/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern minimum(a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) PiDP (7) UsableRulesProof [EQUIVALENT, 0 ms] (8) PiDP (9) PiDPToQDPProof [SOUND, 1 ms] (10) QDP (11) PrologToPiTRSProof [SOUND, 0 ms] (12) PiTRS (13) DependencyPairsProof [EQUIVALENT, 0 ms] (14) PiDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) PiDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) PiDP (19) PiDPToQDPProof [SOUND, 2 ms] (20) QDP (21) PrologToTRSTransformerProof [SOUND, 0 ms] (22) QTRS (23) DependencyPairsProof [EQUIVALENT, 0 ms] (24) QDP (25) DependencyGraphProof [EQUIVALENT, 0 ms] (26) QDP (27) MNOCProof [EQUIVALENT, 0 ms] (28) QDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) QDP (31) QReductionProof [EQUIVALENT, 0 ms] (32) QDP (33) PrologToIRSwTTransformerProof [SOUND, 0 ms] (34) IRSwT (35) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (36) IRSwT (37) IntTRSCompressionProof [EQUIVALENT, 24 ms] (38) IRSwT (39) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (40) IRSwT (41) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (42) IRSwT (43) FilterProof [EQUIVALENT, 0 ms] (44) IntTRS (45) IntTRSPeriodicNontermProof [COMPLETE, 9 ms] (46) NO (47) PrologToDTProblemTransformerProof [SOUND, 0 ms] (48) TRIPLES (49) TriplesToPiDPProof [SOUND, 3 ms] (50) PiDP (51) DependencyGraphProof [EQUIVALENT, 0 ms] (52) PiDP (53) PiDPToQDPProof [SOUND, 0 ms] (54) QDP ---------------------------------------- (0) Obligation: Clauses: minimum(tree(X, void, X1), X). minimum(tree(X2, Left, X3), X) :- minimum(Left, X). Query: minimum(a,g) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: minimum_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X) minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X)) U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X) The argument filtering Pi contains the following mapping: minimum_in_ag(x1, x2) = minimum_in_ag(x2) minimum_out_ag(x1, x2) = minimum_out_ag(x1) tree(x1, x2, x3) = tree(x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X) minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X)) U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X) The argument filtering Pi contains the following mapping: minimum_in_ag(x1, x2) = minimum_in_ag(x2) minimum_out_ag(x1, x2) = minimum_out_ag(x1) tree(x1, x2, x3) = tree(x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x5) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: MINIMUM_IN_AG(tree(X2, Left, X3), X) -> U1_AG(X2, Left, X3, X, minimum_in_ag(Left, X)) MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X) The TRS R consists of the following rules: minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X) minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X)) U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X) The argument filtering Pi contains the following mapping: minimum_in_ag(x1, x2) = minimum_in_ag(x2) minimum_out_ag(x1, x2) = minimum_out_ag(x1) tree(x1, x2, x3) = tree(x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x5) MINIMUM_IN_AG(x1, x2) = MINIMUM_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: MINIMUM_IN_AG(tree(X2, Left, X3), X) -> U1_AG(X2, Left, X3, X, minimum_in_ag(Left, X)) MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X) The TRS R consists of the following rules: minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X) minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X)) U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X) The argument filtering Pi contains the following mapping: minimum_in_ag(x1, x2) = minimum_in_ag(x2) minimum_out_ag(x1, x2) = minimum_out_ag(x1) tree(x1, x2, x3) = tree(x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x5) MINIMUM_IN_AG(x1, x2) = MINIMUM_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X) The TRS R consists of the following rules: minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X) minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X)) U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X) The argument filtering Pi contains the following mapping: minimum_in_ag(x1, x2) = minimum_in_ag(x2) minimum_out_ag(x1, x2) = minimum_out_ag(x1) tree(x1, x2, x3) = tree(x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x5) MINIMUM_IN_AG(x1, x2) = MINIMUM_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X) R is empty. The argument filtering Pi contains the following mapping: tree(x1, x2, x3) = tree(x2) MINIMUM_IN_AG(x1, x2) = MINIMUM_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: MINIMUM_IN_AG(X) -> MINIMUM_IN_AG(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (11) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: minimum_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X) minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X)) U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X) The argument filtering Pi contains the following mapping: minimum_in_ag(x1, x2) = minimum_in_ag(x2) minimum_out_ag(x1, x2) = minimum_out_ag(x1, x2) tree(x1, x2, x3) = tree(x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (12) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X) minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X)) U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X) The argument filtering Pi contains the following mapping: minimum_in_ag(x1, x2) = minimum_in_ag(x2) minimum_out_ag(x1, x2) = minimum_out_ag(x1, x2) tree(x1, x2, x3) = tree(x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) ---------------------------------------- (13) 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: MINIMUM_IN_AG(tree(X2, Left, X3), X) -> U1_AG(X2, Left, X3, X, minimum_in_ag(Left, X)) MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X) The TRS R consists of the following rules: minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X) minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X)) U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X) The argument filtering Pi contains the following mapping: minimum_in_ag(x1, x2) = minimum_in_ag(x2) minimum_out_ag(x1, x2) = minimum_out_ag(x1, x2) tree(x1, x2, x3) = tree(x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) MINIMUM_IN_AG(x1, x2) = MINIMUM_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: MINIMUM_IN_AG(tree(X2, Left, X3), X) -> U1_AG(X2, Left, X3, X, minimum_in_ag(Left, X)) MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X) The TRS R consists of the following rules: minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X) minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X)) U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X) The argument filtering Pi contains the following mapping: minimum_in_ag(x1, x2) = minimum_in_ag(x2) minimum_out_ag(x1, x2) = minimum_out_ag(x1, x2) tree(x1, x2, x3) = tree(x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) MINIMUM_IN_AG(x1, x2) = MINIMUM_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X) The TRS R consists of the following rules: minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X) minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X)) U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X) The argument filtering Pi contains the following mapping: minimum_in_ag(x1, x2) = minimum_in_ag(x2) minimum_out_ag(x1, x2) = minimum_out_ag(x1, x2) tree(x1, x2, x3) = tree(x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) MINIMUM_IN_AG(x1, x2) = MINIMUM_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X) R is empty. The argument filtering Pi contains the following mapping: tree(x1, x2, x3) = tree(x2) MINIMUM_IN_AG(x1, x2) = MINIMUM_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: MINIMUM_IN_AG(X) -> MINIMUM_IN_AG(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 66, "program": { "directives": [], "clauses": [ [ "(minimum (tree X (void) X1) X)", null ], [ "(minimum (tree X2 Left X3) X)", "(minimum