 0, Z) -> U2_GGGA(Y, Z, eq_in_ag(Z, 0)) MULT_IN_GGGA(0, Y, 0, Z) -> EQ_IN_AG(Z, 0) MULT_IN_GGGA(s(X), Y, 0, Z) -> U3_GGGA(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) MULT_IN_GGGA(s(X), Y, 0, Z) -> MULT_IN_GGGA(X, Y, Y, Z) MULT_IN_GGGA(X, Y, W, s(Z)) -> U4_GGGA(X, Y, W, Z, p_in_ga(W, P)) MULT_IN_GGGA(X, Y, W, s(Z)) -> P_IN_GA(W, P) U4_GGGA(X, Y, W, Z, p_out_ga(W, P)) -> U5_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U4_GGGA(X, Y, W, Z, p_out_ga(W, P)) -> MULT_IN_GGGA(X, Y, P, Z) The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, Z) -> U2_ggga(Y, Z, eq_in_ag(Z, 0)) eq_in_ag(X, X) -> eq_out_ag(X, X) U2_ggga(Y, Z, eq_out_ag(Z, 0)) -> mult_out_ggga(0, Y, 0, Z) mult_in_ggga(s(X), Y, 0, Z) -> U3_ggga(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) mult_in_ggga(X, Y, W, s(Z)) -> U4_ggga(X, Y, W, Z, p_in_ga(W, P)) p_in_ga(0, 0) -> p_out_ga(0, 0) p_in_ga(s(X), X) -> p_out_ga(s(X), X) U4_ggga(X, Y, W, Z, p_out_ga(W, P)) -> U5_ggga(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U5_ggga(X, Y, W, Z, mult_out_ggga(X, Y, P, Z)) -> mult_out_ggga(X, Y, W, s(Z)) U3_ggga(X, Y, Z, mult_out_ggga(X, Y, Y, Z)) -> mult_out_ggga(s(X), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 U2_ggga(x1, x2, x3) = U2_ggga(x1, x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1, x2) mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x1, x2, x3, x4) s(x1) = s(x1) U3_ggga(x1, x2, x3, x4) = U3_ggga(x1, x2, x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x3, x5) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x1, x2, x3, x5) times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) MULT_IN_GGGA(x1, x2, x3, x4) = MULT_IN_GGGA(x1, x2, x3) U2_GGGA(x1, x2, x3) = U2_GGGA(x1, x3) EQ_IN_AG(x1, x2) = EQ_IN_AG(x2) U3_GGGA(x1, x2, x3, x4) = U3_GGGA(x1, x2, x4) U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x1, x2, x3, x5) P_IN_GA(x1, x2) = P_IN_GA(x1) U5_GGGA(x1, x2, x3, x4, x5) = U5_GGGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_GGA(X, Y, Z) -> U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) TIMES_IN_GGA(X, Y, Z) -> MULT_IN_GGGA(X, Y, 0, Z) MULT_IN_GGGA(0, Y, 0, Z) -> U2_GGGA(Y, Z, eq_in_ag(Z, 0)) MULT_IN_GGGA(0, Y, 0, Z) -> EQ_IN_AG(Z, 0) MULT_IN_GGGA(s(X), Y, 0, Z) -> U3_GGGA(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) MULT_IN_GGGA(s(X), Y, 0, Z) -> MULT_IN_GGGA(X, Y, Y, Z) MULT_IN_GGGA(X, Y, W, s(Z)) -> U4_GGGA(X, Y, W, Z, p_in_ga(W, P)) MULT_IN_GGGA(X, Y, W, s(Z)) -> P_IN_GA(W, P) U4_GGGA(X, Y, W, Z, p_out_ga(W, P)) -> U5_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U4_GGGA(X, Y, W, Z, p_out_ga(W, P)) -> MULT_IN_GGGA(X, Y, P, Z) The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, Z) -> U2_ggga(Y, Z, eq_in_ag(Z, 0)) eq_in_ag(X, X) -> eq_out_ag(X, X) U2_ggga(Y, Z, eq_out_ag(Z, 0)) -> mult_out_ggga(0, Y, 0, Z) mult_in_ggga(s(X), Y, 0, Z) -> U3_ggga(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) mult_in_ggga(X, Y, W, s(Z)) -> U4_ggga(X, Y, W, Z, p_in_ga(W, P)) p_in_ga(0, 0) -> p_out_ga(0, 0) p_in_ga(s(X), X) -> p_out_ga(s(X), X) U4_ggga(X, Y, W, Z, p_out_ga(W, P)) -> U5_ggga(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U5_ggga(X, Y, W, Z, mult_out_ggga(X, Y, P, Z)) -> mult_out_ggga(X, Y, W, s(Z)) U3_ggga(X, Y, Z, mult_out_ggga(X, Y, Y, Z)) -> mult_out_ggga(s(X), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 U2_ggga(x1, x2, x3) = U2_ggga(x1, x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1, x2) mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x1, x2, x3, x4) s(x1) = s(x1) U3_ggga(x1, x2, x3, x4) = U3_ggga(x1, x2, x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x3, x5) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x1, x2, x3, x5) times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) MULT_IN_GGGA(x1, x2, x3, x4) = MULT_IN_GGGA(x1, x2, x3) U2_GGGA(x1, x2, x3) = U2_GGGA(x1, x3) EQ_IN_AG(x1, x2) = EQ_IN_AG(x2) U3_GGGA(x1, x2, x3, x4) = U3_GGGA(x1, x2, x4) U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x1, x2, x3, x5) P_IN_GA(x1, x2) = P_IN_GA(x1) U5_GGGA(x1, x2, x3, x4, x5) = U5_GGGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes. ---------------------------------------- (32) Obligation: Pi DP problem: The TRS P consists of the following rules: MULT_IN_GGGA(X, Y, W, s(Z)) -> U4_GGGA(X, Y, W, Z, p_in_ga(W, P)) U4_GGGA(X, Y, W, Z, p_out_ga(W, P)) -> MULT_IN_GGGA(X, Y, P, Z) MULT_IN_GGGA(s(X), Y, 0, Z) -> MULT_IN_GGGA(X, Y, Y, Z) The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, Z) -> U2_ggga(Y, Z, eq_in_ag(Z, 0)) eq_in_ag(X, X) -> eq_out_ag(X, X) U2_ggga(Y, Z, eq_out_ag(Z, 0)) -> mult_out_ggga(0, Y, 0, Z) mult_in_ggga(s(X), Y, 0, Z) -> U3_ggga(X, Y, Z, mult_in_ggga(X, Y, Y, Z)) mult_in_ggga(X, Y, W, s(Z)) -> U4_ggga(X, Y, W, Z, p_in_ga(W, P)) p_in_ga(0, 0) -> p_out_ga(0, 0) p_in_ga(s(X), X) -> p_out_ga(s(X), X) U4_ggga(X, Y, W, Z, p_out_ga(W, P)) -> U5_ggga(X, Y, W, Z, mult_in_ggga(X, Y, P, Z)) U5_ggga(X, Y, W, Z, mult_out_ggga(X, Y, P, Z)) -> mult_out_ggga(X, Y, W, s(Z)) U3_ggga(X, Y, Z, mult_out_ggga(X, Y, Y, Z)) -> mult_out_ggga(s(X), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 U2_ggga(x1, x2, x3) = U2_ggga(x1, x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1, x2) mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x1, x2, x3, x4) s(x1) = s(x1) U3_ggga(x1, x2, x3, x4) = U3_ggga(x1, x2, x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x3, x5) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x1, x2, x3, x5) times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) MULT_IN_GGGA(x1, x2, x3, x4) = MULT_IN_GGGA(x1, x2, x3) U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (33) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (34) Obligation: Pi DP problem: The TRS P consists of the following rules: MULT_IN_GGGA(X, Y, W, s(Z)) -> U4_GGGA(X, Y, W, Z, p_in_ga(W, P)) U4_GGGA(X, Y, W, Z, p_out_ga(W, P)) -> MULT_IN_GGGA(X, Y, P, Z) MULT_IN_GGGA(s(X), Y, 0, Z) -> MULT_IN_GGGA(X, Y, Y, Z) The TRS R consists of the following rules: p_in_ga(0, 0) -> p_out_ga(0, 0) p_in_ga(s(X), X) -> p_out_ga(s(X), X) The argument filtering Pi contains the following mapping: 0 = 0 s(x1) = s(x1) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) MULT_IN_GGGA(x1, x2, x3, x4) = MULT_IN_GGGA(x1, x2, x3) U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (35) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: MULT_IN_GGGA(X, Y, W) -> U4_GGGA(X, Y, W, p_in_ga(W)) U4_GGGA(X, Y, W, p_out_ga(W, P)) -> MULT_IN_GGGA(X, Y, P) MULT_IN_GGGA(s(X), Y, 0) -> MULT_IN_GGGA(X, Y, Y) The TRS R consists of the following rules: p_in_ga(0) -> p_out_ga(0, 0) p_in_ga(s(X)) -> p_out_ga(s(X), X) The set Q consists of the following terms: p_in_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (37) TransformationProof (SOUND) By narrowing [LPAR04] the rule MULT_IN_GGGA(X, Y, W) -> U4_GGGA(X, Y, W, p_in_ga(W)) at position [3] we obtained the following new rules [LPAR04]: (MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, 0, p_out_ga(0, 0)),MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, 0, p_out_ga(0, 0))) (MULT_IN_GGGA(y0, y1, s(x0)) -> U4_GGGA(y0, y1, s(x0), p_out_ga(s(x0), x0)),MULT_IN_GGGA(y0, y1, s(x0)) -> U4_GGGA(y0, y1, s(x0), p_out_ga(s(x0), x0))) ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GGGA(X, Y, W, p_out_ga(W, P)) -> MULT_IN_GGGA(X, Y, P) MULT_IN_GGGA(s(X), Y, 0) -> MULT_IN_GGGA(X, Y, Y) MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, 0, p_out_ga(0, 0)) MULT_IN_GGGA(y0, y1, s(x0)) -> U4_GGGA(y0, y1, s(x0), p_out_ga(s(x0), x0)) The TRS R consists of the following rules: p_in_ga(0) -> p_out_ga(0, 0) p_in_ga(s(X)) -> p_out_ga(s(X), X) The set Q consists of the following terms: p_in_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (39) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GGGA(X, Y, W, p_out_ga(W, P)) -> MULT_IN_GGGA(X, Y, P) MULT_IN_GGGA(s(X), Y, 0) -> MULT_IN_GGGA(X, Y, Y) MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, 0, p_out_ga(0, 0)) MULT_IN_GGGA(y0, y1, s(x0)) -> U4_GGGA(y0, y1, s(x0), p_out_ga(s(x0), x0)) R is empty. The set Q consists of the following terms: p_in_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (41) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. p_in_ga(x0) ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GGGA(X, Y, W, p_out_ga(W, P)) -> MULT_IN_GGGA(X, Y, P) MULT_IN_GGGA(s(X), Y, 0) -> MULT_IN_GGGA(X, Y, Y) MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, 0, p_out_ga(0, 0)) MULT_IN_GGGA(y0, y1, s(x0)) -> U4_GGGA(y0, y1, s(x0), p_out_ga(s(x0), x0)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (43) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U4_GGGA(X, Y, W, p_out_ga(W, P)) -> MULT_IN_GGGA(X, Y, P) we obtained the following new rules [LPAR04]: (U4_GGGA(z0, z1, 0, p_out_ga(0, 0)) -> MULT_IN_GGGA(z0, z1, 0),U4_GGGA(z0, z1, 0, p_out_ga(0, 0)) -> MULT_IN_GGGA(z0, z1, 0)) (U4_GGGA(z0, z1, s(z2), p_out_ga(s(z2), z2)) -> MULT_IN_GGGA(z0, z1, z2),U4_GGGA(z0, z1, s(z2), p_out_ga(s(z2), z2)) -> MULT_IN_GGGA(z0, z1, z2)) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: MULT_IN_GGGA(s(X), Y, 0) -> MULT_IN_GGGA(X, Y, Y) MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, 0, p_out_ga(0, 0)) MULT_IN_GGGA(y0, y1, s(x0)) -> U4_GGGA(y0, y1, s(x0), p_out_ga(s(x0), x0)) U4_GGGA(z0, z1, 0, p_out_ga(0, 0)) -> MULT_IN_GGGA(z0, z1, 0) U4_GGGA(z0, z1, s(z2), p_out_ga(s(z2), z2)) -> MULT_IN_GGGA(z0, z1, z2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (45) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MULT_IN_GGGA(s(X), Y, 0) -> MULT_IN_GGGA(X, Y, Y) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. MULT_IN_GGGA(x1, x2, x3) = x1 s(x1) = s(x1) U4_GGGA(x1, x2, x3, x4) = x1 Knuth-Bendix order [KBO] with precedence:trivial and weight map: s_1=1 dummyConstant=1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: none ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, 0, p_out_ga(0, 0)) MULT_IN_GGGA(y0, y1, s(x0)) -> U4_GGGA(y0, y1, s(x0), p_out_ga(s(x0), x0)) U4_GGGA(z0, z1, 0, p_out_ga(0, 0)) -> MULT_IN_GGGA(z0, z1, 0) U4_GGGA(z0, z1, s(z2), p_out_ga(s(z2), z2)) -> MULT_IN_GGGA(z0, z1, z2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (47) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (48) Complex Obligation (AND) ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GGGA(z0, z1, 0, p_out_ga(0, 0)) -> MULT_IN_GGGA(z0, z1, 0) MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, 0, p_out_ga(0, 0)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (50) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = MULT_IN_GGGA(y0, y1, 0) evaluates to t =MULT_IN_GGGA(y0, y1, 0) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence MULT_IN_GGGA(y0, y1, 0) -> U4_GGGA(y0, y1, 0, p_out_ga(0, 0)) with rule MULT_IN_GGGA(y0', y1', 0) -> U4_GGGA(y0', y1', 0, p_out_ga(0, 0)) at position [] and matcher [y0' / y0, y1' / y1] U4_GGGA(y0, y1, 0, p_out_ga(0, 0)) -> MULT_IN_GGGA(y0, y1, 0) with rule U4_GGGA(z0, z1, 0, p_out_ga(0, 0)) -> MULT_IN_GGGA(z0, z1, 0) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (51) NO ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: MULT_IN_GGGA(y0, y1, s(x0)) -> U4_GGGA(y0, y1, s(x0), p_out_ga(s(x0), x0)) U4_GGGA(z0, z1, s(z2), p_out_ga(s(z2), z2)) -> MULT_IN_GGGA(z0, z1, z2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (53) 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: *U4_GGGA(z0, z1, s(z2), p_out_ga(s(z2), z2)) -> MULT_IN_GGGA(z0, z1, z2) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 3 *MULT_IN_GGGA(y0, y1, s(x0)) -> U4_GGGA(y0, y1, s(x0), p_out_ga(s(x0), x0)) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3 ---------------------------------------- (54) YES ---------------------------------------- (55) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 3, "program": { "directives": [], "clauses": [ [ "(times X Y Z)", "(mult X Y (0) Z)" ], [ "(mult (0) Y (0) Z)", "(',' (!) (eq Z (0)))" ], [ "(mult (s X) Y (0) Z)", "(',' (!) (mult X Y Y Z))" ], [ "(mult X Y W (s Z))", "(',' (p W P) (mult X Y P Z))" ], [ "(p (0) (0))", null ], [ "(p (s X) X)", null ], [ "(eq X X)", null ] ] }, "graph": { "nodes": { "590": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "470": { "goal": [{ "clause": 6, "scope": 14, "term": "(eq T192 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "351": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (!_7) (mult T116 (s (0)) (s (0)) T119))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T116"], "free": [], "exprvars": [] } }, "352": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, 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"ground": [], "free": [], "exprvars": [] } }, "693": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_25) (eq T376 (0)))" }, { "clause": 2, "scope": 25, "term": "(mult (0) (s (s (s (s (s (s (s (0)))))))) (0) T359)" }, { "clause": 3, "scope": 25, "term": "(mult (0) (s (s (s (s (s (s (s (0)))))))) (0) T359)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "452": { "goal": [{ "clause": -1, "scope": -1, "term": "(mult T172 (s (s (0))) (0) T175)" }], "kb": { "nonunifying": [[ "(mult T172 (s (s (0))) (0) T129)", "(mult (0) X148 (0) X149)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T172"], "free": [ "X148", "X149" ], "exprvars": [] } }, "694": { "goal": [ { "clause": 2, "scope": 25, "term": "(mult T356 (s (s (s (s (s (s (s T367))))))) T367 T359)" }, { "clause": 3, "scope": 25, "term": "(mult T356 (s (s (s (s (s (s (s T367))))))) T367 T359)" } ], "kb": { "nonunifying": [[ "(mult T356 (s (s (s (s (s (s (s T367))))))) T367 T359)", "(mult (0) X408 (0) X409)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T356", "T367" ], "free": [ "X408", "X409" ], "exprvars": [] } }, "453": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "695": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq T376 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "333": { "goal": [ { "clause": 1, "scope": 7, "term": "(mult T80 (s T91) T91 T83)" }, { "clause": 2, "scope": 7, "term": "(mult T80 (s T91) T91 T83)" }, { "clause": 3, "scope": 7, "term": "(mult T80 (s T91) T91 T83)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T80", "T91" ], "free": [], "exprvars": [] } }, "696": { "goal": [{ "clause": 6, "scope": 26, "term": "(eq T376 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "697": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "456": { "goal": [{ "clause": -1, "scope": -1, "term": "(mult T172 (s (s (s T183))) T183 T175)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T172", "T183" ], "free": [], "exprvars": [] } }, "698": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "336": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_7) (eq T100 (0)))" }, { "clause": 2, "scope": 7, "term": "(mult (0) (s (0)) (0) T83)" }, { "clause": 3, "scope": 7, "term": "(mult (0) (s (0)) (0) T83)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "578": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_19) (eq T284 (0)))" }, { "clause": 2, "scope": 19, "term": "(mult (0) (s (s (s (s (s (0)))))) (0) T267)" }, { "clause": 3, "scope": 19, "term": "(mult (0) (s (s (s (s (s (0)))))) (0) T267)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "699": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "338": { "goal": [ { "clause": 2, "scope": 7, "term": "(mult T80 (s T91) T91 T83)" }, { "clause": 3, "scope": 7, "term": "(mult T80 (s T91) T91 T83)" } ], "kb": { "nonunifying": [[ "(mult T80 (s T91) T91 T83)", "(mult (0) X96 (0) X97)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T80", "T91" ], "free": [ "X96", "X97" ], "exprvars": [] } }, "459": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "339": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq T100 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "580": { "goal": [ { "clause": 2, "scope": 19, "term": "(mult T264 (s (s (s (s (s T275))))) T275 T267)" }, { "clause": 3, "scope": 19, "term": "(mult T264 (s (s (s (s (s T275))))) T275 T267)" } ], "kb": { "nonunifying": [[ "(mult T264 (s (s (s (s (s T275))))) T275 T267)", "(mult (0) X304 (0) X305)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T264", "T275" ], "free": [ "X304", "X305" ], "exprvars": [] } }, "340": { "goal": [{ "clause": 6, "scope": 8, "term": "(eq T100 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "582": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq T284 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "341": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "583": { "goal": [{ "clause": 6, "scope": 20, "term": "(eq T284 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "342": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "343": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "464": { "goal": [ { "clause": 1, "scope": 13, "term": "(mult T172 (s (s (s T183))) T183 T175)" }, { "clause": 2, "scope": 13, "term": "(mult T172 (s (s (s T183))) T183 T175)" }, { "clause": 3, "scope": 13, "term": "(mult T172 (s (s (s T183))) T183 T175)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T172", "T183" ], "free": [], "exprvars": [] } }, "345": { "goal": [{ "clause": 2, "scope": 7, "term": "(mult T80 (s T91) T91 T83)" }], "kb": { "nonunifying": [[ "(mult T80 (s T91) T91 T83)", "(mult (0) X96 (0) X97)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T80", "T91" ], "free": [ "X96", "X97" ], "exprvars": [] } }, "467": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_13) (eq T192 (0)))" }, { "clause": 2, "scope": 13, "term": "(mult (0) (s (s (s (0)))) (0) T175)" }, { "clause": 3, "scope": 13, "term": "(mult (0) (s (s (s (0)))) (0) T175)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "588": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "347": { "goal": [{ "clause": 3, "scope": 7, "term": "(mult T80 (s T91) T91 T83)" }], "kb": { "nonunifying": [[ "(mult T80 (s T91) T91 T83)", "(mult (0) X96 (0) X97)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T80", "T91" ], "free": [ "X96", "X97" ], "exprvars": [] } }, "468": { "goal": [ { "clause": 2, "scope": 13, "term": "(mult T172 (s (s (s T183))) T183 T175)" }, { "clause": 3, "scope": 13, "term": "(mult T172 (s (s (s T183))) T183 T175)" } ], "kb": { "nonunifying": [[ "(mult T172 (s (s (s T183))) T183 T175)", "(mult (0) X200 (0) X201)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T172", "T183" ], "free": [ "X200", "X201" ], "exprvars": [] } }, "589": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "469": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq T192 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 4, "label": "CASE" }, { "from": 4, "to": 126, "label": "ONLY EVAL with clause\ntimes(X9, X10, X11) :- mult(X9, X10, 0, X11).\nand substitutionT1 -> T13,\nX9 -> T13,\nT2 -> T14,\nX10 -> T14,\nT3 -> T16,\nX11 -> T16,\nT15 -> T16" }, { "from": 126, "to": 130, "label": "CASE" }, { "from": 130, "to": 132, "label": "EVAL with clause\nmult(0, X18, 0, X19) :- ','(!_2, eq(X19, 0)).\nand substitutionT13 -> 0,\nT14 -> T23,\nX18 -> T23,\nT16 -> T25,\nX19 -> T25,\nT24 -> T25" }, { "from": 130, "to": 136, "label": "EVAL-BACKTRACK" }, { "from": 132, "to": 138, "label": "CUT" }, { "from": 136, "to": 150, "label": "PARALLEL" }, { "from": 136, "to": 151, "label": "PARALLEL" }, { "from": 138, "to": 140, "label": "CASE" }, { "from": 140, "to": 145, "label": "EVAL with clause\neq(X22, X22).\nand substitutionT25 -> 0,\nX22 -> 0,\nT28 -> 0" }, { "from": 140, "to": 147, "label": "EVAL-BACKTRACK" }, { "from": 145, "to": 149, "label": "SUCCESS" }, { "from": 150, "to": 152, "label": "EVAL with clause\nmult(s(X35), X36, 0, X37) :- ','(!_2, mult(X35, X36, X36, X37)).\nand substitutionX35 -> T41,\nT13 -> s(T41),\nT14 -> T42,\nX36 -> T42,\nT16 -> T44,\nX37 -> T44,\nT43 -> T44" }, { "from": 150, "to": 153, "label": "EVAL-BACKTRACK" }, { "from": 151, "to": 852, "label": "EVAL with clause\nmult(X519, X520, X521, s(X522)) :- ','(p(X521, X523), mult(X519, X520, X523, X522)).\nand substitutionT13 -> T477,\nX519 -> T477,\nT14 -> T478,\nX520 -> T478,\nX521 -> 0,\nX522 -> T480,\nT16 -> s(T480),\nT479 -> T480" }, { "from": 151, "to": 854, "label": "EVAL-BACKTRACK" }, { "from": 152, "to": 154, "label": "CUT" }, { "from": 154, "to": 170, "label": "CASE" }, { "from": 170, "to": 237, "label": "EVAL with clause\nmult(0, X44, 0, X45) :- ','(!_4, eq(X45, 0)).\nand substitutionT41 -> 0,\nT42 -> 0,\nX44 -> 0,\nT51 -> 0,\nT44 -> T53,\nX45 -> T53,\nT52 -> T53" }, { "from": 170, "to": 244, "label": "EVAL-BACKTRACK" }, { "from": 237, "to": 246, "label": "CUT" }, { "from": 244, "to": 302, "label": "PARALLEL" }, { "from": 244, "to": 303, "label": "PARALLEL" }, { "from": 246, "to": 249, "label": "CASE" }, { "from": 249, "to": 257, "label": "EVAL with clause\neq(X48, X48).\nand substitutionT53 -> 0,\nX48 -> 0,\nT56 -> 0" }, { "from": 249, "to": 296, "label": "EVAL-BACKTRACK" }, { "from": 257, "to": 297, "label": "SUCCESS" }, { "from": 302, "to": 307, "label": "EVAL with clause\nmult(s(X61), X62, 0, X63) :- ','(!_4, mult(X61, X62, X62, X63)).\nand substitutionX61 -> T69,\nT41 -> s(T69),\nT42 -> 0,\nX62 -> 0,\nT70 -> 0,\nT44 -> T72,\nX63 -> T72,\nT71 -> T72" }, { "from": 302, "to": 308, "label": "EVAL-BACKTRACK" }, { "from": 303, "to": 316, "label": "EVAL with clause\nmult(X75, X76, X77, s(X78)) :- ','(p(X77, X79), mult(X75, X76, X79, X78)).\nand substitutionT41 -> T80,\nX75 -> T80,\nT42 -> T81,\nX76 -> T81,\nX77 -> T81,\nX78 -> T83,\nT44 -> s(T83),\nT82 -> T83" }, { "from": 303, "to": 317, "label": "EVAL-BACKTRACK" }, { "from": 307, "to": 309, "label": "CUT" }, { "from": 309, "to": 126, "label": "INSTANCE with matching:\nT13 -> T69\nT14 -> 0\nT16 -> T72" }, { "from": 316, "to": 320, "label": "CASE" }, { "from": 320, "to": 322, "label": "PARALLEL" }, { "from": 320, "to": 324, "label": "PARALLEL" }, { "from": 322, "to": 326, "label": "EVAL with clause\np(0, 0).\nand substitutionT81 -> 0,\nX79 -> 0" }, { "from": 322, "to": 327, "label": "EVAL-BACKTRACK" }, { "from": 324, "to": 329, "label": "EVAL with clause\np(s(X89), X89).\nand substitutionX89 -> T91,\nT81 -> s(T91),\nX79 -> T91" }, { "from": 324, "to": 330, "label": "EVAL-BACKTRACK" }, { "from": 326, "to": 126, "label": "INSTANCE with matching:\nT13 -> T80\nT14 -> 0\nT16 -> T83" }, { "from": 329, "to": 333, "label": "CASE" }, { "from": 333, "to": 336, "label": "EVAL with clause\nmult(0, X96, 0, X97) :- ','(!_7, eq(X97, 0)).\nand substitutionT80 -> 0,\nT91 -> 0,\nX96 -> s(0),\nT98 -> 0,\nT83 -> T100,\nX97 -> T100,\nT99 -> T100" }, { "from": 333, "to": 338, "label": "EVAL-BACKTRACK" }, { "from": 336, "to": 339, "label": "CUT" }, { "from": 338, "to": 345, "label": "PARALLEL" }, { "from": 338, "to": 347, "label": "PARALLEL" }, { "from": 339, "to": 340, "label": "CASE" }, { "from": 340, "to": 341, "label": "EVAL with clause\neq(X100, X100).\nand substitutionT100 -> 0,\nX100 -> 0,\nT103 -> 0" }, { "from": 340, "to": 342, "label": "EVAL-BACKTRACK" }, { "from": 341, "to": 343, "label": "SUCCESS" }, { "from": 345, "to": 351, "label": "EVAL with clause\nmult(s(X113), X114, 0, X115) :- ','(!_7, mult(X113, X114, X114, X115)).\nand substitutionX113 -> T116,\nT80 -> s(T116),\nT91 -> 0,\nX114 -> s(0),\nT117 -> 0,\nT83 -> T119,\nX115 -> T119,\nT118 -> T119" }, { "from": 345, "to": 352, "label": "EVAL-BACKTRACK" }, { "from": 347, "to": 362, "label": "EVAL with clause\nmult(X127, X128, X129, s(X130)) :- ','(p(X129, X131), mult(X127, X128, X131, X130)).\nand substitutionT80 -> T126,\nX127 -> T126,\nT91 -> T127,\nX128 -> s(T127),\nX129 -> T127,\nX130 -> T129,\nT83 -> s(T129),\nT128 -> T129" }, { "from": 347, "to": 364, "label": "EVAL-BACKTRACK" }, { "from": 351, "to": 353, "label": "CUT" }, { "from": 353, "to": 154, "label": "INSTANCE with matching:\nT41 -> T116\nT42 -> s(0)\nT44 -> T119" }, { "from": 362, "to": 365, "label": "CASE" }, { "from": 365, "to": 366, "label": "PARALLEL" }, { "from": 365, "to": 368, "label": "PARALLEL" }, { "from": 366, "to": 371, "label": "EVAL with clause\np(0, 0).