MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern p(g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS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, 0 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) QTRSRRRProof [EQUIVALENT, 18 ms] (24) QTRS (25) QTRSRRRProof [EQUIVALENT, 9 ms] (26) QTRS (27) QTRSRRRProof [EQUIVALENT, 5 ms] (28) QTRS (29) Overlay + Local Confluence [EQUIVALENT, 0 ms] (30) QTRS (31) DependencyPairsProof [EQUIVALENT, 0 ms] (32) QDP (33) UsableRulesProof [EQUIVALENT, 0 ms] (34) QDP (35) QReductionProof [EQUIVALENT, 0 ms] (36) QDP (37) PrologToDTProblemTransformerProof [SOUND, 0 ms] (38) TRIPLES (39) TriplesToPiDPProof [SOUND, 0 ms] (40) PiDP (41) DependencyGraphProof [EQUIVALENT, 0 ms] (42) PiDP (43) PiDPToQDPProof [SOUND, 0 ms] (44) QDP (45) PrologToIRSwTTransformerProof [SOUND, 0 ms] (46) IRSwT (47) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (48) IRSwT (49) IntTRSCompressionProof [EQUIVALENT, 0 ms] (50) IRSwT (51) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (52) IRSwT (53) IRSwTTerminationDigraphProof [EQUIVALENT, 7 ms] (54) IRSwT (55) FilterProof [EQUIVALENT, 0 ms] (56) IntTRS (57) IntTRSPeriodicNontermProof [COMPLETE, 7 ms] (58) NO ---------------------------------------- (0) Obligation: Clauses: p(b). p(a) :- p1(X). p1(b). p1(a) :- p1(X). Query: p(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: p_in_1: (b) p1_in_1: (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) ---------------------------------------- (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: P_IN_G(a) -> U1_G(p1_in_a(X)) P_IN_G(a) -> P1_IN_A(X) P1_IN_A(a) -> U2_A(p1_in_a(X)) P1_IN_A(a) -> P1_IN_A(X) The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) P_IN_G(x1) = P_IN_G(x1) U1_G(x1) = U1_G(x1) P1_IN_A(x1) = P1_IN_A U2_A(x1) = U2_A(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_G(a) -> U1_G(p1_in_a(X)) P_IN_G(a) -> P1_IN_A(X) P1_IN_A(a) -> U2_A(p1_in_a(X)) P1_IN_A(a) -> P1_IN_A(X) The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) P_IN_G(x1) = P_IN_G(x1) U1_G(x1) = U1_G(x1) P1_IN_A(x1) = P1_IN_A U2_A(x1) = U2_A(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: P1_IN_A(a) -> P1_IN_A(X) The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) P1_IN_A(x1) = P1_IN_A 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: P1_IN_A(a) -> P1_IN_A(X) R is empty. The argument filtering Pi contains the following mapping: a = a P1_IN_A(x1) = P1_IN_A 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: P1_IN_A -> P1_IN_A 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: p_in_1: (b) p1_in_1: (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g(x1) a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (12) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g(x1) a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) ---------------------------------------- (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: P_IN_G(a) -> U1_G(p1_in_a(X)) P_IN_G(a) -> P1_IN_A(X) P1_IN_A(a) -> U2_A(p1_in_a(X)) P1_IN_A(a) -> P1_IN_A(X) The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g(x1) a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) P_IN_G(x1) = P_IN_G(x1) U1_G(x1) = U1_G(x1) P1_IN_A(x1) = P1_IN_A U2_A(x1) = U2_A(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_G(a) -> U1_G(p1_in_a(X)) P_IN_G(a) -> P1_IN_A(X) P1_IN_A(a) -> U2_A(p1_in_a(X)) P1_IN_A(a) -> P1_IN_A(X) The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g(x1) a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) P_IN_G(x1) = P_IN_G(x1) U1_G(x1) = U1_G(x1) P1_IN_A(x1) = P1_IN_A U2_A(x1) = U2_A(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: P1_IN_A(a) -> P1_IN_A(X) The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g(x1) a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) P1_IN_A(x1) = P1_IN_A 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: P1_IN_A(a) -> P1_IN_A(X) R is empty. The argument filtering Pi contains the following mapping: a = a P1_IN_A(x1) = P1_IN_A 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: P1_IN_A -> P1_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 21, "program": { "directives": [], "clauses": [ [ "(p (b))", null ], [ "(p (a))", "(p1 X)" ], [ "(p1 (b))", null ], [ "(p1 (a))", "(p1 X)" ] ] }, "graph": { "nodes": { "22": { "goal": [ { "clause": 0, "scope": 1, "term": "(p T1)" }, { "clause": 1, "scope": 1, "term": "(p