MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern goal(g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2)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) PrologToDTProblemTransformerProof [SOUND, 0 ms] (12) TRIPLES (13) TriplesToPiDPProof [SOUND, 2 ms] (14) PiDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) PrologToTRSTransformerProof [SOUND, 0 ms] (20) QTRS (21) QTRSRRRProof [EQUIVALENT, 89 ms] (22) QTRS (23) QTRSRRRProof [EQUIVALENT, 0 ms] (24) QTRS (25) QTRSRRRProof [EQUIVALENT, 0 ms] (26) QTRS (27) Overlay + Local Confluence [EQUIVALENT, 0 ms] (28) QTRS (29) DependencyPairsProof [EQUIVALENT, 0 ms] (30) QDP (31) UsableRulesProof [EQUIVALENT, 0 ms] (32) QDP (33) QReductionProof [EQUIVALENT, 0 ms] (34) QDP (35) PrologToIRSwTTransformerProof [SOUND, 0 ms] (36) IRSwT (37) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (38) IRSwT (39) IntTRSCompressionProof [EQUIVALENT, 20 ms] (40) IRSwT (41) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (42) IRSwT (43) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms] (44) IRSwT (45) FilterProof [EQUIVALENT, 0 ms] (46) IntTRS (47) IntTRSNonPeriodicNontermProof [COMPLETE, 8 ms] (48) NO (49) PrologToPiTRSProof [SOUND, 0 ms] (50) PiTRS (51) DependencyPairsProof [EQUIVALENT, 0 ms] (52) PiDP (53) DependencyGraphProof [EQUIVALENT, 1 ms] (54) PiDP (55) UsableRulesProof [EQUIVALENT, 0 ms] (56) PiDP (57) PiDPToQDPProof [SOUND, 0 ms] (58) QDP ---------------------------------------- (0) Obligation: Clauses: p(a). p(X) :- p(Y). q(b). goal(X) :- ','(p(X), q(X)). Query: goal(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: goal_in_1: (b) p_in_1: (b) (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x1, x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g(x1) goal_out_g(x1) = goal_out_g(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: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x1, x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g(x1) goal_out_g(x1) = goal_out_g(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: GOAL_IN_G(X) -> U2_G(X, p_in_g(X)) GOAL_IN_G(X) -> P_IN_G(X) P_IN_G(X) -> U1_G(X, p_in_a(Y)) P_IN_G(X) -> P_IN_A(Y) P_IN_A(X) -> U1_A(X, p_in_a(Y)) P_IN_A(X) -> P_IN_A(Y) U2_G(X, p_out_g(X)) -> U3_G(X, q_in_g(X)) U2_G(X, p_out_g(X)) -> Q_IN_G(X) The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x1, x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g(x1) goal_out_g(x1) = goal_out_g(x1) GOAL_IN_G(x1) = GOAL_IN_G(x1) U2_G(x1, x2) = U2_G(x1, x2) P_IN_G(x1) = P_IN_G(x1) U1_G(x1, x2) = U1_G(x1, x2) P_IN_A(x1) = P_IN_A U1_A(x1, x2) = U1_A(x2) U3_G(x1, x2) = U3_G(x1, x2) Q_IN_G(x1) = Q_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: GOAL_IN_G(X) -> U2_G(X, p_in_g(X)) GOAL_IN_G(X) -> P_IN_G(X) P_IN_G(X) -> U1_G(X, p_in_a(Y)) P_IN_G(X) -> P_IN_A(Y) P_IN_A(X) -> U1_A(X, p_in_a(Y)) P_IN_A(X) -> P_IN_A(Y) U2_G(X, p_out_g(X)) -> U3_G(X, q_in_g(X)) U2_G(X, p_out_g(X)) -> Q_IN_G(X) The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x1, x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g(x1) goal_out_g(x1) = goal_out_g(x1) GOAL_IN_G(x1) = GOAL_IN_G(x1) U2_G(x1, x2) = U2_G(x1, x2) P_IN_G(x1) = P_IN_G(x1) U1_G(x1, x2) = U1_G(x1, x2) P_IN_A(x1) = P_IN_A U1_A(x1, x2) = U1_A(x2) U3_G(x1, x2) = U3_G(x1, x2) Q_IN_G(x1) = Q_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_A(X) -> P_IN_A(Y) The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x1, x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g(x1) goal_out_g(x1) = goal_out_g(x1) P_IN_A(x1) = P_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: P_IN_A(X) -> P_IN_A(Y) R is empty. The argument filtering Pi contains the following mapping: P_IN_A(x1) = P_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: P_IN_A -> P_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (11) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 2, "program": { "directives": [], "clauses": [ [ "(p (a))", null ], [ "(p X)", "(p Y)" ], [ "(q (b))", null ], [ "(goal X)", "(',' (p X) (q X))" ] ] }, "graph": { "nodes": { "11": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T3) (q T3))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "22": { "goal": [{ "clause": 1, "scope": 2, "term": "(',' (p T3) (q T3))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "33": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "12": { "goal": [ { "clause": 0, "scope": 2, "term": "(',' (p T3) (q T3))" }, { "clause": 1, "scope": 2, "term": "(',' (p T3) (q T3))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "23": { "goal": [{ "clause": -1, "scope": -1, "term": "(q (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "34": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "24": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "57": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "25": { "goal": [{ "clause": 2, "scope": 3, "term": "(q (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "26": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "27": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p X7) (q