\nand substitutionT127 -> 0,\nX131 -> 0" }, { "from": 366, "to": 372, "label": "EVAL-BACKTRACK" }, { "from": 368, "to": 373, "label": "EVAL with clause\np(s(X141), X141).\nand substitutionX141 -> T137,\nT127 -> s(T137),\nX131 -> T137" }, { "from": 368, "to": 374, "label": "EVAL-BACKTRACK" }, { "from": 371, "to": 126, "label": "INSTANCE with matching:\nT13 -> T126\nT14 -> s(0)\nT16 -> T129" }, { "from": 373, "to": 378, "label": "CASE" }, { "from": 378, "to": 383, "label": "EVAL with clause\nmult(0, X148, 0, X149) :- ','(!_10, eq(X149, 0)).\nand substitutionT126 -> 0,\nT137 -> 0,\nX148 -> s(s(0)),\nT144 -> 0,\nT129 -> T146,\nX149 -> T146,\nT145 -> T146" }, { "from": 378, "to": 394, "label": "EVAL-BACKTRACK" }, { "from": 383, "to": 395, "label": "CUT" }, { "from": 394, "to": 407, "label": "PARALLEL" }, { "from": 394, "to": 408, "label": "PARALLEL" }, { "from": 395, "to": 396, "label": "CASE" }, { "from": 396, "to": 397, "label": "EVAL with clause\neq(X152, X152).\nand substitutionT146 -> 0,\nX152 -> 0,\nT149 -> 0" }, { "from": 396, "to": 399, "label": "EVAL-BACKTRACK" }, { "from": 397, "to": 400, "label": "SUCCESS" }, { "from": 407, "to": 411, "label": "EVAL with clause\nmult(s(X165), X166, 0, X167) :- ','(!_10, mult(X165, X166, X166, X167)).\nand substitutionX165 -> T162,\nT126 -> s(T162),\nT137 -> 0,\nX166 -> s(s(0)),\nT163 -> 0,\nT129 -> T165,\nX167 -> T165,\nT164 -> T165" }, { "from": 407, "to": 412, "label": "EVAL-BACKTRACK" }, { "from": 408, "to": 445, "label": "EVAL with clause\nmult(X179, X180, X181, s(X182)) :- ','(p(X181, X183), mult(X179, X180, X183, X182)).\nand substitutionT126 -> T172,\nX179 -> T172,\nT137 -> T173,\nX180 -> s(s(T173)),\nX181 -> T173,\nX182 -> T175,\nT129 -> s(T175),\nT174 -> T175" }, { "from": 408, "to": 446, "label": "EVAL-BACKTRACK" }, { "from": 411, "to": 413, "label": "CUT" }, { "from": 413, "to": 154, "label": "INSTANCE with matching:\nT41 -> T162\nT42 -> s(s(0))\nT44 -> T165" }, { "from": 445, "to": 447, "label": "CASE" }, { "from": 447, "to": 448, "label": "PARALLEL" }, { "from": 447, "to": 449, "label": "PARALLEL" }, { "from": 448, "to": 452, "label": "EVAL with clause\np(0, 0).\nand substitutionT173 -> 0,\nX183 -> 0" }, { "from": 448, "to": 453, "label": "EVAL-BACKTRACK" }, { "from": 449, "to": 456, "label": "EVAL with clause\np(s(X193), X193).\nand substitutionX193 -> T183,\nT173 -> s(T183),\nX183 -> T183" }, { "from": 449, "to": 459, "label": "EVAL-BACKTRACK" }, { "from": 452, "to": 126, "label": "INSTANCE with matching:\nT13 -> T172\nT14 -> s(s(0))\nT16 -> T175" }, { "from": 456, "to": 464, "label": "CASE" }, { "from": 464, "to": 467, "label": "EVAL with clause\nmult(0, X200, 0, X201) :- ','(!_13, eq(X201, 0)).\nand substitutionT172 -> 0,\nT183 -> 0,\nX200 -> s(s(s(0))),\nT190 -> 0,\nT175 -> T192,\nX201 -> T192,\nT191 -> T192" }, { "from": 464, "to": 468, "label": "EVAL-BACKTRACK" }, { "from": 467, "to": 469, "label": "CUT" }, { "from": 468, "to": 477, "label": "PARALLEL" }, { "from": 468, "to": 478, "label": "PARALLEL" }, { "from": 469, "to": 470, "label": "CASE" }, { "from": 470, "to": 474, "label": "EVAL with clause\neq(X204, X204).\nand substitutionT192 -> 0,\nX204 -> 0,\nT195 -> 0" }, { "from": 470, "to": 475, "label": "EVAL-BACKTRACK" }, { "from": 474, "to": 476, "label": "SUCCESS" }, { "from": 477, "to": 482, "label": "EVAL with clause\nmult(s(X217), X218, 0, X219) :- ','(!_13, mult(X217, X218, X218, X219)).\nand substitutionX217 -> T208,\nT172 -> s(T208),\nT183 -> 0,\nX218 -> s(s(s(0))),\nT209 -> 0,\nT175 -> T211,\nX219 -> T211,\nT210 -> T211" }, { "from": 477, "to": 484, "label": "EVAL-BACKTRACK" }, { "from": 478, "to": 486, "label": "EVAL with clause\nmult(X231, X232, X233, s(X234)) :- ','(p(X233, X235), mult(X231, X232, X235, X234)).\nand substitutionT172 -> T218,\nX231 -> T218,\nT183 -> T219,\nX232 -> s(s(s(T219))),\nX233 -> T219,\nX234 -> T221,\nT175 -> s(T221),\nT220 -> T221" }, { "from": 478, "to": 491, "label": "EVAL-BACKTRACK" }, { "from": 482, "to": 485, "label": "CUT" }, { "from": 485, "to": 154, "label": "INSTANCE with matching:\nT41 -> T208\nT42 -> s(s(s(0)))\nT44 -> T211" }, { "from": 486, "to": 495, "label": "CASE" }, { "from": 495, "to": 496, "label": "PARALLEL" }, { "from": 495, "to": 497, "label": "PARALLEL" }, { "from": 496, "to": 498, "label": "EVAL with clause\np(0, 0).\nand substitutionT219 -> 0,\nX235 -> 0" }, { "from": 496, "to": 500, "label": "EVAL-BACKTRACK" }, { "from": 497, "to": 505, "label": "EVAL with clause\np(s(X245), X245).\nand substitutionX245 -> T229,\nT219 -> s(T229),\nX235 -> T229" }, { "from": 497, "to": 506, "label": "EVAL-BACKTRACK" }, { "from": 498, "to": 126, "label": "INSTANCE with matching:\nT13 -> T218\nT14 -> s(s(s(0)))\nT16 -> T221" }, { "from": 505, "to": 507, "label": "CASE" }, { "from": 507, "to": 510, "label": "EVAL with clause\nmult(0, X252, 0, X253) :- ','(!_16, eq(X253, 0)).\nand substitutionT218 -> 0,\nT229 -> 0,\nX252 -> s(s(s(s(0)))),\nT236 -> 0,\nT221 -> T238,\nX253 -> T238,\nT237 -> T238" }, { "from": 507, "to": 512, "label": "EVAL-BACKTRACK" }, { "from": 510, "to": 513, "label": "CUT" }, { "from": 512, "to": 522, "label": "PARALLEL" }, { "from": 512, "to": 523, "label": "PARALLEL" }, { "from": 513, "to": 514, "label": "CASE" }, { "from": 514, "to": 516, "label": "EVAL with clause\neq(X256, X256).\nand substitutionT238 -> 0,\nX256 -> 0,\nT241 -> 0" }, { "from": 514, "to": 517, "label": "EVAL-BACKTRACK" }, { "from": 516, "to": 519, "label": "SUCCESS" }, { "from": 522, "to": 524, "label": "EVAL with clause\nmult(s(X269), X270, 0, X271) :- ','(!_16, mult(X269, X270, X270, X271)).\nand substitutionX269 -> T254,\nT218 -> s(T254),\nT229 -> 0,\nX270 -> s(s(s(s(0)))),\nT255 -> 0,\nT221 -> T257,\nX271 -> T257,\nT256 -> T257" }, { "from": 522, "to": 525, "label": "EVAL-BACKTRACK" }, { "from": 523, "to": 530, "label": "EVAL with clause\nmult(X283, X284, X285, s(X286)) :- ','(p(X285, X287), mult(X283, X284, X287, X286)).\nand substitutionT218 -> T264,\nX283 -> T264,\nT229 -> T265,\nX284 -> s(s(s(s(T265)))),\nX285 -> T265,\nX286 -> T267,\nT221 -> s(T267),\nT266 -> T267" }, { "from": 523, "to": 531, "label": "EVAL-BACKTRACK" }, { "from": 524, "to": 526, "label": "CUT" }, { "from": 526, "to": 154, "label": "INSTANCE with matching:\nT41 -> T254\nT42 -> s(s(s(s(0))))\nT44 -> T257" }, { "from": 530, "to": 532, "label": "CASE" }, { "from": 532, "to": 534, "label": "PARALLEL" }, { "from": 532, "to": 535, "label": "PARALLEL" }, { "from": 534, "to": 539, "label": "EVAL with clause\np(0, 0).\nand substitutionT265 -> 0,\nX287 -> 0" }, { "from": 534, "to": 540, "label": "EVAL-BACKTRACK" }, { "from": 535, "to": 546, "label": "EVAL with clause\np(s(X297), X297).\nand substitutionX297 -> T275,\nT265 -> s(T275),\nX287 -> T275" }, { "from": 535, "to": 547, "label": "EVAL-BACKTRACK" }, { "from": 539, "to": 126, "label": "INSTANCE with matching:\nT13 -> T264\nT14 -> s(s(s(s(0))))\nT16 -> T267" }, { "from": 546, "to": 552, "label": "CASE" }, { "from": 552, "to": 578, "label": "EVAL with clause\nmult(0, X304, 0, X305) :- ','(!_19, eq(X305, 0)).\nand substitutionT264 -> 0,\nT275 -> 0,\nX304 -> s(s(s(s(s(0))))),\nT282 -> 0,\nT267 -> T284,\nX305 -> T284,\nT283 -> T284" }, { "from": 552, "to": 580, "label": "EVAL-BACKTRACK" }, { "from": 578, "to": 582, "label": "CUT" }, { "from": 580, "to": 594, "label": "PARALLEL" }, { "from": 580, "to": 597, "label": "PARALLEL" }, { "from": 582, "to": 583, "label": "CASE" }, { "from": 583, "to": 588, "label": "EVAL with clause\neq(X308, X308).\nand substitutionT284 -> 0,\nX308 -> 0,\nT287 -> 0" }, { "from": 583, "to": 589, "label": "EVAL-BACKTRACK" }, { "from": 588, "to": 590, "label": "SUCCESS" }, { "from": 594, "to": 601, "label": "EVAL with clause\nmult(s(X321), X322, 0, X323) :- ','(!_19, mult(X321, X322, X322, X323)).\nand substitutionX321 -> T300,\nT264 -> s(T300),\nT275 -> 0,\nX322 -> s(s(s(s(s(0))))),\nT301 -> 0,\nT267 -> T303,\nX323 -> T303,\nT302 -> T303" }, { "from": 594, "to": 602, "label": "EVAL-BACKTRACK" }, { "from": 597, "to": 633, "label": "EVAL with clause\nmult(X335, X336, X337, s(X338)) :- ','(p(X337, X339), mult(X335, X336, X339, X338)).\nand substitutionT264 -> T310,\nX335 -> T310,\nT275 -> T311,\nX336 -> s(s(s(s(s(T311))))),\nX337 -> T311,\nX338 -> T313,\nT267 -> s(T313),\nT312 -> T313" }, { "from": 597, "to": 634, "label": "EVAL-BACKTRACK" }, { "from": 601, "to": 603, "label": "CUT" }, { "from": 603, "to": 154, "label": "INSTANCE with matching:\nT41 -> T300\nT42 -> s(s(s(s(s(0)))))\nT44 -> T303" }, { "from": 633, "to": 639, "label": "CASE" }, { "from": 639, "to": 640, "label": "PARALLEL" }, { "from": 639, "to": 641, "label": "PARALLEL" }, { "from": 640, "to": 645, "label": "EVAL with clause\np(0, 0).\nand substitutionT311 -> 0,\nX339 -> 0" }, { "from": 640, "to": 646, "label": "EVAL-BACKTRACK" }, { "from": 641, "to": 659, "label": "EVAL with clause\np(s(X349), X349).\nand substitutionX349 -> T321,\nT311 -> s(T321),\nX339 -> T321" }, { "from": 641, "to": 660, "label": "EVAL-BACKTRACK" }, { "from": 645, "to": 126, "label": "INSTANCE with matching:\nT13 -> T310\nT14 -> s(s(s(s(s(0)))))\nT16 -> T313" }, { "from": 659, "to": 662, "label": "CASE" }, { "from": 662, "to": 665, "label": "EVAL with clause\nmult(0, X356, 0, X357) :- ','(!_22, eq(X357, 0)).\nand substitutionT310 -> 0,\nT321 -> 0,\nX356 -> s(s(s(s(s(s(0)))))),\nT328 -> 0,\nT313 -> T330,\nX357 -> T330,\nT329 -> T330" }, { "from": 662, "to": 666, "label": "EVAL-BACKTRACK" }, { "from": 665, "to": 667, "label": "CUT" }, { "from": 666, "to": 672, "label": "PARALLEL" }, { "from": 666, "to": 673, "label": "PARALLEL" }, { "from": 667, "to": 668, "label": "CASE" }, { "from": 668, "to": 669, "label": "EVAL with clause\neq(X360, X360).\nand substitutionT330 -> 0,\nX360 -> 0,\nT333 -> 0" }, { "from": 668, "to": 670, "label": "EVAL-BACKTRACK" }, { "from": 669, "to": 671, "label": "SUCCESS" }, { "from": 672, "to": 674, "label": "EVAL with clause\nmult(s(X373), X374, 0, X375) :- ','(!_22, mult(X373, X374, X374, X375)).\nand substitutionX373 -> T346,\nT310 -> s(T346),\nT321 -> 0,\nX374 -> s(s(s(s(s(s(0)))))),\nT347 -> 0,\nT313 -> T349,\nX375 -> T349,\nT348 -> T349" }, { "from": 672, "to": 675, "label": "EVAL-BACKTRACK" }, { "from": 673, "to": 679, "label": "EVAL with clause\nmult(X387, X388, X389, s(X390)) :- ','(p(X389, X391), mult(X387, X388, X391, X390)).\nand substitutionT310 -> T356,\nX387 -> T356,\nT321 -> T357,\nX388 -> s(s(s(s(s(s(T357)))))),\nX389 -> T357,\nX390 -> T359,\nT313 -> s(T359),\nT358 -> T359" }, { "from": 673, "to": 680, "label": "EVAL-BACKTRACK" }, { "from": 674, "to": 676, "label": "CUT" }, { "from": 676, "to": 154, "label": "INSTANCE with matching:\nT41 -> T346\nT42 -> s(s(s(s(s(s(0))))))\nT44 -> T349" }, { "from": 679, "to": 681, "label": "CASE" }, { "from": 681, "to": 682, "label": "PARALLEL" }, { "from": 681, "to": 683, "label": "PARALLEL" }, { "from": 682, "to": 684, "label": "EVAL with clause\np(0, 0).\nand substitutionT357 -> 0,\nX391 -> 0" }, { "from": 682, "to": 685, "label": "EVAL-BACKTRACK" }, { "from": 683, "to": 689, "label": "EVAL with clause\np(s(X401), X401).\nand substitutionX401 -> T367,\nT357 -> s(T367),\nX391 -> T367" }, { "from": 683, "to": 690, "label": "EVAL-BACKTRACK" }, { "from": 684, "to": 126, "label": "INSTANCE with matching:\nT13 -> T356\nT14 -> s(s(s(s(s(s(0))))))\nT16 -> T359" }, { "from": 689, "to": 692, "label": "CASE" }, { "from": 692, "to": 693, "label": "EVAL with clause\nmult(0, X408, 0, X409) :- ','(!_25, eq(X409, 0)).\nand substitutionT356 -> 0,\nT367 -> 0,\nX408 -> s(s(s(s(s(s(s(0))))))),\nT374 -> 0,\nT359 -> T376,\nX409 -> T376,\nT375 -> T376" }, { "from": 692, "to": 694, "label": "EVAL-BACKTRACK" }, { "from": 693, "to": 695, "label": "CUT" }, { "from": 694, "to": 706, "label": "PARALLEL" }, { "from": 694, "to": 708, "label": "PARALLEL" }, { "from": 695, "to": 696, "label": "CASE" }, { "from": 696, "to": 697, "label": "EVAL with clause\neq(X412, X412).\nand substitutionT376 -> 0,\nX412 -> 0,\nT379 -> 0" }, { "from": 696, "to": 698, "label": "EVAL-BACKTRACK" }, { "from": 697, "to": 699, "label": "SUCCESS" }, { "from": 706, "to": 733, "label": "EVAL with clause\nmult(s(X425), X426, 0, X427) :- ','(!_25, mult(X425, X426, X426, X427)).\nand substitutionX425 -> T392,\nT356 -> s(T392),\nT367 -> 0,\nX426 -> s(s(s(s(s(s(s(0))))))),\nT393 -> 0,\nT359 -> T395,\nX427 -> T395,\nT394 -> T395" }, { "from": 706, "to": 734, "label": "EVAL-BACKTRACK" }, { "from": 708, "to": 736, "label": "EVAL with clause\nmult(X439, X440, X441, s(X442)) :- ','(p(X441, X443), mult(X439, X440, X443, X442)).\nand substitutionT356 -> T402,\nX439 -> T402,\nT367 -> T403,\nX440 -> s(s(s(s(s(s(s(T403))))))),\nX441 -> T403,\nX442 -> T405,\nT359 -> s(T405),\nT404 -> T405" }, { "from": 708, "to": 737, "label": "EVAL-BACKTRACK" }, { "from": 733, "to": 735, "label": "CUT" }, { "from": 735, "to": 154, "label": "INSTANCE with matching:\nT41 -> T392\nT42 -> s(s(s(s(s(s(s(0)))))))\nT44 -> T395" }, { "from": 736, "to": 738, "label": "CASE" }, { "from": 738, "to": 739, "label": "PARALLEL" }, { "from": 738, "to": 740, "label": "PARALLEL" }, { "from": 739, "to": 741, "label": "EVAL with clause\np(0, 0).\nand substitutionT403 -> 0,\nX443 -> 0" }, { "from": 739, "to": 742, "label": "EVAL-BACKTRACK" }, { "from": 740, "to": 745, "label": "EVAL with clause\np(s(X453), X453).\nand substitutionX453 -> T413,\nT403 -> s(T413),\nX443 -> T413" }, { "from": 740, "to": 746, "label": "EVAL-BACKTRACK" }, { "from": 741, "to": 126, "label": "INSTANCE with matching:\nT13 -> T402\nT14 -> s(s(s(s(s(s(s(0)))))))\nT16 -> T405" }, { "from": 745, "to": 797, "label": "GENERALIZATION\nT416 <-- s(s(s(s(s(s(s(T413)))))))\n\nNew Knowledge:\nT416 is ground" }, { "from": 797, "to": 803, "label": "CASE" }, { "from": 803, "to": 804, "label": "EVAL with clause\nmult(0, X462, 0, X463) :- ','(!_28, eq(X463, 0)).\nand substitutionT402 -> 0,\nT416 -> T423,\nX462 -> s(T423),\nT413 -> 0,\nT405 -> T425,\nX463 -> T425,\nT424 -> T425" }, { "from": 803, "to": 805, "label": "EVAL-BACKTRACK" }, { "from": 804, "to": 806, "label": "CUT" }, { "from": 805, "to": 818, "label": "PARALLEL" }, { "from": 805, "to": 819, "label": "PARALLEL" }, { "from": 806, "to": 807, "label": "CASE" }, { "from": 807, "to": 811, "label": "EVAL with clause\neq(X466, X466).\nand substitutionT425 -> 0,\nX466 -> 0,\nT428 -> 0" }, { "from": 807, "to": 812, "label": "EVAL-BACKTRACK" }, { "from": 811, "to": 813, "label": "SUCCESS" }, { "from": 818, "to": 821, "label": "EVAL with clause\nmult(s(X479), X480, 0, X481) :- ','(!_28, mult(X479, X480, X480, X481)).\nand substitutionX479 -> T441,\nT402 -> s(T441),\nT416 -> T442,\nX480 -> s(T442),\nT413 -> 0,\nT405 -> T444,\nX481 -> T444,\nT443 -> T444" }, { "from": 818, "to": 822, "label": "EVAL-BACKTRACK" }, { "from": 819, "to": 824, "label": "EVAL with clause\nmult(X493, X494, X495, s(X496)) :- ','(p(X495, X497), mult(X493, X494, X497, X496)).\nand substitutionT402 -> T454,\nX493 -> T454,\nT416 -> T455,\nX494 -> s(T455),\nT413 -> T456,\nX495 -> T456,\nX496 -> T458,\nT405 -> s(T458),\nT457 -> T458" }, { "from": 819, "to": 826, "label": "EVAL-BACKTRACK" }, { "from": 821, "to": 823, "label": "CUT" }, { "from": 823, "to": 154, "label": "INSTANCE with matching:\nT41 -> T441\nT42 -> s(T442)\nT44 -> T444" }, { "from": 824, "to": 828, "label": "CASE" }, { "from": 828, "to": 829, "label": "PARALLEL" }, { "from": 828, "to": 830, "label": "PARALLEL" }, { "from": 829, "to": 831, "label": "EVAL with clause\np(0, 0).\nand substitutionT456 -> 0,\nX497 -> 0" }, { "from": 829, "to": 832, "label": "EVAL-BACKTRACK" }, { "from": 830, "to": 844, "label": "EVAL with clause\np(s(X507), X507).\nand substitutionX507 -> T468,\nT456 -> s(T468),\nX497 -> T468" }, { "from": 830, "to": 845, "label": "EVAL-BACKTRACK" }, { "from": 831, "to": 126, "label": "INSTANCE with matching:\nT13 -> T454\nT14 -> s(T455)\nT16 -> T458" }, { "from": 844, "to": 797, "label": "INSTANCE with matching:\nT402 -> T454\nT416 -> T455\nT413 -> T468\nT405 -> T458" }, { "from": 852, "to": 858, "label": "CASE" }, { "from": 858, "to": 860, "label": "PARALLEL" }, { "from": 858, "to": 861, "label": "PARALLEL" }, { "from": 860, "to": 863, "label": "ONLY EVAL with clause\np(0, 0).\nand substitutionX523 -> 0" }, { "from": 861, "to": 865, "label": "BACKTRACK\nfor clause: p(s(X), X)because of non-unification" }, { "from": 863, "to": 126, "label": "INSTANCE with matching:\nT13 -> T477\nT14 -> T478\nT16 -> T480" } ], "type": "Graph" } } ---------------------------------------- (56) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f3_in(T13, T14) -> U1(f126_in(T13, T14), T13, T14) U1(f126_out1(T16), T13, T14) -> f3_out1(T16) f126_in(0, T23) -> f126_out1(0) f126_in(s(T41), T42) -> U2(f154_in(T41, T42), s(T41), T42) U2(f154_out1(T44), s(T41), T42) -> f126_out1(T44) f126_in(T477, T478) -> U3(f126_in(T477, T478), T477, T478) U3(f126_out1(T480), T477, T478) -> f126_out1(s(T480)) f154_in(0, 0) -> f154_out1(0) f154_in(s(T69), 0) -> U4(f126_in(T69, 0), s(T69), 0) U4(f126_out1(T72), s(T69), 0) -> f154_out1(T72) f154_in(T80, 0) -> U5(f126_in(T80, 0), T80, 0) U5(f126_out1(T83), T80, 0) -> f154_out1(s(T83)) f154_in(0, s(0)) -> f154_out1(s(0)) f154_in(s(T116), s(0)) -> U6(f154_in(T116, s(0)), s(T116), s(0)) U6(f154_out1(T119), s(T116), s(0)) -> f154_out1(s(T119)) f154_in(T126, s(0)) -> U7(f126_in(T126, s(0)), T126, s(0)) U7(f126_out1(T129), T126, s(0)) -> f154_out1(s(s(T129))) f154_in(0, s(s(0))) -> f154_out1(s(s(0))) f154_in(s(T162), s(s(0))) -> U8(f154_in(T162, s(s(0))), s(T162), s(s(0))) U8(f154_out1(T165), s(T162), s(s(0))) -> f154_out1(s(s(T165))) f154_in(T172, s(s(0))) -> U9(f126_in(T172, s(s(0))), T172, s(s(0))) U9(f126_out1(T175), T172, s(s(0))) -> f154_out1(s(s(s(T175)))) f154_in(0, s(s(s(0)))) -> f154_out1(s(s(s(0)))) f154_in(s(T208), s(s(s(0)))) -> U10(f154_in(T208, s(s(s(0)))), s(T208), s(s(s(0)))) U10(f154_out1(T211), s(T208), s(s(s(0)))) -> f154_out1(s(s(s(T211)))) f154_in(T218, s(s(s(0)))) -> U11(f126_in(T218, s(s(s(0)))), T218, s(s(s(0)))) U11(f126_out1(T221), T218, s(s(s(0)))) -> f154_out1(s(s(s(s(T221))))) f154_in(0, s(s(s(s(0))))) -> f154_out1(s(s(s(s(0))))) f154_in(s(T254), s(s(s(s(0))))) -> U12(f154_in(T254, s(s(s(s(0))))), s(T254), s(s(s(s(0))))) U12(f154_out1(T257), s(T254), s(s(s(s(0))))) -> f154_out1(s(s(s(s(T257))))) f154_in(T264, s(s(s(s(0))))) -> U13(f126_in(T264, s(s(s(s(0))))), T264, s(s(s(s(0))))) U13(f126_out1(T267), T264, s(s(s(s(0))))) -> f154_out1(s(s(s(s(s(T267)))))) f154_in(0, s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(0)))))) f154_in(s(T300), s(s(s(s(s(0)))))) -> U14(f154_in(T300, s(s(s(s(s(0)))))), s(T300), s(s(s(s(s(0)))))) U14(f154_out1(T303), s(T300), s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(T303)))))) f154_in(T310, s(s(s(s(s(0)))))) -> U15(f126_in(T310, s(s(s(s(s(0)))))), T310, s(s(s(s(s(0)))))) U15(f126_out1(T313), T310, s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(s(T313))))))) f154_in(0, s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(0))))))) f154_in(s(T346), s(s(s(s(s(s(0))))))) -> U16(f154_in(T346, s(s(s(s(s(s(0))))))), s(T346), s(s(s(s(s(s(0))))))) U16(f154_out1(T349), s(T346), s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(T349))))))) f154_in(T356, s(s(s(s(s(s(0))))))) -> U17(f126_in(T356, s(s(s(s(s(s(0))))))), T356, s(s(s(s(s(s(0))))))) U17(f126_out1(T359), T356, s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(s(T359)))))))) f154_in(0, s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(0)))))))) f154_in(s(T392), s(s(s(s(s(s(s(0)))))))) -> U18(f154_in(T392, s(s(s(s(s(s(s(0)))))))), s(T392), s(s(s(s(s(s(s(0)))))))) U18(f154_out1(T395), s(T392), s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(T395)))))))) f154_in(T402, s(s(s(s(s(s(s(0)))))))) -> U19(f126_in(T402, s(s(s(s(s(s(s(0)))))))), T402, s(s(s(s(s(s(s(0)))))))) U19(f126_out1(T405), T402, s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(s(T405))))))))) f154_in(T402, s(s(s(s(s(s(s(s(T413))))))))) -> U20(f797_in(T402, s(s(s(s(s(s(s(T413))))))), T413), T402, s(s(s(s(s(s(s(s(T413))))))))) U20(f797_out1(T405), T402, s(s(s(s(s(s(s(s(T413))))))))) -> f154_out1(s(s(s(s(s(s(s(s(T405))))))))) f797_in(0, T423, 0) -> f797_out1(0) f797_in(s(T441), T442, 0) -> U21(f154_in(T441, s(T442)), s(T441), T442, 0) U21(f154_out1(T444), s(T441), T442, 0) -> f797_out1(T444) f797_in(T454, T455, 0) -> U22(f126_in(T454, s(T455)), T454, T455, 0) U22(f126_out1(T458), T454, T455, 0) -> f797_out1(s(T458)) f797_in(T454, T455, s(T468)) -> U23(f797_in(T454, T455, T468), T454, T455, s(T468)) U23(f797_out1(T458), T454, T455, s(T468)) -> f797_out1(s(T458)) Q is empty. ---------------------------------------- (57) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: F3_IN(T13, T14) -> U1^1(f126_in(T13, T14), T13, T14) F3_IN(T13, T14) -> F126_IN(T13, T14) F126_IN(s(T41), T42) -> U2^1(f154_in(T41, T42), s(T41), T42) F126_IN(s(T41), T42) -> F154_IN(T41, T42) F126_IN(T477, T478) -> U3^1(f126_in(T477, T478), T477, T478) F126_IN(T477, T478) -> F126_IN(T477, T478) F154_IN(s(T69), 0) -> U4^1(f126_in(T69, 0), s(T69), 0) F154_IN(s(T69), 0) -> F126_IN(T69, 0) F154_IN(T80, 0) -> U5^1(f126_in(T80, 0), T80, 0) F154_IN(T80, 0) -> F126_IN(T80, 0) F154_IN(s(T116), s(0)) -> U6^1(f154_in(T116, s(0)), s(T116), s(0)) F154_IN(s(T116), s(0)) -> F154_IN(T116, s(0)) F154_IN(T126, s(0)) -> U7^1(f126_in(T126, s(0)), T126, s(0)) F154_IN(T126, s(0)) -> F126_IN(T126, s(0)) F154_IN(s(T162), s(s(0))) -> U8^1(f154_in(T162, s(s(0))), s(T162), s(s(0))) F154_IN(s(T162), s(s(0))) -> F154_IN(T162, s(s(0))) F154_IN(T172, s(s(0))) -> U9^1(f126_in(T172, s(s(0))), T172, s(s(0))) F154_IN(T172, s(s(0))) -> F126_IN(T172, s(s(0))) F154_IN(s(T208), s(s(s(0)))) -> U10^1(f154_in(T208, s(s(s(0)))), s(T208), s(s(s(0)))) F154_IN(s(T208), s(s(s(0)))) -> F154_IN(T208, s(s(s(0)))) F154_IN(T218, s(s(s(0)))) -> U11^1(f126_in(T218, s(s(s(0)))), T218, s(s(s(0)))) F154_IN(T218, s(s(s(0)))) -> F126_IN(T218, s(s(s(0)))) F154_IN(s(T254), s(s(s(s(0))))) -> U12^1(f154_in(T254, s(s(s(s(0))))), s(T254), s(s(s(s(0))))) F154_IN(s(T254), s(s(s(s(0))))) -> F154_IN(T254, s(s(s(s(0))))) F154_IN(T264, s(s(s(s(0))))) -> U13^1(f126_in(T264, s(s(s(s(0))))), T264, s(s(s(s(0))))) F154_IN(T264, s(s(s(s(0))))) -> F126_IN(T264, s(s(s(s(0))))) F154_IN(s(T300), s(s(s(s(s(0)))))) -> U14^1(f154_in(T300, s(s(s(s(s(0)))))), s(T300), s(s(s(s(s(0)))))) F154_IN(s(T300), s(s(s(s(s(0)))))) -> F154_IN(T300, s(s(s(s(s(0)))))) F154_IN(T310, s(s(s(s(s(0)))))) -> U15^1(f126_in(T310, s(s(s(s(s(0)))))), T310, s(s(s(s(s(0)))))) F154_IN(T310, s(s(s(s(s(0)))))) -> F126_IN(T310, s(s(s(s(s(0)))))) F154_IN(s(T346), s(s(s(s(s(s(0))))))) -> U16^1(f154_in(T346, s(s(s(s(s(s(0))))))), s(T346), s(s(s(s(s(s(0))))))) F154_IN(s(T346), s(s(s(s(s(s(0))))))) -> F154_IN(T346, s(s(s(s(s(s(0))))))) F154_IN(T356, s(s(s(s(s(s(0))))))) -> U17^1(f126_in(T356, s(s(s(s(s(s(0))))))), T356, s(s(s(s(s(s(0))))))) F154_IN(T356, s(s(s(s(s(s(0))))))) -> F126_IN(T356, s(s(s(s(s(s(0))))))) F154_IN(s(T392), s(s(s(s(s(s(s(0)))))))) -> U18^1(f154_in(T392, s(s(s(s(s(s(s(0)))))))), s(T392), s(s(s(s(s(s(s(0)))))))) F154_IN(s(T392), s(s(s(s(s(s(s(0)))))))) -> F154_IN(T392, s(s(s(s(s(s(s(0)))))))) F154_IN(T402, s(s(s(s(s(s(s(0)))))))) -> U19^1(f126_in(T402, s(s(s(s(s(s(s(0)))))))), T402, s(s(s(s(s(s(s(0)))))))) F154_IN(T402, s(s(s(s(s(s(s(0)))))))) -> F126_IN(T402, s(s(s(s(s(s(s(0)))))))) F154_IN(T402, s(s(s(s(s(s(s(s(T413))))))))) -> U20^1(f797_in(T402, s(s(s(s(s(s(s(T413))))))), T413), T402, s(s(s(s(s(s(s(s(T413))))))))) F154_IN(T402, s(s(s(s(s(s(s(s(T413))))))))) -> F797_IN(T402, s(s(s(s(s(s(s(T413))))))), T413) F797_IN(s(T441), T442, 0) -> U21^1(f154_in(T441, s(T442)), s(T441), T442, 0) F797_IN(s(T441), T442, 0) -> F154_IN(T441, s(T442)) F797_IN(T454, T455, 