Left X)" ] ] }, "graph": { "nodes": { "66": { "goal": [{ "clause": -1, "scope": -1, "term": "(minimum T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "88": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "99": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "67": { "goal": [ { "clause": 0, "scope": 1, "term": "(minimum T1 T2)" }, { "clause": 1, "scope": 1, "term": "(minimum T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "100": { "goal": [{ "clause": -1, "scope": -1, "term": "(minimum T25 T24)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T24"], "free": [], "exprvars": [] } }, "68": { "goal": [{ "clause": 0, "scope": 1, "term": "(minimum T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "101": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "69": { "goal": [{ "clause": 1, "scope": 1, "term": "(minimum T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "type": "Nodes", "87": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 66, "to": 67, "label": "CASE" }, { "from": 67, "to": 68, "label": "PARALLEL" }, { "from": 67, "to": 69, "label": "PARALLEL" }, { "from": 68, "to": 87, "label": "EVAL with clause\nminimum(tree(X12, void, X13), X12).\nand substitutionX12 -> T11,\nX13 -> T12,\nT1 -> tree(T11, void, T12),\nT2 -> T11" }, { "from": 68, "to": 88, "label": "EVAL-BACKTRACK" }, { "from": 69, "to": 100, "label": "EVAL with clause\nminimum(tree(X22, X23, X24), X25) :- minimum(X23, X25).\nand substitutionX22 -> T21,\nX23 -> T25,\nX24 -> T23,\nT1 -> tree(T21, T25, T23),\nT2 -> T24,\nX25 -> T24,\nT22 -> T25" }, { "from": 69, "to": 101, "label": "EVAL-BACKTRACK" }, { "from": 87, "to": 99, "label": "SUCCESS" }, { "from": 100, "to": 66, "label": "INSTANCE with matching:\nT1 -> T25\nT2 -> T24" } ], "type": "Graph" } } ---------------------------------------- (22) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f66_in(T11) -> f66_out1 f66_in(T24) -> U1(f66_in(T24), T24) U1(f66_out1, T24) -> f66_out1 Q is empty. ---------------------------------------- (23) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: F66_IN(T24) -> U1^1(f66_in(T24), T24) F66_IN(T24) -> F66_IN(T24) The TRS R consists of the following rules: f66_in(T11) -> f66_out1 f66_in(T24) -> U1(f66_in(T24), T24) U1(f66_out1, T24) -> f66_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: F66_IN(T24) -> F66_IN(T24) The TRS R consists of the following rules: f66_in(T11) -> f66_out1 f66_in(T24) -> U1(f66_in(T24), T24) U1(f66_out1, T24) -> f66_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) MNOCProof (EQUIVALENT) We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: F66_IN(T24) -> F66_IN(T24) The TRS R consists of the following rules: f66_in(T11) -> f66_out1 f66_in(T24) -> U1(f66_in(T24), T24) U1(f66_out1, T24) -> f66_out1 The set Q consists of the following terms: f66_in(x0) U1(f66_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) 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. ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: F66_IN(T24) -> F66_IN(T24) R is empty. The set Q consists of the following terms: f66_in(x0) U1(f66_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) 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]. f66_in(x0) U1(f66_out1, x0) ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: F66_IN(T24) -> F66_IN(T24) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 12, "program": { "directives": [], "clauses": [ [ "(minimum (tree X (void) X1) X)", null ], [ "(minimum (tree X2 Left X3) X)", "(minimum Left X)" ] ] }, "graph": { "nodes": { "12": { "goal": [{ "clause": -1, "scope": -1, "term": "(minimum T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "13": { "goal": [ { "clause": 0, "scope": 1, "term": "(minimum T1 T2)" }, { "clause": 1, "scope": 1, "term": "(minimum T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "102": { "goal": [{ "clause": -1, "scope": -1, "term": "(minimum T25 