/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern p(a,a,a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) CutEliminatorProof [SOUND, 0 ms] (2)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, 0 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) PiDP (9) UsableRulesProof [EQUIVALENT, 0 ms] (10) PiDP (11) PiDPToQDPProof [SOUND, 2 ms] (12) QDP (13) PrologToPiTRSProof [SOUND, 0 ms] (14) PiTRS (15) DependencyPairsProof [EQUIVALENT, 0 ms] (16) PiDP (17) DependencyGraphProof [EQUIVALENT, 0 ms] (18) PiDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) PiDP (21) PiDPToQDPProof [SOUND, 1 ms] (22) QDP (23) PrologToDTProblemTransformerProof [SOUND, 0 ms] (24) TRIPLES (25) TriplesToPiDPProof [SOUND, 0 ms] (26) PiDP (27) DependencyGraphProof [EQUIVALENT, 0 ms] (28) PiDP (29) PiDPToQDPProof [SOUND, 1 ms] (30) QDP (31) PrologToTRSTransformerProof [SOUND, 0 ms] (32) QTRS (33) QTRSRRRProof [EQUIVALENT, 54 ms] (34) QTRS (35) QTRSRRRProof [EQUIVALENT, 6 ms] (36) QTRS (37) QTRSRRRProof [EQUIVALENT, 3 ms] (38) QTRS (39) QTRSRRRProof [EQUIVALENT, 5 ms] (40) QTRS (41) QTRSRRRProof [EQUIVALENT, 4 ms] (42) QTRS (43) QTRSRRRProof [EQUIVALENT, 0 ms] (44) QTRS (45) QTRSRRRProof [EQUIVALENT, 3 ms] (46) QTRS (47) QTRSRRRProof [EQUIVALENT, 10 ms] (48) QTRS (49) PrologToIRSwTTransformerProof [SOUND, 0 ms] (50) IRSwT (51) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (52) IRSwT (53) IntTRSCompressionProof [EQUIVALENT, 11 ms] (54) IRSwT (55) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (56) IRSwT (57) IRSwTTerminationDigraphProof [EQUIVALENT, 7 ms] (58) IRSwT (59) FilterProof [EQUIVALENT, 0 ms] (60) IntTRS (61) IntTRSPeriodicNontermProof [COMPLETE, 8 ms] (62) NO ---------------------------------------- (0) Obligation: Clauses: p(X, Y, Z) :- ','(append(X, Y, Z), !). append([], Y, Y). append(.(H, Xs), Ys, .(H, Zs)) :- append(Xs, Ys, Zs). Query: p(a,a,a) ---------------------------------------- (1) CutEliminatorProof (SOUND) Eliminated all cuts by simply ignoring them[PROLOG]. ---------------------------------------- (2) Obligation: Clauses: p(X, Y, Z) :- append(X, Y, Z). append([], Y, Y). append(.(H, Xs), Ys, .(H, Zs)) :- append(Xs, Ys, Zs). Query: p(a,a,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: p_in_3: (f,f,f) append_in_3: (f,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(X, Y, Z)) append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y) append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs)) U1_aaa(X, Y, Z, append_out_aaa(X, Y, Z)) -> p_out_aaa(X, Y, Z) The argument filtering Pi contains the following mapping: p_in_aaa(x1, x2, x3) = p_in_aaa U1_aaa(x1, x2, x3, x4) = U1_aaa(x4) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5) .(x1, x2) = .(x2) p_out_aaa(x1, x2, x3) = p_out_aaa(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (4) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(X, Y, Z)) append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y) append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs)) U1_aaa(X, Y, Z, append_out_aaa(X, Y, Z)) -> p_out_aaa(X, Y, Z) The argument filtering Pi contains the following mapping: p_in_aaa(x1, x2, x3) = p_in_aaa U1_aaa(x1, x2, x3, x4) = U1_aaa(x4) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5) .(x1, x2) = .(x2) p_out_aaa(x1, x2, x3) = p_out_aaa(x1) ---------------------------------------- (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: P_IN_AAA(X, Y, Z) -> U1_AAA(X, Y, Z, append_in_aaa(X, Y, Z)) P_IN_AAA(X, Y, Z) -> APPEND_IN_AAA(X, Y, Z) APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> U2_AAA(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) The TRS R consists of the following rules: p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(X, Y, Z)) append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y) append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs)) U1_aaa(X, Y, Z, append_out_aaa(X, Y, Z)) -> p_out_aaa(X, Y, Z) The argument filtering Pi contains the following mapping: p_in_aaa(x1, x2, x3) = p_in_aaa U1_aaa(x1, x2, x3, x4) = U1_aaa(x4) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5) .