0) -> U22^1(f126_in(T454, s(T455)), T454, T455, 0) F797_IN(T454, T455, 0) -> F126_IN(T454, s(T455)) F797_IN(T454, T455, s(T468)) -> U23^1(f797_in(T454, T455, T468), T454, T455, s(T468)) F797_IN(T454, T455, s(T468)) -> F797_IN(T454, T455, T468) The TRS R consists of the following rules: f3_in(T13, T14) -> U1(f126_in(T13, T14), T13, T14) U1(f126_out1(T16), T13, T14) -> f3_out1(T16) f126_in(0, T23) -> f126_out1(0) f126_in(s(T41), T42) -> U2(f154_in(T41, T42), s(T41), T42) U2(f154_out1(T44), s(T41), T42) -> f126_out1(T44) f126_in(T477, T478) -> U3(f126_in(T477, T478), T477, T478) U3(f126_out1(T480), T477, T478) -> f126_out1(s(T480)) f154_in(0, 0) -> f154_out1(0) f154_in(s(T69), 0) -> U4(f126_in(T69, 0), s(T69), 0) U4(f126_out1(T72), s(T69), 0) -> f154_out1(T72) f154_in(T80, 0) -> U5(f126_in(T80, 0), T80, 0) U5(f126_out1(T83), T80, 0) -> f154_out1(s(T83)) f154_in(0, s(0)) -> f154_out1(s(0)) f154_in(s(T116), s(0)) -> U6(f154_in(T116, s(0)), s(T116), s(0)) U6(f154_out1(T119), s(T116), s(0)) -> f154_out1(s(T119)) f154_in(T126, s(0)) -> U7(f126_in(T126, s(0)), T126, s(0)) U7(f126_out1(T129), T126, s(0)) -> f154_out1(s(s(T129))) f154_in(0, s(s(0))) -> f154_out1(s(s(0))) f154_in(s(T162), s(s(0))) -> U8(f154_in(T162, s(s(0))), s(T162), s(s(0))) U8(f154_out1(T165), s(T162), s(s(0))) -> f154_out1(s(s(T165))) f154_in(T172, s(s(0))) -> U9(f126_in(T172, s(s(0))), T172, s(s(0))) U9(f126_out1(T175), T172, s(s(0))) -> f154_out1(s(s(s(T175)))) f154_in(0, s(s(s(0)))) -> f154_out1(s(s(s(0)))) f154_in(s(T208), s(s(s(0)))) -> U10(f154_in(T208, s(s(s(0)))), s(T208), s(s(s(0)))) U10(f154_out1(T211), s(T208), s(s(s(0)))) -> f154_out1(s(s(s(T211)))) f154_in(T218, s(s(s(0)))) -> U11(f126_in(T218, s(s(s(0)))), T218, s(s(s(0)))) U11(f126_out1(T221), T218, s(s(s(0)))) -> f154_out1(s(s(s(s(T221))))) f154_in(0, s(s(s(s(0))))) -> f154_out1(s(s(s(s(0))))) f154_in(s(T254), s(s(s(s(0))))) -> U12(f154_in(T254, s(s(s(s(0))))), s(T254), s(s(s(s(0))))) U12(f154_out1(T257), s(T254), s(s(s(s(0))))) -> f154_out1(s(s(s(s(T257))))) f154_in(T264, s(s(s(s(0))))) -> U13(f126_in(T264, s(s(s(s(0))))), T264, s(s(s(s(0))))) U13(f126_out1(T267), T264, s(s(s(s(0))))) -> f154_out1(s(s(s(s(s(T267)))))) f154_in(0, s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(0)))))) f154_in(s(T300), s(s(s(s(s(0)))))) -> U14(f154_in(T300, s(s(s(s(s(0)))))), s(T300), s(s(s(s(s(0)))))) U14(f154_out1(T303), s(T300), s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(T303)))))) f154_in(T310, s(s(s(s(s(0)))))) -> U15(f126_in(T310, s(s(s(s(s(0)))))), T310, s(s(s(s(s(0)))))) U15(f126_out1(T313), T310, s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(s(T313))))))) f154_in(0, s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(0))))))) f154_in(s(T346), s(s(s(s(s(s(0))))))) -> U16(f154_in(T346, s(s(s(s(s(s(0))))))), s(T346), s(s(s(s(s(s(0))))))) U16(f154_out1(T349), s(T346), s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(T349))))))) f154_in(T356, s(s(s(s(s(s(0))))))) -> U17(f126_in(T356, s(s(s(s(s(s(0))))))), T356, s(s(s(s(s(s(0))))))) U17(f126_out1(T359), T356, s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(s(T359)))))))) f154_in(0, s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(0)))))))) f154_in(s(T392), s(s(s(s(s(s(s(0)))))))) -> U18(f154_in(T392, s(s(s(s(s(s(s(0)))))))), s(T392), s(s(s(s(s(s(s(0)))))))) U18(f154_out1(T395), s(T392), s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(T395)))))))) f154_in(T402, s(s(s(s(s(s(s(0)))))))) -> U19(f126_in(T402, s(s(s(s(s(s(s(0)))))))), T402, s(s(s(s(s(s(s(0)))))))) U19(f126_out1(T405), T402, s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(s(T405))))))))) f154_in(T402, s(s(s(s(s(s(s(s(T413))))))))) -> U20(f797_in(T402, s(s(s(s(s(s(s(T413))))))), T413), T402, s(s(s(s(s(s(s(s(T413))))))))) U20(f797_out1(T405), T402, s(s(s(s(s(s(s(s(T413))))))))) -> f154_out1(s(s(s(s(s(s(s(s(T405))))))))) f797_in(0, T423, 0) -> f797_out1(0) f797_in(s(T441), T442, 0) -> U21(f154_in(T441, s(T442)), s(T441), T442, 0) U21(f154_out1(T444), s(T441), T442, 0) -> f797_out1(T444) f797_in(T454, T455, 0) -> U22(f126_in(T454, s(T455)), T454, T455, 0) U22(f126_out1(T458), T454, T455, 0) -> f797_out1(s(T458)) f797_in(T454, T455, s(T468)) -> U23(f797_in(T454, T455, T468), T454, T455, s(T468)) U23(f797_out1(T458), T454, T455, s(T468)) -> f797_out1(s(T458)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 24 less nodes. ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: F126_IN(s(T41), T42) -> F154_IN(T41, T42) F154_IN(s(T69), 0) -> F126_IN(T69, 0) F126_IN(T477, T478) -> F126_IN(T477, T478) F154_IN(T80, 0) -> F126_IN(T80, 0) F154_IN(s(T116), s(0)) -> F154_IN(T116, s(0)) F154_IN(T126, s(0)) -> F126_IN(T126, s(0)) F154_IN(s(T162), s(s(0))) -> F154_IN(T162, s(s(0))) F154_IN(T172, s(s(0))) -> F126_IN(T172, s(s(0))) F154_IN(s(T208), s(s(s(0)))) -> F154_IN(T208, s(s(s(0)))) F154_IN(T218, s(s(s(0)))) -> F126_IN(T218, s(s(s(0)))) F154_IN(s(T254), s(s(s(s(0))))) -> F154_IN(T254, s(s(s(s(0))))) F154_IN(T264, s(s(s(s(0))))) -> F126_IN(T264, s(s(s(s(0))))) F154_IN(s(T300), s(s(s(s(s(0)))))) -> F154_IN(T300, s(s(s(s(s(0)))))) F154_IN(T310, s(s(s(s(s(0)))))) -> F126_IN(T310, s(s(s(s(s(0)))))) F154_IN(s(T346), s(s(s(s(s(s(0))))))) -> F154_IN(T346, s(s(s(s(s(s(0))))))) F154_IN(T356, s(s(s(s(s(s(0))))))) -> F126_IN(T356, s(s(s(s(s(s(0))))))) F154_IN(s(T392), s(s(s(s(s(s(s(0)))))))) -> F154_IN(T392, s(s(s(s(s(s(s(0)))))))) F154_IN(T402, s(s(s(s(s(s(s(0)))))))) -> F126_IN(T402, s(s(s(s(s(s(s(0)))))))) F154_IN(T402, s(s(s(s(s(s(s(s(T413))))))))) -> F797_IN(T402, s(s(s(s(s(s(s(T413))))))), T413) F797_IN(s(T441), T442, 0) -> F154_IN(T441, s(T442)) F797_IN(T454, T455, 0) -> F126_IN(T454, s(T455)) F797_IN(T454, T455, s(T468)) -> F797_IN(T454, T455, T468) The TRS R consists of the following rules: f3_in(T13, T14) -> U1(f126_in(T13, T14), T13, T14) U1(f126_out1(T16), T13, T14) -> f3_out1(T16) f126_in(0, T23) -> f126_out1(0) f126_in(s(T41), T42) -> U2(f154_in(T41, T42), s(T41), T42) U2(f154_out1(T44), s(T41), T42) -> f126_out1(T44) f126_in(T477, T478) -> U3(f126_in(T477, T478), T477, T478) U3(f126_out1(T480), T477, T478) -> f126_out1(s(T480)) f154_in(0, 0) -> f154_out1(0) f154_in(s(T69), 0) -> U4(f126_in(T69, 0), s(T69), 0) U4(f126_out1(T72), s(T69), 0) -> f154_out1(T72) f154_in(T80, 0) -> U5(f126_in(T80, 0), T80, 0) U5(f126_out1(T83), T80, 0) -> f154_out1(s(T83)) f154_in(0, s(0)) -> f154_out1(s(0)) f154_in(s(T116), s(0)) -> U6(f154_in(T116, s(0)), s(T116), s(0)) U6(f154_out1(T119), s(T116), s(0)) -> f154_out1(s(T119)) f154_in(T126, s(0)) -> U7(f126_in(T126, s(0)), T126, s(0)) U7(f126_out1(T129), T126, s(0)) -> f154_out1(s(s(T129))) f154_in(0, s(s(0))) -> f154_out1(s(s(0))) f154_in(s(T162), s(s(0))) -> U8(f154_in(T162, s(s(0))), s(T162), s(s(0))) U8(f154_out1(T165), s(T162), s(s(0))) -> f154_out1(s(s(T165))) f154_in(T172, s(s(0))) -> U9(f126_in(T172, s(s(0))), T172, s(s(0))) U9(f126_out1(T175), T172, s(s(0))) -> f154_out1(s(s(s(T175)))) f154_in(0, s(s(s(0)))) -> f154_out1(s(s(s(0)))) f154_in(s(T208), s(s(s(0)))) -> U10(f154_in(T208, s(s(s(0)))), s(T208), s(s(s(0)))) U10(f154_out1(T211), s(T208), s(s(s(0)))) -> f154_out1(s(s(s(T211)))) f154_in(T218, s(s(s(0)))) -> U11(f126_in(T218, s(s(s(0)))), T218, s(s(s(0)))) U11(f126_out1(T221), T218, s(s(s(0)))) -> f154_out1(s(s(s(s(T221))))) f154_in(0, s(s(s(s(0))))) -> f154_out1(s(s(s(s(0))))) f154_in(s(T254), s(s(s(s(0))))) -> U12(f154_in(T254, s(s(s(s(0))))), s(T254), s(s(s(s(0))))) U12(f154_out1(T257), s(T254), s(s(s(s(0))))) -> f154_out1(s(s(s(s(T257))))) f154_in(T264, s(s(s(s(0))))) -> U13(f126_in(T264, s(s(s(s(0))))), T264, s(s(s(s(0))))) U13(f126_out1(T267), T264, s(s(s(s(0))))) -> f154_out1(s(s(s(s(s(T267)))))) f154_in(0, s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(0)))))) f154_in(s(T300), s(s(s(s(s(0)))))) -> U14(f154_in(T300, s(s(s(s(s(0)))))), s(T300), s(s(s(s(s(0)))))) U14(f154_out1(T303), s(T300), s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(T303)))))) f154_in(T310, s(s(s(s(s(0)))))) -> U15(f126_in(T310, s(s(s(s(s(0)))))), T310, s(s(s(s(s(0)))))) U15(f126_out1(T313), T310, s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(s(T313))))))) f154_in(0, s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(0))))))) f154_in(s(T346), s(s(s(s(s(s(0))))))) -> U16(f154_in(T346, s(s(s(s(s(s(0))))))), s(T346), s(s(s(s(s(s(0))))))) U16(f154_out1(T349), s(T346), s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(T349))))))) f154_in(T356, s(s(s(s(s(s(0))))))) -> U17(f126_in(T356, s(s(s(s(s(s(0))))))), T356, s(s(s(s(s(s(0))))))) U17(f126_out1(T359), T356, s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(s(T359)))))))) f154_in(0, s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(0)))))))) f154_in(s(T392), s(s(s(s(s(s(s(0)))))))) -> U18(f154_in(T392, s(s(s(s(s(s(s(0)))))))), s(T392), s(s(s(s(s(s(s(0)))))))) U18(f154_out1(T395), s(T392), s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(T395)))))))) f154_in(T402, s(s(s(s(s(s(s(0)))))))) -> U19(f126_in(T402, s(s(s(s(s(s(s(0)))))))), T402, s(s(s(s(s(s(s(0)))))))) U19(f126_out1(T405), T402, s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(s(T405))))))))) f154_in(T402, s(s(s(s(s(s(s(s(T413))))))))) -> U20(f797_in(T402, s(s(s(s(s(s(s(T413))))))), T413), T402, s(s(s(s(s(s(s(s(T413))))))))) U20(f797_out1(T405), T402, s(s(s(s(s(s(s(s(T413))))))))) -> f154_out1(s(s(s(s(s(s(s(s(T405))))))))) f797_in(0, T423, 0) -> f797_out1(0) f797_in(s(T441), T442, 0) -> U21(f154_in(T441, s(T442)), s(T441), T442, 0) U21(f154_out1(T444), s(T441), T442, 0) -> f797_out1(T444) f797_in(T454, T455, 0) -> U22(f126_in(T454, s(T455)), T454, T455, 0) U22(f126_out1(T458), T454, T455, 0) -> f797_out1(s(T458)) f797_in(T454, T455, s(T468)) -> U23(f797_in(T454, T455, T468), T454, T455, s(T468)) U23(f797_out1(T458), T454, T455, s(T468)) -> f797_out1(s(T458)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. F126_IN(s(T41), T42) -> F154_IN(T41, T42) F154_IN(s(T69), 0) -> F126_IN(T69, 0) F154_IN(s(T116), s(0)) -> F154_IN(T116, s(0)) F154_IN(s(T162), s(s(0))) -> F154_IN(T162, s(s(0))) F154_IN(s(T208), s(s(s(0)))) -> F154_IN(T208, s(s(s(0)))) F154_IN(s(T254), s(s(s(s(0))))) -> F154_IN(T254, s(s(s(s(0))))) F154_IN(s(T300), s(s(s(s(s(0)))))) -> F154_IN(T300, s(s(s(s(s(0)))))) F154_IN(s(T346), s(s(s(s(s(s(0))))))) -> F154_IN(T346, s(s(s(s(s(s(0))))))) F154_IN(s(T392), s(s(s(s(s(s(s(0)))))))) -> F154_IN(T392, s(s(s(s(s(s(s(0)))))))) F797_IN(s(T441), T442, 0) -> F154_IN(T441, s(T442)) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. F126_IN(x1, x2) = x1 s(x1) = s(x1) F154_IN(x1, x2) = x1 F797_IN(x1, x2, x3) = x1 Knuth-Bendix order [KBO] with precedence:trivial and weight map: s_1=1 dummyConstant=1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: none ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: F126_IN(T477, T478) -> F126_IN(T477, T478) F154_IN(T80, 0) -> F126_IN(T80, 0) F154_IN(T126, s(0)) -> F126_IN(T126, s(0)) F154_IN(T172, s(s(0))) -> F126_IN(T172, s(s(0))) F154_IN(T218, s(s(s(0)))) -> F126_IN(T218, s(s(s(0)))) F154_IN(T264, s(s(s(s(0))))) -> F126_IN(T264, s(s(s(s(0))))) F154_IN(T310, s(s(s(s(s(0)))))) -> F126_IN(T310, s(s(s(s(s(0)))))) F154_IN(T356, s(s(s(s(s(s(0))))))) -> F126_IN(T356, s(s(s(s(s(s(0))))))) F154_IN(T402, s(s(s(s(s(s(s(0)))))))) -> F126_IN(T402, s(s(s(s(s(s(s(0)))))))) F154_IN(T402, s(s(s(s(s(s(s(s(T413))))))))) -> F797_IN(T402, s(s(s(s(s(s(s(T413))))))), T413) F797_IN(T454, T455, 0) -> F126_IN(T454, s(T455)) F797_IN(T454, T455, s(T468)) -> F797_IN(T454, T455, T468) The TRS R consists of the following rules: f3_in(T13, T14) -> U1(f126_in(T13, T14), T13, T14) U1(f126_out1(T16), T13, T14) -> f3_out1(T16) f126_in(0, T23) -> f126_out1(0) f126_in(s(T41), T42) -> U2(f154_in(T41, T42), s(T41), T42) U2(f154_out1(T44), s(T41), T42) -> f126_out1(T44) f126_in(T477, T478) -> U3(f126_in(T477, T478), T477, T478) U3(f126_out1(T480), T477, T478) -> f126_out1(s(T480)) f154_in(0, 0) -> f154_out1(0) f154_in(s(T69), 0) -> U4(f126_in(T69, 0), s(T69), 0) U4(f126_out1(T72), s(T69), 0) -> f154_out1(T72) f154_in(T80, 0) -> U5(f126_in(T80, 0), T80, 0) U5(f126_out1(T83), T80, 0) -> f154_out1(s(T83)) f154_in(0, s(0)) -> f154_out1(s(0)) f154_in(s(T116), s(0)) -> U6(f154_in(T116, s(0)), s(T116), s(0)) U6(f154_out1(T119), s(T116), s(0)) -> f154_out1(s(T119)) f154_in(T126, s(0)) -> U7(f126_in(T126, s(0)), T126, s(0)) U7(f126_out1(T129), T126, s(0)) -> f154_out1(s(s(T129))) f154_in(0, s(s(0))) -> f154_out1(s(s(0))) f154_in(s(T162), s(s(0))) -> U8(f154_in(T162, s(s(0))), s(T162), s(s(0))) U8(f154_out1(T165), s(T162), s(s(0))) -> f154_out1(s(s(T165))) f154_in(T172, s(s(0))) -> U9(f126_in(T172, s(s(0))), T172, s(s(0))) U9(f126_out1(T175), T172, s(s(0))) -> f154_out1(s(s(s(T175)))) f154_in(0, s(s(s(0)))) -> f154_out1(s(s(s(0)))) f154_in(s(T208), s(s(s(0)))) -> U10(f154_in(T208, s(s(s(0)))), s(T208), s(s(s(0)))) U10(f154_out1(T211), s(T208), s(s(s(0)))) -> f154_out1(s(s(s(T211)))) f154_in(T218, s(s(s(0)))) -> U11(f126_in(T218, s(s(s(0)))), T218, s(s(s(0)))) U11(f126_out1(T221), T218, s(s(s(0)))) -> f154_out1(s(s(s(s(T221))))) f154_in(0, s(s(s(s(0))))) -> f154_out1(s(s(s(s(0))))) f154_in(s(T254), s(s(s(s(0))))) -> U12(f154_in(T254, s(s(s(s(0))))), s(T254), s(s(s(s(0))))) U12(f154_out1(T257), s(T254), s(s(s(s(0))))) -> f154_out1(s(s(s(s(T257))))) f154_in(T264, s(s(s(s(0))))) -> U13(f126_in(T264, s(s(s(s(0))))), T264, s(s(s(s(0))))) U13(f126_out1(T267), T264, s(s(s(s(0))))) -> f154_out1(s(s(s(s(s(T267)))))) f154_in(0, s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(0)))))) f154_in(s(T300), s(s(s(s(s(0)))))) -> U14(f154_in(T300, s(s(s(s(s(0)))))), s(T300), s(s(s(s(s(0)))))) U14(f154_out1(T303), s(T300), s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(T303)))))) f154_in(T310, s(s(s(s(s(0)))))) -> U15(f126_in(T310, s(s(s(s(s(0)))))), T310, s(s(s(s(s(0)))))) U15(f126_out1(T313), T310, s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(s(T313))))))) f154_in(0, s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(0))))))) f154_in(s(T346), s(s(s(s(s(s(0))))))) -> U16(f154_in(T346, s(s(s(s(s(s(0))))))), s(T346), s(s(s(s(s(s(0))))))) U16(f154_out1(T349), s(T346), s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(T349))))))) f154_in(T356, s(s(s(s(s(s(0))))))) -> U17(f126_in(T356, s(s(s(s(s(s(0))))))), T356, s(s(s(s(s(s(0))))))) U17(f126_out1(T359), T356, s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(s(T359)))))))) f154_in(0, s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(0)))))))) f154_in(s(T392), s(s(s(s(s(s(s(0)))))))) -> U18(f154_in(T392, s(s(s(s(s(s(s(0)))))))), s(T392), s(s(s(s(s(s(s(0)))))))) U18(f154_out1(T395), s(T392), s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(T395)))))))) f154_in(T402, s(s(s(s(s(s(s(0)))))))) -> U19(f126_in(T402, s(s(s(s(s(s(s(0)))))))), T402, s(s(s(s(s(s(s(0)))))))) U19(f126_out1(T405), T402, s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(s(T405))))))))) f154_in(T402, s(s(s(s(s(s(s(s(T413))))))))) -> U20(f797_in(T402, s(s(s(s(s(s(s(T413))))))), T413), T402, s(s(s(s(s(s(s(s(T413))))))))) U20(f797_out1(T405), T402, s(s(s(s(s(s(s(s(T413))))))))) -> f154_out1(s(s(s(s(s(s(s(s(T405))))))))) f797_in(0, T423, 0) -> f797_out1(0) f797_in(s(T441), T442, 0) -> U21(f154_in(T441, s(T442)), s(T441), T442, 0) U21(f154_out1(T444), s(T441), T442, 0) -> f797_out1(T444) f797_in(T454, T455, 0) -> U22(f126_in(T454, s(T455)), T454, T455, 0) U22(f126_out1(T458), T454, T455, 0) -> f797_out1(s(T458)) f797_in(T454, T455, s(T468)) -> U23(f797_in(T454, T455, T468), T454, T455, s(T468)) U23(f797_out1(T458), T454, T455, s(T468)) -> f797_out1(s(T458)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 10 less nodes. ---------------------------------------- (64) Complex Obligation (AND) ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: F126_IN(T477, T478) -> F126_IN(T477, T478) The TRS R consists of the following rules: f3_in(T13, T14) -> U1(f126_in(T13, T14), T13, T14) U1(f126_out1(T16), T13, T14) -> f3_out1(T16) f126_in(0, T23) -> f126_out1(0) f126_in(s(T41), T42) -> U2(f154_in(T41, T42), s(T41), T42) U2(f154_out1(T44), s(T41), T42) -> f126_out1(T44) f126_in(T477, T478) -> U3(f126_in(T477, T478), T477, T478) U3(f126_out1(T480), T477, T478) -> f126_out1(s(T480)) f154_in(0, 0) -> f154_out1(0) f154_in(s(T69), 0) -> U4(f126_in(T69, 0), s(T69), 0) U4(f126_out1(T72), s(T69), 0) -> f154_out1(T72) f154_in(T80, 0) -> U5(f126_in(T80, 0), T80, 0) U5(f126_out1(T83), T80, 0) -> f154_out1(s(T83)) f154_in(0, s(0)) -> f154_out1(s(0)) f154_in(s(T116), s(0)) -> U6(f154_in(T116, s(0)), s(T116), s(0)) U6(f154_out1(T119), s(T116), s(0)) -> f154_out1(s(T119)) f154_in(T126, s(0)) -> U7(f126_in(T126, s(0)), T126, s(0)) U7(f126_out1(T129), T126, s(0)) -> f154_out1(s(s(T129))) f154_in(0, s(s(0))) -> f154_out1(s(s(0))) f154_in(s(T162), s(s(0))) -> U8(f154_in(T162, s(s(0))), s(T162), s(s(0))) U8(f154_out1(T165), s(T162), s(s(0))) -> f154_out1(s(s(T165))) f154_in(T172, s(s(0))) -> U9(f126_in(T172, s(s(0))), T172, s(s(0))) U9(f126_out1(T175), T172, s(s(0))) -> f154_out1(s(s(s(T175)))) f154_in(0, s(s(s(0)))) -> f154_out1(s(s(s(0)))) f154_in(s(T208), s(s(s(0)))) -> U10(f154_in(T208, s(s(s(0)))), s(T208), s(s(s(0)))) U10(f154_out1(T211), s(T208), s(s(s(0)))) -> f154_out1(s(s(s(T211)))) f154_in(T218, s(s(s(0)))) -> U11(f126_in(T218, s(s(s(0)))), T218, s(s(s(0)))) U11(f126_out1(T221), T218, s(s(s(0)))) -> f154_out1(s(s(s(s(T221))))) f154_in(0, s(s(s(s(0))))) -> f154_out1(s(s(s(s(0))))) f154_in(s(T254), s(s(s(s(0))))) -> U12(f154_in(T254, s(s(s(s(0))))), s(T254), s(s(s(s(0))))) U12(f154_out1(T257), s(T254), s(s(s(s(0))))) -> f154_out1(s(s(s(s(T257))))) f154_in(T264, s(s(s(s(0))))) -> U13(f126_in(T264, s(s(s(s(0))))), T264, s(s(s(s(0))))) U13(f126_out1(T267), T264, s(s(s(s(0))))) -> f154_out1(s(s(s(s(s(T267)))))) f154_in(0, s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(0)))))) f154_in(s(T300), s(s(s(s(s(0)))))) -> U14(f154_in(T300, s(s(s(s(s(0)))))), s(T300), s(s(s(s(s(0)))))) U14(f154_out1(T303), s(T300), s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(T303)))))) f154_in(T310, s(s(s(s(s(0)))))) -> U15(f126_in(T310, s(s(s(s(s(0)))))), T310, s(s(s(s(s(0)))))) U15(f126_out1(T313), T310, s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(s(T313))))))) f154_in(0, s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(0))))))) f154_in(s(T346), s(s(s(s(s(s(0))))))) -> U16(f154_in(T346, s(s(s(s(s(s(0))))))), s(T346), s(s(s(s(s(s(0))))))) U16(f154_out1(T349), s(T346), s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(T349))))))) f154_in(T356, s(s(s(s(s(s(0))))))) -> U17(f126_in(T356, s(s(s(s(s(s(0))))))), T356, s(s(s(s(s(s(0))))))) U17(f126_out1(T359), T356, s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(s(T359)))))))) f154_in(0, s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(0)))))))) f154_in(s(T392), s(s(s(s(s(s(s(0)))))))) -> U18(f154_in(T392, s(s(s(s(s(s(s(0)))))))), s(T392), s(s(s(s(s(s(s(0)))))))) U18(f154_out1(T395), s(T392), s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(T395)))))))) f154_in(T402, s(s(s(s(s(s(s(0)))))))) -> U19(f126_in(T402, s(s(s(s(s(s(s(0)))))))), T402, s(s(s(s(s(s(s(0)))))))) U19(f126_out1(T405), T402, s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(s(T405))))))))) f154_in(T402, s(s(s(s(s(s(s(s(T413))))))))) -> U20(f797_in(T402, s(s(s(s(s(s(s(T413))))))), T413), T402, s(s(s(s(s(s(s(s(T413))))))))) U20(f797_out1(T405), T402, s(s(s(s(s(s(s(s(T413))))))))) -> f154_out1(s(s(s(s(s(s(s(s(T405))))))))) f797_in(0, T423, 0) -> f797_out1(0) f797_in(s(T441), T442, 0) -> U21(f154_in(T441, s(T442)), s(T441), T442, 0) U21(f154_out1(T444), s(T441), T442, 0) -> f797_out1(T444) f797_in(T454, T455, 0) -> U22(f126_in(T454, s(T455)), T454, T455, 0) U22(f126_out1(T458), T454, T455, 0) -> f797_out1(s(T458)) f797_in(T454, T455, s(T468)) -> U23(f797_in(T454, T455, T468), T454, T455, s(T468)) U23(f797_out1(T458), T454, T455, s(T468)) -> f797_out1(s(T458)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (66) 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. ---------------------------------------- (67) Obligation: Q DP problem: The TRS P consists of the following rules: F126_IN(T477, T478) -> F126_IN(T477, T478) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (68) 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 = F126_IN(T477, T478) evaluates to t =F126_IN(T477, T478) 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 F126_IN(T477, T478) to F126_IN(T477, T478). ---------------------------------------- (69) NO ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: F797_IN(T454, T455, s(T468)) -> F797_IN(T454, T455, T468) The TRS R consists of the following rules: f3_in(T13, T14) -> U1(f126_in(T13, T14), T13, T14) U1(f126_out1(T16), T13, T14) -> f3_out1(T16) f126_in(0, T23) -> f126_out1(0) f126_in(s(T41), T42) -> U2(f154_in(T41, T42), s(T41), T42) U2(f154_out1(T44), s(T41), T42) -> f126_out1(T44) f126_in(T477, T478) -> U3(f126_in(T477, T478), T477, T478) U3(f126_out1(T480), T477, T478) -> f126_out1(s(T480)) f154_in(0, 0) -> f154_out1(0) f154_in(s(T69), 0) -> U4(f126_in(T69, 0), s(T69), 0) U4(f126_out1(T72), s(T69), 0) -> f154_out1(T72) f154_in(T80, 0) -> U5(f126_in(T80, 0), T80, 0) U5(f126_out1(T83), T80, 0) -> f154_out1(s(T83)) f154_in(0, s(0)) -> f154_out1(s(0)) f154_in(s(T116), s(0)) -> U6(f154_in(T116, s(0)), s(T116), s(0)) U6(f154_out1(T119), s(T116), s(0)) -> f154_out1(s(T119)) f154_in(T126, s(0)) -> U7(f126_in(T126, s(0)), T126, s(0)) U7(f126_out1(T129), T126, s(0)) -> f154_out1(s(s(T129))) f154_in(0, s(s(0))) -> f154_out1(s(s(0))) f154_in(s(T162), s(s(0))) -> U8(f154_in(T162, s(s(0))), s(T162), s(s(0))) U8(f154_out1(T165), s(T162), s(s(0))) -> f154_out1(s(s(T165))) f154_in(T172, s(s(0))) -> U9(f126_in(T172, s(s(0))), T172, s(s(0))) U9(f126_out1(T175), T172, s(s(0))) -> f154_out1(s(s(s(T175)))) f154_in(0, s(s(s(0)))) -> f154_out1(s(s(s(0)))) f154_in(s(T208), s(s(s(0)))) -> U10(f154_in(T208, s(s(s(0)))), s(T208), s(s(s(0)))) U10(f154_out1(T211), s(T208), s(s(s(0)))) -> f154_out1(s(s(s(T211)))) f154_in(T218, s(s(s(0)))) -> U11(f126_in(T218, s(s(s(0)))), T218, s(s(s(0)))) U11(f126_out1(T221), T218, s(s(s(0)))) -> f154_out1(s(s(s(s(T221))))) f154_in(0, s(s(s(s(0))))) -> f154_out1(s(s(s(s(0))))) f154_in(s(T254), s(s(s(s(0))))) -> U12(f154_in(T254, s(s(s(s(0))))), s(T254), s(s(s(s(0))))) U12(f154_out1(T257), s(T254), s(s(s(s(0))))) -> f154_out1(s(s(s(s(T257))))) f154_in(T264, s(s(s(s(0))))) -> U13(f126_in(T264, s(s(s(s(0))))), T264, s(s(s(s(0))))) U13(f126_out1(T267), T264, s(s(s(s(0))))) -> f154_out1(s(s(s(s(s(T267)))))) f154_in(0, s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(0)))))) f154_in(s(T300), s(s(s(s(s(0)))))) -> U14(f154_in(T300, s(s(s(s(s(0)))))), s(T300), s(s(s(s(s(0)))))) U14(f154_out1(T303), s(T300), s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(T303)))))) f154_in(T310, s(s(s(s(s(0)))))) -> U15(f126_in(T310, s(s(s(s(s(0)))))), T310, s(s(s(s(s(0)))))) U15(f126_out1(T313), T310, s(s(s(s(s(0)))))) -> f154_out1(s(s(s(s(s(s(T313))))))) f154_in(0, s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(0))))))) f154_in(s(T346), s(s(s(s(s(s(0))))))) -> U16(f154_in(T346, s(s(s(s(s(s(0))))))), s(T346), s(s(s(s(s(s(0))))))) U16(f154_out1(T349), s(T346), s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(T349))))))) f154_in(T356, s(s(s(s(s(s(0))))))) -> U17(f126_in(T356, s(s(s(s(s(s(0))))))), T356, s(s(s(s(s(s(0))))))) U17(f126_out1(T359), T356, s(s(s(s(s(s(0))))))) -> f154_out1(s(s(s(s(s(s(s(T359)))))))) f154_in(0, s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(0)))))))) f154_in(s(T392), s(s(s(s(s(s(s(0)))))))) -> U18(f154_in(T392, s(s(s(s(s(s(s(0)))))))), s(T392), s(s(s(s(s(s(s(0)))))))) U18(f154_out1(T395), s(T392), s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(T395)))))))) f154_in(T402, s(s(s(s(s(s(s(0)))))))) -> U19(f126_in(T402, s(s(s(s(s(s(s(0)))))))), T402, s(s(s(s(s(s(s(0)))))))) U19(f126_out1(T405), T402, s(s(s(s(s(s(s(0)))))))) -> f154_out1(s(s(s(s(s(s(s(s(T405))))))))) f154_in(T402, s(s(s(s(s(s(s(s(T413))))))))) -> U20(f797_in(T402, s(s(s(s(s(s(s(T413))))))), T413), T402, s(s(s(s(s(s(s(s(T413))))))))) U20(f797_out1(T405), T402, s(s(s(s(s(s(s(s(T413))))))))) -> f154_out1(s(s(s(s(s(s(s(s(T405))))))))) f797_in(0, T423, 0) -> f797_out1(0) f797_in(s(T441), T442, 0) -> U21(f154_in(T441, s(T442)), s(T441), T442, 0) U21(f154_out1(T444), s(T441), T442, 0) -> f797_out1(T444) f797_in(T454, T455, 0) -> U22(f126_in(T454, s(T455)), T454, T455, 0) U22(f126_out1(T458), T454, T455, 0) -> f797_out1(s(T458)) f797_in(T454, T455, s(T468)) -> U23(f797_in(T454, T455, T468), T454, T455, s(T468)) U23(f797_out1(T458), T454, T455, s(T468)) -> f797_out1(s(T458)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) 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. ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: F797_IN(T454, T455, s(T468)) -> F797_IN(T454, T455, T468) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) 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: *F797_IN(T454, T455, s(T468)) -> F797_IN(T454, T455, T468) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (74) YES ---------------------------------------- (75) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 5, "program": { "directives": [], "clauses": [ [ "(times X Y Z)", "(mult X Y (0) Z)" ], [ "(mult (0) Y (0) Z)", "(',' (!) (eq Z (0)))" ], [ "(mult (s X) Y (0) Z)", "(',' (!) (mult X Y Y Z))" ], [ "(mult X Y W (s Z))", "(',' (p W P) (mult X Y P Z))" ], [ "(p (0) (0))", null ], [ "(p (s X) X)", null ], [ "(eq X X)", null ] ] }, "graph": { "nodes": { "908": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "471": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (!_10) (mult T162 (s (s (0))) (s (s (0))) T165))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T162"], "free": [], "exprvars": [] } }, "472": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "473": { "goal": [{ "clause": -1, "scope": -1, "term": "(mult T162 (s (s (0))) (s (s (0))) T165)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T162"], "free": [], "exprvars": [] } }, "235": { "goal": [ { "clause": 1, "scope": 2, "term": "(mult T13 T14 (0) T16)" }, { "clause": 2, "scope": 2, "term": "(mult T13 T14 (0) T16)" }, { "clause": 3, "scope": 2, "term": "(mult T13 T14 (0) T16)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T13", "T14" ], "free": [], "exprvars": [] } }, "479": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T173 X183) (mult T172 (s (s T173)) X183 T175))" }], "kb": { "nonunifying": [[ "(mult T172 (s (s T173)) T173 T129)", "(mult (0) X148 (0) X149)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T172", "T173" ], "free": [ "X148", "X149", "X183" ], "exprvars": [] } }, "480": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "481": { "goal": [ { "clause": 4, "scope": 12, "term": "(',' (p T173 X183) (mult T172 (s (s T173)) X183 T175))" }, { "clause": 5, "scope": 12, "term": "(',' (p T173 X183) (mult T172 (s (s T173)) X183 T175))" } ], "kb": { "nonunifying": [[ "(mult T172 (s (s T173)) T173 T129)", "(mult (0) X148 (0) X149)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T172", "T173" ], "free": [ "X148", "X149", "X183" ], "exprvars": [] } }, "483": { "goal": [{ "clause": 4, "scope": 12, "term": "(',' (p T173 X183) (mult T172 (s (s T173)) X183 T175))" }], "kb": { "nonunifying": [[ "(mult T172 (s (s T173)) T173 T129)", "(mult (0) X148 (0) X149)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T172", "T173" ], "free": [ "X148", "X149", "X183" ], "exprvars": [] } }, "5": { "goal": [{ "clause": -1, "scope": -1, "term": "(times T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "6": { "goal": [{ "clause": 0, "scope": 1, "term": "(times T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "808": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T357 X391) (mult T356 (s (s (s (s (s (s T357)))))) X391 T359))" }], "kb": { "nonunifying": [[ "(mult T356 (s (s (s (s (s (s T357)))))) T357 T313)", "(mult (0) X356 (0) X357)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T356", "T357" ], "free": [ "X356", "X357", "X391" ], "exprvars": [] } }, "252": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_2) (eq T25 (0)))" }, { "clause": 2, "scope": 2, "term": "(mult (0) T23 (0) T16)" }, { "clause": 3, "scope": 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"scope": 13, "term": "(mult T172 (s (s (s T183))) T183 T175)" }], "kb": { "nonunifying": [[ "(mult T172 (s (s (s T183))) T183 T175)", "(mult (0) X200 (0) X201)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T172", "T183" ], "free": [ "X200", "X201" ], "exprvars": [] } }, "454": { "goal": [ { "clause": 1, "scope": 10, "term": "(mult T126 (s (s T137)) T137 T129)" }, { "clause": 2, "scope": 10, "term": "(mult T126 (s (s T137)) T137 T129)" }, { "clause": 3, "scope": 10, "term": "(mult T126 (s (s T137)) T137 T129)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T126", "T137" ], "free": [], "exprvars": [] } }, "575": { "goal": [{ "clause": 3, "scope": 13, "term": "(mult T172 (s (s (s T183))) T183 T175)" }], "kb": { "nonunifying": [[ "(mult T172 (s (s (s T183))) T183 T175)", "(mult (0) X200 (0) X201)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T172", "T183" ], "free": [ "X200", "X201" ], "exprvars": [] } }, "334": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (!_4) (mult T69 (0) (0) T72))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T69"], "free": [], "exprvars": [] } }, "455": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_10) (eq T146 (0)))" }, { "clause": 2, "scope": 10, "term": "(mult (0) (s (s (0))) (0) T129)" }, { "clause": 3, "scope": 10, "term": "(mult (0) (s (s (0))) (0) T129)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "335": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "577": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (!_13) (mult T208 (s (s (s (0)))) (s (s (s (0)))) T211))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T208"], "free": [], "exprvars": [] } }, "457": { "goal": [ { "clause": 2, "scope": 10, "term": "(mult T126 (s (s T137)) T137 T129)" }, { "clause": 3, "scope": 10, "term": "(mult T126 (s (s T137)) T137 T129)" } ], "kb": { "nonunifying": [[ "(mult T126 (s (s T137)) T137 T129)", "(mult (0) X148 (0) X149)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T126", "T137" ], "free": [ "X148", "X149" ], "exprvars": [] } }, "337": { "goal": [{ "clause": -1, "scope": -1, "term": "(mult T69 (0) (0) T72)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T69"], "free": [], "exprvars": [] } }, "458": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq T146 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "579": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "460": { "goal": [{ "clause": 6, "scope": 11, "term": "(eq T146 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "581": { "goal": [{ "clause": -1, "scope": -1, "term": "(mult T208 (s (s (s (0)))) (s (s (s (0)))) T211)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T208"], "free": [], "exprvars": [] } }, "461": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "462": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "463": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "344": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T81 X79) (mult T80 T81 X79 T83))" }], "kb": { "nonunifying": [[ "(mult T80 T81 T81 T44)", "(mult (0) X44 (0) X45)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T80", "T81" ], "free": [ "X44", "X45", "X79" ], "exprvars": [] } }, "465": { "goal": [{ "clause": 2, "scope": 10, "term": "(mult T126 (s (s T137)) T137 T129)" }], "kb": { "nonunifying": [[ "(mult T126 (s (s T137)) T137 T129)", "(mult (0) X148 (0) X149)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T126", "T137" ], "free": [ "X148", "X149" ], "exprvars": [] } }, "466": { "goal": [{ "clause": 3, "scope": 10, "term": "(mult T126 (s (s T137)) T137 T129)" }], "kb": { "nonunifying": [[ "(mult T126 (s (s T137)) T137 T129)", "(mult (0) X148 (0) X149)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T126", "T137" ], "free": [ "X148", "X149" ], "exprvars": [] } }, "346": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "901": { "goal": [ { "clause": 4, "scope": 31, "term": "(',' (p (0) X523) (mult T477 T478 X523 T480))" }, { "clause": 5, "scope": 31, "term": "(',' (p (0) X523) (mult T477 T478 X523 T480))" } ], "kb": { "nonunifying": [[ "(mult T477 T478 (0) T16)", "(mult (0) X18 (0) X19)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T477", "T478" ], "free": [ "X18", "X19", "X523" ], "exprvars": [] } }, "902": { "goal": [{ "clause": 4, "scope": 31, "term": "(',' (p (0) X523) (mult T477 T478 X523 T480))" }], "kb": { "nonunifying": [[ "(mult T477 T478 (0) T16)", "(mult (0) X18 (0) X19)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T477", "T478" ], "free": [ "X18", "X19", "X523" ], "exprvars": [] } }, "903": { "goal": [{ "clause": 5, "scope": 31, "term": "(',' (p (0) X523) (mult T477 T478 X523 T480))" }], "kb": { "nonunifying": [[ "(mult T477 T478 (0) T16)", "(mult (0) X18 (0) X19)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T477", "T478" ], "free": [ "X18", "X19", "X523" ], "exprvars": [] } }, "904": { "goal": [{ "clause": -1, "scope": -1, "term": "(mult T477 T478 (0) T480)" }], "kb": { "nonunifying": [[ "(mult T477 T478 (0) T16)", "(mult (0) X18 (0) X19)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T477", "T478" ], "free": [ "X18", "X19" ], "exprvars": [] } } }, "edges": [ { "from": 5, "to": 6, "label": "CASE" }, { "from": 6, "to": 208, "label": "ONLY EVAL with clause\ntimes(X9, X10, X11) :- mult(X9, X10, 0, X11).\nand substitutionT1 -> T13,\nX9 -> T13,\nT2 -> T14,\nX10 -> T14,\nT3 -> T16,\nX11 -> T16,\nT15 -> T16" }, { "from": 208, "to": 235, "label": "CASE" }, { "from": 235, "to": 252, "label": "EVAL with clause\nmult(0, X18, 0, X19) :- ','(!_2, eq(X19, 0)).\nand substitutionT13 -> 0,\nT14 -> T23,\nX18 -> T23,\nT16 -> T25,\nX19 -> T25,\nT24 -> T25" }, { "from": 235, "to": 258, "label": "EVAL-BACKTRACK" }, { "from": 252, "to": 260, "label": "CUT" }, { "from": 258, "to": 282, "label": "PARALLEL" }, { "from": 258, "to": 284, "label": "PARALLEL" }, { "from": 260, "to": 263, "label": "CASE" }, { "from": 263, "to": 268, "label": "EVAL with clause\neq(X22, X22).\nand substitutionT25 -> 0,\nX22 -> 0,\nT28 -> 0" }, { "from": 263, "to": 270, "label": "EVAL-BACKTRACK" }, { "from": 268, "to": 271, "label": "SUCCESS" }, { "from": 282, "to": 288, "label": "EVAL with clause\nmult(s(X35), X36, 0, X37) :- ','(!_2, mult(X35, X36, X36, X37)).\nand substitutionX35 -> T41,\nT13 -> s(T41),\nT14 -> T42,\nX36 -> T42,\nT16 -> T44,\nX37 -> T44,\nT43 -> T44" }, { "from": 282, "to": 290, "label": "EVAL-BACKTRACK" }, { "from": 284, "to": 898, "label": "EVAL with clause\nmult(X519, X520, X521, s(X522)) :- ','(p(X521, X523), mult(X519, X520, X523, X522)).\nand substitutionT13 -> T477,\nX519 -> T477,\nT14 -> T478,\nX520 -> T478,\nX521 -> 0,\nX522 -> T480,\nT16 -> s(T480),\nT479 -> T480" }, { "from": 284, "to": 899, "label": "EVAL-BACKTRACK" }, { "from": 288, "to": 291, "label": "CUT" }, { "from": 291, "to": 295, "label": "CASE" }, { "from": 295, "to": 298, "label": "EVAL with clause\nmult(0, X44, 0, X45) :- ','(!_4, eq(X45, 0)).\nand substitutionT41 -> 0,\nT42 -> 0,\nX44 -> 0,\nT51 -> 0,\nT44 -> T53,\nX45 -> T53,\nT52 -> T53" }, { "from": 295, "to": 299, "label": "EVAL-BACKTRACK" }, { "from": 298, "to": 300, "label": "CUT" }, { "from": 299, "to": 331, "label": "PARALLEL" }, { "from": 299, "to": 332, "label": "PARALLEL" }, { "from": 300, "to": 301, "label": "CASE" }, { "from": 301, "to": 304, "label": "EVAL with clause\neq(X48, X48).\nand substitutionT53 -> 0,\nX48 -> 0,\nT56 -> 0" }, { "from": 301, "to": 305, "label": "EVAL-BACKTRACK" }, { "from": 304, "to": 306, "label": "SUCCESS" }, { "from": 331, "to": 334, "label": "EVAL with clause\nmult(s(X61), X62, 0, X63) :- ','(!_4, mult(X61, X62, X62, X63)).\nand substitutionX61 -> T69,\nT41 -> s(T69),\nT42 -> 0,\nX62 -> 0,\nT70 -> 0,\nT44 -> T72,\nX63 -> T72,\nT71 -> T72" }, { "from": 331, "to": 335, "label": "EVAL-BACKTRACK" }, { "from": 332, "to": 344, "label": "EVAL with clause\nmult(X75, X76, X77, s(X78)) :- ','(p(X77, X79), mult(X75, X76, X79, X78)).\nand substitutionT41 -> T80,\nX75 -> T80,\nT42 -> T81,\nX76 -> T81,\nX77 -> T81,\nX78 -> T83,\nT44 -> s(T83),\nT82 -> T83" }, { "from": 332, "to": 346, "label": "EVAL-BACKTRACK" }, { "from": 334, "to": 337, "label": "CUT" }, { "from": 337, "to": 208, "label": "INSTANCE with matching:\nT13 -> T69\nT14 -> 0\nT16 -> T72" }, { "from": 344, "to": 398, "label": "CASE" }, { "from": 398, "to": 402, "label": "PARALLEL" }, { "from": 398, "to": 404, "label": "PARALLEL" }, { "from": 402, "to": 405, "label": "EVAL with clause\np(0, 0).\nand substitutionT81 -> 0,\nX79 -> 0" }, { "from": 402, "to": 406, "label": "EVAL-BACKTRACK" }, { "from": 404, "to": 409, "label": "EVAL with clause\np(s(X89), X89).\nand substitutionX89 -> T91,\nT81 -> s(T91),\nX79 -> T91" }, { "from": 404, "to": 410, "label": "EVAL-BACKTRACK" }, { "from": 405, "to": 208, "label": "INSTANCE with matching:\nT13 -> T80\nT14 -> 0\nT16 -> T83" }, { "from": 409, "to": 414, "label": "CASE" }, { "from": 414, "to": 415, "label": "EVAL with clause\nmult(0, X96, 0, X97) :- ','(!_7, eq(X97, 0)).\nand substitutionT80 -> 0,\nT91 -> 0,\nX96 -> s(0),\nT98 -> 0,\nT83 -> T100,\nX97 -> T100,\nT99 -> T100" }, { "from": 414, "to": 416, "label": "EVAL-BACKTRACK" }, { "from": 415, "to": 417, "label": "CUT" }, { "from": 416, "to": 432, "label": "PARALLEL" }, { "from": 416, "to": 433, "label": "PARALLEL" }, { "from": 417, "to": 428, "label": "CASE" }, { "from": 428, "to": 429, "label": "EVAL with clause\neq(X100, X100).\nand substitutionT100 -> 0,\nX100 -> 0,\nT103 -> 0" }, { "from": 428, "to": 430, "label": "EVAL-BACKTRACK" }, { "from": 429, "to": 431, "label": "SUCCESS" }, { "from": 432, "to": 434, "label": "EVAL with clause\nmult(s(X113), X114, 0, X115) :- ','(!_7, mult(X113, X114, X114, X115)).\nand substitutionX113 -> T116,\nT80 -> s(T116),\nT91 -> 0,\nX114 -> s(0),\nT117 -> 0,\nT83 -> T119,\nX115 -> T119,\nT118 -> T119" }, { "from": 432, "to": 435, "label": "EVAL-BACKTRACK" }, { "from": 433, "to": 438, "label": "EVAL with clause\nmult(X127, X128, X129, s(X130)) :- ','(p(X129, X131), mult(X127, X128, X131, X130)).\nand substitutionT80 -> T126,\nX127 -> T126,\nT91 -> T127,\nX128 -> s(T127),\nX129 -> T127,\nX130 -> T129,\nT83 -> s(T129),\nT128 -> T129" }, { "from": 433, "to": 439, "label": "EVAL-BACKTRACK" }, { "from": 434, "to": 436, "label": "CUT" }, { "from": 436, "to": 291, "label": "INSTANCE with matching:\nT41 -> T116\nT42 -> s(0)\nT44 -> T119" }, { "from": 438, "to": 440, "label": "CASE" }, { "from": 440, "to": 441, "label": "PARALLEL" }, { "from": 440, "to": 442, "label": "PARALLEL" }, { "from": 441, "to": 443, "label": "EVAL with clause\np(0, 0).\nand substitutionT127 -> 0,\nX131 -> 0" }, { "from": 441, "to": 444, "label": "EVAL-BACKTRACK" }, { "from": 442, "to": 450, "label": "EVAL with clause\np(s(X141), X141).\nand substitutionX141 -> T137,\nT127 -> s(T137),\nX131 -> T137" }, { "from": 442, "to": 451, "label": "EVAL-BACKTRACK" }, { "from": 443, "to": 208, "label": "INSTANCE with matching:\nT13 -> T126\nT14 -> s(0)\nT16 -> T129" }, { "from": 450, "to": 454, "label": "CASE" }, { "from": 454, "to": 455, "label": "EVAL with clause\nmult(0, X148, 0, X149) :- ','(!_10, eq(X149, 0)).\nand substitutionT126 -> 0,\nT137 -> 0,\nX148 -> s(s(0)),\nT144 -> 0,\nT129 -> T146,\nX149 -> T146,\nT145 -> T146" }, { "from": 454, "to": 457, "label": "EVAL-BACKTRACK" }, { "from": 455, "to": 458, "label": "CUT" }, { "from": 457, "to": 465, "label": "PARALLEL" }, { "from": 457, "to": 466, "label": "PARALLEL" }, { "from": 458, "to": 460, "label": "CASE" }, { "from": 460, "to": 461, "label": "EVAL with clause\neq(X152, X152).\nand substitutionT146 -> 0,\nX152 -> 0,\nT149 -> 0" }, { "from": 460, "to": 462, "label": "EVAL-BACKTRACK" }, { "from": 461, "to": 463, "label": "SUCCESS" }, { "from": 465, "to": 471, "label": "EVAL with clause\nmult(s(X165), X166, 0, X167) :- ','(!_10, mult(X165, X166, X166, X167)).\nand substitutionX165 -> T162,\nT126 -> s(T162),\nT137 -> 0,\nX166 -> s(s(0)),\nT163 -> 0,\nT129 -> T165,\nX167 -> T165,\nT164 -> T165" }, { "from": 465, "to": 472, "label": "EVAL-BACKTRACK" }, { "from": 466, "to": 479, "label": "EVAL with clause\nmult(X179, X180, X181, s(X182)) :- ','(p(X181, X183), mult(X179, X180, X183, X182)).\nand substitutionT126 -> T172,\nX179 -> T172,\nT137 -> T173,\nX180 -> s(s(T173)),\nX181 -> T173,\nX182 -> T175,\nT129 -> s(T175),\nT174 -> T175" }, { "from": 466, "to": 480, "label": "EVAL-BACKTRACK" }, { "from": 471, "to": 473, "label": "CUT" }, { "from": 473, "to": 291, "label": "INSTANCE with matching:\nT41 -> T162\nT42 -> s(s(0))\nT44 -> T165" }, { "from": 479, "to": 481, "label": "CASE" }, { "from": 481, "to": 483, "label": "PARALLEL" }, { "from": 481, "to": 548, "label": "PARALLEL" }, { "from": 483, "to": 550, "label": "EVAL with clause\np(0, 0).\nand substitutionT173 -> 0,\nX183 -> 0" }, { "from": 483, "to": 551, "label": "EVAL-BACKTRACK" }, { "from": 548, "to": 555, "label": "EVAL with clause\np(s(X193), X193).\nand substitutionX193 -> T183,\nT173 -> s(T183),\nX183 -> T183" }, { "from": 548, "to": 556, "label": "EVAL-BACKTRACK" }, { "from": 550, "to": 208, "label": "INSTANCE with matching:\nT13 -> T172\nT14 -> s(s(0))\nT16 -> T175" }, { "from": 555, "to": 558, "label": "CASE" }, { "from": 558, "to": 563, "label": "EVAL with clause\nmult(0, X200, 0, X201) :- ','(!_13, eq(X201, 0)).\nand substitutionT172 -> 0,\nT183 -> 0,\nX200 -> s(s(s(0))),\nT190 -> 0,\nT175 -> T192,\nX201 -> T192,\nT191 -> T192" }, { "from": 558, "to": 565, "label": "EVAL-BACKTRACK" }, { "from": 563, "to": 567, "label": "CUT" }, { "from": 565, "to": 574, "label": "PARALLEL" }, { "from": 565, "to": 575, "label": "PARALLEL" }, { "from": 567, "to": 568, "label": "CASE" }, { "from": 568, "to": 569, "label": "EVAL with clause\neq(X204, X204).\nand substitutionT192 -> 0,\nX204 -> 0,\nT195 -> 0" }, { "from": 568, "to": 570, "label": "EVAL-BACKTRACK" }, { "from": 569, "to": 571, "label": "SUCCESS" }, { "from": 574, "to": 577, "label": "EVAL with clause\nmult(s(X217), X218, 0, X219) :- ','(!_13, mult(X217, X218, X218, X219)).\nand substitutionX217 -> T208,\nT172 -> s(T208),\nT183 -> 0,\nX218 -> s(s(s(0))),\nT209 -> 0,\nT175 -> T211,\nX219 -> T211,\nT210 -> T211" }, { "from": 574, "to": 579, "label": "EVAL-BACKTRACK" }, { "from": 575, "to": 604, "label": "EVAL with clause\nmult(X231, X232, X233, s(X234)) :- ','(p(X233, X235), mult(X231, X232, X235, X234)).\nand substitutionT172 -> T218,\nX231 -> T218,\nT183 -> T219,\nX232 -> s(s(s(T219))),\nX233 -> T219,\nX234 -> T221,\nT175 -> s(T221),\nT220 -> T221" }, { "from": 575, "to": 605, "label": "EVAL-BACKTRACK" }, { "from": 577, "to": 581, "label": "CUT" }, { "from": 581, "to": 291, "label": "INSTANCE with matching:\nT41 -> T208\nT42 -> s(s(s(0)))\nT44 -> T211" }, { "from": 604, "to": 606, "label": "CASE" }, { "from": 606, "to": 607, "label": "PARALLEL" }, { "from": 606, "to": 608, "label": "PARALLEL" }, { "from": 607, "to": 609, "label": "EVAL with clause\np(0, 0).\nand substitutionT219 -> 0,\nX235 -> 0" }, { "from": 607, "to": 610, "label": "EVAL-BACKTRACK" }, { "from": 608, "to": 613, "label": "EVAL with clause\np(s(X245), X245).\nand substitutionX245 -> T229,\nT219 -> s(T229),\nX235 -> T229" }, { "from": 608, "to": 614, "label": "EVAL-BACKTRACK" }, { "from": 609, "to": 208, "label": "INSTANCE with matching:\nT13 -> T218\nT14 -> s(s(s(0)))\nT16 -> T221" }, { "from": 613, "to": 618, "label": "CASE" }, { "from": 618, "to": 621, "label": "EVAL with clause\nmult(0, X252, 0, X253) :- ','(!_16, eq(X253, 0)).\nand substitutionT218 -> 0,\nT229 -> 0,\nX252 -> s(s(s(s(0)))),\nT236 -> 0,\nT221 -> T238,\nX253 -> T238,\nT237 -> T238" }, { "from": 618, "to": 622, "label": "EVAL-BACKTRACK" }, { "from": 621, "to": 623, "label": "CUT" }, { "from": 622, "to": 631, "label": "PARALLEL" }, { "from": 622, "to": 632, "label": "PARALLEL" }, { "from": 623, "to": 624, "label": "CASE" }, { "from": 624, "to": 626, "label": "EVAL with clause\neq(X256, X256).\nand substitutionT238 -> 0,\nX256 -> 0,\nT241 -> 0" }, { "from": 624, "to": 628, "label": "EVAL-BACKTRACK" }, { "from": 626, "to": 629, "label": "SUCCESS" }, { "from": 631, "to": 642, "label": "EVAL with clause\nmult(s(X269), X270, 0, X271) :- ','(!_16, mult(X269, X270, X270, X271)).\nand substitutionX269 -> T254,\nT218 -> s(T254),\nT229 -> 0,\nX270 -> s(s(s(s(0)))),\nT255 -> 0,\nT221 -> T257,\nX271 -> T257,\nT256 -> T257" }, { "from": 631, "to": 643, "label": "EVAL-BACKTRACK" }, { "from": 632, "to": 650, "label": "EVAL with clause\nmult(X283, X284, X285, s(X286)) :- ','(p(X285, X287), mult(X283, X284, X287, X286)).\nand substitutionT218 -> T264,\nX283 -> T264,\nT229 -> T265,\nX284 -> s(s(s(s(T265)))),\nX285 -> T265,\nX286 -> T267,\nT221 -> s(T267),\nT266 -> T267" }, { "from": 632, "to": 651, "label": "EVAL-BACKTRACK" }, { "from": 642, "to": 644, "label": "CUT" }, { "from": 644, "to": 291, "label": "INSTANCE with matching:\nT41 -> T254\nT42 -> s(s(s(s(0))))\nT44 -> T257" }, { "from": 650, "to": 652, "label": "CASE" }, { "from": 652, "to": 653, "label": "PARALLEL" }, { "from": 652, "to": 655, "label": "PARALLEL" }, { "from": 653, "to": 657, "label": "EVAL with clause\np(0, 0).\nand substitutionT265 -> 0,\nX287 -> 0" }, { "from": 653, "to": 658, "label": "EVAL-BACKTRACK" }, { "from": 655, "to": 711, "label": "EVAL with clause\np(s(X297), X297).\nand substitutionX297 -> T275,\nT265 -> s(T275),\nX287 -> T275" }, { "from": 655, "to": 712, "label": "EVAL-BACKTRACK" }, { "from": 657, "to": 208, "label": "INSTANCE with matching:\nT13 -> T264\nT14 -> s(s(s(s(0))))\nT16 -> T267" }, { "from": 711, "to": 714, "label": "CASE" }, { "from": 714, "to": 715, "label": "EVAL with clause\nmult(0, X304, 0, X305) :- ','(!_19, eq(X305, 0)).\nand substitutionT264 -> 0,\nT275 -> 0,\nX304 -> s(s(s(s(s(0))))),\nT282 -> 0,\nT267 -> T284,\nX305 -> T284,\nT283 -> T284" }, { "from": 714, "to": 717, "label": "EVAL-BACKTRACK" }, { "from": 715, "to": 718, "label": "CUT" }, { "from": 717, "to": 728, "label": "PARALLEL" }, { "from": 717, "to": 729, "label": "PARALLEL" }, { "from": 718, "to": 719, "label": "CASE" }, { "from": 719, "to": 720, "label": "EVAL with clause\neq(X308, X308).\nand substitutionT284 -> 0,\nX308 -> 0,\nT287 -> 0" }, { "from": 719, "to": 722, "label": "EVAL-BACKTRACK" }, { "from": 720, "to": 723, "label": "SUCCESS" }, { "from": 728, "to": 747, "label": "EVAL with clause\nmult(s(X321), X322, 0, X323) :- ','(!_19, mult(X321, X322, X322, X323)).\nand substitutionX321 -> T300,\nT264 -> s(T300),\nT275 -> 0,\nX322 -> s(s(s(s(s(0))))),\nT301 -> 0,\nT267 -> T303,\nX323 -> T303,\nT302 -> T303" }, { "from": 728, "to": 749, "label": "EVAL-BACKTRACK" }, { "from": 729, "to": 755, "label": "EVAL with clause\nmult(X335, X336, X337, s(X338)) :- ','(p(X337, X339), mult(X335, X336, X339, X338)).\nand substitutionT264 -> T310,\nX335 -> T310,\nT275 -> T311,\nX336 -> s(s(s(s(s(T311))))),\nX337 -> T311,\nX338 -> T313,\nT267 -> s(T313),\nT312 -> T313" }, { "from": 729, "to": 756, "label": "EVAL-BACKTRACK" }, { "from": 747, "to": 750, "label": "CUT" }, { "from": 750, "to": 291, "label": "INSTANCE with matching:\nT41 -> T300\nT42 -> s(s(s(s(s(0)))))\nT44 -> T303" }, { "from": 755, "to": 758, "label": "CASE" }, { "from": 758, "to": 761, "label": "PARALLEL" }, { "from": 758, "to": 762, "label": "PARALLEL" }, { "from": 761, "to": 765, "label": "EVAL with clause\np(0, 0).\nand substitutionT311 -> 0,\nX339 -> 0" }, { "from": 761, "to": 766, "label": "EVAL-BACKTRACK" }, { "from": 762, "to": 767, "label": "EVAL with clause\np(s(X349), X349).