T24)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T24"], "free": [], "exprvars": [] } }, "103": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "92": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "62": { "goal": [{ "clause": 0, "scope": 1, "term": "(minimum T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "85": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "64": { "goal": [{ "clause": 1, "scope": 1, "term": "(minimum T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "86": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 12, "to": 13, "label": "CASE" }, { "from": 13, "to": 62, "label": "PARALLEL" }, { "from": 13, "to": 64, "label": "PARALLEL" }, { "from": 62, "to": 85, "label": "EVAL with clause\nminimum(tree(X12, void, X13), X12).\nand substitutionX12 -> T11,\nX13 -> T12,\nT1 -> tree(T11, void, T12),\nT2 -> T11" }, { "from": 62, "to": 86, "label": "EVAL-BACKTRACK" }, { "from": 64, "to": 102, "label": "EVAL with clause\nminimum(tree(X22, X23, X24), X25) :- minimum(X23, X25).\nand substitutionX22 -> T21,\nX23 -> T25,\nX24 -> T23,\nT1 -> tree(T21, T25, T23),\nT2 -> T24,\nX25 -> T24,\nT22 -> T25" }, { "from": 64, "to": 103, "label": "EVAL-BACKTRACK" }, { "from": 85, "to": 92, "label": "SUCCESS" }, { "from": 102, "to": 12, "label": "INSTANCE with matching:\nT1 -> T25\nT2 -> T24" } ], "type": "Graph" } } ---------------------------------------- (34) Obligation: Rules: f103_out -> f64_out(T2) :|: TRUE f102_out(T24) -> f64_out(T24) :|: TRUE f64_in(x) -> f102_in(x) :|: TRUE f64_in(x1) -> f103_in :|: TRUE f64_out(x2) -> f13_out(x2) :|: TRUE f13_in(x3) -> f62_in(x3) :|: TRUE f62_out(x4) -> f13_out(x4) :|: TRUE f13_in(x5) -> f64_in(x5) :|: TRUE f13_out(x6) -> f12_out(x6) :|: TRUE f12_in(x7) -> f13_in(x7) :|: TRUE f102_in(x8) -> f12_in(x8) :|: TRUE f12_out(x9) -> f102_out(x9) :|: TRUE Start term: f12_in(T2) ---------------------------------------- (35) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f64_in(x) -> f102_in(x) :|: TRUE f13_in(x5) -> f64_in(x5) :|: TRUE f12_in(x7) -> f13_in(x7) :|: TRUE f102_in(x8) -> f12_in(x8) :|: TRUE ---------------------------------------- (36) Obligation: Rules: f64_in(x) -> f102_in(x) :|: TRUE f13_in(x5) -> f64_in(x5) :|: TRUE f12_in(x7) -> f13_in(x7) :|: TRUE f102_in(x8) -> f12_in(x8) :|: TRUE ---------------------------------------- (37) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (38) Obligation: Rules: f12_in(x7:0) -> f12_in(x7:0) :|: TRUE ---------------------------------------- (39) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (40) Obligation: Rules: f12_in(x7:0) -> f12_in(x7:0) :|: TRUE ---------------------------------------- (41) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f12_in(x7:0) -> f12_in(x7:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (42) Obligation: Termination digraph: Nodes: (1) f12_in(x7:0) -> f12_in(x7:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (43) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f12_in(VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (44) Obligation: Rules: f12_in(x7:0) -> f12_in(x7:0) :|: TRUE ---------------------------------------- (45) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, x7:0) -> f(1, x7:0) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1, -8) ---------------------------------------- (46) NO ---------------------------------------- (47) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 9, "program": { "directives": [], "clauses": [ [ "(minimum (tree X (void) X1) X)", null ], [ "(minimum (tree X2 Left X3) X)", "(minimum Left X)" ] ] }, "graph": { "nodes": { "89": { "goal": [ { "clause": 0, "scope": 1, "term": "(minimum T1 T2)" }, { "clause": 1, "scope": 1, "term": "(minimum T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "type": "Nodes", "111": { "goal": [{ "clause": -1, "scope": -1, "term": "(minimum T51 T50)" }], "kb": { "nonunifying": [[ "(minimum T1 T50)", "(minimum (tree X6 (void) X7) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T50"], "free": [ "X6", "X7" ], "exprvars": [] } }, "112": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "113": { "goal": [ { "clause": 0, "scope": 3, "term": "(minimum T51 T50)" }, { "clause": 1, "scope": 3, "term": "(minimum T51 T50)" } ], "kb": { "nonunifying": [[ "(minimum T1 T50)", "(minimum (tree X6 (void) X7) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T50"], "free": [ "X6", "X7" ], "exprvars": [] } }, "114": { "goal": [{ "clause": 0, "scope": 3, "term": "(minimum T51 T50)" }], "kb": { "nonunifying": [[ "(minimum T1 T50)", "(minimum (tree X6 (void) X7) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T50"], "free": [ "X6", "X7" ], "exprvars": [] } }, "115": { "goal": [{ "clause": 1, "scope": 3, "term": "(minimum T51 T50)" }], "kb": { "nonunifying": [[ "(minimum T1 T50)", "(minimum (tree X6 (void) X7) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T50"], "free": [ "X6", "X7" ], "exprvars": [] } }, "90": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(minimum T1 T5)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "91": { "goal": [{ "clause": 1, "scope": 1, "term": "(minimum T1 T2)" }], "kb": { "nonunifying": [[ "(minimum T1 T2)", "(minimum (tree X6 (void) X7) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [ "X6", "X7" ], "exprvars": [] } }, "118": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "119": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "93": { "goal": [{ "clause": 1, "scope": 1, "term": "(minimum T1 T5)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "94": { "goal": [{ "clause": -1, "scope": -1, "term": "(minimum T15 T14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "95": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "96": { "goal": [ { "clause": 0, "scope": 2, "term": "(minimum T15 T14)" }, { "clause": 1, "scope": 2, "term": "(minimum T15 T14)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "97": { "goal": [{ "clause": 0, "scope": 2, "term": "(minimum T15 T14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "98": { "goal": [{ "clause": 1, "scope": 2, "term": "(minimum T15 T14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "120": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "121": { "goal": [{ "clause": -1, "scope": -1, "term": "(minimum T74 T73)" }], "kb": { "nonunifying": [[ "(minimum T1 T73)", "(minimum (tree X6 (void) X7) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T73"], "free": [ "X6", "X7" ], "exprvars": [] } }, "122": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "104": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "105": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "106": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "107": { "goal": [{ "clause": -1, "scope": -1, "term": "(minimum T38 T37)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T37"], "free": [], "exprvars": [] } }, "9": { "goal": [{ "clause": -1, "scope": -1, "term": "(minimum T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "108": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 9, "to": 89, "label": "CASE" }, { "from": 89, "to": 90, "label": "EVAL with clause\nminimum(tree(X6, void, X7), X6).\nand substitutionX6 -> T5,\nX7 -> T6,\nT1 -> tree(T5, void, T6),\nT2 -> T5" }, { "from": 89, "to": 91, "label": "EVAL-BACKTRACK" }, { "from": 90, "to": 93, "label": "SUCCESS" }, { "from": 91, "to": 111, "label": "EVAL with clause\nminimum(tree(X46, X47, X48), X49) :- minimum(X47, X49).\nand substitutionX46 -> T47,\nX47 -> T51,\nX48 -> T49,\nT1 -> tree(T47, T51, T49),\nT2 -> T50,\nX49 -> T50,\nT48 -> T51" }, { "from": 91, "to": 112, "label": "EVAL-BACKTRACK" }, { "from": 93, "to": 94, "label": "EVAL with clause\nminimum(tree(X12, X13, X14), X15) :- minimum(X13, X15).