(x1, x2) = .(x2) p_out_aaa(x1, x2, x3) = p_out_aaa(x1) P_IN_AAA(x1, x2, x3) = P_IN_AAA U1_AAA(x1, x2, x3, x4) = U1_AAA(x4) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U2_AAA(x1, x2, x3, x4, x5) = U2_AAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_AAA(X, Y, Z) -> U1_AAA(X, Y, Z, append_in_aaa(X, Y, Z)) P_IN_AAA(X, Y, Z) -> APPEND_IN_AAA(X, Y, Z) APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> U2_AAA(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) The TRS R consists of the following rules: p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(X, Y, Z)) append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y) append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs)) U1_aaa(X, Y, Z, append_out_aaa(X, Y, Z)) -> p_out_aaa(X, Y, Z) The argument filtering Pi contains the following mapping: p_in_aaa(x1, x2, x3) = p_in_aaa U1_aaa(x1, x2, x3, x4) = U1_aaa(x4) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5) .(x1, x2) = .(x2) p_out_aaa(x1, x2, x3) = p_out_aaa(x1) P_IN_AAA(x1, x2, x3) = P_IN_AAA U1_AAA(x1, x2, x3, x4) = U1_AAA(x4) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U2_AAA(x1, x2, x3, x4, x5) = U2_AAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) The TRS R consists of the following rules: p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(X, Y, Z)) append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y) append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs)) U1_aaa(X, Y, Z, append_out_aaa(X, Y, Z)) -> p_out_aaa(X, Y, Z) The argument filtering Pi contains the following mapping: p_in_aaa(x1, x2, x3) = p_in_aaa U1_aaa(x1, x2, x3, x4) = U1_aaa(x4) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5) .(x1, x2) = .(x2) p_out_aaa(x1, x2, x3) = p_out_aaa(x1) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA 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: APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA 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: APPEND_IN_AAA -> APPEND_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (13) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: p_in_3: (f,f,f) append_in_3: (f,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(X, Y, Z)) append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y) append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs)) U1_aaa(X, Y, Z, append_out_aaa(X, Y, Z)) -> p_out_aaa(X, Y, Z) The argument filtering Pi contains the following mapping: p_in_aaa(x1, x2, x3) = p_in_aaa U1_aaa(x1, x2, x3, x4) = U1_aaa(x4) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5) .(x1, x2) = .(x2) p_out_aaa(x1, x2, x3) = p_out_aaa(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (14) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(X, Y, Z)) append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y) append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs)) U1_aaa(X, Y, Z, append_out_aaa(X, Y, Z)) -> p_out_aaa(X, Y, Z) The argument filtering Pi contains the following mapping: p_in_aaa(x1, x2, x3) = p_in_aaa U1_aaa(x1, x2, x3, x4) = U1_aaa(x4) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5) .(x1, x2) = .(x2) p_out_aaa(x1, x2, x3) = p_out_aaa(x1) ---------------------------------------- (15) 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: P_IN_AAA(X, Y, Z) -> U1_AAA(X, Y, Z, append_in_aaa(X, Y, Z)) P_IN_AAA(X, Y, Z) -> APPEND_IN_AAA(X, Y, Z) APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> U2_AAA(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) The TRS R consists of the following rules: p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(X, Y, Z)) append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y) append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs)) U1_aaa(X, Y, Z, append_out_aaa(X, Y, Z)) -> p_out_aaa(X, Y, Z) The argument filtering Pi contains the following mapping: p_in_aaa(x1, x2, x3) = p_in_aaa U1_aaa(x1, x2, x3, x4) = U1_aaa(x4) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5) .