\nand substitutionX349 -> T321,\nT311 -> s(T321),\nX339 -> T321" }, { "from": 762, "to": 768, "label": "EVAL-BACKTRACK" }, { "from": 765, "to": 208, "label": "INSTANCE with matching:\nT13 -> T310\nT14 -> s(s(s(s(s(0)))))\nT16 -> T313" }, { "from": 767, "to": 769, "label": "CASE" }, { "from": 769, "to": 778, "label": "EVAL with clause\nmult(0, X356, 0, X357) :- ','(!_22, eq(X357, 0)).\nand substitutionT310 -> 0,\nT321 -> 0,\nX356 -> s(s(s(s(s(s(0)))))),\nT328 -> 0,\nT313 -> T330,\nX357 -> T330,\nT329 -> T330" }, { "from": 769, "to": 779, "label": "EVAL-BACKTRACK" }, { "from": 778, "to": 780, "label": "CUT" }, { "from": 779, "to": 788, "label": "PARALLEL" }, { "from": 779, "to": 789, "label": "PARALLEL" }, { "from": 780, "to": 781, "label": "CASE" }, { "from": 781, "to": 782, "label": "EVAL with clause\neq(X360, X360).\nand substitutionT330 -> 0,\nX360 -> 0,\nT333 -> 0" }, { "from": 781, "to": 784, "label": "EVAL-BACKTRACK" }, { "from": 782, "to": 785, "label": "SUCCESS" }, { "from": 788, "to": 794, "label": "EVAL with clause\nmult(s(X373), X374, 0, X375) :- ','(!_22, mult(X373, X374, X374, X375)).\nand substitutionX373 -> T346,\nT310 -> s(T346),\nT321 -> 0,\nX374 -> s(s(s(s(s(s(0)))))),\nT347 -> 0,\nT313 -> T349,\nX375 -> T349,\nT348 -> T349" }, { "from": 788, "to": 795, "label": "EVAL-BACKTRACK" }, { "from": 789, "to": 808, "label": "EVAL with clause\nmult(X387, X388, X389, s(X390)) :- ','(p(X389, X391), mult(X387, X388, X391, X390)).\nand substitutionT310 -> T356,\nX387 -> T356,\nT321 -> T357,\nX388 -> s(s(s(s(s(s(T357)))))),\nX389 -> T357,\nX390 -> T359,\nT313 -> s(T359),\nT358 -> T359" }, { "from": 789, "to": 810, "label": "EVAL-BACKTRACK" }, { "from": 794, "to": 796, "label": "CUT" }, { "from": 796, "to": 291, "label": "INSTANCE with matching:\nT41 -> T346\nT42 -> s(s(s(s(s(s(0))))))\nT44 -> T349" }, { "from": 808, "to": 814, "label": "CASE" }, { "from": 814, "to": 815, "label": "PARALLEL" }, { "from": 814, "to": 816, "label": "PARALLEL" }, { "from": 815, "to": 817, "label": "EVAL with clause\np(0, 0).\nand substitutionT357 -> 0,\nX391 -> 0" }, { "from": 815, "to": 820, "label": "EVAL-BACKTRACK" }, { "from": 816, "to": 825, "label": "EVAL with clause\np(s(X401), X401).\nand substitutionX401 -> T367,\nT357 -> s(T367),\nX391 -> T367" }, { "from": 816, "to": 827, "label": "EVAL-BACKTRACK" }, { "from": 817, "to": 208, "label": "INSTANCE with matching:\nT13 -> T356\nT14 -> s(s(s(s(s(s(0))))))\nT16 -> T359" }, { "from": 825, "to": 834, "label": "CASE" }, { "from": 834, "to": 835, "label": "EVAL with clause\nmult(0, X408, 0, X409) :- ','(!_25, eq(X409, 0)).\nand substitutionT356 -> 0,\nT367 -> 0,\nX408 -> s(s(s(s(s(s(s(0))))))),\nT374 -> 0,\nT359 -> T376,\nX409 -> T376,\nT375 -> T376" }, { "from": 834, "to": 836, "label": "EVAL-BACKTRACK" }, { "from": 835, "to": 837, "label": "CUT" }, { "from": 836, "to": 842, "label": "PARALLEL" }, { "from": 836, "to": 843, "label": "PARALLEL" }, { "from": 837, "to": 838, "label": "CASE" }, { "from": 838, "to": 839, "label": "EVAL with clause\neq(X412, X412).\nand substitutionT376 -> 0,\nX412 -> 0,\nT379 -> 0" }, { "from": 838, "to": 840, "label": "EVAL-BACKTRACK" }, { "from": 839, "to": 841, "label": "SUCCESS" }, { "from": 842, "to": 846, "label": "EVAL with clause\nmult(s(X425), X426, 0, X427) :- ','(!_25, mult(X425, X426, X426, X427)).\nand substitutionX425 -> T392,\nT356 -> s(T392),\nT367 -> 0,\nX426 -> s(s(s(s(s(s(s(0))))))),\nT393 -> 0,\nT359 -> T395,\nX427 -> T395,\nT394 -> T395" }, { "from": 842, "to": 847, "label": "EVAL-BACKTRACK" }, { "from": 843, "to": 849, "label": "EVAL with clause\nmult(X439, X440, X441, s(X442)) :- ','(p(X441, X443), mult(X439, X440, X443, X442)).\nand substitutionT356 -> T402,\nX439 -> T402,\nT367 -> T403,\nX440 -> s(s(s(s(s(s(s(T403))))))),\nX441 -> T403,\nX442 -> T405,\nT359 -> s(T405),\nT404 -> T405" }, { "from": 843, "to": 850, "label": "EVAL-BACKTRACK" }, { "from": 846, "to": 848, "label": "CUT" }, { "from": 848, "to": 291, "label": "INSTANCE with matching:\nT41 -> T392\nT42 -> s(s(s(s(s(s(s(0)))))))\nT44 -> T395" }, { "from": 849, "to": 851, "label": "CASE" }, { "from": 851, "to": 853, "label": "PARALLEL" }, { "from": 851, "to": 855, "label": "PARALLEL" }, { "from": 853, "to": 856, "label": "EVAL with clause\np(0, 0).\nand substitutionT403 -> 0,\nX443 -> 0" }, { "from": 853, "to": 857, "label": "EVAL-BACKTRACK" }, { "from": 855, "to": 859, "label": "EVAL with clause\np(s(X453), X453).\nand substitutionX453 -> T413,\nT403 -> s(T413),\nX443 -> T413" }, { "from": 855, "to": 862, "label": "EVAL-BACKTRACK" }, { "from": 856, "to": 208, "label": "INSTANCE with matching:\nT13 -> T402\nT14 -> s(s(s(s(s(s(s(0)))))))\nT16 -> T405" }, { "from": 859, "to": 864, "label": "GENERALIZATION\nT416 <-- s(s(s(s(s(s(s(T413)))))))\n\nNew Knowledge:\nT416 is ground" }, { "from": 864, "to": 866, "label": "CASE" }, { "from": 866, "to": 867, "label": "EVAL with clause\nmult(0, X462, 0, X463) :- ','(!_28, eq(X463, 0)).\nand substitutionT402 -> 0,\nT416 -> T423,\nX462 -> s(T423),\nT413 -> 0,\nT405 -> T425,\nX463 -> T425,\nT424 -> T425" }, { "from": 866, "to": 868, "label": "EVAL-BACKTRACK" }, { "from": 867, "to": 869, "label": "CUT" }, { "from": 868, "to": 874, "label": "PARALLEL" }, { "from": 868, "to": 875, "label": "PARALLEL" }, { "from": 869, "to": 870, "label": "CASE" }, { "from": 870, "to": 871, "label": "EVAL with clause\neq(X466, X466).\nand substitutionT425 -> 0,\nX466 -> 0,\nT428 -> 0" }, { "from": 870, "to": 872, "label": "EVAL-BACKTRACK" }, { "from": 871, "to": 873, "label": "SUCCESS" }, { "from": 874, "to": 876, "label": "EVAL with clause\nmult(s(X479), X480, 0, X481) :- ','(!_28, mult(X479, X480, X480, X481)).\nand substitutionX479 -> T441,\nT402 -> s(T441),\nT416 -> T442,\nX480 -> s(T442),\nT413 -> 0,\nT405 -> T444,\nX481 -> T444,\nT443 -> T444" }, { "from": 874, "to": 877, "label": "EVAL-BACKTRACK" }, { "from": 875, "to": 879, "label": "EVAL with clause\nmult(X493, X494, X495, s(X496)) :- ','(p(X495, X497), mult(X493, X494, X497, X496)).\nand substitutionT402 -> T454,\nX493 -> T454,\nT416 -> T455,\nX494 -> s(T455),\nT413 -> T456,\nX495 -> T456,\nX496 -> T458,\nT405 -> s(T458),\nT457 -> T458" }, { "from": 875, "to": 880, "label": "EVAL-BACKTRACK" }, { "from": 876, "to": 878, "label": "CUT" }, { "from": 878, "to": 291, "label": "INSTANCE with matching:\nT41 -> T441\nT42 -> s(T442)\nT44 -> T444" }, { "from": 879, "to": 881, "label": "CASE" }, { "from": 881, "to": 882, "label": "PARALLEL" }, { "from": 881, "to": 883, "label": "PARALLEL" }, { "from": 882, "to": 885, "label": "EVAL with clause\np(0, 0).\nand substitutionT456 -> 0,\nX497 -> 0" }, { "from": 882, "to": 886, "label": "EVAL-BACKTRACK" }, { "from": 883, "to": 887, "label": "EVAL with clause\np(s(X507), X507).\nand substitutionX507 -> T468,\nT456 -> s(T468),\nX497 -> T468" }, { "from": 883, "to": 888, "label": "EVAL-BACKTRACK" }, { "from": 885, "to": 208, "label": "INSTANCE with matching:\nT13 -> T454\nT14 -> s(T455)\nT16 -> T458" }, { "from": 887, "to": 864, "label": "INSTANCE with matching:\nT402 -> T454\nT416 -> T455\nT413 -> T468\nT405 -> T458" }, { "from": 898, "to": 901, "label": "CASE" }, { "from": 901, "to": 902, "label": "PARALLEL" }, { "from": 901, "to": 903, "label": "PARALLEL" }, { "from": 902, "to": 904, "label": "ONLY EVAL with clause\np(0, 0).\nand substitutionX523 -> 0" }, { "from": 903, "to": 908, "label": "BACKTRACK\nfor clause: p(s(X), X)because of non-unification" }, { "from": 904, "to": 208, "label": "INSTANCE with matching:\nT13 -> T477\nT14 -> T478\nT16 -> T480" } ], "type": "Graph" } } ---------------------------------------- (76) Obligation: Rules: f758_out(T311, T310) -> f755_out(T311, T310) :|: TRUE f755_in(x, x1) -> f758_in(x, x1) :|: TRUE f808_in(T357, T356) -> f814_in(T357, T356) :|: TRUE f814_out(x2, x3) -> f808_out(x2, x3) :|: TRUE f581_in(T208) -> f291_in(T208, s(s(s(0)))) :|: TRUE f291_out(x4, s(s(s(0)))) -> f581_out(x4) :|: TRUE f855_in(T403, T402) -> f862_in :|: TRUE f862_out -> f855_out(x5, x6) :|: TRUE f855_in(s(x7), x8) -> f859_in(x8, x7) :|: TRUE f859_out(x9, x10) -> f855_out(s(x10), x9) :|: TRUE f455_out -> f454_out(0, 0) :|: TRUE f454_in(T126, T137) -> f457_in(T126, T137) :|: TRUE f457_out(x11, x12) -> f454_out(x11, x12) :|: TRUE f454_in(0, 0) -> f455_in :|: TRUE f879_in(T456, T454, T455) -> f881_in(T456, T454, T455) :|: TRUE f881_out(x13, x14, x15) -> f879_out(x13, x14, x15) :|: TRUE f337_in(T69) -> f208_in(T69, 0) :|: TRUE f208_out(x16, 0) -> f337_out(x16) :|: TRUE f416_in(T80, T91) -> f433_in(T80, T91) :|: TRUE f433_out(x17, x18) -> f416_out(x17, x18) :|: TRUE f416_in(x19, x20) -> f432_in(x19, x20) :|: TRUE f432_out(x21, x22) -> f416_out(x21, x22) :|: TRUE f788_in(x23, x24) -> f795_in :|: TRUE f788_in(s(T346), 0) -> f794_in(T346) :|: TRUE f794_out(x25) -> f788_out(s(x25), 0) :|: TRUE f795_out -> f788_out(x26, x27) :|: TRUE f291_out(x28, s(s(s(s(s(s(0))))))) -> f796_out(x28) :|: TRUE f796_in(x29) -> f291_in(x29, s(s(s(s(s(s(0))))))) :|: TRUE f252_out(T23) -> f235_out(0, T23) :|: TRUE f235_in(T13, T14) -> f258_in(T13, T14) :|: TRUE f258_out(x30, x31) -> f235_out(x30, x31) :|: TRUE f235_in(0, x32) -> f252_in(x32) :|: TRUE f565_in(T172, T183) -> f575_in(T172, T183) :|: TRUE f574_out(x33, x34) -> f565_out(x33, x34) :|: TRUE f565_in(x35, x36) -> f574_in(x35, x36) :|: TRUE f575_out(x37, x38) -> f565_out(x37, x38) :|: TRUE f769_out(x39, x40) -> f767_out(x39, x40) :|: TRUE f767_in(x41, x42) -> f769_in(x41, x42) :|: TRUE f879_out(x43, x44, x45) -> f875_out(x44, x45, x43) :|: TRUE f875_in(x46, x47, x48) -> f879_in(x48, x46, x47) :|: TRUE f880_out -> f875_out(x49, x50, x51) :|: TRUE f875_in(x52, x53, x54) -> f880_in :|: TRUE f604_in(T219, T218) -> f606_in(T219, T218) :|: TRUE f606_out(x55, x56) -> f604_out(x55, x56) :|: TRUE f778_out -> f769_out(0, 0) :|: TRUE f769_in(x57, x58) -> f779_in(x57, x58) :|: TRUE f779_out(x59, x60) -> f769_out(x59, x60) :|: TRUE f769_in(0, 0) -> f778_in :|: TRUE f885_in(x61, x62) -> f208_in(x61, s(x62)) :|: TRUE f208_out(x63, s(x64)) -> f885_out(x63, x64) :|: TRUE f432_in(s(T116), 0) -> f434_in(T116) :|: TRUE f434_out(x65) -> f432_out(s(x65), 0) :|: TRUE f435_out -> f432_out(x66, x67) :|: TRUE f432_in(x68, x69) -> f435_in :|: TRUE f288_out(T41, T42) -> f282_out(s(T41), T42) :|: TRUE f282_in(s(x70), x71) -> f288_in(x70, x71) :|: TRUE f282_in(x72, x73) -> f290_in :|: TRUE f290_out -> f282_out(x74, x75) :|: TRUE f439_out -> f433_out(x76, x77) :|: TRUE f438_out(x78, x79) -> f433_out(x79, x78) :|: TRUE f433_in(x80, x81) -> f439_in :|: TRUE f433_in(x82, x83) -> f438_in(x83, x82) :|: TRUE f606_in(x84, x85) -> f607_in(x84, x85) :|: TRUE f608_out(x86, x87) -> f606_out(x86, x87) :|: TRUE f606_in(x88, x89) -> f608_in(x88, x89) :|: TRUE f607_out(x90, x91) -> f606_out(x90, x91) :|: TRUE f902_out(T477, T478) -> f901_out(T477, T478) :|: TRUE f903_out(x92, x93) -> f901_out(x92, x93) :|: TRUE f901_in(x94, x95) -> f902_in(x94, x95) :|: TRUE f901_in(x96, x97) -> f903_in(x96, x97) :|: TRUE f756_out -> f729_out(T264, T275) :|: TRUE f729_in(x98, x99) -> f756_in :|: TRUE f729_in(x100, x101) -> f755_in(x101, x100) :|: TRUE f755_out(x102, x103) -> f729_out(x103, x102) :|: TRUE f765_in(x104) -> f208_in(x104, s(s(s(s(s(0)))))) :|: TRUE f208_out(x105, s(s(s(s(s(0)))))) -> f765_out(x105) :|: TRUE f749_out -> f728_out(x106, x107) :|: TRUE f747_out(T300) -> f728_out(s(T300), 0) :|: TRUE f728_in(x108, x109) -> f749_in :|: TRUE f728_in(s(x110), 0) -> f747_in(x110) :|: TRUE f473_in(T162) -> f291_in(T162, s(s(0))) :|: TRUE f291_out(x111, s(s(0))) -> f473_out(x111) :|: TRUE f405_out(x112) -> f402_out(0, x112) :|: TRUE f402_in(x113, x114) -> f406_in :|: TRUE f402_in(0, x115) -> f405_in(x115) :|: TRUE f406_out -> f402_out(x116, x117) :|: TRUE f866_out(x118, x119, x120) -> f864_out(x118, x119, x120) :|: TRUE f864_in(x121, x122, x123) -> f866_in(x121, x122, x123) :|: TRUE f288_in(x124, x125) -> f291_in(x124, x125) :|: TRUE f291_out(x126, x127) -> f288_out(x126, x127) :|: TRUE f608_in(s(x128), x129) -> f613_in(x129, x128) :|: TRUE f608_in(x130, x131) -> f614_in :|: TRUE f614_out -> f608_out(x132, x133) :|: TRUE f613_out(x134, x135) -> f608_out(s(x135), x134) :|: TRUE f714_in(0, 0) -> f715_in :|: TRUE f717_out(x136, x137) -> f714_out(x136, x137) :|: TRUE f715_out -> f714_out(0, 0) :|: TRUE f714_in(x138, x139) -> f717_in(x138, x139) :|: TRUE f901_out(x140, x141) -> f898_out(x140, x141) :|: TRUE f898_in(x142, x143) -> f901_in(x142, x143) :|: TRUE f454_out(x144, x145) -> f450_out(x144, x145) :|: TRUE f450_in(x146, x147) -> f454_in(x146, x147) :|: TRUE f850_out -> f843_out(x148, x149) :|: TRUE f843_in(x150, x151) -> f849_in(x151, x150) :|: TRUE f849_out(x152, x153) -> f843_out(x153, x152) :|: TRUE f843_in(x154, x155) -> f850_in :|: TRUE f867_out(T423) -> f866_out(0, T423, 0) :|: TRUE f868_out(x156, x157, x158) -> f866_out(x156, x157, x158) :|: TRUE f866_in(0, x159, 0) -> f867_in(x159) :|: TRUE f866_in(x160, x161, x162) -> f868_in(x160, x161, x162) :|: TRUE f883_out(x163, x164, x165) -> f881_out(x163, x164, x165) :|: TRUE f881_in(x166, x167, x168) -> f883_in(x166, x167, x168) :|: TRUE f881_in(x169, x170, x171) -> f882_in(x169, x170, x171) :|: TRUE f882_out(x172, x173, x174) -> f881_out(x172, x173, x174) :|: TRUE f817_out(x175) -> f815_out(0, x175) :|: TRUE f820_out -> f815_out(x176, x177) :|: TRUE f815_in(x178, x179) -> f820_in :|: TRUE f815_in(0, x180) -> f817_in(x180) :|: TRUE f284_out(x181, x182) -> f258_out(x181, x182) :|: TRUE f258_in(x183, x184) -> f282_in(x183, x184) :|: TRUE f282_out(x185, x186) -> f258_out(x185, x186) :|: TRUE f258_in(x187, x188) -> f284_in(x187, x188) :|: TRUE f208_out(x189, s(s(s(s(s(s(0))))))) -> f817_out(x189) :|: TRUE f817_in(x190) -> f208_in(x190, s(s(s(s(s(s(0))))))) :|: TRUE f750_in(x191) -> f291_in(x191, s(s(s(s(s(0)))))) :|: TRUE f291_out(x192, s(s(s(s(s(0)))))) -> f750_out(x192) :|: TRUE f291_in(x193, x194) -> f295_in(x193, x194) :|: TRUE f295_out(x195, x196) -> f291_out(x195, x196) :|: TRUE f415_out -> f414_out(0, 0) :|: TRUE f414_in(0, 0) -> f415_in :|: TRUE f414_in(x197, x198) -> f416_in(x197, x198) :|: TRUE f416_out(x199, x200) -> f414_out(x199, x200) :|: TRUE f574_in(x201, x202) -> f579_in :|: TRUE f574_in(s(x203), 0) -> f577_in(x203) :|: TRUE f577_out(x204) -> f574_out(s(x204), 0) :|: TRUE f579_out -> f574_out(x205, x206) :|: TRUE f851_out(x207, x208) -> f849_out(x207, x208) :|: TRUE f849_in(x209, x210) -> f851_in(x209, x210) :|: TRUE f434_in(x211) -> f436_in(x211) :|: TRUE f436_out(x212) -> f434_out(x212) :|: TRUE f652_in(x213, x214) -> f655_in(x213, x214) :|: TRUE f655_out(x215, x216) -> f652_out(x215, x216) :|: TRUE f652_in(x217, x218) -> f653_in(x217, x218) :|: TRUE f653_out(x219, x220) -> f652_out(x219, x220) :|: TRUE f472_out -> f465_out(x221, x222) :|: TRUE f465_in(x223, x224) -> f472_in :|: TRUE f471_out(x225) -> f465_out(s(x225), 0) :|: TRUE f465_in(s(x226), 0) -> f471_in(x226) :|: TRUE f644_out(T254) -> f642_out(T254) :|: TRUE f642_in(x227) -> f644_in(x227) :|: TRUE f816_out(x228, x229) -> f814_out(x228, x229) :|: TRUE f814_in(x230, x231) -> f815_in(x230, x231) :|: TRUE f814_in(x232, x233) -> f816_in(x232, x233) :|: TRUE f815_out(x234, x235) -> f814_out(x234, x235) :|: TRUE f631_in(s(x236), 0) -> f642_in(x236) :|: TRUE f631_in(x237, x238) -> f643_in :|: TRUE f642_out(x239) -> f631_out(s(x239), 0) :|: TRUE f643_out -> f631_out(x240, x241) :|: TRUE f208_out(x242, s(s(0))) -> f550_out(x242) :|: TRUE f550_in(x243) -> f208_in(x243, s(s(0))) :|: TRUE f618_in(x244, x245) -> f622_in(x244, x245) :|: TRUE f621_out -> f618_out(0, 0) :|: TRUE f618_in(0, 0) -> f621_in :|: TRUE f622_out(x246, x247) -> f618_out(x246, x247) :|: TRUE f208_out(x248, s(s(s(s(s(s(s(0)))))))) -> f856_out(x248) :|: TRUE f856_in(x249) -> f208_in(x249, s(s(s(s(s(s(s(0)))))))) :|: TRUE f558_in(0, 0) -> f563_in :|: TRUE f563_out -> f558_out(0, 0) :|: TRUE f565_out(x250, x251) -> f558_out(x250, x251) :|: TRUE f558_in(x252, x253) -> f565_in(x252, x253) :|: TRUE f344_in(x254, x255) -> f398_in(x254, x255) :|: TRUE f398_out(x256, x257) -> f344_out(x256, x257) :|: TRUE f851_in(x258, x259) -> f855_in(x258, x259) :|: TRUE f853_out(x260, x261) -> f851_out(x260, x261) :|: TRUE f855_out(x262, x263) -> f851_out(x262, x263) :|: TRUE f851_in(x264, x265) -> f853_in(x264, x265) :|: TRUE f883_in(x266, x267, x268) -> f888_in :|: TRUE f888_out -> f883_out(x269, x270, x271) :|: TRUE f883_in(s(x272), x273, x274) -> f887_in(x273, x274, x272) :|: TRUE f887_out(x275, x276, x277) -> f883_out(s(x277), x275, x276) :|: TRUE f291_out(T392, s(s(s(s(s(s(s(0)))))))) -> f848_out(T392) :|: TRUE f848_in(x278) -> f291_in(x278, s(s(s(s(s(s(s(0)))))))) :|: TRUE f443_in(x279) -> f208_in(x279, s(0)) :|: TRUE f208_out(x280, s(0)) -> f443_out(x280) :|: TRUE f409_in(x281, x282) -> f414_in(x281, x282) :|: TRUE f414_out(x283, x284) -> f409_out(x283, x284) :|: TRUE f836_in(x285, x286) -> f843_in(x285, x286) :|: TRUE f836_in(x287, x288) -> f842_in(x287, x288) :|: TRUE f842_out(x289, x290) -> f836_out(x289, x290) :|: TRUE f843_out(x291, x292) -> f836_out(x291, x292) :|: TRUE f657_out(x293) -> f653_out(0, x293) :|: TRUE f653_in(x294, x295) -> f658_in :|: TRUE f658_out -> f653_out(x296, x297) :|: TRUE f653_in(0, x298) -> f657_in(x298) :|: TRUE f436_in(x299) -> f291_in(x299, s(0)) :|: TRUE f291_out(x300, s(0)) -> f436_out(x300) :|: TRUE f652_out(x301, x302) -> f650_out(x301, x302) :|: TRUE f650_in(x303, x304) -> f652_in(x303, x304) :|: TRUE f441_in(0, x305) -> f443_in(x305) :|: TRUE f444_out -> f441_out(x306, x307) :|: TRUE f443_out(x308) -> f441_out(0, x308) :|: TRUE f441_in(x309, x310) -> f444_in :|: TRUE f847_out -> f842_out(x311, x312) :|: TRUE f846_out(x313) -> f842_out(s(x313), 0) :|: TRUE f842_in(x314, x315) -> f847_in :|: TRUE f842_in(s(x316), 0) -> f846_in(x316) :|: TRUE f607_in(x317, x318) -> f610_in :|: TRUE f607_in(0, x319) -> f609_in(x319) :|: TRUE f610_out -> f607_out(x320, x321) :|: TRUE f609_out(x322) -> f607_out(0, x322) :|: TRUE f442_out(x323, x324) -> f440_out(x323, x324) :|: TRUE f440_in(x325, x326) -> f441_in(x325, x326) :|: TRUE f441_out(x327, x328) -> f440_out(x327, x328) :|: TRUE f440_in(x329, x330) -> f442_in(x329, x330) :|: TRUE f337_out(x331) -> f334_out(x331) :|: TRUE f334_in(x332) -> f337_in(x332) :|: TRUE f410_out -> f404_out(x333, x334) :|: TRUE f404_in(s(x335), x336) -> f409_in(x336, x335) :|: TRUE f409_out(x337, x338) -> f404_out(s(x338), x337) :|: TRUE f404_in(x339, x340) -> f410_in :|: TRUE f483_out(x341, x342) -> f481_out(x341, x342) :|: TRUE f481_in(x343, x344) -> f483_in(x343, x344) :|: TRUE f481_in(x345, x346) -> f548_in(x345, x346) :|: TRUE f548_out(x347, x348) -> f481_out(x347, x348) :|: TRUE f604_out(x349, x350) -> f575_out(x350, x349) :|: TRUE f605_out -> f575_out(x351, x352) :|: TRUE f575_in(x353, x354) -> f605_in :|: TRUE f575_in(x355, x356) -> f604_in(x356, x355) :|: TRUE f711_out(x357, x358) -> f655_out(s(x358), x357) :|: TRUE f655_in(s(x359), x360) -> f711_in(x360, x359) :|: TRUE f712_out -> f655_out(x361, x362) :|: TRUE f655_in(x363, x364) -> f712_in :|: TRUE f344_out(x365, x366) -> f332_out(x366, x365) :|: TRUE f332_in(x367, x368) -> f344_in(x368, x367) :|: TRUE f332_in(x369, x370) -> f346_in :|: TRUE f346_out -> f332_out(x371, x372) :|: TRUE f864_out(x373, s(s(s(s(s(s(s(x374))))))), x374) -> f859_out(x373, x374) :|: TRUE f859_in(x375, x376) -> f864_in(x375, s(s(s(s(s(s(s(x376))))))), x376) :|: TRUE f864_out(x377, x378, x379) -> f887_out(x377, x378, x379) :|: TRUE f887_in(x380, x381, x382) -> f864_in(x380, x381, x382) :|: TRUE f609_in(x383) -> f208_in(x383, s(s(s(0)))) :|: TRUE f208_out(x384, s(s(s(0)))) -> f609_out(x384) :|: TRUE f796_out(x385) -> f794_out(x385) :|: TRUE f794_in(x386) -> f796_in(x386) :|: TRUE f856_out(x387) -> f853_out(0, x387) :|: TRUE f857_out -> f853_out(x388, x389) :|: TRUE f853_in(0, x390) -> f856_in(x390) :|: TRUE f853_in(x391, x392) -> f857_in :|: TRUE f334_out(x393) -> f331_out(s(x393), 0) :|: TRUE f331_in(x394, x395) -> f335_in :|: TRUE f335_out -> f331_out(x396, x397) :|: TRUE f331_in(s(x398), 0) -> f334_in(x398) :|: TRUE f481_out(x399, x400) -> f479_out(x399, x400) :|: TRUE f479_in(x401, x402) -> f481_in(x401, x402) :|: TRUE f868_in(x403, x404, x405) -> f874_in(x403, x404, x405) :|: TRUE f874_out(x406, x407, x408) -> f868_out(x406, x407, x408) :|: TRUE f868_in(x409, x410, x411) -> f875_in(x409, x410, x411) :|: TRUE f875_out(x412, x413, x414) -> f868_out(x412, x413, x414) :|: TRUE f581_out(x415) -> f577_out(x415) :|: TRUE f577_in(x416) -> f581_in(x416) :|: TRUE f442_in(x417, x418) -> f451_in :|: TRUE f451_out -> f442_out(x419, x420) :|: TRUE f442_in(s(x421), x422) -> f450_in(x422, x421) :|: TRUE f450_out(x423, x424) -> f442_out(s(x424), x423) :|: TRUE f761_in(0, x425) -> f765_in(x425) :|: TRUE f765_out(x426) -> f761_out(0, x426) :|: TRUE f761_in(x427, x428) -> f766_in :|: TRUE f766_out -> f761_out(x429, x430) :|: TRUE f208_in(x431, x432) -> f235_in(x431, x432) :|: TRUE f235_out(x433, x434) -> f208_out(x433, x434) :|: TRUE f825_in(x435, x436) -> f834_in(x435, x436) :|: TRUE f834_out(x437, x438) -> f825_out(x437, x438) :|: TRUE f762_in(x439, x440) -> f768_in :|: TRUE f767_out(x441, x442) -> f762_out(s(x442), x441) :|: TRUE f762_in(s(x443), x444) -> f767_in(x444, x443) :|: TRUE f768_out -> f762_out(x445, x446) :|: TRUE f834_in(x447, x448) -> f836_in(x447, x448) :|: TRUE f835_out -> f834_out(0, 0) :|: TRUE f836_out(x449, x450) -> f834_out(x449, x450) :|: TRUE f834_in(0, 0) -> f835_in :|: TRUE f555_out(x451, x452) -> f548_out(s(x452), x451) :|: TRUE f556_out -> f548_out(x453, x454) :|: TRUE f548_in(s(x455), x456) -> f555_in(x456, x455) :|: TRUE f548_in(x457, x458) -> f556_in :|: TRUE f398_in(x459, x460) -> f402_in(x459, x460) :|: TRUE f398_in(x461, x462) -> f404_in(x461, x462) :|: TRUE f404_out(x463, x464) -> f398_out(x463, x464) :|: TRUE f402_out(x465, x466) -> f398_out(x465, x466) :|: TRUE f651_out -> f632_out(x467, x468) :|: TRUE f650_out(x469, x470) -> f632_out(x470, x469) :|: TRUE f632_in(x471, x472) -> f651_in :|: TRUE f632_in(x473, x474) -> f650_in(x474, x473) :|: TRUE f295_in(x475, x476) -> f299_in(x475, x476) :|: TRUE f299_out(x477, x478) -> f295_out(x477, x478) :|: TRUE f295_in(0, 0) -> f298_in :|: TRUE f298_out -> f295_out(0, 0) :|: TRUE f882_in(x479, x480, x481) -> f886_in :|: TRUE f886_out -> f882_out(x482, x483, x484) :|: TRUE f882_in(0, x485, x486) -> f885_in(x485, x486) :|: TRUE f885_out(x487, x488) -> f882_out(0, x487, x488) :|: TRUE f711_in(x489, x490) -> f714_in(x489, x490) :|: TRUE f714_out(x491, x492) -> f711_out(x491, x492) :|: TRUE f291_out(T441, s(T442)) -> f878_out(T441, T442) :|: TRUE f878_in(x493, x494) -> f291_in(x493, s(x494)) :|: TRUE f717_in(x495, x496) -> f728_in(x495, x496) :|: TRUE f728_out(x497, x498) -> f717_out(x497, x498) :|: TRUE f717_in(x499, x500) -> f729_in(x499, x500) :|: TRUE f729_out(x501, x502) -> f717_out(x501, x502) :|: TRUE f747_in(x503) -> f750_in(x503) :|: TRUE f750_out(x504) -> f747_out(x504) :|: TRUE f878_out(x505, x506) -> f876_out(x505, x506) :|: TRUE f876_in(x507, x508) -> f878_in(x507, x508) :|: TRUE f208_out(x509, x510) -> f904_out(x509, x510) :|: TRUE f904_in(x511, x512) -> f208_in(x511, x512) :|: TRUE f779_in(x513, x514) -> f788_in(x513, x514) :|: TRUE f779_in(x515, x516) -> f789_in(x515, x516) :|: TRUE f788_out(x517, x518) -> f779_out(x517, x518) :|: TRUE f789_out(x519, x520) -> f779_out(x519, x520) :|: TRUE f299_in(x521, x522) -> f332_in(x521, x522) :|: TRUE f332_out(x523, x524) -> f299_out(x523, x524) :|: TRUE f331_out(x525, x526) -> f299_out(x525, x526) :|: TRUE f299_in(x527, x528) -> f331_in(x527, x528) :|: TRUE f789_in(x529, x530) -> f808_in(x530, x529) :|: TRUE f789_in(x531, x532) -> f810_in :|: TRUE f808_out(x533, x534) -> f789_out(x534, x533) :|: TRUE f810_out -> f789_out(x535, x536) :|: TRUE f761_out(x537, x538) -> f758_out(x537, x538) :|: TRUE