/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern less(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 0 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) PiDPToQDPProof [EQUIVALENT, 0 ms] (9) QDP (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] (11) YES (12) PiDP (13) PiDPToQDPProof [EQUIVALENT, 0 ms] (14) QDP (15) QDPSizeChangeProof [EQUIVALENT, 0 ms] (16) YES (17) PiDP (18) PiDPToQDPProof [SOUND, 0 ms] (19) QDP (20) QDPSizeChangeProof [EQUIVALENT, 0 ms] (21) YES ---------------------------------------- (0) Obligation: Clauses: less(0, s(X2)). less(X, Y) :- ','(no(zero(X)), ','(p(X, X1), ','(p(Y, Y1), less(X1, Y1)))). p(0, 0). p(s(X), X). zero(0). no(X) :- ','(X, ','(!, failure(a))). no(X3). failure(b). Query: less(g,a) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(less (0) (s X2))", null ], [ "(less X Y)", "(',' (no (zero X)) (',' (p X X1) (',' (p Y Y1) (less X1 Y1))))" ], [ "(p (0) (0))", null ], [ "(p (s X) X)", null ], [ "(zero (0))", null ], [ "(no X)", "(',' X (',' (!) (failure (a))))" ], [ "(no X3)", null ], [ "(failure (b))", null ] ] }, "graph": { "nodes": { "22": { "goal": [ { "clause": 5, "scope": 2, "term": "(',' (no (zero (0))) (',' (p (0) X10) (',' (p T7 X11) (less X10 X11))))" }, { "clause": 6, "scope": 2, "term": "(',' (no (zero (0))) (',' (p (0) X10) (',' (p T7 X11) (less X10 X11))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X10", "X11" ], "exprvars": [] } }, "88": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "24": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (call (zero (0))) (',' (!_2) (failure (a)))) (',' (p (0) X10) (',' (p T7 X11) (less X10 X11))))" }, { "clause": 6, "scope": 2, "term": "(',' (no (zero (0))) (',' (p (0) X10) (',' (p T7 X11) (less X10 X11))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X10", "X11" ], "exprvars": [] } }, "25": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (zero (0)) (',' (',' (!_2) (failure (a))) (',' (p (0) X10) (',' (p T7 X11) (less X10 X11)))))" }, { "clause": -1, "scope": 3, "term": null }, { "clause": 6, "scope": 2, "term": "(',' (no (zero (0))) (',' (p (0) X10) (',' (p T7 X11) (less X10 X11))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X10", "X11" ], "exprvars": [] } }, "26": { "goal": [ { "clause": 4, "scope": 4, "term": "(',' (zero (0)) (',' (',' (!_2) (failure (a))) (',' (p (0) X10) (',' (p T7 X11) (less X10 X11)))))" }, { "clause": -1, "scope": 4, "term": null }, { "clause": -1, "scope": 3, "term": null }, { "clause": 6, "scope": 2, "term": "(',' (no (zero (0))) (',' (p (0) X10) (',' (p T7 X11) (less X10 X11))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X10", "X11" ], "exprvars": [] } }, "type": "Nodes", "110": { "goal": [{ "clause": 2, "scope": 11, "term": "(',' (p T12 X22) (less T21 X22))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": ["X22"], "exprvars": [] } }, "111": { "goal": [{ "clause": 3, "scope": 11, "term": "(',' (p T12 X22) (less T21 X22))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": ["X22"], "exprvars": [] } }, "112": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T21 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": [], "exprvars": [] } }, "114": { "goal": [], "kb": { 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"exprvars": [] } }, "84": { "goal": [ { "clause": -1, "scope": 8, "term": null }, { "clause": -1, "scope": 7, "term": null }, { "clause": 6, "scope": 6, "term": "(',' (no (zero T15)) (',' (p T15 X21) (',' (p T12 X22) (less X21 X22))))" } ], "kb": { "nonunifying": [ [ "(less