(x1, x2) = .(x2) p_out_aaa(x1, x2, x3) = p_out_aaa(x1) P_IN_AAA(x1, x2, x3) = P_IN_AAA U1_AAA(x1, x2, x3, x4) = U1_AAA(x4) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U2_AAA(x1, x2, x3, x4, x5) = U2_AAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_AAA(X, Y, Z) -> U1_AAA(X, Y, Z, append_in_aaa(X, Y, Z)) P_IN_AAA(X, Y, Z) -> APPEND_IN_AAA(X, Y, Z) APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> U2_AAA(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) The TRS R consists of the following rules: p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(X, Y, Z)) append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y) append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs)) U1_aaa(X, Y, Z, append_out_aaa(X, Y, Z)) -> p_out_aaa(X, Y, Z) The argument filtering Pi contains the following mapping: p_in_aaa(x1, x2, x3) = p_in_aaa U1_aaa(x1, x2, x3, x4) = U1_aaa(x4) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5) .(x1, x2) = .(x2) p_out_aaa(x1, x2, x3) = p_out_aaa(x1) P_IN_AAA(x1, x2, x3) = P_IN_AAA U1_AAA(x1, x2, x3, x4) = U1_AAA(x4) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U2_AAA(x1, x2, x3, x4, x5) = U2_AAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) The TRS R consists of the following rules: p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(X, Y, Z)) append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y) append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs)) U1_aaa(X, Y, Z, append_out_aaa(X, Y, Z)) -> p_out_aaa(X, Y, Z) The argument filtering Pi contains the following mapping: p_in_aaa(x1, x2, x3) = p_in_aaa U1_aaa(x1, x2, x3, x4) = U1_aaa(x4) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5) .(x1, x2) = .(x2) p_out_aaa(x1, x2, x3) = p_out_aaa(x1) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (20) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (21) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: APPEND_IN_AAA -> APPEND_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(p X Y Z)", "(',' (append X Y Z) (!))" ], [ "(append ([]) Y Y)", null ], [ "(append (. H Xs) Ys (. H Zs))", "(append Xs Ys Zs)" ] ] }, "graph": { "nodes": { "143": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "124": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T10 T11 T12) (!_1))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "135": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "4": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "126": { "goal": [ { "clause": 1, "scope": 2, "term": "(',' (append T10 T11 T12) (!_1))" }, { "clause": 2, "scope": 2, "term": "(',' (append T10 T11 T12) (!_1))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "129": { "goal": [ { "clause": -1, "scope": -1, "term": "(!_1)" }, { "clause": 2, "scope": 2, "term": "(',' (append T10 T11 T12) (!_1))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "140": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T28 T29 T30) (!_1))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "130": { "goal": [{ "clause": 2, "scope": 2, "term": "(',' (append T10 T11 T12) (!_1))" }], "kb": { "nonunifying": [[ "(append T10 T11 T12)", "(append ([]) X9 X9)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X9"], "exprvars": [] } }, "131": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 4, "label": "CASE" }, { "from": 4, "to": 124, "label": "ONLY EVAL with clause\np(X4, X5, X6) :- ','(append(X4, X5, X6), !_1).