f762_out(x539, x540) -> f758_out(x539, x540) :|: TRUE f758_in(x541, x542) -> f762_in(x541, x542) :|: TRUE f758_in(x543, x544) -> f761_in(x543, x544) :|: TRUE f284_in(x545, x546) -> f898_in(x545, x546) :|: TRUE f284_in(x547, x548) -> f899_in :|: TRUE f899_out -> f284_out(x549, x550) :|: TRUE f898_out(x551, x552) -> f284_out(x551, x552) :|: TRUE f208_out(x553, 0) -> f405_out(x553) :|: TRUE f405_in(x554) -> f208_in(x554, 0) :|: TRUE f846_in(x555) -> f848_in(x555) :|: TRUE f848_out(x556) -> f846_out(x556) :|: TRUE f632_out(x557, x558) -> f622_out(x557, x558) :|: TRUE f622_in(x559, x560) -> f632_in(x559, x560) :|: TRUE f631_out(x561, x562) -> f622_out(x561, x562) :|: TRUE f622_in(x563, x564) -> f631_in(x563, x564) :|: TRUE f440_out(x565, x566) -> f438_out(x565, x566) :|: TRUE f438_in(x567, x568) -> f440_in(x567, x568) :|: TRUE f208_out(x569, s(s(s(s(0))))) -> f657_out(x569) :|: TRUE f657_in(x570) -> f208_in(x570, s(s(s(s(0))))) :|: TRUE f618_out(x571, x572) -> f613_out(x571, x572) :|: TRUE f613_in(x573, x574) -> f618_in(x573, x574) :|: TRUE f825_out(x575, x576) -> f816_out(s(x576), x575) :|: TRUE f816_in(x577, x578) -> f827_in :|: TRUE f816_in(s(x579), x580) -> f825_in(x580, x579) :|: TRUE f827_out -> f816_out(x581, x582) :|: TRUE f876_out(x583, x584) -> f874_out(s(x583), x584, 0) :|: TRUE f877_out -> f874_out(x585, x586, x587) :|: TRUE f874_in(s(x588), x589, 0) -> f876_in(x588, x589) :|: TRUE f874_in(x590, x591, x592) -> f877_in :|: TRUE f558_out(x593, x594) -> f555_out(x593, x594) :|: TRUE f555_in(x595, x596) -> f558_in(x595, x596) :|: TRUE f466_in(x597, x598) -> f479_in(x598, x597) :|: TRUE f479_out(x599, x600) -> f466_out(x600, x599) :|: TRUE f466_in(x601, x602) -> f480_in :|: TRUE f480_out -> f466_out(x603, x604) :|: TRUE f457_in(x605, x606) -> f466_in(x605, x606) :|: TRUE f457_in(x607, x608) -> f465_in(x607, x608) :|: TRUE f465_out(x609, x610) -> f457_out(x609, x610) :|: TRUE f466_out(x611, x612) -> f457_out(x611, x612) :|: TRUE f902_in(x613, x614) -> f904_in(x613, x614) :|: TRUE f904_out(x615, x616) -> f902_out(x615, x616) :|: TRUE f550_out(x617) -> f483_out(0, x617) :|: TRUE f483_in(x618, x619) -> f551_in :|: TRUE f551_out -> f483_out(x620, x621) :|: TRUE f483_in(0, x622) -> f550_in(x622) :|: TRUE f471_in(x623) -> f473_in(x623) :|: TRUE f473_out(x624) -> f471_out(x624) :|: TRUE f644_in(x625) -> f291_in(x625, s(s(s(s(0))))) :|: TRUE f291_out(x626, s(s(s(s(0))))) -> f644_out(x626) :|: TRUE f5_in(T1, T2) -> f6_in(T1, T2) :|: TRUE f6_out(x627, x628) -> f5_out(x627, x628) :|: TRUE f6_in(x629, x630) -> f208_in(x629, x630) :|: TRUE f208_out(x631, x632) -> f6_out(x631, x632) :|: TRUE Start term: f5_in(T1, T2) ---------------------------------------- (77) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f755_in(x, x1) -> f758_in(x, x1) :|: TRUE f808_in(T357, T356) -> f814_in(T357, T356) :|: TRUE f581_in(T208) -> f291_in(T208, s(s(s(0)))) :|: TRUE f855_in(s(x7), x8) -> f859_in(x8, x7) :|: TRUE f454_in(T126, T137) -> f457_in(T126, T137) :|: TRUE f879_in(T456, T454, T455) -> f881_in(T456, T454, T455) :|: TRUE f337_in(T69) -> f208_in(T69, 0) :|: TRUE f416_in(T80, T91) -> f433_in(T80, T91) :|: TRUE f416_in(x19, x20) -> f432_in(x19, x20) :|: TRUE f788_in(s(T346), 0) -> f794_in(T346) :|: TRUE f796_in(x29) -> f291_in(x29, s(s(s(s(s(s(0))))))) :|: TRUE f235_in(T13, T14) -> f258_in(T13, T14) :|: TRUE f565_in(T172, T183) -> f575_in(T172, T183) :|: TRUE f565_in(x35, x36) -> f574_in(x35, x36) :|: TRUE f767_in(x41, x42) -> f769_in(x41, x42) :|: TRUE f875_in(x46, x47, x48) -> f879_in(x48, x46, x47) :|: TRUE f604_in(T219, T218) -> f606_in(T219, T218) :|: TRUE f769_in(x57, x58) -> f779_in(x57, x58) :|: TRUE f885_in(x61, x62) -> f208_in(x61, s(x62)) :|: TRUE f432_in(s(T116), 0) -> f434_in(T116) :|: TRUE f282_in(s(x70), x71) -> f288_in(x70, x71) :|: TRUE f433_in(x82, x83) -> f438_in(x83, x82) :|: TRUE f606_in(x84, x85) -> f607_in(x84, x85) :|: TRUE f606_in(x88, x89) -> f608_in(x88, x89) :|: TRUE f901_in(x94, x95) -> f902_in(x94, x95) :|: TRUE f729_in(x100, x101) -> f755_in(x101, x100) :|: TRUE f765_in(x104) -> f208_in(x104, s(s(s(s(s(0)))))) :|: TRUE f728_in(s(x110), 0) -> f747_in(x110) :|: TRUE f473_in(T162) -> f291_in(T162, s(s(0))) :|: TRUE f402_in(0, x115) -> f405_in(x115) :|: TRUE f864_in(x121, x122, x123) -> f866_in(x121, x122, x123) :|: TRUE f288_in(x124, x125) -> f291_in(x124, x125) :|: TRUE f608_in(s(x128), x129) -> f613_in(x129, x128) :|: TRUE f714_in(x138, x139) -> f717_in(x138, x139) :|: TRUE f898_in(x142, x143) -> f901_in(x142, x143) :|: TRUE f450_in(x146, x147) -> f454_in(x146, x147) :|: TRUE f843_in(x150, x151) -> f849_in(x151, x150) :|: TRUE f866_in(x160, x161, x162) -> f868_in(x160, x161, x162) :|: TRUE f881_in(x166, x167, x168) -> f883_in(x166, x167, x168) :|: TRUE f881_in(x169, x170, x171) -> f882_in(x169, x170, x171) :|: TRUE f815_in(0, x180) -> f817_in(x180) :|: TRUE f258_in(x183, x184) -> f282_in(x183, x184) :|: TRUE f258_in(x187, x188) -> f284_in(x187, x188) :|: TRUE f817_in(x190) -> f208_in(x190, s(s(s(s(s(s(0))))))) :|: TRUE f750_in(x191) -> f291_in(x191, s(s(s(s(s(0)))))) :|: TRUE f291_in(x193, x194) -> f295_in(x193, x194) :|: TRUE f414_in(x197, x198) -> f416_in(x197, x198) :|: TRUE f574_in(s(x203), 0) -> f577_in(x203) :|: TRUE f849_in(x209, x210) -> f851_in(x209, x210) :|: TRUE f434_in(x211) -> f436_in(x211) :|: TRUE f652_in(x213, x214) -> f655_in(x213, x214) :|: TRUE f652_in(x217, x218) -> f653_in(x217, x218) :|: TRUE f465_in(s(x226), 0) -> f471_in(x226) :|: TRUE f642_in(x227) -> f644_in(x227) :|: TRUE f814_in(x230, x231) -> f815_in(x230, x231) :|: TRUE f814_in(x232, x233) -> f816_in(x232, x233) :|: TRUE f631_in(s(x236), 0) -> f642_in(x236) :|: TRUE f550_in(x243) -> f208_in(x243, s(s(0))) :|: TRUE f618_in(x244, x245) -> f622_in(x244, x245) :|: TRUE f856_in(x249) -> f208_in(x249, s(s(s(s(s(s(s(0)))))))) :|: TRUE f558_in(x252, x253) -> f565_in(x252, x253) :|: TRUE f344_in(x254, x255) -> f398_in(x254, x255) :|: TRUE f851_in(x258, x259) -> f855_in(x258, x259) :|: TRUE f851_in(x264, x265) -> f853_in(x264, x265) :|: TRUE f883_in(s(x272), x273, x274) -> f887_in(x273, x274, x272) :|: TRUE f848_in(x278) -> f291_in(x278, s(s(s(s(s(s(s(0)))))))) :|: TRUE f443_in(x279) -> f208_in(x279, s(0)) :|: TRUE f409_in(x281, x282) -> f414_in(x281, x282) :|: TRUE f836_in(x285, x286) -> f843_in(x285, x286) :|: TRUE f836_in(x287, x288) -> f842_in(x287, x288) :|: TRUE f653_in(0, x298) -> f657_in(x298) :|: TRUE f436_in(x299) -> f291_in(x299, s(0)) :|: TRUE f650_in(x303, x304) -> f652_in(x303, x304) :|: TRUE f441_in(0, x305) -> f443_in(x305) :|: TRUE f842_in(s(x316), 0) -> f846_in(x316) :|: TRUE f607_in(0, x319) -> f609_in(x319) :|: TRUE f440_in(x325, x326) -> f441_in(x325, x326) :|: TRUE f440_in(x329, x330) -> f442_in(x329, x330) :|: TRUE f334_in(x332) -> f337_in(x332) :|: TRUE f404_in(s(x335), x336) -> f409_in(x336, x335) :|: TRUE f481_in(x343, x344) -> f483_in(x343, x344) :|: TRUE f481_in(x345, x346) -> f548_in(x345, x346) :|: TRUE f575_in(x355, x356) -> f604_in(x356, x355) :|: TRUE f655_in(s(x359), x360) -> f711_in(x360, x359) :|: TRUE f332_in(x367, x368) -> f344_in(x368, x367) :|: TRUE f859_in(x375, x376) -> f864_in(x375, s(s(s(s(s(s(s(x376))))))), x376) :|: TRUE f887_in(x380, x381, x382) -> f864_in(x380, x381, x382) :|: TRUE f609_in(x383) -> f208_in(x383, s(s(s(0)))) :|: TRUE f794_in(x386) -> f796_in(x386) :|: TRUE f853_in(0, x390) -> f856_in(x390) :|: TRUE f331_in(s(x398), 0) -> f334_in(x398) :|: TRUE f479_in(x401, x402) -> f481_in(x401, x402) :|: TRUE f868_in(x403, x404, x405) -> f874_in(x403, x404, x405) :|: TRUE f868_in(x409, x410, x411) -> f875_in(x409, x410, x411) :|: TRUE f577_in(x416) -> f581_in(x416) :|: TRUE f442_in(s(x421), x422) -> f450_in(x422, x421) :|: TRUE f761_in(0, x425) -> f765_in(x425) :|: TRUE f208_in(x431, x432) -> f235_in(x431, x432) :|: TRUE f825_in(x435, x436) -> f834_in(x435, x436) :|: TRUE f762_in(s(x443), x444) -> f767_in(x444, x443) :|: TRUE f834_in(x447, x448) -> f836_in(x447, x448) :|: TRUE f548_in(s(x455), x456) -> f555_in(x456, x455) :|: TRUE f398_in(x459, x460) -> f402_in(x459, x460) :|: TRUE f398_in(x461, x462) -> f404_in(x461, x462) :|: TRUE f632_in(x473, x474) -> f650_in(x474, x473) :|: TRUE f295_in(x475, x476) -> f299_in(x475, x476) :|: TRUE f882_in(0, x485, x486) -> f885_in(x485, x486) :|: TRUE f711_in(x489, x490) -> f714_in(x489, x490) :|: TRUE f878_in(x493, x494) -> f291_in(x493, s(x494)) :|: TRUE f717_in(x495, x496) -> f728_in(x495, x496) :|: TRUE f717_in(x499, x500) -> f729_in(x499, x500) :|: TRUE f747_in(x503) -> f750_in(x503) :|: TRUE f876_in(x507, x508) -> f878_in(x507, x508) :|: TRUE f904_in(x511, x512) -> f208_in(x511, x512) :|: TRUE f779_in(x513, x514) -> f788_in(x513, x514) :|: TRUE f779_in(x515, x516) -> f789_in(x515, x516) :|: TRUE f299_in(x521, x522) -> f332_in(x521, x522) :|: TRUE f299_in(x527, x528) -> f331_in(x527, x528) :|: TRUE f789_in(x529, x530) -> f808_in(x530, x529) :|: TRUE f758_in(x541, x542) -> f762_in(x541, x542) :|: TRUE f758_in(x543, x544) -> f761_in(x543, x544) :|: TRUE f284_in(x545, x546) -> f898_in(x545, x546) :|: TRUE f405_in(x554) -> f208_in(x554, 0) :|: TRUE f846_in(x555) -> f848_in(x555) :|: TRUE f622_in(x559, x560) -> f632_in(x559, x560) :|: TRUE f622_in(x563, x564) -> f631_in(x563, x564) :|: TRUE f438_in(x567, x568) -> f440_in(x567, x568) :|: TRUE f657_in(x570) -> f208_in(x570, s(s(s(s(0))))) :|: TRUE f613_in(x573, x574) -> f618_in(x573, x574) :|: TRUE f816_in(s(x579), x580) -> f825_in(x580, x579) :|: TRUE f874_in(s(x588), x589, 0) -> f876_in(x588, x589) :|: TRUE f555_in(x595, x596) -> f558_in(x595, x596) :|: TRUE f466_in(x597, x598) -> f479_in(x598, x597) :|: TRUE f457_in(x605, x606) -> f466_in(x605, x606) :|: TRUE f457_in(x607, x608) -> f465_in(x607, x608) :|: TRUE f902_in(x613, x614) -> f904_in(x613, x614) :|: TRUE f483_in(0, x622) -> f550_in(x622) :|: TRUE f471_in(x623) -> f473_in(x623) :|: TRUE f644_in(x625) -> f291_in(x625, s(s(s(s(0))))) :|: TRUE ---------------------------------------- (78) Obligation: Rules: f755_in(x, x1) -> f758_in(x, x1) :|: TRUE f808_in(T357, T356) -> f814_in(T357, T356) :|: TRUE f581_in(T208) -> f291_in(T208, s(s(s(0)))) :|: TRUE f855_in(s(x7), x8) -> f859_in(x8, x7) :|: TRUE f454_in(T126, T137) -> f457_in(T126, T137) :|: TRUE f879_in(T456, T454, T455) -> f881_in(T456, T454, T455) :|: TRUE f337_in(T69) -> f208_in(T69, 0) :|: TRUE f416_in(T80, T91) -> f433_in(T80, T91) :|: TRUE f416_in(x19, x20) -> f432_in(x19, x20) :|: TRUE f788_in(s(T346), 0) -> f794_in(T346) :|: TRUE f796_in(x29) -> f291_in(x29, s(s(s(s(s(s(0))))))) :|: TRUE f235_in(T13, T14) -> f258_in(T13, T14) :|: TRUE f565_in(T172, T183) -> f575_in(T172, T183) :|: TRUE f565_in(x35, x36) -> f574_in(x35, x36) :|: TRUE f767_in(x41, x42) -> f769_in(x41, x42) :|: TRUE f875_in(x46, x47, x48) -> f879_in(x48, x46, x47) :|: TRUE f604_in(T219, T218) -> f606_in(T219, T218) :|: TRUE f769_in(x57, x58) -> f779_in(x57, x58) :|: TRUE f885_in(x61, x62) -> f208_in(x61, s(x62)) :|: TRUE f432_in(s(T116), 0) -> f434_in(T116) :|: TRUE f282_in(s(x70), x71) -> f288_in(x70, x71) :|: TRUE f433_in(x82, x83) -> f438_in(x83, x82) :|: TRUE f606_in(x84, x85) -> f607_in(x84, x85) :|: TRUE f606_in(x88, x89) -> f608_in(x88, x89) :|: TRUE f901_in(x94, x95) -> f902_in(x94, x95) :|: TRUE f729_in(x100, x101) -> f755_in(x101, x100) :|: TRUE f765_in(x104) -> f208_in(x104, s(s(s(s(s(0)))))) :|: TRUE f728_in(s(x110), 0) -> f747_in(x110) :|: TRUE f473_in(T162) -> f291_in(T162, s(s(0))) :|: TRUE f402_in(0, x115) -> f405_in(x115) :|: TRUE f864_in(x121, x122, x123) -> f866_in(x121, x122, x123) :|: TRUE f288_in(x124, x125) -> f291_in(x124, x125) :|: TRUE f608_in(s(x128), x129) -> f613_in(x129, x128) :|: TRUE f714_in(x138, x139) -> f717_in(x138, x139) :|: TRUE f898_in(x142, x143) -> f901_in(x142, x143) :|: TRUE f450_in(x146, x147) -> f454_in(x146, x147) :|: TRUE f843_in(x150, x151) -> f849_in(x151, x150) :|: TRUE f866_in(x160, x161, x162) -> f868_in(x160, x161, x162) :|: TRUE f881_in(x166, x167, x168) -> f883_in(x166, x167, x168) :|: TRUE f881_in(x169, x170, x171) -> f882_in(x169, x170, x171) :|: TRUE f815_in(0, x180) -> f817_in(x180) :|: TRUE f258_in(x183, x184) -> f282_in(x183, x184) :|: TRUE f258_in(x187, x188) -> f284_in(x187, x188) :|: TRUE f817_in(x190) -> f208_in(x190, s(s(s(s(s(s(0))))))) :|: TRUE f750_in(x191) -> f291_in(x191, s(s(s(s(s(0)))))) :|: TRUE f291_in(x193, x194) -> f295_in(x193, x194) :|: TRUE f414_in(x197, x198) -> f416_in(x197, x198) :|: TRUE f574_in(s(x203), 0) -> f577_in(x203) :|: TRUE f849_in(x209, x210) -> f851_in(x209, x210) :|: TRUE f434_in(x211) -> f436_in(x211) :|: TRUE f652_in(x213, x214) -> f655_in(x213, x214) :|: TRUE f652_in(x217, x218) -> f653_in(x217, x218) :|: TRUE f465_in(s(x226), 0) -> f471_in(x226) :|: TRUE f642_in(x227) -> f644_in(x227) :|: TRUE f814_in(x230, x231) -> f815_in(x230, x231) :|: TRUE f814_in(x232, x233) -> f816_in(x232, x233) :|: TRUE f631_in(s(x236), 0) -> f642_in(x236) :|: TRUE f550_in(x243) -> f208_in(x243, s(s(0))) :|: TRUE f618_in(x244, x245) -> f622_in(x244, x245) :|: TRUE f856_in(x249) -> f208_in(x249, s(s(s(s(s(s(s(0)))))))) :|: TRUE f558_in(x252, x253) -> f565_in(x252, x253) :|: TRUE f344_in(x254, x255) -> f398_in(x254, x255) :|: TRUE f851_in(x258, x259) -> f855_in(x258, x259) :|: TRUE f851_in(x264, x265) -> f853_in(x264, x265) :|: TRUE f883_in(s(x272), x273, x274) -> f887_in(x273, x274, x272) :|: TRUE f848_in(x278) -> f291_in(x278, s(s(s(s(s(s(s(0)))))))) :|: TRUE f443_in(x279) -> f208_in(x279, s(0)) :|: TRUE f409_in(x281, x282) -> f414_in(x281, x282) :|: TRUE f836_in(x285, x286) -> f843_in(x285, x286) :|: TRUE f836_in(x287, x288) -> f842_in(x287, x288) :|: TRUE f653_in(0, x298) -> f657_in(x298) :|: TRUE f436_in(x299) -> f291_in(x299, s(0)) :|: TRUE f650_in(x303, x304) -> f652_in(x303, x304) :|: TRUE f441_in(0, x305) -> f443_in(x305) :|: TRUE f842_in(s(x316), 0) -> f846_in(x316) :|: TRUE f607_in(0, x319) -> f609_in(x319) :|: TRUE f440_in(x325, x326) -> f441_in(x325, x326) :|: TRUE f440_in(x329, x330) -> f442_in(x329, x330) :|: TRUE f334_in(x332) -> f337_in(x332) :|: TRUE f404_in(s(x335), x336) -> f409_in(x336, x335) :|: TRUE f481_in(x343, x344) -> f483_in(x343, x344) :|: TRUE f481_in(x345, x346) -> f548_in(x345, x346) :|: TRUE f575_in(x355, x356) -> f604_in(x356, x355) :|: TRUE f655_in(s(x359), x360) -> f711_in(x360, x359) :|: TRUE f332_in(x367, x368) -> f344_in(x368, x367) :|: TRUE f859_in(x375, x376) -> f864_in(x375, s(s(s(s(s(s(s(x376))))))), x376) :|: TRUE f887_in(x380, x381, x382) -> f864_in(x380, x381, x382) :|: TRUE f609_in(x383) -> f208_in(x383, s(s(s(0)))) :|: TRUE f794_in(x386) -> f796_in(x386) :|: TRUE f853_in(0, x390) -> f856_in(x390) :|: TRUE f331_in(s(x398), 0) -> f334_in(x398) :|: TRUE f479_in(x401, x402) -> f481_in(x401, x402) :|: TRUE f868_in(x403, x404, x405) -> f874_in(x403, x404, x405) :|: TRUE f868_in(x409, x410, x411) -> f875_in(x409, x410, x411) :|: TRUE f577_in(x416) -> f581_in(x416) :|: TRUE f442_in(s(x421), x422) -> f450_in(x422, x421) :|: TRUE f761_in(0, x425) -> f765_in(x425) :|: TRUE f208_in(x431, x432) -> f235_in(x431, x432) :|: TRUE f825_in(x435, x436) -> f834_in(x435, x436) :|: TRUE f762_in(s(x443), x444) -> f767_in(x444, x443) :|: TRUE f834_in(x447, x448) -> f836_in(x447, x448) :|: TRUE f548_in(s(x455), x456) -> f555_in(x456, x455) :|: TRUE f398_in(x459, x460) -> f402_in(x459, x460) :|: TRUE f398_in(x461, x462) -> f404_in(x461, x462) :|: TRUE f632_in(x473, x474) -> f650_in(x474, x473) :|: TRUE f295_in(x475, x476) -> f299_in(x475, x476) :|: TRUE f882_in(0, x485, x486) -> f885_in(x485, x486) :|: TRUE f711_in(x489, x490) -> f714_in(x489, x490) :|: TRUE f878_in(x493, x494) -> f291_in(x493, s(x494)) :|: TRUE f717_in(x495, x496) -> f728_in(x495, x496) :|: TRUE f717_in(x499, x500) -> f729_in(x499, x500) :|: TRUE f747_in(x503) -> f750_in(x503) :|: TRUE f876_in(x507, x508) -> f878_in(x507, x508) :|: TRUE f904_in(x511, x512) -> f208_in(x511, x512) :|: TRUE f779_in(x513, x514) -> f788_in(x513, x514) :|: TRUE f779_in(x515, x516) -> f789_in(x515, x516) :|: TRUE f299_in(x521, x522) -> f332_in(x521, x522) :|: TRUE f299_in(x527, x528) -> f331_in(x527, x528) :|: TRUE f789_in(x529, x530) -> f808_in(x530, x529) :|: TRUE f758_in(x541, x542) -> f762_in(x541, x542) :|: TRUE f758_in(x543, x544) -> f761_in(x543, x544) :|: TRUE f284_in(x545, x546) -> f898_in(x545, x546) :|: TRUE f405_in(x554) -> f208_in(x554, 0) :|: TRUE f846_in(x555) -> f848_in(x555) :|: TRUE f622_in(x559, x560) -> f632_in(x559, x560) :|: TRUE f622_in(x563, x564) -> f631_in(x563, x564) :|: TRUE f438_in(x567, x568) -> f440_in(x567, x568) :|: TRUE f657_in(x570) -> f208_in(x570, s(s(s(s(0))))) :|: TRUE f613_in(x573, x574) -> f618_in(x573, x574) :|: TRUE f816_in(s(x579), x580) -> f825_in(x580, x579) :|: TRUE f874_in(s(x588), x589, 0) -> f876_in(x588, x589) :|: TRUE f555_in(x595, x596) -> f558_in(x595, x596) :|: TRUE f466_in(x597, x598) -> f479_in(x598, x597) :|: TRUE f457_in(x605, x606) -> f466_in(x605, x606) :|: TRUE f457_in(x607, x608) -> f465_in(x607, x608) :|: TRUE f902_in(x613, x614) -> f904_in(x613, x614) :|: TRUE f483_in(0, x622) -> f550_in(x622) :|: TRUE f471_in(x623) -> f473_in(x623) :|: TRUE f644_in(x625) -> f291_in(x625, s(s(s(s(0))))) :|: TRUE ---------------------------------------- (79) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (80) Obligation: Rules: f864_in(x121:0, x122:0, cons_0) -> f235_in(x121:0, s(x122:0)) :|: TRUE && cons_0 = 0 f299_in(s(x), s(x1)) -> f299_in(x, s(0)) :|: TRUE && x1 = 0 f299_in(x2, s(s(s(s(s(x3)))))) -> f235_in(x2, s(s(s(s(s(0)))))) :|: TRUE && x3 = 0 f299_in(x4, s(s(s(s(s(s(s(x5)))))))) -> f235_in(x4, s(s(s(s(s(s(s(0)))))))) :|: TRUE && x5 = 0 f299_in(x6, s(s(s(s(x7))))) -> f235_in(x6, s(s(s(s(0))))) :|: TRUE && x7 = 0 f235_in(s(x70:0), T14:0) -> f299_in(x70:0, T14:0) :|: TRUE f299_in(s(x8), s(s(s(s(s(s(x9))))))) -> f299_in(x8, s(s(s(s(s(s(0))))))) :|: TRUE && x9 = 0 f299_in(s(x10), s(s(s(s(x11))))) -> f299_in(x10, s(s(s(s(0))))) :|: TRUE && x11 = 0 f235_in(x12, x13) -> f235_in(x12, x13) :|: TRUE f299_in(x14, s(s(x15))) -> f235_in(x14, s(s(0))) :|: TRUE && x15 = 0 f864_in(s(x16), x17, x18) -> f299_in(x16, s(x17)) :|: TRUE && x18 = 0 f299_in(s(x19), s(s(s(s(s(x20)))))) -> f299_in(x19, s(s(s(s(s(0)))))) :|: TRUE && x20 = 0 f299_in(s(x21), x22) -> f235_in(x21, 0) :|: TRUE && x22 = 0 f864_in(x23, x24, s(x25)) -> f864_in(x23, x24, x25) :|: TRUE f299_in(s(x26), s(s(s(s(s(s(s(x27)))))))) -> f299_in(x26, s(s(s(s(s(s(s(0)))))))) :|: TRUE && x27 = 0 f299_in(s(x28), s(s(x29))) -> f299_in(x28, s(s(0))) :|: TRUE && x29 = 0 f299_in(x30, x31) -> f235_in(x30, 0) :|: TRUE && x31 = 0 f299_in(x32, s(s(s(s(s(s(x33))))))) -> f235_in(x32, s(s(s(s(s(s(0))))))) :|: TRUE && x33 = 0 f299_in(x521:0, s(s(s(s(s(s(s(s(x7:0))))))))) -> f864_in(x521:0, s(s(s(s(s(s(s(x7:0))))))), x7:0) :|: TRUE f299_in(x34, s(x35)) -> f235_in(x34, s(0)) :|: TRUE && x35 = 0 f299_in(x36, s(s(s(x37)))) -> f235_in(x36, s(s(s(0)))) :|: TRUE && x37 = 0 f299_in(s(x38), s(s(s(x39)))) -> f299_in(x38, s(s(s(0)))) :|: TRUE && x39 = 0 ---------------------------------------- (81) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (82) Obligation: Rules: f864_in(x121:0, x122:0, cons_0) -> f235_in(x121:0, s(x122:0)) :|: TRUE && cons_0 = 0 f299_in(s(x), s(x1)) -> f299_in(x, s(0)) :|: TRUE && x1 = 0 f299_in(x2, s(s(s(s(s(x3)))))) -> f235_in(x2, s(s(s(s(s(0)))))) :|: TRUE && x3 = 0 f299_in(x4, s(s(s(s(s(s(s(x5)))))))) -> f235_in(x4, s(s(s(s(s(s(s(0)))))))) :|: TRUE && x5 = 0 f299_in(x6, s(s(s(s(x7))))) -> f235_in(x6, s(s(s(s(0))))) :|: TRUE && x7 = 0 f235_in(s(x70:0), T14:0) -> f299_in(x70:0, T14:0) :|: TRUE f299_in(s(x8), s(s(s(s(s(s(x9))))))) -> f299_in(x8, s(s(s(s(s(s(0))))))) :|: TRUE && x9 = 0 f299_in(s(x10), s(s(s(s(x11))))) -> f299_in(x10, s(s(s(s(0))))) :|: TRUE && x11 = 0 f235_in(x12, x13) -> f235_in(x12, x13) :|: TRUE f299_in(x14, s(s(x15))) -> f235_in(x14, s(s(0))) :|: TRUE && x15 = 0 f864_in(s(x16), x17, x18) -> f299_in(x16, s(x17)) :|: TRUE && x18 = 0 f299_in(s(x19), s(s(s(s(s(x20)))))) -> f299_in(x19, s(s(s(s(s(0)))))) :|: TRUE && x20 = 0 f299_in(s(x21), x22) -> f235_in(x21, 0) :|: TRUE && x22 = 0 f864_in(x23, x24, s(x25)) -> f864_in(x23, x24, x25) :|: TRUE f299_in(s(x26), s(s(s(s(s(s(s(x27)))))))) -> f299_in(x26, s(s(s(s(s(s(s(0)))))))) :|: TRUE && x27 = 0 f299_in(s(x28), s(s(x29))) -> f299_in(x28, s(s(0))) :|: TRUE && x29 = 0 f299_in(x30, x31) -> f235_in(x30, 0) :|: TRUE && x31 = 0 f299_in(x32, s(s(s(s(s(s(x33))))))) -> f235_in(x32, s(s(s(s(s(s(0))))))) :|: TRUE && x33 = 0 f299_in(x521:0, s(s(s(s(s(s(s(s(x7:0))))))))) -> f864_in(x521:0, s(s(s(s(s(s(s(x7:0))))))), x7:0) :|: TRUE f299_in(x34, s(x35)) -> f235_in(x34, s(0)) :|: TRUE && x35 = 0 f299_in(x36, s(s(s(x37)))) -> f235_in(x36, s(s(s(0)))) :|: TRUE && x37 = 0 f299_in(s(x38), s(s(s(x39)))) -> f299_in(x38, s(s(s(0)))) :|: TRUE && x39 = 0 ---------------------------------------- (83) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(times X Y Z)", "(mult X Y (0) Z)" ], [ "(mult (0) Y (0) Z)", "(',' (!) (eq Z (0)))" ], [ "(mult (s X) Y (0) Z)", "(',' (!) (mult X Y Y Z))" ], [ "(mult X Y W (s Z))", "(',' (p W P) (mult X Y P Z))" ], [ "(p (0) (0))", null ], [ "(p (s X) X)", null ], [ "(eq X X)", null ] ] }, "graph": { "nodes": { "907": { "goal": [ { "clause": 2, "scope": 28, "term": "(mult T394 (s T408) T405 T397)" }, { "clause": 3, "scope": 28, "term": "(mult T394 (s T408) T405 T397)" } ], "kb": { "nonunifying": [[ "(mult T394 (s T408) T405 T397)", "(mult (0) X455 (0) X456)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T394", "T405", "T408" ], "free": [ "X455", "X456" ], "exprvars": [] } }, "909": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq T417 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "591": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "350": { "goal": [ { "clause": 1, "scope": 7, "term": "(mult T72 (s T83) T83 T75)" }, { "clause": 2, "scope": 7, "term": "(mult T72 (s T83) T83 T75)" }, { "clause": 3, "scope": 7, "term": "(mult T72 (s T83) T83 T75)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T72", "T83" ], "free": [], "exprvars": [] } }, "592": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "593": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "595": { "goal": [{ "clause": 2, "scope": 16, "term": "(mult T210 (s (s (s (s T221)))) T221 T213)" }], "kb": { "nonunifying": [[ "(mult T210 (s (s (s (s T221)))) T221 T213)", "(mult (0) X245 (0) X246)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T210", "T221" ], "free": [ "X245", "X246" ], "exprvars": [] } }, "354": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_7) (eq T92 (0)))" }, { "clause": 2, "scope": 7, "term": "(mult (0) (s (0)) (0) T75)" }, { "clause": 3, "scope": 7, "term": "(mult (0) (s (0)) (0) T75)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "596": { "goal": [{ "clause": 3, "scope": 16, "term": "(mult T210 (s (s (s (s T221)))) T221 T213)" }], "kb": { "nonunifying": [[ "(mult T210 (s (s (s (s T221)))) T221 T213)", "(mult (0) X245 (0) X246)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T210", "T221" ], "free": [ "X245", "X246" ], "exprvars": [] } }, "355": { "goal": [ { "clause": 2, "scope": 7, "term": "(mult T72 (s T83) T83 T75)" }, { "clause": 3, "scope": 7, "term": "(mult T72 (s T83) T83 T75)" } ], "kb": { "nonunifying": [[ "(mult T72 (s T83) T83 T75)", "(mult (0) X89 (0) X90)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T72", "T83" ], "free": [ "X89", "X90" ], "exprvars": [] } }, "356": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq T92 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "598": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (!_16) (mult T246 (s (s (s (s (0))))) (s (s (s (s (0))))) T249))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T246"], "free": [], "exprvars": [] } }, "357": { "goal": [{ "clause": 6, "scope": 8, "term": "(eq T92 (0))" }], "kb": { 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"clause": -1, "scope": -1, "term": "(mult T210 (s (s (s (0)))) (0) T213)" }], "kb": { "nonunifying": [[ "(mult T210 (s (s (s (0)))) (0) T167)", "(mult (0) X193 (0) X194)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T210"], "free": [ "X193", "X194" ], "exprvars": [] } }, "323": { "goal": [{ "clause": 5, "scope": 6, "term": "(',' (p T73 X72) (mult T72 T73 X72 T75))" }], "kb": { "nonunifying": [[ "(mult T72 T73 T73 T36)", "(mult (0) X37 (0) X38)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T72", "T73" ], "free": [ "X37", "X38", "X72" ], "exprvars": [] } }, "566": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "325": { "goal": [{ "clause": -1, "scope": -1, "term": "(mult T72 (0) (0) T75)" }], "kb": { "nonunifying": [[ "(mult T72 (0) (0) T36)", "(mult (0) X37 (0) X38)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T72"], "free": [ "X37", "X38" ], "exprvars": [] } }, "328": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "572": { "goal": [{ "clause": -1, "scope": -1, "term": "(mult T210 (s (s (s (s T221)))) T221 T213)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T210", "T221" ], "free": [], "exprvars": [] } }, "573": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "576": { "goal": [ { "clause": 1, "scope": 16, "term": "(mult T210 (s (s (s (s T221)))) T221 T213)" }, { "clause": 2, "scope": 16, "term": "(mult T210 (s (s (s (s T221)))) T221 T213)" }, { "clause": 3, "scope": 16, "term": "(mult T210 (s (s (s (s T221)))) T221 T213)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T210", "T221" ], "free": [], "exprvars": [] } }, "584": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_16) (eq T230 (0)))" }, { "clause": 2, "scope": 16, "term": "(mult (0) (s (s (s (s (0))))) (0) T213)" }, { "clause": 3, "scope": 16, "term": "(mult (0) (s (s (s (s (0))))) (0) T213)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "585": { "goal": [ { "clause": 2, "scope": 16, "term": "(mult T210 (s (s (s (s T221)))) T221 T213)" }, { "clause": 3, "scope": 16, "term": "(mult T210 (s (s (s (s T221)))) T221 T213)" } ], "kb": { "nonunifying": [[ "(mult T210 (s (s (s (s T221)))) T221 T213)", "(mult (0) X245 (0) X246)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T210", "T221" ], "free": [ "X245", "X246" ], "exprvars": [] } }, "586": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq T230 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "587": { "goal": [{ "clause": 6, "scope": 17, "term": "(eq T230 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "348": { "goal": [{ "clause": -1, "scope": -1, "term": "(mult T72 (s T83) T83 T75)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T72", "T83" ], "free": [], "exprvars": [] } }, "349": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "900": { "goal": [{ "clause": -1, "scope": -1, "term": "(mult T394 (s T408) T405 T397)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T394", "T405", "T408" ], "free": [], "exprvars": [] } }, "905": { "goal": [ { "clause": 1, "scope": 28, "term": "(mult T394 (s T408) T405 T397)" }, { "clause": 2, "scope": 28, "term": "(mult T394 (s T408) T405 T397)" }, { "clause": 3, "scope": 28, "term": "(mult T394 (s T408) T405 T397)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T394", "T405", "T408" ], "free": [], "exprvars": [] } }, "906": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_28) (eq T417 (0)))" }, { "clause": 2, "scope": 28, "term": "(mult (0) (s T415) (0) T397)" }, { "clause": 3, "scope": 28, "term": "(mult (0) (s T415) (0) T397)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T415"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 2, "label": "CASE" }, { "from": 2, "to": 25, "label": "ONLY EVAL with clause\ntimes(X4, X5, X6) :- mult(X4, X5, 0, X6).\nand substitutionT1 -> T7,\nX4 -> T7,\nT2 -> T8,\nX5 -> T8,\nT3 -> T10,\nX6 -> T10,\nT9 -> T10" }, { "from": 25, "to": 34, "label": "CASE" }, { "from": 34, "to": 35, "label": "EVAL with clause\nmult(0, X11, 0, X12) :- ','(!_2, eq(X12, 0)).\nand substitutionT7 -> 0,\nT8 -> T15,\nX11 -> T15,\nT10 -> T17,\nX12 -> T17,\nT16 -> T17" }, { "from": 34, "to": 36, "label": "EVAL-BACKTRACK" }, { "from": 35, "to": 37, "label": "CUT" }, { "from": 36, "to": 51, "label": "PARALLEL" }, { "from": 36, "to": 60, "label": "PARALLEL" }, { "from": 37, "to": 38, "label": "CASE" }, { "from": 38, "to": 39, "label": "EVAL with clause\neq(X15, X15).\nand substitutionT17 -> 0,\nX15 -> 0,\nT20 -> 0" }, { "from": 38, "to": 40, "label": "EVAL-BACKTRACK" }, { "from": 39, "to": 41, "label": "SUCCESS" }, { "from": 51, "to": 273, "label": "EVAL with clause\nmult(s(X28), X29, 0, X30) :- ','(!_2, mult(X28, X29, X29, X30)).\nand substitutionX28 -> T33,\nT7 -> s(T33),\nT8 -> T34,\nX29 -> T34,\nT10 -> T36,\nX30 -> T36,\nT35 -> T36" }, { "from": 51, "to": 277, "label": "EVAL-BACKTRACK" }, { "from": 60, "to": 928, "label": "EVAL with clause\nmult(X512, X513, X514, s(X515)) :- ','(p(X514, X516), mult(X512, X513, X516, X515)).\nand substitutionT7 -> T469,\nX512 -> T469,\nT8 -> T470,\nX513 -> T470,\nX514 -> 0,\nX515 -> T472,\nT10 -> s(T472),\nT471 -> T472" }, { "from": 60, "to": 929, "label": "EVAL-BACKTRACK" }, { "from": 273, "to": 279, "label": "CUT" }, { "from": 279, "to": 283, "label": "CASE" }, { "from": 283, "to": 285, "label": "EVAL with clause\nmult(0, X37, 0, X38) :- ','(!_4, eq(X38, 0)).\nand substitutionT33 -> 0,\nT34 -> 0,\nX37 -> 0,\nT43 -> 0,\nT36 -> T45,\nX38 -> T45,\nT44 -> T45" }, { "from": 283, "to": 286, "label": "EVAL-BACKTRACK" }, { "from": 285, "to": 287, "label": "CUT" }, { "from": 286, "to": 310, "label": "PARALLEL" }, { "from": 286, "to": 311, "label": "PARALLEL" }, { "from": 287, "to": 289, "label": "CASE" }, { "from": 289, "to": 292, "label": "EVAL with clause\neq(X41, X41).\nand substitutionT45 -> 0,\nX41 -> 0,\nT48 -> 0" }, { "from": 289, "to": 293, "label": "EVAL-BACKTRACK" }, { "from": 292, "to": 294, "label": "SUCCESS" }, { "from": 310, "to": 312, "label": "EVAL with clause\nmult(s(X54), X55, 0, X56) :- ','(!_4, mult(X54, X55, X55, X56)).\nand substitutionX54 -> T61,\nT33 -> s(T61),\nT34 -> 0,\nX55 -> 0,\nT62 -> 0,\nT36 -> T64,\nX56 -> T64,\nT63 -> T64" }, { "from": 310, "to": 313, "label": "EVAL-BACKTRACK" }, { "from": 311, "to": 315, "label": "EVAL with clause\nmult(X68, X69, X70, s(X71)) :- ','(p(X70, X72), mult(X68, X69, X72, X71)).\nand substitutionT33 -> T72,\nX68 -> T72,\nT34 -> T73,\nX69 -> T73,\nX70 -> T73,\nX71 -> T75,\nT36 -> s(T75),\nT74 -> T75" }, { "from": 311, "to": 318, "label": "EVAL-BACKTRACK" }, { "from": 312, "to": 314, "label": "CUT" }, { "from": 314, "to": 25, "label": "INSTANCE with matching:\nT7 -> T61\nT8 -> 0\nT10 -> T64" }, { "from": 315, "to": 319, "label": "CASE" }, { "from": 319, "to": 321, "label": "PARALLEL" }, { "from": 319, "to": 323, "label": "PARALLEL" }, { "from": 321, "to": 325, "label": "EVAL with clause\np(0, 0).\nand substitutionT73 -> 0,\nX72 -> 0" }, { "from": 321, "to": 328, "label": "EVAL-BACKTRACK" }, { "from": 323, "to": 348, "label": "EVAL with clause\np(s(X82), X82).\nand substitutionX82 -> T83,\nT73 -> s(T83),\nX72 -> T83" }, { "from": 323, "to": 349, "label": "EVAL-BACKTRACK" }, { "from": 325, "to": 25, "label": "INSTANCE with matching:\nT7 -> T72\nT8 -> 0\nT10 -> T75" }, { "from": 348, "to": 350, "label": "CASE" }, { "from": 350, "to": 354, "label": "EVAL with clause\nmult(0, X89, 0, X90) :- ','(!_7, eq(X90, 0)).\nand substitutionT72 -> 0,\nT83 -> 0,\nX89 -> s(0),\nT90 -> 0,\nT75 -> T92,\nX90 -> T92,\nT91 -> T92" }, { "from": 350, "to": 355, "label": "EVAL-BACKTRACK" }, { "from": 354, "to": 356, "label": "CUT" }, { "from": 355, "to": 361, "label": "PARALLEL" }, { "from": 355, "to": 363, "label": "PARALLEL" }, { "from": 356, "to": 357, "label": "CASE" }, { "from": 357, "to": 358, "label": "EVAL with clause\neq(X93, X93).\nand substitutionT92 -> 0,\nX93 -> 0,\nT95 -> 0" }, { "from": 357, "to": 359, "label": "EVAL-BACKTRACK" }, { "from": 358, "to": 360, "label": "SUCCESS" }, { "from": 361, "to": 367, "label": "EVAL with clause\nmult(s(X106), X107, 0, X108) :- ','(!_7, mult(X106, X107, X107, X108)).\nand substitutionX106 -> T108,\nT72 -> s(T108),\nT83 -> 0,\nX107 -> s(0),\nT109 -> 0,\nT75 -> T111,\nX108 -> T111,\nT110 -> T111" }, { "from": 361, "to": 369, "label": "EVAL-BACKTRACK" }, { "from": 363, "to": 375, "label": "EVAL with clause\nmult(X120, X121, X122, s(X123)) :- ','(p(X122, X124), mult(X120, X121, X124, X123)).\nand substitutionT72 -> T118,\nX120 -> T118,\nT83 -> T119,\nX121 -> s(T119),\nX122 -> T119,\nX123 -> T121,\nT75 -> s(T121),\nT120 -> T121" }, { "from": 363, "to": 376, "label": "EVAL-BACKTRACK" }, { "from": 367, "to": 370, "label": "CUT" }, { "from": 370, "to": 279, "label": "INSTANCE with matching:\nT33 -> T108\nT34 -> s(0)\nT36 -> T111" }, { "from": 375, "to": 377, "label": "CASE" }, { "from": 377, "to": 379, "label": "PARALLEL" }, { "from": 377, "to": 380, "label": "PARALLEL" }, { "from": 379, "to": 381, "label": "EVAL with clause\np(0, 0).\nand substitutionT119 -> 0,\nX124 -> 0" }, { "from": 379, "to": 382, "label": "EVAL-BACKTRACK" }, { "from": 380, "to": 401, "label": "EVAL with clause\np(s(X134), X134).\nand substitutionX134 -> T129,\nT119 -> s(T129),\nX124 -> T129" }, { "from": 380, "to": 403, "label": "EVAL-BACKTRACK" }, { "from": 381, "to": 25, "label": "INSTANCE with matching:\nT7 -> T118\nT8 -> s(0)\nT10 -> T121" }, { "from": 401, "to": 437, "label": "CASE" }, { "from": 437, "to": 487, "label": "EVAL with clause\nmult(0, X141, 0, X142) :- ','(!_10, eq(X142, 0)).\nand substitutionT118 -> 0,\nT129 -> 0,\nX141 -> s(s(0)),\nT136 -> 0,\nT121 -> T138,\nX142 -> T138,\nT137 -> T138" }, { "from": 437, "to": 488, "label": "EVAL-BACKTRACK" }, { "from": 487, "to": 489, "label": "CUT" }, { "from": 488, "to": 499, "label": "PARALLEL" }, { "from": 488, "to": 501, "label": "PARALLEL" }, { "from": 489, "to": 490, "label": "CASE" }, { "from": 490, "to": 492, "label": "EVAL with clause\neq(X145, X145).\nand substitutionT138 -> 0,\nX145 -> 0,\nT141 -> 0" }, { "from": 490, "to": 493, "label": "EVAL-BACKTRACK" }, { "from": 492, "to": 494, "label": "SUCCESS" }, { "from": 499, "to": 502, "label": "EVAL with clause\nmult(s(X158), X159, 0, X160) :- ','(!_10, mult(X158, X159, X159, X160)).\nand substitutionX158 -> T154,\nT118 -> s(T154),\nT129 -> 0,\nX159 -> s(s(0)),\nT155 -> 0,\nT121 -> T157,\nX160 -> T157,\nT156 -> T157" }, { "from": 499, "to": 503, "label": "EVAL-BACKTRACK" }, { "from": 501, "to": 508, "label": "EVAL with clause\nmult(X172, X173, X174, s(X175)) :- ','(p(X174, X176), mult(X172, X173, X176, X175)).\nand substitutionT118 -> T164,\nX172 -> T164,\nT129 -> T165,\nX173 -> s(s(T165)),\nX174 -> T165,\nX175 -> T167,\nT121 -> s(T167),\nT166 -> T167" }, { "from": 501, "to": 509, "label": "EVAL-BACKTRACK" }, { "from": 502, "to": 504, "label": "CUT" }, { "from": 504, "to": 279, "label": "INSTANCE with matching:\nT33 -> T154\nT34 -> s(s(0))\nT36 -> T157" }, { "from": 508, "to": 511, "label": "CASE" }, { "from": 511, "to": 515, "label": "PARALLEL" }, { "from": 511, "to": 518, "label": "PARALLEL" }, { "from": 515, "to": 520, "label": "EVAL with clause\np(0, 0).\nand substitutionT165 -> 0,\nX176 -> 0" }, { "from": 515, "to": 521, "label": "EVAL-BACKTRACK" }, { "from": 518, "to": 527, "label": "EVAL with clause\np(s(X186), X186).\nand substitutionX186 -> T175,\nT165 -> s(T175),\nX176 -> T175" }, { "from": 518, "to": 528, "label": "EVAL-BACKTRACK" }, { "from": 520, "to": 25, "label": "INSTANCE with matching:\nT7 -> T164\nT8 -> s(s(0))\nT10 -> T167" }, { "from": 527, "to": 529, "label": "CASE" }, { "from": 529, "to": 533, "label": "EVAL with clause\nmult(0, X193, 0, X194) :- ','(!_13, eq(X194, 0)).\nand substitutionT164 -> 0,\nT175 -> 0,\nX193 -> s(s(s(0))),\nT182 -> 0,\nT167 -> T184,\nX194 -> T184,\nT183 -> T184" }, { "from": 529, "to": 536, "label": "EVAL-BACKTRACK" }, { "from": 533, "to": 537, "label": "CUT" }, { "from": 536, "to": 544, "label": "PARALLEL" }, { "from": 536, "to": 545, "label": "PARALLEL" }, { "from": 537, "to": 538, "label": "CASE" }, { "from": 538, "to": 541, "label": "EVAL with clause\neq(X197, X197).\nand substitutionT184 -> 0,\nX197 -> 0,\nT187 -> 0" }, { "from": 538, "to": 542, "label": "EVAL-BACKTRACK" }, { "from": 541, "to": 543, "label": "SUCCESS" }, { "from": 544, "to": 549, "label": "EVAL with clause\nmult(s(X210), X211, 0, X212) :- ','(!_13, mult(X210, X211, X211, X212)).\nand substitutionX210 -> T200,\nT164 -> s(T200),\nT175 -> 0,\nX211 -> s(s(s(0))),\nT201 -> 0,\nT167 -> T203,\nX212 -> T203,\nT202 -> T203" }, { "from": 544, "to": 553, "label": "EVAL-BACKTRACK" }, { "from": 545, "to": 557, "label": "EVAL with clause\nmult(X224, X225, X226, s(X227)) :- ','(p(X226, X228), mult(X224, X225, X228, X227)).\nand substitutionT164 -> T210,\nX224 -> T210,\nT175 -> T211,\nX225 -> s(s(s(T211))),\nX226 -> T211,\nX227 -> T213,\nT167 -> s(T213),\nT212 -> T213" }, { "from": 545, "to": 559, "label": "EVAL-BACKTRACK" }, { "from": 549, "to": 554, "label": "CUT" }, { "from": 554, "to": 279, "label": "INSTANCE with matching:\nT33 -> T200\nT34 -> s(s(s(0)))\nT36 -> T203" }, { "from": 557, "to": 560, "label": "CASE" }, { "from": 560, "to": 561, "label": "PARALLEL" }, { "from": 560, "to": 562, "label": "PARALLEL" }, { "from": 561, "to": 564, "label": "EVAL with clause\np(0, 0).\nand substitutionT211 -> 0,\nX228 -> 0" }, { "from": 561, "to": 566, "label": "EVAL-BACKTRACK" }, { "from": 562, "to": 572, "label": "EVAL with clause\np(s(X238), X238).\nand substitutionX238 -> T221,\nT211 -> s(T221),\nX228 -> T221" }, { "from": 562, "to": 573, "label": "EVAL-BACKTRACK" }, { "from": 564, "to": 25, "label": "INSTANCE with matching:\nT7 -> T210\nT8 -> s(s(s(0)))\nT10 -> T213" }, { "from": 572, "to": 576, "label": "CASE" }, { "from": 576, "to": 584, "label": "EVAL with clause\nmult(0, X245, 0, X246) :- ','(!_16, eq(X246, 0)).\nand substitutionT210 -> 0,\nT221 -> 0,\nX245 -> s(s(s(s(0)))),\nT228 -> 0,\nT213 -> T230,\nX246 -> T230,\nT229 -> T230" }, { "from": 576, "to": 585, "label": "EVAL-BACKTRACK" }, { "from": 584, "to": 586, "label": "CUT" }, { "from": 585, "to": 595, "label": "PARALLEL" }, { "from": 585, "to": 596, "label": "PARALLEL" }, { "from": 586, "to": 587, "label": "CASE" }, { "from": 587, "to": 591, "label": "EVAL with clause\neq(X249, X249).\nand substitutionT230 -> 0,\nX249 -> 0,\nT233 -> 0" }, { "from": 587, "to": 592, "label": "EVAL-BACKTRACK" }, { "from": 591, "to": 593, "label": "SUCCESS" }, { "from": 595, "to": 598, "label": "EVAL with clause\nmult(s(X262), X263, 0, X264) :- ','(!_16, mult(X262, X263, X263, X264)).\nand substitutionX262 -> T246,\nT210 -> s(T246),\nT221 -> 0,\nX263 -> s(s(s(s(0)))),\nT247 -> 0,\nT213 -> T249,\nX264 -> T249,\nT248 -> T249" }, { "from": 595, "to": 599, "label": "EVAL-BACKTRACK" }, { "from": 596, "to": 611, "label": "EVAL with clause\nmult(X276, X277, X278, s(X279)) :- ','(p(X278, X280), mult(X276, X277, X280, X279)).\nand substitutionT210 -> T256,\nX276 -> T256,\nT221 -> T257,\nX277 -> s(s(s(s(T257)))),\nX278 -> T257,\nX279 -> T259,\nT213 -> s(T259),\nT258 -> T259" }, { "from": 596, "to": 612, "label": "EVAL-BACKTRACK" }, { "from": 598, "to": 600, "label": "CUT" }, { "from": 600, "to": 279, "label": "INSTANCE with matching:\nT33 -> T246\nT34 -> s(s(s(s(0))))\nT36 -> T249" }, { "from": 611, "to": 615, "label": "CASE" }, { "from": 615, "to": 616, "label": "PARALLEL" }, { "from": 615, "to": 617, "label": "PARALLEL" }, { "from": 616, "to": 619, "label": "EVAL with clause\np(0, 0).\nand substitutionT257 -> 0,\nX280 -> 0" }, { "from": 616, "to": 620, "label": "EVAL-BACKTRACK" }, { "from": 617, "to": 625, "label": "EVAL with clause\np(s(X290), X290).\nand substitutionX290 -> T267,\nT257 -> s(T267),\nX280 -> T267" }, { "from": 617, "to": 627, "label": "EVAL-BACKTRACK" }, { "from": 619, "to": 25, "label": "INSTANCE with matching:\nT7 -> T256\nT8 -> s(s(s(s(0))))\nT10 -> T259" }, { "from": 625, "to": 630, "label": "CASE" }, { "from": 630, "to": 635, "label": "EVAL with clause\nmult(0, X297, 0, X298) :- ','(!_19, eq(X298, 0)).\nand substitutionT256 -> 0,\nT267 -> 0,\nX297 -> s(s(s(s(s(0))))),\nT274 -> 0,\nT259 -> T276,\nX298 -> T276,\nT275 -> T276" }, { "from": 630, "to": 636, "label": "EVAL-BACKTRACK" }, { "from": 635, "to": 637, "label": "CUT" }, { "from": 636, "to": 654, "label": "PARALLEL" }, { "from": 636, "to": 656, "label": "PARALLEL" }, { "from": 637, "to": 638, "label": "CASE" }, { "from": 638, "to": 647, "label": "EVAL with clause\neq(X301, X301).\nand substitutionT276 -> 0,\nX301 -> 0,\nT279 -> 0" }, { "from": 638, "to": 648, "label": "EVAL-BACKTRACK" }, { "from": 647, "to": 649, "label": "SUCCESS" }, { "from": 654, "to": 661, "label": "EVAL with clause\nmult(s(X314), X315, 0, X316) :- ','(!_19, mult(X314, X315, X315, X316)).\nand substitutionX314 -> T292,\nT256 -> s(T292),\nT267 -> 0,\nX315 -> s(s(s(s(s(0))))),\nT293 -> 0,\nT259 -> T295,\nX316 -> T295,\nT294 -> T295" }, { "from": 654, "to": 663, "label": "EVAL-BACKTRACK" }, { "from": 656, "to": 700, "label": "EVAL with clause\nmult(X328, X329, X330, s(X331)) :- ','(p(X330, X332), mult(X328, X329, X332, X331)).\nand substitutionT256 -> T302,\nX328 -> T302,\nT267 -> T303,\nX329 -> s(s(s(s(s(T303))))),\nX330 -> T303,\nX331 -> T305,\nT259 -> s(T305),\nT304 -> T305" }, { "from": 656, "to": 701, "label": "EVAL-BACKTRACK" }, { "from": 661, "to": 664, "label": "CUT" }, { "from": 664, "to": 279, "label": "INSTANCE with matching:\nT33 -> T292\nT34 -> s(s(s(s(s(0)))))\nT36 -> T295" }, { "from": 700, "to": 702, "label": "CASE" }, { "from": 702, "to": 703, "label": "PARALLEL" }, { "from": 702, "to": 704, "label": "PARALLEL" }, { "from": 703, "to": 705, "label": "EVAL with clause\np(0, 0).\nand substitutionT303 -> 0,\nX332 -> 0" }, { "from": 703, "to": 707, "label": "EVAL-BACKTRACK" }, { "from": 704, "to": 709, "label": "EVAL with clause\np(s(X342), X342).\nand substitutionX342 -> T313,\nT303 -> s(T313),\nX332 -> T313" }, { "from": 704, "to": 710, "label": "EVAL-BACKTRACK" }, { "from": 705, "to": 25, "label": "INSTANCE with matching:\nT7 -> T302\nT8 -> s(s(s(s(s(0)))))\nT10 -> T305" }, { "from": 709, "to": 713, "label": "CASE" }, { "from": 713, "to": 716, "label": "EVAL with clause\nmult(0, X349, 0, X350) :- ','(!_22, eq(X350, 0)).\nand substitutionT302 -> 0,\nT313 -> 0,\nX349 -> s(s(s(s(s(s(0)))))),\nT320 -> 0,\nT305 -> T322,\nX350 -> T322,\nT321 -> T322" }, { "from": 713, "to": 725, "label": "EVAL-BACKTRACK" }, { "from": 716, "to": 726, "label": "CUT" }, { "from": 725, "to": 743, "label": "PARALLEL" }, { "from": 725, "to": 744, "label": "PARALLEL" }, { "from": 726, "to": 727, "label": "CASE" }, { "from": 727, "to": 730, "label": "EVAL with clause\neq(X353, X353).\nand substitutionT322 -> 0,\nX353 -> 0,\nT325 -> 0" }, { "from": 727, "to": 731, "label": "EVAL-BACKTRACK" }, { "from": 730, "to": 732, "label": "SUCCESS" }, { "from": 743, "to": 748, "label": "EVAL with clause\nmult(s(X366), X367, 0, X368) :- ','(!_22, mult(X366, X367, X367, X368)).\nand substitutionX366 -> T338,\nT302 -> s(T338),\nT313 -> 0,\nX367 -> s(s(s(s(s(s(0)))))),\nT339 -> 0,\nT305 -> T341,\nX368 -> T341,\nT340 -> T341" }, { "from": 743, "to": 751, "label": "EVAL-BACKTRACK" }, { "from": 744, "to": 753, "label": "EVAL with clause\nmult(X380, X381, X382, s(X383)) :- ','(p(X382, X384), mult(X380, X381, X384, X383)).\nand substitutionT302 -> T348,\nX380 -> T348,\nT313 -> T349,\nX381 -> s(s(s(s(s(s(T349)))))),\nX382 -> T349,\nX383 -> T351,\nT305 -> s(T351),\nT350 -> T351" }, { "from": 744, "to": 754, "label": "EVAL-BACKTRACK" }, { "from": 748, "to": 752, "label": "CUT" }, { "from": 752, "to": 279, "label": "INSTANCE with matching:\nT33 -> T338\nT34 -> s(s(s(s(s(s(0))))))\nT36 -> T341" }, { "from": 753, "to": 757, "label": "CASE" }, { "from": 757, "to": 759, "label": "PARALLEL" }, { "from": 757, "to": 760, "label": "PARALLEL" }, { "from": 759, "to": 763, "label": "EVAL with clause\np(0, 0).\nand substitutionT349 -> 0,\nX384 -> 0" }, { "from": 759, "to": 764, "label": "EVAL-BACKTRACK" }, { "from": 760, "to": 783, "label": "EVAL with clause\np(s(X394), X394).\nand substitutionX394 -> T359,\nT349 -> s(T359),\nX384 -> T359" }, { "from": 760, "to": 786, "label": "EVAL-BACKTRACK" }, { "from": 763, "to": 25, "label": "INSTANCE with matching:\nT7 -> T348\nT8 -> s(s(s(s(s(s(0))))))\nT10 -> T351" }, { "from": 783, "to": 787, "label": "CASE" }, { "from": 787, "to": 790, "label": "EVAL with clause\nmult(0, X401, 0, X402) :- ','(!_25, eq(X402, 0)).\nand substitutionT348 -> 0,\nT359 -> 0,\nX401 -> s(s(s(s(s(s(s(0))))))),\nT366 -> 0,\nT351 -> T368,\nX402 -> T368,\nT367 -> T368" }, { "from": 787, "to": 791, "label": "EVAL-BACKTRACK" }, { "from": 790, "to": 792, "label": "CUT" }, { "from": 791, "to": 801, "label": "PARALLEL" }, { "from": 791, "to": 802, "label": "PARALLEL" }, { "from": 792, "to": 793, "label": "CASE" }, { "from": 793, "to": 798, "label": "EVAL with clause\neq(X405, X405).\nand substitutionT368 -> 0,\nX405 -> 0,\nT371 -> 0" }, { "from": 793, "to": 799, "label": "EVAL-BACKTRACK" }, { "from": 798, "to": 800, "label": "SUCCESS" }, { "from": 801, "to": 809, "label": "EVAL with clause\nmult(s(X418), X419, 0, X420) :- ','(!_25, mult(X418, X419, X419, X420)).\nand substitutionX418 -> T384,\nT348 -> s(T384),\nT359 -> 0,\nX419 -> s(s(s(s(s(s(s(0))))))),\nT385 -> 0,\nT351 -> T387,\nX420 -> T387,\nT386 -> T387" }, { "from": 801, "to": 833, "label": "EVAL-BACKTRACK" }, { "from": 802, "to": 889, "label": "EVAL with clause\nmult(X432, X433, X434, s(X435)) :- ','(p(X434, X436), mult(X432, X433, X436, X435)).\nand substitutionT348 -> T394,\nX432 -> T394,\nT359 -> T395,\nX433 -> s(s(s(s(s(s(s(T395))))))),\nX434 -> T395,\nX435 -> T397,\nT351 -> s(T397),\nT396 -> T397" }, { "from": 802, "to": 890, "label": "EVAL-BACKTRACK" }, { "from": 809, "to": 884, "label": "CUT" }, { "from": 884, "to": 279, "label": "INSTANCE with matching:\nT33 -> T384\nT34 -> s(s(s(s(s(s(s(0)))))))\nT36 -> T387" }, { "from": 889, "to": 891, "label": "CASE" }, { "from": 891, "to": 892, "label": "PARALLEL" }, { "from": 891, "to": 893, "label": "PARALLEL" }, { "from": 892, "to": 894, "label": "EVAL with clause\np(0, 0).\nand substitutionT395 -> 0,\nX436 -> 0" }, { "from": 892, "to": 895, "label": "EVAL-BACKTRACK" }, { "from": 893, "to": 896, "label": "EVAL with clause\np(s(X446), X446).\nand substitutionX446 -> T405,\nT395 -> s(T405),\nX436 -> T405" }, { "from": 893, "to": 897, "label": "EVAL-BACKTRACK" }, { "from": 894, "to": 25, "label": "INSTANCE with matching:\nT7 -> T394\nT8 -> s(s(s(s(s(s(s(0)))))))\nT10 -> T397" }, { "from": 896, "to": 900, "label": "GENERALIZATION\nT408 <-- s(s(s(s(s(s(s(T405)))))))\n\nNew Knowledge:\nT408 is ground" }, { "from": 900, "to": 905, "label": "CASE" }, { "from": 905, "to": 906, "label": "EVAL with clause\nmult(0, X455, 0, X456) :- ','(!_28, eq(X456, 0)).\nand substitutionT394 -> 0,\nT408 -> T415,\nX455 -> s(T415),\nT405 -> 0,\nT397 -> T417,\nX456 -> T417,\nT416 -> T417" }, { "from": 905, "to": 907, "label": "EVAL-BACKTRACK" }, { "from": 906, "to": 909, "label": "CUT" }, { "from": 907, "to": 914, "label": "PARALLEL" }, { "from": 907, "to": 915, "label": "PARALLEL" }, { "from": 909, "to": 910, "label": "CASE" }, { "from": 910, "to": 911, "label": "EVAL with clause\neq(X459, X459).\nand substitutionT417 -> 0,\nX459 -> 0,\nT420 -> 0" }, { "from": 910, "to": 912, "label": "EVAL-BACKTRACK" }, { "from": 911, "to": 913, "label": "SUCCESS" }, { "from": 914, "to": 916, "label": "EVAL with clause\nmult(s(X472), X473, 0, X474) :- ','(!_28, mult(X472, X473, X473, X474)).\nand substitutionX472 -> T433,\nT394 -> s(T433),\nT408 -> T434,\nX473 -> s(T434),\nT405 -> 0,\nT397 -> T436,\nX474 -> T436,\nT435 -> T436" }, { "from": 914, "to": 917, "label": "EVAL-BACKTRACK" }, { "from": 915, "to": 919, "label": "EVAL with clause\nmult(X486, X487, X488, s(X489)) :- ','(p(X488, X490), mult(X486, X487, X490, X489)).