T15 T2)", "(less (0) (s X5))" ], [ "(zero T15)", "(zero (0))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": [ "X5", "X21", "X22" ], "exprvars": [] } }, "20": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (no (zero (0))) (',' (p (0) X10) (',' (p T7 X11) (less X10 X11))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X10", "X11" ], "exprvars": [] } }, "86": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (failure (a)) (',' (p (0) X21) (',' (p T12 X22) (less X21 X22))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X21", "X22" ], "exprvars": [] } }, "87": { "goal": [{ "clause": 7, "scope": 9, "term": "(',' (failure (a)) (',' (p (0) X21) (',' (p T12 X22) (less X21 X22))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X21", "X22" ], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 6, "label": "CASE" }, { "from": 6, "to": 17, "label": "EVAL with clause\nless(0, s(X5)).\nand substitutionT1 -> 0,\nX5 -> T4,\nT2 -> s(T4)" }, { "from": 6, "to": 18, "label": "EVAL-BACKTRACK" }, { "from": 17, "to": 19, "label": "SUCCESS" }, { "from": 18, "to": 75, "label": "ONLY EVAL with clause\nless(X19, X20) :- ','(no(zero(X19)), ','(p(X19, X21), ','(p(X20, X22), less(X21, X22)))).\nand substitutionT1 -> T10,\nX19 -> T10,\nT2 -> T12,\nX20 -> T12,\nT11 -> T12" }, { "from": 19, "to": 20, "label": "ONLY EVAL with clause\nless(X8, X9) :- ','(no(zero(X8)), ','(p(X8, X10), ','(p(X9, X11), less(X10, X11)))).\nand substitutionX8 -> 0,\nT2 -> T7,\nX9 -> T7,\nT6 -> T7" }, { "from": 20, "to": 22, "label": "CASE" }, { "from": 22, "to": 24, "label": "ONLY EVAL with clause\nno(X14) :- ','(call(X14), ','(!_2, failure(a))).\nand substitutionX14 -> zero(0)" }, { "from": 24, "to": 25, "label": "CALL" }, { "from": 25, "to": 26, "label": "CASE" }, { "from": 26, "to": 56, "label": "ONLY EVAL with clause\nzero(0).\nand substitution" }, { "from": 56, "to": 57, "label": "CUT" }, { "from": 57, "to": 58, "label": "CASE" }, { "from": 58, "to": 59, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 75, "to": 76, "label": "CASE" }, { "from": 76, "to": 79, "label": "ONLY EVAL with clause\nno(X25) :- ','(call(X25), ','(!_6, failure(a))).\nand substitutionT10 -> T15,\nX25 -> zero(T15)" }, { "from": 79, "to": 80, "label": "CALL" }, { "from": 80, "to": 81, "label": "CASE" }, { "from": 81, "to": 83, "label": "EVAL with clause\nzero(0).\nand substitutionT15 -> 0" }, { "from": 81, "to": 84, "label": "EVAL-BACKTRACK" }, { "from": 83, "to": 86, "label": "CUT" }, { "from": 84, "to": 90, "label": "FAILURE" }, { "from": 86, "to": 87, "label": "CASE" }, { "from": 87, "to": 88, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 90, "to": 92, "label": "FAILURE" }, { "from": 92, "to": 97, "label": "ONLY EVAL with clause\nno(X32).\nand substitutionT15 -> T18,\nX32 -> zero(T18)" }, { "from": 97, "to": 102, "label": "CASE" }, { "from": 102, "to": 103, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T18), zero(0))" }, { "from": 103, "to": 105, "label": "EVAL with clause\np(s(X37), X37).\nand substitutionX37 -> T21,\nT18 -> s(T21),\nX21 -> T21" }, { "from": 103, "to": 106, "label": "EVAL-BACKTRACK" }, { "from": 105, "to": 108, "label": "CASE" }, { "from": 108, "to": 110, "label": "PARALLEL" }, { "from": 108, "to": 111, "label": "PARALLEL" }, { "from": 110, "to": 112, "label": "EVAL with clause\np(0, 0).\nand substitutionT12 -> 0,\nX22 -> 0" }, { "from": 110, "to": 114, "label": "EVAL-BACKTRACK" }, { "from": 111, "to": 136, "label": "EVAL with clause\np(s(X46), X46).