\nand substitutionT1 -> T10,\nX4 -> T10,\nT2 -> T11,\nX5 -> T11,\nT3 -> T12,\nX6 -> T12,\nT7 -> T10,\nT8 -> T11,\nT9 -> T12" }, { "from": 124, "to": 126, "label": "CASE" }, { "from": 126, "to": 129, "label": "EVAL with clause\nappend([], X9, X9).\nand substitutionT10 -> [],\nT11 -> T15,\nX9 -> T15,\nT12 -> T15" }, { "from": 126, "to": 130, "label": "EVAL-BACKTRACK" }, { "from": 129, "to": 131, "label": "CUT" }, { "from": 130, "to": 140, "label": "EVAL with clause\nappend(.(X18, X19), X20, .(X18, X21)) :- append(X19, X20, X21).\nand substitutionX18 -> T24,\nX19 -> T28,\nT10 -> .(T24, T28),\nT11 -> T29,\nX20 -> T29,\nX21 -> T30,\nT12 -> .(T24, T30),\nT25 -> T28,\nT26 -> T29,\nT27 -> T30" }, { "from": 130, "to": 143, "label": "EVAL-BACKTRACK" }, { "from": 131, "to": 135, "label": "SUCCESS" }, { "from": 140, "to": 124, "label": "INSTANCE with matching:\nT10 -> T28\nT11 -> T29\nT12 -> T30" } ], "type": "Graph" } } ---------------------------------------- (24) Obligation: Triples: pA(.(X1, X2), X3, .(X1, X4)) :- pA(X2, X3, X4). pB(X1, X2, X3) :- pA(X1, X2, X3). Clauses: qcA([], X1, X1). qcA(.(X1, X2), X3, .(X1, X4)) :- qcA(X2, X3, X4). Afs: pB(x1, x2, x3) = pB ---------------------------------------- (25) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: pB_in_3: (f,f,f) pA_in_3: (f,f,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: PB_IN_AAA(X1, X2, X3) -> U2_AAA(X1, X2, X3, pA_in_aaa(X1, X2, X3)) PB_IN_AAA(X1, X2, X3) -> PA_IN_AAA(X1, X2, X3) PA_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> U1_AAA(X1, X2, X3, X4, pA_in_aaa(X2, X3, X4)) PA_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> PA_IN_AAA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: pA_in_aaa(x1, x2, x3) = pA_in_aaa .(x1, x2) = .(x2) PB_IN_AAA(x1, x2, x3) = PB_IN_AAA U2_AAA(x1, x2, x3, x4) = U2_AAA(x4) PA_IN_AAA(x1, x2, x3) = PA_IN_AAA U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (26) Obligation: Pi DP problem: The TRS P consists of the following rules: PB_IN_AAA(X1, X2, X3) -> U2_AAA(X1, X2, X3, pA_in_aaa(X1, X2, X3)) PB_IN_AAA(X1, X2, X3) -> PA_IN_AAA(X1, X2, X3) PA_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> U1_AAA(X1, X2, X3, X4, pA_in_aaa(X2, X3, X4)) PA_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> PA_IN_AAA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: pA_in_aaa(x1, x2, x3) = pA_in_aaa .(x1, x2) = .(x2) PB_IN_AAA(x1, x2, x3) = PB_IN_AAA U2_AAA(x1, x2, x3, x4) = U2_AAA(x4) PA_IN_AAA(x1, x2, x3) = PA_IN_AAA U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (27) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: PA_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> PA_IN_AAA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) PA_IN_AAA(x1, x2, x3) = PA_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: PA_IN_AAA -> PA_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (31) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 17, "program": { "directives": [], "clauses": [ [ "(p X Y Z)", "(',' (append X Y Z) (!))" ], [ "(append ([]) Y Y)", null ], [ "(append (. H Xs) Ys (. H Zs))", "(append Xs Ys Zs)" ] ] }, "graph": { "nodes": { "88": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T18 T19 T20) (!_1))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "17": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "18": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "150": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "120": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T42 T43 T44)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "110": { "goal": [{ "clause": -1, "scope": -1, "term": "(!