\nand substitutionT394 -> T446,\nX486 -> T446,\nT408 -> T447,\nX487 -> s(T447),\nT405 -> T448,\nX488 -> T448,\nX489 -> T450,\nT397 -> s(T450),\nT449 -> T450" }, { "from": 915, "to": 920, "label": "EVAL-BACKTRACK" }, { "from": 916, "to": 918, "label": "CUT" }, { "from": 918, "to": 279, "label": "INSTANCE with matching:\nT33 -> T433\nT34 -> s(T434)\nT36 -> T436" }, { "from": 919, "to": 921, "label": "CASE" }, { "from": 921, "to": 922, "label": "PARALLEL" }, { "from": 921, "to": 923, "label": "PARALLEL" }, { "from": 922, "to": 924, "label": "EVAL with clause\np(0, 0).\nand substitutionT448 -> 0,\nX490 -> 0" }, { "from": 922, "to": 925, "label": "EVAL-BACKTRACK" }, { "from": 923, "to": 926, "label": "EVAL with clause\np(s(X500), X500).\nand substitutionX500 -> T460,\nT448 -> s(T460),\nX490 -> T460" }, { "from": 923, "to": 927, "label": "EVAL-BACKTRACK" }, { "from": 924, "to": 25, "label": "INSTANCE with matching:\nT7 -> T446\nT8 -> s(T447)\nT10 -> T450" }, { "from": 926, "to": 900, "label": "INSTANCE with matching:\nT394 -> T446\nT408 -> T447\nT405 -> T460\nT397 -> T450" }, { "from": 928, "to": 930, "label": "CASE" }, { "from": 930, "to": 931, "label": "PARALLEL" }, { "from": 930, "to": 932, "label": "PARALLEL" }, { "from": 931, "to": 933, "label": "ONLY EVAL with clause\np(0, 0).\nand substitutionX516 -> 0" }, { "from": 932, "to": 934, "label": "BACKTRACK\nfor clause: p(s(X), X)because of non-unification" }, { "from": 933, "to": 25, "label": "INSTANCE with matching:\nT7 -> T469\nT8 -> T470\nT10 -> T472" } ], "type": "Graph" } } ---------------------------------------- (84) Obligation: Triples: multA(s(X1), X2, X3) :- multB(X1, X2, X3). multA(X1, X2, s(X3)) :- multA(X1, X2, X3). multB(s(X1), 0, X2) :- multA(X1, 0, X2). multB(X1, 0, s(X2)) :- multA(X1, 0, X2). multB(s(X1), s(0), s(X2)) :- multB(X1, s(0), X2). multB(X1, s(0), s(s(X2))) :- multA(X1, s(0), X2). multB(s(X1), s(s(0)), s(s(X2))) :- multB(X1, s(s(0)), X2). multB(X1, s(s(0)), s(s(s(X2)))) :- multA(X1, s(s(0)), X2). multB(s(X1), s(s(s(0))), s(s(s(X2)))) :- multB(X1, s(s(s(0))), X2). multB(X1, s(s(s(0))), s(s(s(s(X2))))) :- multA(X1, s(s(s(0))), X2). multB(s(X1), s(s(s(s(0)))), s(s(s(s(X2))))) :- multB(X1, s(s(s(s(0)))), X2). multB(X1, s(s(s(s(0)))), s(s(s(s(s(X2)))))) :- multA(X1, s(s(s(s(0)))), X2). multB(s(X1), s(s(s(s(s(0))))), s(s(s(s(s(X2)))))) :- multB(X1, s(s(s(s(s(0))))), X2). multB(X1, s(s(s(s(s(0))))), s(s(s(s(s(s(X2))))))) :- multA(X1, s(s(s(s(s(0))))), X2). multB(s(X1), s(s(s(s(s(s(0)))))), s(s(s(s(s(s(X2))))))) :- multB(X1, s(s(s(s(s(s(0)))))), X2). multB(X1, s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X2)))))))) :- multA(X1, s(s(s(s(s(s(0)))))), X2). multB(s(X1), s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(X2)))))))) :- multB(X1, s(s(s(s(s(s(s(0))))))), X2). multB(X1, s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(X2))))))))) :- multA(X1, s(s(s(s(s(s(s(0))))))), X2). multB(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) :- multC(X1, s(s(s(s(s(s(s(X2))))))), X2, X3). multC(s(X1), X2, 0, X3) :- multB(X1, s(X2), X3). multC(X1, X2, 0, s(X3)) :- multA(X1, s(X2), X3). multC(X1, X2, s(X3), s(X4)) :- multC(X1, X2, X3, X4). timesD(X1, X2, X3) :- multA(X1, X2, X3). Clauses: multcA(0, X1, 0). multcA(s(X1), X2, X3) :- multcB(X1, X2, X3). multcA(X1, X2, s(X3)) :- multcA(X1, X2, X3). multcB(0, 0, 0). multcB(s(X1), 0, X2) :- multcA(X1, 0, X2). multcB(X1, 0, s(X2)) :- multcA(X1, 0, X2). multcB(0, s(0), s(0)). multcB(s(X1), s(0), s(X2)) :- multcB(X1, s(0), X2). multcB(X1, s(0), s(s(X2))) :- multcA(X1, s(0), X2). multcB(0, s(s(0)), s(s(0))). multcB(s(X1), s(s(0)), s(s(X2))) :- multcB(X1, s(s(0)), X2). multcB(X1, s(s(0)), s(s(s(X2)))) :- multcA(X1, s(s(0)), X2). multcB(0, s(s(s(0))), s(s(s(0)))). multcB(s(X1), s(s(s(0))), s(s(s(X2)))) :- multcB(X1, s(s(s(0))), X2). multcB(X1, s(s(s(0))), s(s(s(s(X2))))) :- multcA(X1, s(s(s(0))), X2). multcB(0, s(s(s(s(0)))), s(s(s(s(0))))). multcB(s(X1), s(s(s(s(0)))), s(s(s(s(X2))))) :- multcB(X1, s(s(s(s(0)))), X2). multcB(X1, s(s(s(s(0)))), s(s(s(s(s(X2)))))) :- multcA(X1, s(s(s(s(0)))), X2). multcB(0, s(s(s(s(s(0))))), s(s(s(s(s(0)))))). multcB(s(X1), s(s(s(s(s(0))))), s(s(s(s(s(X2)))))) :- multcB(X1, s(s(s(s(s(0))))), X2). multcB(X1, s(s(s(s(s(0))))), s(s(s(s(s(s(X2))))))) :- multcA(X1, s(s(s(s(s(0))))), X2). multcB(0, s(s(s(s(s(s(0)))))), s(s(s(s(s(s(0))))))). multcB(s(X1), s(s(s(s(s(s(0)))))), s(s(s(s(s(s(X2))))))) :- multcB(X1, s(s(s(s(s(s(0)))))), X2). multcB(X1, s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X2)))))))) :- multcA(X1, s(s(s(s(s(s(0)))))), X2). multcB(0, s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(0)))))))). multcB(s(X1), s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(X2)))))))) :- multcB(X1, s(s(s(s(s(s(s(0))))))), X2). multcB(X1, s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(X2))))))))) :- multcA(X1, s(s(s(s(s(s(s(0))))))), X2). multcB(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) :- multcC(X1, s(s(s(s(s(s(s(X2))))))), X2, X3). multcC(0, X1, 0, 0). multcC(s(X1), X2, 0, X3) :- multcB(X1, s(X2), X3). multcC(X1, X2, 0, s(X3)) :- multcA(X1, s(X2), X3). multcC(X1, X2, s(X3), s(X4)) :- multcC(X1, X2, X3, X4). Afs: timesD(x1, x2, x3) = timesD(x1, x2) ---------------------------------------- (85) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: timesD_in_3: (b,b,f) multA_in_3: (b,b,f) multB_in_3: (b,b,f) multC_in_4: (b,b,b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: TIMESD_IN_GGA(X1, X2, X3) -> U23_GGA(X1, X2, X3, multA_in_gga(X1, X2, X3)) TIMESD_IN_GGA(X1, X2, X3) -> MULTA_IN_GGA(X1, X2, X3) MULTA_IN_GGA(s(X1), X2, X3) -> U1_GGA(X1, X2, X3, multB_in_gga(X1, X2, X3)) MULTA_IN_GGA(s(X1), X2, X3) -> MULTB_IN_GGA(X1, X2, X3) MULTB_IN_GGA(s(X1), 0, X2) -> U3_GGA(X1, X2, multA_in_gga(X1, 0, X2)) MULTB_IN_GGA(s(X1), 0, X2) -> MULTA_IN_GGA(X1, 0, X2) MULTA_IN_GGA(X1, X2, s(X3)) -> U2_GGA(X1, X2, X3, multA_in_gga(X1, X2, X3)) MULTA_IN_GGA(X1, X2, s(X3)) -> MULTA_IN_GGA(X1, X2, X3) MULTB_IN_GGA(X1, 0, s(X2)) -> U4_GGA(X1, X2, multA_in_gga(X1, 0, X2)) MULTB_IN_GGA(X1, 0, s(X2)) -> MULTA_IN_GGA(X1, 0, X2) MULTB_IN_GGA(s(X1), s(0), s(X2)) -> U5_GGA(X1, X2, multB_in_gga(X1, s(0), X2)) MULTB_IN_GGA(s(X1), s(0), s(X2)) -> MULTB_IN_GGA(X1, s(0), X2) MULTB_IN_GGA(X1, s(0), s(s(X2))) -> U6_GGA(X1, X2, multA_in_gga(X1, s(0), X2)) MULTB_IN_GGA(X1, s(0), s(s(X2))) -> MULTA_IN_GGA(X1, s(0), X2) MULTB_IN_GGA(s(X1), s(s(0)), s(s(X2))) -> U7_GGA(X1, X2, multB_in_gga(X1, s(s(0)), X2)) MULTB_IN_GGA(s(X1), s(s(0)), s(s(X2))) -> MULTB_IN_GGA(X1, s(s(0)), X2) MULTB_IN_GGA(X1, s(s(0)), s(s(s(X2)))) -> U8_GGA(X1, X2, multA_in_gga(X1, s(s(0)), X2)) MULTB_IN_GGA(X1, s(s(0)), s(s(s(X2)))) -> MULTA_IN_GGA(X1, s(s(0)), X2) MULTB_IN_GGA(s(X1), s(s(s(0))), s(s(s(X2)))) -> U9_GGA(X1, X2, multB_in_gga(X1, s(s(s(0))), X2)) MULTB_IN_GGA(s(X1), s(s(s(0))), s(s(s(X2)))) -> MULTB_IN_GGA(X1, s(s(s(0))), X2) MULTB_IN_GGA(X1, s(s(s(0))), s(s(s(s(X2))))) -> U10_GGA(X1, X2, multA_in_gga(X1, s(s(s(0))), X2)) MULTB_IN_GGA(X1, s(s(s(0))), s(s(s(s(X2))))) -> MULTA_IN_GGA(X1, s(s(s(0))), X2) MULTB_IN_GGA(s(X1), s(s(s(s(0)))), s(s(s(s(X2))))) -> U11_GGA(X1, X2, multB_in_gga(X1, s(s(s(s(0)))), X2)) MULTB_IN_GGA(s(X1), s(s(s(s(0)))), s(s(s(s(X2))))) -> MULTB_IN_GGA(X1, s(s(s(s(0)))), X2) MULTB_IN_GGA(X1, s(s(s(s(0)))), s(s(s(s(s(X2)))))) -> U12_GGA(X1, X2, multA_in_gga(X1, s(s(s(s(0)))), X2)) MULTB_IN_GGA(X1, s(s(s(s(0)))), s(s(s(s(s(X2)))))) -> MULTA_IN_GGA(X1, s(s(s(s(0)))), X2) MULTB_IN_GGA(s(X1), s(s(s(s(s(0))))), s(s(s(s(s(X2)))))) -> U13_GGA(X1, X2, multB_in_gga(X1, s(s(s(s(s(0))))), X2)) MULTB_IN_GGA(s(X1), s(s(s(s(s(0))))), s(s(s(s(s(X2)))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(0))))), X2) MULTB_IN_GGA(X1, s(s(s(s(s(0))))), s(s(s(s(s(s(X2))))))) -> U14_GGA(X1, X2, multA_in_gga(X1, s(s(s(s(s(0))))), X2)) MULTB_IN_GGA(X1, s(s(s(s(s(0))))), s(s(s(s(s(s(X2))))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(0))))), X2) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(0)))))), s(s(s(s(s(s(X2))))))) -> U15_GGA(X1, X2, multB_in_gga(X1, s(s(s(s(s(s(0)))))), X2)) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(0)))))), s(s(s(s(s(s(X2))))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(s(0)))))), X2) MULTB_IN_GGA(X1, s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X2)))))))) -> U16_GGA(X1, X2, multA_in_gga(X1, s(s(s(s(s(s(0)))))), X2)) MULTB_IN_GGA(X1, s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X2)))))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(s(0)))))), X2) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(X2)))))))) -> U17_GGA(X1, X2, multB_in_gga(X1, s(s(s(s(s(s(s(0))))))), X2)) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(X2)))))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), X2) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(X2))))))))) -> U18_GGA(X1, X2, multA_in_gga(X1, s(s(s(s(s(s(s(0))))))), X2)) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(X2))))))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), X2) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) -> U19_GGA(X1, X2, X3, multC_in_ggga(X1, s(s(s(s(s(s(s(X2))))))), X2, X3)) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) -> MULTC_IN_GGGA(X1, s(s(s(s(s(s(s(X2))))))), X2, X3) MULTC_IN_GGGA(s(X1), X2, 0, X3) -> U20_GGGA(X1, X2, X3, multB_in_gga(X1, s(X2), X3)) MULTC_IN_GGGA(s(X1), X2, 0, X3) -> MULTB_IN_GGA(X1, s(X2), X3) MULTC_IN_GGGA(X1, X2, 0, s(X3)) -> U21_GGGA(X1, X2, X3, multA_in_gga(X1, s(X2), X3)) MULTC_IN_GGGA(X1, X2, 0, s(X3)) -> MULTA_IN_GGA(X1, s(X2), X3) MULTC_IN_GGGA(X1, X2, s(X3), s(X4)) -> U22_GGGA(X1, X2, X3, X4, multC_in_ggga(X1, X2, X3, X4)) MULTC_IN_GGGA(X1, X2, s(X3), s(X4)) -> MULTC_IN_GGGA(X1, X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: multA_in_gga(x1, x2, x3) = multA_in_gga(x1, x2) s(x1) = s(x1) multB_in_gga(x1, x2, x3) = multB_in_gga(x1, x2) 0 = 0 multC_in_ggga(x1, x2, x3, x4) = multC_in_ggga(x1, x2, x3) TIMESD_IN_GGA(x1, x2, x3) = TIMESD_IN_GGA(x1, x2) U23_GGA(x1, x2, x3, x4) = U23_GGA(x1, x2, x4) MULTA_IN_GGA(x1, x2, x3) = MULTA_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) MULTB_IN_GGA(x1, x2, x3) = MULTB_IN_GGA(x1, x2) U3_GGA(x1, x2, x3) = U3_GGA(x1, x3) U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) U4_GGA(x1, x2, x3) = U4_GGA(x1, x3) U5_GGA(x1, x2, x3) = U5_GGA(x1, x3) U6_GGA(x1, x2, x3) = U6_GGA(x1, x3) U7_GGA(x1, x2, x3) = U7_GGA(x1, x3) U8_GGA(x1, x2, x3) = U8_GGA(x1, x3) U9_GGA(x1, x2, x3) = U9_GGA(x1, x3) U10_GGA(x1, x2, x3) = U10_GGA(x1, x3) U11_GGA(x1, x2, x3) = U11_GGA(x1, x3) U12_GGA(x1, x2, x3) = U12_GGA(x1, x3) U13_GGA(x1, x2, x3) = U13_GGA(x1, x3) U14_GGA(x1, x2, x3) = U14_GGA(x1, x3) U15_GGA(x1, x2, x3) = U15_GGA(x1, x3) U16_GGA(x1, x2, x3) = U16_GGA(x1, x3) U17_GGA(x1, x2, x3) = U17_GGA(x1, x3) U18_GGA(x1, x2, x3) = U18_GGA(x1, x3) U19_GGA(x1, x2, x3, x4) = U19_GGA(x1, x2, x4) MULTC_IN_GGGA(x1, x2, x3, x4) = MULTC_IN_GGGA(x1, x2, x3) U20_GGGA(x1, x2, x3, x4) = U20_GGGA(x1, x2, x4) U21_GGGA(x1, x2, x3, x4) = U21_GGGA(x1, x2, x4) U22_GGGA(x1, x2, x3, x4, x5) = U22_GGGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (86) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMESD_IN_GGA(X1, X2, X3) -> U23_GGA(X1, X2, X3, multA_in_gga(X1, X2, X3)) TIMESD_IN_GGA(X1, X2, X3) -> MULTA_IN_GGA(X1, X2, X3) MULTA_IN_GGA(s(X1), X2, X3) -> U1_GGA(X1, X2, X3, multB_in_gga(X1, X2, X3)) MULTA_IN_GGA(s(X1), X2, X3) -> MULTB_IN_GGA(X1, X2, X3) MULTB_IN_GGA(s(X1), 0, X2) -> U3_GGA(X1, X2, multA_in_gga(X1, 0, X2)) MULTB_IN_GGA(s(X1), 0, X2) -> MULTA_IN_GGA(X1, 0, X2) MULTA_IN_GGA(X1, X2, s(X3)) -> U2_GGA(X1, X2, X3, multA_in_gga(X1, X2, X3)) MULTA_IN_GGA(X1, X2, s(X3)) -> MULTA_IN_GGA(X1, X2, X3) MULTB_IN_GGA(X1, 0, s(X2)) -> U4_GGA(X1, X2, multA_in_gga(X1, 0, X2)) MULTB_IN_GGA(X1, 0, s(X2)) -> MULTA_IN_GGA(X1, 0, X2) MULTB_IN_GGA(s(X1), s(0), s(X2)) -> U5_GGA(X1, X2, multB_in_gga(X1, s(0), X2)) MULTB_IN_GGA(s(X1), s(0), s(X2)) -> MULTB_IN_GGA(X1, s(0), X2) MULTB_IN_GGA(X1, s(0), s(s(X2))) -> U6_GGA(X1, X2, multA_in_gga(X1, s(0), X2)) MULTB_IN_GGA(X1, s(0), s(s(X2))) -> MULTA_IN_GGA(X1, s(0), X2) MULTB_IN_GGA(s(X1), s(s(0)), s(s(X2))) -> U7_GGA(X1, X2, multB_in_gga(X1, s(s(0)), X2)) MULTB_IN_GGA(s(X1), s(s(0)), s(s(X2))) -> MULTB_IN_GGA(X1, s(s(0)), X2) MULTB_IN_GGA(X1, s(s(0)), s(s(s(X2)))) -> U8_GGA(X1, X2, multA_in_gga(X1, s(s(0)), X2)) MULTB_IN_GGA(X1, s(s(0)), s(s(s(X2)))) -> MULTA_IN_GGA(X1, s(s(0)), X2) MULTB_IN_GGA(s(X1), s(s(s(0))), s(s(s(X2)))) -> U9_GGA(X1, X2, multB_in_gga(X1, s(s(s(0))), X2)) MULTB_IN_GGA(s(X1), s(s(s(0))), s(s(s(X2)))) -> MULTB_IN_GGA(X1, s(s(s(0))), X2) MULTB_IN_GGA(X1, s(s(s(0))), s(s(s(s(X2))))) -> U10_GGA(X1, X2, multA_in_gga(X1, s(s(s(0))), X2)) MULTB_IN_GGA(X1, s(s(s(0))), s(s(s(s(X2))))) -> MULTA_IN_GGA(X1, s(s(s(0))), X2) MULTB_IN_GGA(s(X1), s(s(s(s(0)))), s(s(s(s(X2))))) -> U11_GGA(X1, X2, multB_in_gga(X1, s(s(s(s(0)))), X2)) MULTB_IN_GGA(s(X1), s(s(s(s(0)))), s(s(s(s(X2))))) -> MULTB_IN_GGA(X1, s(s(s(s(0)))), X2) MULTB_IN_GGA(X1, s(s(s(s(0)))), s(s(s(s(s(X2)))))) -> U12_GGA(X1, X2, multA_in_gga(X1, s(s(s(s(0)))), X2)) MULTB_IN_GGA(X1, s(s(s(s(0)))), s(s(s(s(s(X2)))))) -> MULTA_IN_GGA(X1, s(s(s(s(0)))), X2) MULTB_IN_GGA(s(X1), s(s(s(s(s(0))))), s(s(s(s(s(X2)))))) -> U13_GGA(X1, X2, multB_in_gga(X1, s(s(s(s(s(0))))), X2)) MULTB_IN_GGA(s(X1), s(s(s(s(s(0))))), s(s(s(s(s(X2)))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(0))))), X2) MULTB_IN_GGA(X1, s(s(s(s(s(0))))), s(s(s(s(s(s(X2))))))) -> U14_GGA(X1, X2, multA_in_gga(X1, s(s(s(s(s(0))))), X2)) MULTB_IN_GGA(X1, s(s(s(s(s(0))))), s(s(s(s(s(s(X2))))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(0))))), X2) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(0)))))), s(s(s(s(s(s(X2))))))) -> U15_GGA(X1, X2, multB_in_gga(X1, s(s(s(s(s(s(0)))))), X2)) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(0)))))), s(s(s(s(s(s(X2))))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(s(0)))))), X2) MULTB_IN_GGA(X1, s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X2)))))))) -> U16_GGA(X1, X2, multA_in_gga(X1, s(s(s(s(s(s(0)))))), X2)) MULTB_IN_GGA(X1, s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X2)))))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(s(0)))))), X2) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(X2)))))))) -> U17_GGA(X1, X2, multB_in_gga(X1, s(s(s(s(s(s(s(0))))))), X2)) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(X2)))))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), X2) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(X2))))))))) -> U18_GGA(X1, X2, multA_in_gga(X1, s(s(s(s(s(s(s(0))))))), X2)) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(X2))))))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), X2) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) -> U19_GGA(X1, X2, X3, multC_in_ggga(X1, s(s(s(s(s(s(s(X2))))))), X2, X3)) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) -> MULTC_IN_GGGA(X1, s(s(s(s(s(s(s(X2))))))), X2, X3) MULTC_IN_GGGA(s(X1), X2, 0, X3) -> U20_GGGA(X1, X2, X3, multB_in_gga(X1, s(X2), X3)) MULTC_IN_GGGA(s(X1), X2, 0, X3) -> MULTB_IN_GGA(X1, s(X2), X3) MULTC_IN_GGGA(X1, X2, 0, s(X3)) -> U21_GGGA(X1, X2, X3, multA_in_gga(X1, s(X2), X3)) MULTC_IN_GGGA(X1, X2, 0, s(X3)) -> MULTA_IN_GGA(X1, s(X2), X3) MULTC_IN_GGGA(X1, X2, s(X3), s(X4)) -> U22_GGGA(X1, X2, X3, X4, multC_in_ggga(X1, X2, X3, X4)) MULTC_IN_GGGA(X1, X2, s(X3), s(X4)) -> MULTC_IN_GGGA(X1, X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: multA_in_gga(x1, x2, x3) = multA_in_gga(x1, x2) s(x1) = s(x1) multB_in_gga(x1, x2, x3) = multB_in_gga(x1, x2) 0 = 0 multC_in_ggga(x1, x2, x3, x4) = multC_in_ggga(x1, x2, x3) TIMESD_IN_GGA(x1, x2, x3) = TIMESD_IN_GGA(x1, x2) U23_GGA(x1, x2, x3, x4) = U23_GGA(x1, x2, x4) MULTA_IN_GGA(x1, x2, x3) = MULTA_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) MULTB_IN_GGA(x1, x2, x3) = MULTB_IN_GGA(x1, x2) U3_GGA(x1, x2, x3) = U3_GGA(x1, x3) U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) U4_GGA(x1, x2, x3) = U4_GGA(x1, x3) U5_GGA(x1, x2, x3) = U5_GGA(x1, x3) U6_GGA(x1, x2, x3) = U6_GGA(x1, x3) U7_GGA(x1, x2, x3) = U7_GGA(x1, x3) U8_GGA(x1, x2, x3) = U8_GGA(x1, x3) U9_GGA(x1, x2, x3) = U9_GGA(x1, x3) U10_GGA(x1, x2, x3) = U10_GGA(x1, x3) U11_GGA(x1, x2, x3) = U11_GGA(x1, x3) U12_GGA(x1, x2, x3) = U12_GGA(x1, x3) U13_GGA(x1, x2, x3) = U13_GGA(x1, x3) U14_GGA(x1, x2, x3) = U14_GGA(x1, x3) U15_GGA(x1, x2, x3) = U15_GGA(x1, x3) U16_GGA(x1, x2, x3) = U16_GGA(x1, x3) U17_GGA(x1, x2, x3) = U17_GGA(x1, x3) U18_GGA(x1, x2, x3) = U18_GGA(x1, x3) U19_GGA(x1, x2, x3, x4) = U19_GGA(x1, x2, x4) MULTC_IN_GGGA(x1, x2, x3, x4) = MULTC_IN_GGGA(x1, x2, x3) U20_GGGA(x1, x2, x3, x4) = U20_GGGA(x1, x2, x4) U21_GGGA(x1, x2, x3, x4) = U21_GGGA(x1, x2, x4) U22_GGGA(x1, x2, x3, x4, x5) = U22_GGGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (87) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 24 less nodes. ---------------------------------------- (88) Obligation: Pi DP problem: The TRS P consists of the following rules: MULTA_IN_GGA(s(X1), X2, X3) -> MULTB_IN_GGA(X1, X2, X3) MULTB_IN_GGA(s(X1), 0, X2) -> MULTA_IN_GGA(X1, 0, X2) MULTA_IN_GGA(X1, X2, s(X3)) -> MULTA_IN_GGA(X1, X2, X3) MULTB_IN_GGA(X1, 0, s(X2)) -> MULTA_IN_GGA(X1, 0, X2) MULTB_IN_GGA(s(X1), s(0), s(X2)) -> MULTB_IN_GGA(X1, s(0), X2) MULTB_IN_GGA(X1, s(0), s(s(X2))) -> MULTA_IN_GGA(X1, s(0), X2) MULTB_IN_GGA(s(X1), s(s(0)), s(s(X2))) -> MULTB_IN_GGA(X1, s(s(0)), X2) MULTB_IN_GGA(X1, s(s(0)), s(s(s(X2)))) -> MULTA_IN_GGA(X1, s(s(0)), X2) MULTB_IN_GGA(s(X1), s(s(s(0))), s(s(s(X2)))) -> MULTB_IN_GGA(X1, s(s(s(0))), X2) MULTB_IN_GGA(X1, s(s(s(0))), s(s(s(s(X2))))) -> MULTA_IN_GGA(X1, s(s(s(0))), X2) MULTB_IN_GGA(s(X1), s(s(s(s(0)))), s(s(s(s(X2))))) -> MULTB_IN_GGA(X1, s(s(s(s(0)))), X2) MULTB_IN_GGA(X1, s(s(s(s(0)))), s(s(s(s(s(X2)))))) -> MULTA_IN_GGA(X1, s(s(s(s(0)))), X2) MULTB_IN_GGA(s(X1), s(s(s(s(s(0))))), s(s(s(s(s(X2)))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(0))))), X2) MULTB_IN_GGA(X1, s(s(s(s(s(0))))), s(s(s(s(s(s(X2))))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(0))))), X2) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(0)))))), s(s(s(s(s(s(X2))))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(s(0)))))), X2) MULTB_IN_GGA(X1, s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X2)))))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(s(0)))))), X2) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(X2)))))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), X2) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(X2))))))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), X2) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) -> MULTC_IN_GGGA(X1, s(s(s(s(s(s(s(X2))))))), X2, X3) MULTC_IN_GGGA(s(X1), X2, 0, X3) -> MULTB_IN_GGA(X1, s(X2), X3) MULTC_IN_GGGA(X1, X2, 0, s(X3)) -> MULTA_IN_GGA(X1, s(X2), X3) MULTC_IN_GGGA(X1, X2, s(X3), s(X4)) -> MULTC_IN_GGGA(X1, X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) 0 = 0 MULTA_IN_GGA(x1, x2, x3) = MULTA_IN_GGA(x1, x2) MULTB_IN_GGA(x1, x2, x3) = MULTB_IN_GGA(x1, x2) MULTC_IN_GGGA(x1, x2, x3, x4) = MULTC_IN_GGGA(x1, x2, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (89) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: MULTA_IN_GGA(s(X1), X2) -> MULTB_IN_GGA(X1, X2) MULTB_IN_GGA(s(X1), 0) -> MULTA_IN_GGA(X1, 0) MULTA_IN_GGA(X1, X2) -> MULTA_IN_GGA(X1, X2) MULTB_IN_GGA(X1, 0) -> MULTA_IN_GGA(X1, 0) MULTB_IN_GGA(s(X1), s(0)) -> MULTB_IN_GGA(X1, s(0)) MULTB_IN_GGA(X1, s(0)) -> MULTA_IN_GGA(X1, s(0)) MULTB_IN_GGA(s(X1), s(s(0))) -> MULTB_IN_GGA(X1, s(s(0))) MULTB_IN_GGA(X1, s(s(0))) -> MULTA_IN_GGA(X1, s(s(0))) MULTB_IN_GGA(s(X1), s(s(s(0)))) -> MULTB_IN_GGA(X1, s(s(s(0)))) MULTB_IN_GGA(X1, s(s(s(0)))) -> MULTA_IN_GGA(X1, s(s(s(0)))) MULTB_IN_GGA(s(X1), s(s(s(s(0))))) -> MULTB_IN_GGA(X1, s(s(s(s(0))))) MULTB_IN_GGA(X1, s(s(s(s(0))))) -> MULTA_IN_GGA(X1, s(s(s(s(0))))) MULTB_IN_GGA(s(X1), s(s(s(s(s(0)))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(0)))))) MULTB_IN_GGA(X1, s(s(s(s(s(0)))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(0)))))) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(0))))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(s(0))))))) MULTB_IN_GGA(X1, s(s(s(s(s(s(0))))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(s(0))))))) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(s(0)))))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(s(s(0)))))))) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(0)))))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(s(s(0)))))))) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(s(X2))))))))) -> MULTC_IN_GGGA(X1, s(s(s(s(s(s(s(X2))))))), X2) MULTC_IN_GGGA(s(X1), X2, 0) -> MULTB_IN_GGA(X1, s(X2)) MULTC_IN_GGGA(X1, X2, 0) -> MULTA_IN_GGA(X1, s(X2)) MULTC_IN_GGGA(X1, X2, s(X3)) -> MULTC_IN_GGGA(X1, X2, X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (91) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MULTA_IN_GGA(s(X1), X2) -> MULTB_IN_GGA(X1, X2) MULTB_IN_GGA(s(X1), 0) -> MULTA_IN_GGA(X1, 0) MULTB_IN_GGA(s(X1), s(0)) -> MULTB_IN_GGA(X1, s(0)) MULTB_IN_GGA(s(X1), s(s(0))) -> MULTB_IN_GGA(X1, s(s(0))) MULTB_IN_GGA(s(X1), s(s(s(0)))) -> MULTB_IN_GGA(X1, s(s(s(0)))) MULTB_IN_GGA(s(X1), s(s(s(s(0))))) -> MULTB_IN_GGA(X1, s(s(s(s(0))))) MULTB_IN_GGA(s(X1), s(s(s(s(s(0)))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(0)))))) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(0))))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(s(0))))))) MULTB_IN_GGA(s(X1), s(s(s(s(s(s(s(0)))))))) -> MULTB_IN_GGA(X1, s(s(s(s(s(s(s(0)))))))) MULTC_IN_GGGA(s(X1), X2, 0) -> MULTB_IN_GGA(X1, s(X2)) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. MULTA_IN_GGA(x1, x2) = x1 s(x1) = s(x1) MULTB_IN_GGA(x1, x2) = x1 MULTC_IN_GGGA(x1, x2, x3) = x1 Knuth-Bendix order [KBO] with precedence:trivial and weight map: s_1=1 dummyConstant=1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: none ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: MULTA_IN_GGA(X1, X2) -> MULTA_IN_GGA(X1, X2) MULTB_IN_GGA(X1, 0) -> MULTA_IN_GGA(X1, 0) MULTB_IN_GGA(X1, s(0)) -> MULTA_IN_GGA(X1, s(0)) MULTB_IN_GGA(X1, s(s(0))) -> MULTA_IN_GGA(X1, s(s(0))) MULTB_IN_GGA(X1, s(s(s(0)))) -> MULTA_IN_GGA(X1, s(s(s(0)))) MULTB_IN_GGA(X1, s(s(s(s(0))))) -> MULTA_IN_GGA(X1, s(s(s(s(0))))) MULTB_IN_GGA(X1, s(s(s(s(s(0)))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(0)))))) MULTB_IN_GGA(X1, s(s(s(s(s(s(0))))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(s(0))))))) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(0)))))))) -> MULTA_IN_GGA(X1, s(s(s(s(s(s(s(0)))))))) MULTB_IN_GGA(X1, s(s(s(s(s(s(s(s(X2))))))))) -> MULTC_IN_GGGA(X1, s(s(s(s(s(s(s(X2))))))), X2) MULTC_IN_GGGA(X1, X2, 0) -> MULTA_IN_GGA(X1, s(X2)) MULTC_IN_GGGA(X1, X2, s(X3)) -> MULTC_IN_GGGA(X1, X2, X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (93) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 10 less nodes. ---------------------------------------- (94) Complex Obligation (AND) ---------------------------------------- (95) Obligation: Q DP problem: The TRS P consists of the following rules: MULTA_IN_GGA(X1, X2) -> MULTA_IN_GGA(X1, X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (96) 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 = MULTA_IN_GGA(X1, X2) evaluates to t =MULTA_IN_GGA(X1, X2) 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 MULTA_IN_GGA(X1, X2) to MULTA_IN_GGA(X1, X2). ---------------------------------------- (97) NO ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: MULTC_IN_GGGA(X1, X2, s(X3)) -> MULTC_IN_GGGA(X1, X2, X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (99) 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: *MULTC_IN_GGGA(X1, X2, s(X3)) -> MULTC_IN_GGGA(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (100) YES