\nand substitutionX12 -> T11,\nX13 -> T15,\nX14 -> T13,\nT1 -> tree(T11, T15, T13),\nT5 -> T14,\nX15 -> T14,\nT12 -> T15" }, { "from": 93, "to": 95, "label": "EVAL-BACKTRACK" }, { "from": 94, "to": 96, "label": "CASE" }, { "from": 96, "to": 97, "label": "PARALLEL" }, { "from": 96, "to": 98, "label": "PARALLEL" }, { "from": 97, "to": 104, "label": "EVAL with clause\nminimum(tree(X24, void, X25), X24).\nand substitutionX24 -> T24,\nX25 -> T25,\nT15 -> tree(T24, void, T25),\nT14 -> T24" }, { "from": 97, "to": 105, "label": "EVAL-BACKTRACK" }, { "from": 98, "to": 107, "label": "EVAL with clause\nminimum(tree(X34, X35, X36), X37) :- minimum(X35, X37).\nand substitutionX34 -> T34,\nX35 -> T38,\nX36 -> T36,\nT15 -> tree(T34, T38, T36),\nT14 -> T37,\nX37 -> T37,\nT35 -> T38" }, { "from": 98, "to": 108, "label": "EVAL-BACKTRACK" }, { "from": 104, "to": 106, "label": "SUCCESS" }, { "from": 107, "to": 9, "label": "INSTANCE with matching:\nT1 -> T38\nT2 -> T37" }, { "from": 111, "to": 113, "label": "CASE" }, { "from": 113, "to": 114, "label": "PARALLEL" }, { "from": 113, "to": 115, "label": "PARALLEL" }, { "from": 114, "to": 118, "label": "EVAL with clause\nminimum(tree(X58, void, X59), X58).\nand substitutionX58 -> T60,\nX59 -> T61,\nT51 -> tree(T60, void, T61),\nT50 -> T60" }, { "from": 114, "to": 119, "label": "EVAL-BACKTRACK" }, { "from": 115, "to": 121, "label": "EVAL with clause\nminimum(tree(X68, X69, X70), X71) :- minimum(X69, X71).\nand substitutionX68 -> T70,\nX69 -> T74,\nX70 -> T72,\nT51 -> tree(T70, T74, T72),\nT50 -> T73,\nX71 -> T73,\nT71 -> T74" }, { "from": 115, "to": 122, "label": "EVAL-BACKTRACK" }, { "from": 118, "to": 120, "label": "SUCCESS" }, { "from": 121, "to": 9, "label": "INSTANCE with matching:\nT1 -> T74\nT2 -> T73" } ], "type": "Graph" } } ---------------------------------------- (48) Obligation: Triples: minimumA(tree(X1, tree(X2, X3, X4), X5), X6) :- minimumA(X3, X6). minimumA(tree(X1, tree(X2, X3, X4), X5), X6) :- minimumA(X3, X6). Clauses: minimumcA(tree(X1, void, X2), X1). minimumcA(tree(X1, tree(X2, void, X3), X4), X2). minimumcA(tree(X1, tree(X2, X3, X4), X5), X6) :- minimumcA(X3, X6). minimumcA(tree(X1, tree(X2, void, X3), X4), X2). minimumcA(tree(X1, tree(X2, X3, X4), X5), X6) :- minimumcA(X3, X6). Afs: minimumA(x1, x2) = minimumA(x2) ---------------------------------------- (49) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: minimumA_in_2: (f,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: MINIMUMA_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> U1_AG(X1, X2, X3, X4, X5, X6, minimumA_in_ag(X3, X6)) MINIMUMA_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> MINIMUMA_IN_AG(X3, X6) R is empty. The argument filtering Pi contains the following mapping: minimumA_in_ag(x1, x2) = minimumA_in_ag(x2) tree(x1, x2, x3) = tree(x2) MINIMUMA_IN_AG(x1, x2) = MINIMUMA_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5, x6, x7) = U1_AG(x6, x7) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (50) Obligation: Pi DP problem: The TRS P consists of the following rules: MINIMUMA_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> U1_AG(X1, X2, X3, X4, X5, X6, minimumA_in_ag(X3, X6)) MINIMUMA_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> MINIMUMA_IN_AG(X3, X6) R is empty. The argument filtering Pi contains the following mapping: minimumA_in_ag(x1, x2) = minimumA_in_ag(x2) tree(x1, x2, x3) = tree(x2) MINIMUMA_IN_AG(x1, x2) = MINIMUMA_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5, x6, x7) = U1_AG(x6, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (51) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (52) Obligation: Pi DP problem: The TRS P consists of the following rules: MINIMUMA_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> MINIMUMA_IN_AG(X3, X6) R is empty. The argument filtering Pi contains the following mapping: tree(x1, x2, x3) = tree(x2) MINIMUMA_IN_AG(x1, x2) = MINIMUMA_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (53) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: MINIMUMA_IN_AG(X6) -> MINIMUMA_IN_AG(X6) R is empty. Q is empty. We have to consider all (P,Q,R)-chains.