\nand substitutionX46 -> T25,\nT12 -> s(T25),\nX22 -> T25,\nT24 -> T25" }, { "from": 111, "to": 137, "label": "EVAL-BACKTRACK" }, { "from": 112, "to": 3, "label": "INSTANCE with matching:\nT1 -> T21\nT2 -> 0" }, { "from": 136, "to": 3, "label": "INSTANCE with matching:\nT1 -> T21\nT2 -> T25" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: lessA(s(X1), 0) :- lessA(X1, 0). lessA(s(X1), s(X2)) :- lessA(X1, X2). Clauses: lesscA(0, s(X1)). lesscA(s(X1), 0) :- lesscA(X1, 0). lesscA(s(X1), s(X2)) :- lesscA(X1, X2). Afs: lessA(x1, x2) = lessA(x1) ---------------------------------------- (3) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: lessA_in_2: (b,f) (b,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: LESSA_IN_GA(s(X1), 0) -> U1_GA(X1, lessA_in_gg(X1, 0)) LESSA_IN_GA(s(X1), 0) -> LESSA_IN_GG(X1, 0) LESSA_IN_GG(s(X1), 0) -> U1_GG(X1, lessA_in_gg(X1, 0)) LESSA_IN_GG(s(X1), 0) -> LESSA_IN_GG(X1, 0) LESSA_IN_GG(s(X1), s(X2)) -> U2_GG(X1, X2, lessA_in_gg(X1, X2)) LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) LESSA_IN_GA(s(X1), s(X2)) -> U2_GA(X1, X2, lessA_in_ga(X1, X2)) LESSA_IN_GA(s(X1), s(X2)) -> LESSA_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: lessA_in_ga(x1, x2) = lessA_in_ga(x1) s(x1) = s(x1) lessA_in_gg(x1, x2) = lessA_in_gg(x1, x2) 0 = 0 LESSA_IN_GA(x1, x2) = LESSA_IN_GA(x1) U1_GA(x1, x2) = U1_GA(x1, x2) LESSA_IN_GG(x1, x2) = LESSA_IN_GG(x1, x2) U1_GG(x1, x2) = U1_GG(x1, x2) U2_GG(x1, x2, x3) = U2_GG(x1, x2, x3) U2_GA(x1, x2, x3) = U2_GA(x1, x3) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSA_IN_GA(s(X1), 0) -> U1_GA(X1, lessA_in_gg(X1, 0)) LESSA_IN_GA(s(X1), 0) -> LESSA_IN_GG(X1, 0) LESSA_IN_GG(s(X1), 0) -> U1_GG(X1, lessA_in_gg(X1, 0)) LESSA_IN_GG(s(X1), 0) -> LESSA_IN_GG(X1, 0) LESSA_IN_GG(s(X1), s(X2)) -> U2_GG(X1, X2, lessA_in_gg(X1, X2)) LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) LESSA_IN_GA(s(X1), s(X2)) -> U2_GA(X1, X2, lessA_in_ga(X1, X2)) LESSA_IN_GA(s(X1), s(X2)) -> LESSA_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: lessA_in_ga(x1, x2) = lessA_in_ga(x1) s(x1) = s(x1) lessA_in_gg(x1, x2) = lessA_in_gg(x1, x2) 0 = 0 LESSA_IN_GA(x1, x2) = LESSA_IN_GA(x1) U1_GA(x1, x2) = U1_GA(x1, x2) LESSA_IN_GG(x1, x2) = LESSA_IN_GG(x1, x2) U1_GG(x1, x2) = U1_GG(x1, x2) U2_GG(x1, x2, x3) = U2_GG(x1, x2, x3) U2_GA(x1, x2, x3) = U2_GA(x1, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSA_IN_GG(s(X1), 0) -> LESSA_IN_GG(X1, 0) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: LESSA_IN_GG(s(X1), 0) -> LESSA_IN_GG(X1, 0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (10) 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: *LESSA_IN_GG(s(X1), 0) -> LESSA_IN_GG(X1, 0) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (11) YES ---------------------------------------- (12) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (13) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (15) 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: *LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (16) YES ---------------------------------------- (17) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSA_IN_GA(s(X1), s(X2)) -> LESSA_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESSA_IN_GA(x1, x2) = LESSA_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (18) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: LESSA_IN_GA(s(X1)) -> LESSA_IN_GA(X1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (20) 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: *LESSA_IN_GA(s(X1)) -> LESSA_IN_GA(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (21) YES