T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "55": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "23": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "24": { "goal": [{ "clause": 1, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "57": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "59": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "104": { "goal": [{ "clause": -1, "scope": -1, "term": "(p1 X4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X4"], "exprvars": [] } }, "82": { "goal": [ { "clause": 2, "scope": 2, "term": "(p1 X2)" }, { "clause": 3, "scope": 2, "term": "(p1 X2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "61": { "goal": [{ "clause": -1, "scope": -1, "term": "(p1 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "83": { "goal": [{ "clause": 2, "scope": 2, "term": "(p1 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "84": { "goal": [{ "clause": 3, "scope": 2, "term": "(p1 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "63": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "85": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "86": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "21": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 21, "to": 22, "label": "CASE" }, { "from": 22, "to": 23, "label": "PARALLEL" }, { "from": 22, "to": 24, "label": "PARALLEL" }, { "from": 23, "to": 55, "label": "EVAL with clause\np(b).\nand substitutionT1 -> b" }, { "from": 23, "to": 57, "label": "EVAL-BACKTRACK" }, { "from": 24, "to": 61, "label": "EVAL with clause\np(a) :- p1(X2).\nand substitutionT1 -> a" }, { "from": 24, "to": 63, "label": "EVAL-BACKTRACK" }, { "from": 55, "to": 59, "label": "SUCCESS" }, { "from": 61, "to": 82, "label": "CASE" }, { "from": 82, "to": 83, "label": "PARALLEL" }, { "from": 82, "to": 84, "label": "PARALLEL" }, { "from": 83, "to": 85, "label": "ONLY EVAL with clause\np1(b).\nand substitutionX2 -> b" }, { "from": 84, "to": 104, "label": "ONLY EVAL with clause\np1(a) :- p1(X4).\nand substitutionX2 -> a" }, { "from": 85, "to": 86, "label": "SUCCESS" }, { "from": 104, "to": 61, "label": "INSTANCE with matching:\nX2 -> X4" } ], "type": "Graph" } } ---------------------------------------- (22) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f21_in(b) -> f21_out1 f21_in(a) -> U1(f61_in, a) U1(f61_out1(X2), a) -> f21_out1 f61_in -> f61_out1(b) f61_in -> U2(f61_in) U2(f61_out1(X4)) -> f61_out1(a) Q is empty. ---------------------------------------- (23) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1, x_2)) = x_1 + 2*x_2 POL(U2(x_1)) = 2*x_1 POL(a) = 0 POL(b) = 0 POL(f21_in(x_1)) = 1 + 2*x_1 POL(f21_out1) = 0 POL(f61_in) = 0 POL(f61_out1(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f21_in(b) -> f21_out1 f21_in(a) -> U1(f61_in, a) ---------------------------------------- (24) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U1(f61_out1(X2), a) -> f21_out1 f61_in -> f61_out1(b) f61_in -> U2(f61_in) U2(f61_out1(X4)) -> f61_out1(a) Q is empty. ---------------------------------------- (25) QTRSRRRProof (EQUIVALENT) Used ordering: U1/2(YES,YES) f61_out1/1(YES) a/0) f21_out1/0) f61_in/0) b/0) U2/1)YES( Quasi precedence: [U1_2, f21_out1] > [f61_out1_1, a] f61_in > b > [f61_out1_1, a] Status: U1_2: [2,1] f61_out1_1: [1] a: multiset status f21_out1: multiset status f61_in: multiset status b: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f61_out1(X2), a) -> f21_out1 f61_in -> f61_out1(b) ---------------------------------------- (26) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f61_in -> U2(f61_in) U2(f61_out1(X4)) -> f61_out1(a) Q is empty. ---------------------------------------- (27) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = 2*x_1 POL(a) = 0 POL(f61_in) = 0 POL(f61_out1(x_1)) = 1 + 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U2(f61_out1(X4)) -> f61_out1(a) ---------------------------------------- (28) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f61_in -> U2(f61_in) Q is empty. ---------------------------------------- (29) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (30) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f61_in -> U2(f61_in) The set Q consists of the following terms: f61_in ---------------------------------------- (31) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: F61_IN -> F61_IN The TRS R consists of the following rules: f61_in -> U2(f61_in) The set Q consists of the following terms: f61_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) 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. ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: F61_IN -> F61_IN R is empty. The set Q consists of the following terms: f61_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) 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]. f61_in ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: F61_IN -> F61_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(p (b))", null ], [ "(p (a))", "(p1 X)" ], [ "(p1 (b))", null ], [ "(p1 (a))", "(p1 X)" ] ] }, "graph": { "nodes": { "78": { "goal": [{ "clause": -1, "scope": -1, "term": "(p1 X3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X3"], "exprvars": [] } }, "47": { "goal": [{ "clause": 1, "scope": 1, "term": "(p (b))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "37": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(p (b))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "48": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "49": { "goal": [{ "clause": -1, "scope": -1, "term": "(p1 X1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X1"], "exprvars": [] } }, "type": "Nodes", "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "4": { "goal": [ { "clause": 0, "scope": 1, "term": "(p T1)" }, { "clause": 1, "scope": 1, "term": "(p T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "50": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "51": { "goal": [ { "clause": 2, "scope": 2, "term": "(p1 X1)" }, { "clause": 3, "scope": 2, "term": "(p1 X1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X1"], "exprvars": [] } }, "41": { "goal": [{ "clause": 1, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [[ "(p T1)", "(p (b))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "52": { "goal": [{ "clause": 2, "scope": 2, "term": "(p1 X1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X1"], "exprvars": [] } }, "53": { "goal": [{ "clause": 3, "scope": 2, "term": "(p1 X1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X1"], "exprvars": [] } }, "54": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "65": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 4, "label": "CASE" }, { "from": 4, "to": 37, "label": "EVAL with clause\np(b).\nand substitutionT1 -> b" }, { "from": 4, "to": 41, "label": "EVAL-BACKTRACK" }, { "from": 37, "to": 47, "label": "SUCCESS" }, { "from": 41, "to": 49, "label": "EVAL with clause\np(a) :- p1(X1).\nand substitutionT1 -> a" }, { "from": 41, "to": 50, "label": "EVAL-BACKTRACK" }, { "from": 47, "to": 48, "label": "BACKTRACK\nfor clause: p(a) :- p1(X)because of non-unification" }, { "from": 49, "to": 51, "label": "CASE" }, { "from": 51, "to": 52, "label": "PARALLEL" }, { "from": 51, "to": 53, "label": "PARALLEL" }, { "from": 52, "to": 54, "label": "ONLY EVAL with clause\np1(b).\nand substitutionX1 -> b" }, { "from": 53, "to": 78, "label": "ONLY EVAL with clause\np1(a) :- p1(X3).\nand substitutionX1 -> a" }, { "from": 54, "to": 65, "label": "SUCCESS" }, { "from": 78, "to": 49, "label": "INSTANCE with matching:\nX1 -> X3" } ], "type": "Graph" } } ---------------------------------------- (38) Obligation: Triples: p1A(a) :- p1A(X1). pB(a) :- p1A(X1). Clauses: p1cA(b). p1cA(a) :- p1cA(X1). Afs: pB(x1) = pB(x1) ---------------------------------------- (39) 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_1: (b) p1A_in_1: (f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: PB_IN_G(a) -> U2_G(p1A_in_a(X1)) PB_IN_G(a) -> P1A_IN_A(X1) P1A_IN_A(a) -> U1_A(p1A_in_a(X1)) P1A_IN_A(a) -> P1A_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: a = a p1A_in_a(x1) = p1A_in_a PB_IN_G(x1) = PB_IN_G(x1) U2_G(x1) = U2_G(x1) P1A_IN_A(x1) = P1A_IN_A U1_A(x1) = U1_A(x1) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (40) Obligation: Pi DP problem: The TRS P consists of the following rules: PB_IN_G(a) -> U2_G(p1A_in_a(X1)) PB_IN_G(a) -> P1A_IN_A(X1) P1A_IN_A(a) -> U1_A(p1A_in_a(X1)) P1A_IN_A(a) -> P1A_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: a = a p1A_in_a(x1) = p1A_in_a PB_IN_G(x1) = PB_IN_G(x1) U2_G(x1) = U2_G(x1) P1A_IN_A(x1) = P1A_IN_A U1_A(x1) = U1_A(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (41) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (42) Obligation: Pi DP problem: The TRS P consists of the following rules: P1A_IN_A(a) -> P1A_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: a = a P1A_IN_A(x1) = P1A_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (43) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: P1A_IN_A -> P1A_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (45) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 17, "program": { "directives": [], "clauses": [ [ "(p (b))", null ], [ "(p (a))", "(p1 X)" ], [ "(p1 (b))", null ], [ "(p1 (a))", "(p1 X)" ] ] }, "graph": { "nodes": { "99": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "56": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "58": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "17": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "18": { "goal": [ { "clause": 0, "scope": 1, "term": "(p T1)" }, { "clause": 1, "scope": 1, "term": "(p T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "19": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "type": "Nodes", "100": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "103": { "goal": [{ "clause": -1, "scope": -1, "term": "(p1 X4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X4"], "exprvars": [] } }, "90": { "goal": [ { "clause": 2, "scope": 2, "term": "(p1 X2)" }, { "clause": 3, "scope": 2, "term": "(p1 X2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "60": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "93": { "goal": [{ "clause": 2, "scope": 2, "term": "(p1 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "94": { "goal": [{ "clause": 3, "scope": 2, "term": "(p1 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "62": { "goal": [{ "clause": -1, "scope": -1, "term": "(p1 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "20": { "goal": [{ "clause": 1, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "64": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 17, "to": 18, "label": "CASE" }, { "from": 18, "to": 19, "label": "PARALLEL" }, { "from": 18, "to": 20, "label": "PARALLEL" }, { "from": 19, "to": 56, "label": "EVAL with clause\np(b).\nand substitutionT1 -> b" }, { "from": 19, "to": 58, "label": "EVAL-BACKTRACK" }, { "from": 20, "to": 62, "label": "EVAL with clause\np(a) :- p1(X2).\nand substitutionT1 -> a" }, { "from": 20, "to": 64, "label": "EVAL-BACKTRACK" }, { "from": 56, "to": 60, "label": "SUCCESS" }, { "from": 62, "to": 90, "label": "CASE" }, { "from": 90, "to": 93, "label": "PARALLEL" }, { "from": 90, "to": 94, "label": "PARALLEL" }, { "from": 93, "to": 99, "label": "ONLY EVAL with clause\np1(b).\nand substitutionX2 -> b" }, { "from": 94, "to": 103, "label": "ONLY EVAL with clause\np1(a) :- p1(X4).\nand substitutionX2 -> a" }, { "from": 99, "to": 100, "label": "SUCCESS" }, { "from": 103, "to": 62, "label": "INSTANCE with matching:\nX2 -> X4" } ], "type": "Graph" } } ---------------------------------------- (46) Obligation: Rules: f94_in -> f103_in :|: TRUE f103_out -> f94_out :|: TRUE f90_in -> f94_in :|: TRUE f93_out -> f90_out :|: TRUE f94_out -> f90_out :|: TRUE f90_in -> f93_in :|: TRUE f103_in -> f62_in :|: TRUE f62_out -> f103_out :|: TRUE f62_in -> f90_in :|: TRUE f90_out -> f62_out :|: TRUE f18_out(T1) -> f17_out(T1) :|: TRUE f17_in(x) -> f18_in(x) :|: TRUE f20_out(x1) -> f18_out(x1) :|: TRUE f18_in(x2) -> f19_in(x2) :|: TRUE f19_out(x3) -> f18_out(x3) :|: TRUE f18_in(x4) -> f20_in(x4) :|: TRUE f20_in(x5) -> f64_in :|: TRUE f64_out -> f20_out(x6) :|: TRUE f62_out -> f20_out(a) :|: TRUE f20_in(a) -> f62_in :|: TRUE Start term: f17_in(T1) ---------------------------------------- (47) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f94_in -> f103_in :|: TRUE f90_in -> f94_in :|: TRUE f103_in -> f62_in :|: TRUE f62_in -> f90_in :|: TRUE ---------------------------------------- (48) Obligation: Rules: f94_in -> f103_in :|: TRUE f90_in -> f94_in :|: TRUE f103_in -> f62_in :|: TRUE f62_in -> f90_in :|: TRUE ---------------------------------------- (49) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (50) Obligation: Rules: f90_in -> f90_in :|: TRUE ---------------------------------------- (51) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (52) Obligation: Rules: f90_in -> f90_in :|: TRUE ---------------------------------------- (53) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f90_in -> f90_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (54) Obligation: Termination digraph: Nodes: (1) f90_in -> f90_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (55) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f90_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (56) Obligation: Rules: f90_in -> f90_in :|: TRUE ---------------------------------------- (57) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (58) NO