_1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "121": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "114": { "goal": [ { "clause": 1, "scope": 2, "term": "(append T18 T19 T20)" }, { "clause": 2, "scope": 2, "term": "(append T18 T19 T20)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "115": { "goal": [{ "clause": 1, "scope": 2, "term": "(append T18 T19 T20)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "148": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "116": { "goal": [{ "clause": 2, "scope": 2, "term": "(append T18 T19 T20)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "117": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "118": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "119": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "109": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T18 T19 T20)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 17, "to": 18, "label": "CASE" }, { "from": 18, "to": 88, "label": "ONLY EVAL with clause\np(X9, X10, X11) :- ','(append(X9, X10, X11), !_1).\nand substitutionT1 -> T18,\nX9 -> T18,\nT2 -> T19,\nX10 -> T19,\nT3 -> T20,\nX11 -> T20,\nT15 -> T18,\nT16 -> T19,\nT17 -> T20" }, { "from": 88, "to": 109, "label": "SPLIT 1" }, { "from": 88, "to": 110, "label": "SPLIT 2" }, { "from": 109, "to": 114, "label": "CASE" }, { "from": 110, "to": 148, "label": "CUT" }, { "from": 114, "to": 115, "label": "PARALLEL" }, { "from": 114, "to": 116, "label": "PARALLEL" }, { "from": 115, "to": 117, "label": "EVAL with clause\nappend([], X20, X20).\nand substitutionT18 -> [],\nT19 -> T29,\nX20 -> T29,\nT20 -> T29" }, { "from": 115, "to": 118, "label": "EVAL-BACKTRACK" }, { "from": 116, "to": 120, "label": "EVAL with clause\nappend(.(X29, X30), X31, .(X29, X32)) :- append(X30, X31, X32).\nand substitutionX29 -> T38,\nX30 -> T42,\nT18 -> .(T38, T42),\nT19 -> T43,\nX31 -> T43,\nX32 -> T44,\nT20 -> .(T38, T44),\nT39 -> T42,\nT40 -> T43,\nT41 -> T44" }, { "from": 116, "to": 121, "label": "EVAL-BACKTRACK" }, { "from": 117, "to": 119, "label": "SUCCESS" }, { "from": 120, "to": 109, "label": "INSTANCE with matching:\nT18 -> T42\nT19 -> T43\nT20 -> T44" }, { "from": 148, "to": 150, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (32) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f17_in -> U1(f88_in) U1(f88_out1) -> f17_out1 f109_in -> f109_out1 f109_in -> U2(f109_in) U2(f109_out1) -> f109_out1 f110_in -> f110_out1 f88_in -> U3(f109_in) U3(f109_out1) -> U4(f110_in) U4(f110_out1) -> f88_out1 Q is empty. ---------------------------------------- (33) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1)) = 2*x_1 POL(U2(x_1)) = 2*x_1 POL(U3(x_1)) = 2*x_1 POL(U4(x_1)) = x_1 POL(f109_in) = 0 POL(f109_out1) = 0 POL(f110_in) = 0 POL(f110_out1) = 0 POL(f17_in) = 2 POL(f17_out1) = 0 POL(f88_in) = 0 POL(f88_out1) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f17_in -> U1(f88_in) ---------------------------------------- (34) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U1(f88_out1) -> f17_out1 f109_in -> f109_out1 f109_in -> U2(f109_in) U2(f109_out1) -> f109_out1 f110_in -> f110_out1 f88_in -> U3(f109_in) U3(f109_out1) -> U4(f110_in) U4(f110_out1) -> f88_out1 Q is empty. ---------------------------------------- (35) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1)) = 1 + 2*x_1 POL(U2(x_1)) = 2*x_1 POL(U3(x_1)) = 2*x_1 POL(U4(x_1)) = 2*x_1 POL(f109_in) = 0 POL(f109_out1) = 0 POL(f110_in) = 0 POL(f110_out1) = 0 POL(f17_out1) = 0 POL(f88_in) = 1 POL(f88_out1) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f88_out1) -> f17_out1 f88_in -> U3(f109_in) ---------------------------------------- (36) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f109_in -> f109_out1 f109_in -> U2(f109_in) U2(f109_out1) -> f109_out1 f110_in -> f110_out1 U3(f109_out1) -> U4(f110_in) U4(f110_out1) -> f88_out1 Q is empty. ---------------------------------------- (37) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = 2*x_1 POL(U3(x_1)) = 2 + 2*x_1 POL(U4(x_1)) = 1 + 2*x_1 POL(f109_in) = 0 POL(f109_out1) = 0 POL(f110_in) = 0 POL(f110_out1) = 0 POL(f88_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U3(f109_out1) -> U4(f110_in) ---------------------------------------- (38) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f109_in -> f109_out1 f109_in -> U2(f109_in) U2(f109_out1) -> f109_out1 f110_in -> f110_out1 U4(f110_out1) -> f88_out1 Q is empty. ---------------------------------------- (39) QTRSRRRProof (EQUIVALENT) Used ordering: f109_in/0) f109_out1/0) U2/1)YES( f110_in/0) f110_out1/0) U4/1(YES) f88_out1/0) Quasi precedence: [f109_in, f109_out1] f110_in > f110_out1 U4_1 > f88_out1 Status: f109_in: multiset status f109_out1: multiset status f110_in: multiset status f110_out1: multiset status U4_1: [1] f88_out1: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f110_in -> f110_out1 U4(f110_out1) -> f88_out1 ---------------------------------------- (40) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f109_in -> f109_out1 f109_in -> U2(f109_in) U2(f109_out1) -> f109_out1 Q is empty. ---------------------------------------- (41) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = x_1 POL(f109_in) = 2 POL(f109_out1) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f109_in -> f109_out1 ---------------------------------------- (42) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f109_in -> U2(f109_in) U2(f109_out1) -> f109_out1 Q is empty. ---------------------------------------- (43) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = 2*x_1 POL(f109_in) = 0 POL(f109_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U2(f109_out1) -> f109_out1 ---------------------------------------- (44) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f109_in -> U2(f109_in) Q is empty. ---------------------------------------- (45) QTRSRRRProof (EQUIVALENT) Used ordering: f109_in/0) f109_out1/0) U2/1)YES( Quasi precedence: f109_in > f109_out1 Status: f109_in: multiset status f109_out1: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f109_in -> f109_out1 ---------------------------------------- (46) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f109_in -> U2(f109_in) U2(f109_out1) -> f109_out1 Q is empty. ---------------------------------------- (47) QTRSRRRProof (EQUIVALENT) Used ordering: U1/1(YES) f88_out1/0) f17_out1/0) f109_in/0) f109_out1/0) U2/1)YES( f110_in/0) f110_out1/0) f88_in/0) U3/1)YES( U4/1)YES( Quasi precedence: U1_1 > f17_out1 f88_in > [f109_in, f109_out1, f110_in] > [f88_out1, f110_out1] Status: U1_1: multiset status f88_out1: multiset status f17_out1: multiset status f109_in: multiset status f109_out1: multiset status f110_in: multiset status f110_out1: multiset status f88_in: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f88_out1) -> f17_out1 f110_in -> f110_out1 f88_in -> U3(f109_in) ---------------------------------------- (48) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f109_in -> f109_out1 f109_in -> U2(f109_in) U2(f109_out1) -> f109_out1 U3(f109_out1) -> U4(f110_in) U4(f110_out1) -> f88_out1 Q is empty. ---------------------------------------- (49) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 2, "program": { "directives": [], "clauses": [ [ "(p X Y Z)", "(',' (append X Y Z) (!))" ], [ "(append ([]) Y Y)", null ], [ "(append (. H Xs) Ys (. H Zs))", "(append Xs Ys Zs)" ] ] }, "graph": { "nodes": { "16": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "141": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T42 T43 T44)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "142": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "132": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "122": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T18 T19 T20)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "133": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "123": { "goal": [{ "clause": -1, "scope": -1, "term": "(!_1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "134": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "125": { "goal": [ { "clause": 1, "scope": 2, "term": "(append T18 T19 T20)" }, { "clause": 2, "scope": 2, "term": "(append T18 T19 T20)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "147": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "127": { "goal": [{ "clause": 1, "scope": 2, "term": "(append T18 T19 T20)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "149": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "128": { "goal": [{ "clause": 2, "scope": 2, "term": "(append T18 T19 T20)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "53": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T18 T19 T20) (!_1))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 16, "label": "CASE" }, { "from": 16, "to": 53, "label": "ONLY EVAL with clause\np(X9, X10, X11) :- ','(append(X9, X10, X11), !_1).\nand substitutionT1 -> T18,\nX9 -> T18,\nT2 -> T19,\nX10 -> T19,\nT3 -> T20,\nX11 -> T20,\nT15 -> T18,\nT16 -> T19,\nT17 -> T20" }, { "from": 53, "to": 122, "label": "SPLIT 1" }, { "from": 53, "to": 123, "label": "SPLIT 2" }, { "from": 122, "to": 125, "label": "CASE" }, { "from": 123, "to": 147, "label": "CUT" }, { "from": 125, "to": 127, "label": "PARALLEL" }, { "from": 125, "to": 128, "label": "PARALLEL" }, { "from": 127, "to": 132, "label": "EVAL with clause\nappend([], X20, X20).\nand substitutionT18 -> [],\nT19 -> T29,\nX20 -> T29,\nT20 -> T29" }, { "from": 127, "to": 133, "label": "EVAL-BACKTRACK" }, { "from": 128, "to": 141, "label": "EVAL with clause\nappend(.(X29, X30), X31, .(X29, X32)) :- append(X30, X31, X32).\nand substitutionX29 -> T38,\nX30 -> T42,\nT18 -> .(T38, T42),\nT19 -> T43,\nX31 -> T43,\nX32 -> T44,\nT20 -> .(T38, T44),\nT39 -> T42,\nT40 -> T43,\nT41 -> T44" }, { "from": 128, "to": 142, "label": "EVAL-BACKTRACK" }, { "from": 132, "to": 134, "label": "SUCCESS" }, { "from": 141, "to": 122, "label": "INSTANCE with matching:\nT18 -> T42\nT19 -> T43\nT20 -> T44" }, { "from": 147, "to": 149, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (50) Obligation: Rules: f127_out -> f125_out :|: TRUE f128_out -> f125_out :|: TRUE f125_in -> f127_in :|: TRUE f125_in -> f128_in :|: TRUE f122_out -> f141_out :|: TRUE f141_in -> f122_in :|: TRUE f141_out -> f128_out :|: TRUE f128_in -> f142_in :|: TRUE f142_out -> f128_out :|: TRUE f128_in -> f141_in :|: TRUE f125_out -> f122_out :|: TRUE f122_in -> f125_in :|: TRUE f16_out -> f2_out :|: TRUE f2_in -> f16_in :|: TRUE f16_in -> f53_in :|: TRUE f53_out -> f16_out :|: TRUE f53_in -> f122_in :|: TRUE f122_out -> f123_in :|: TRUE f123_out -> f53_out :|: TRUE Start term: f2_in ---------------------------------------- (51) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f125_in -> f128_in :|: TRUE f141_in -> f122_in :|: TRUE f128_in -> f141_in :|: TRUE f122_in -> f125_in :|: TRUE ---------------------------------------- (52) Obligation: Rules: f125_in -> f128_in :|: TRUE f141_in -> f122_in :|: TRUE f128_in -> f141_in :|: TRUE f122_in -> f125_in :|: TRUE ---------------------------------------- (53) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (54) Obligation: Rules: f125_in -> f125_in :|: TRUE ---------------------------------------- (55) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (56) Obligation: Rules: f125_in -> f125_in :|: TRUE ---------------------------------------- (57) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f125_in -> f125_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (58) Obligation: Termination digraph: Nodes: (1) f125_in -> f125_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (59) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f125_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (60) Obligation: Rules: f125_in -> f125_in :|: TRUE ---------------------------------------- (61) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (62) NO