T6))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": ["X7"], "exprvars": [] } }, "38": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "28": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X7"], "exprvars": [] } }, "29": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [], "exprvars": [] } }, "type": "Nodes", "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "5": { "goal": [{ "clause": 3, "scope": 1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "30": { "goal": [ { "clause": 0, "scope": 4, "term": "(p X7)" }, { "clause": 1, "scope": 4, "term": "(p X7)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X7"], "exprvars": [] } }, "20": { "goal": [{ "clause": 0, "scope": 2, "term": "(',' (p T3) (q T3))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "31": { "goal": [{ "clause": 0, "scope": 4, "term": "(p X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X7"], "exprvars": [] } }, "64": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "32": { "goal": [{ "clause": 1, "scope": 4, "term": "(p X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X7"], "exprvars": [] } }, "54": { "goal": [{ "clause": 2, "scope": 5, "term": "(q T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [], "exprvars": [] } }, "65": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 5, "label": "CASE" }, { "from": 5, "to": 11, "label": "ONLY EVAL with clause\ngoal(X2) :- ','(p(X2), q(X2)).\nand substitutionT1 -> T3,\nX2 -> T3" }, { "from": 11, "to": 12, "label": "CASE" }, { "from": 12, "to": 20, "label": "PARALLEL" }, { "from": 12, "to": 22, "label": "PARALLEL" }, { "from": 20, "to": 23, "label": "EVAL with clause\np(a).\nand substitutionT3 -> a" }, { "from": 20, "to": 24, "label": "EVAL-BACKTRACK" }, { "from": 22, "to": 27, "label": "ONLY EVAL with clause\np(X6) :- p(X7).\nand substitutionT3 -> T6,\nX6 -> T6" }, { "from": 23, "to": 25, "label": "CASE" }, { "from": 25, "to": 26, "label": "BACKTRACK\nfor clause: q(b)because of non-unification" }, { "from": 27, "to": 28, "label": "SPLIT 1" }, { "from": 27, "to": 29, "label": "SPLIT 2\nreplacements:X7 -> T7" }, { "from": 28, "to": 30, "label": "CASE" }, { "from": 29, "to": 54, "label": "CASE" }, { "from": 30, "to": 31, "label": "PARALLEL" }, { "from": 30, "to": 32, "label": "PARALLEL" }, { "from": 31, "to": 33, "label": "ONLY EVAL with clause\np(a).\nand substitutionX7 -> a" }, { "from": 32, "to": 38, "label": "ONLY EVAL with clause\np(X13) :- p(X14).\nand substitutionX7 -> X15,\nX13 -> X15" }, { "from": 33, "to": 34, "label": "SUCCESS" }, { "from": 38, "to": 28, "label": "INSTANCE with matching:\nX7 -> X14" }, { "from": 54, "to": 57, "label": "EVAL with clause\nq(b).\nand substitutionT6 -> b" }, { "from": 54, "to": 64, "label": "EVAL-BACKTRACK" }, { "from": 57, "to": 65, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (12) Obligation: Triples: pA(X1) :- pA(X2). goalB(X1) :- pA(X2). Clauses: pcA(a). pcA(X1) :- pcA(X2). Afs: goalB(x1) = goalB(x1) ---------------------------------------- (13) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: goalB_in_1: (b) pA_in_1: (f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: GOALB_IN_G(X1) -> U2_G(X1, pA_in_a(X2)) GOALB_IN_G(X1) -> PA_IN_A(X2) PA_IN_A(X1) -> U1_A(X1, pA_in_a(X2)) PA_IN_A(X1) -> PA_IN_A(X2) R is empty. The argument filtering Pi contains the following mapping: pA_in_a(x1) = pA_in_a GOALB_IN_G(x1) = GOALB_IN_G(x1) U2_G(x1, x2) = U2_G(x1, x2) PA_IN_A(x1) = PA_IN_A U1_A(x1, x2) = U1_A(x2) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: GOALB_IN_G(X1) -> U2_G(X1, pA_in_a(X2)) GOALB_IN_G(X1) -> PA_IN_A(X2) PA_IN_A(X1) -> U1_A(X1, pA_in_a(X2)) PA_IN_A(X1) -> PA_IN_A(X2) R is empty. The argument filtering Pi contains the following mapping: pA_in_a(x1) = pA_in_a GOALB_IN_G(x1) = GOALB_IN_G(x1) U2_G(x1, x2) = U2_G(x1, x2) PA_IN_A(x1) = PA_IN_A U1_A(x1, x2) = U1_A(x2) 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: PA_IN_A(X1) -> PA_IN_A(X2) R is empty. The argument filtering Pi contains the following mapping: PA_IN_A(x1) = PA_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: PA_IN_A -> PA_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 3, "program": { "directives": [], "clauses": [ [ "(p (a))", null ], [ "(p X)", "(p Y)" ], [ "(q (b))", null ], [ "(goal X)", "(',' (p X) (q X))" ] ] }, "graph": { "nodes": { "type": "Nodes", "161": { "goal": [{ "clause": 2, "scope": 4, "term": "(q T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "140": { "goal": [{ "clause": 1, "scope": 3, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "162": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "163": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "120": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "164": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "122": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "114": { "goal": [ { "clause": 0, "scope": 2, "term": "(p T4)" }, { "clause": 1, "scope": 2, "term": "(p T4)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "136": { "goal": [ { "clause": 0, "scope": 3, "term": "(p X8)" }, { "clause": 1, "scope": 3, "term": "(p X8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "147": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "158": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X15)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "115": { "goal": [{ "clause": 0, "scope": 2, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "6": { "goal": [{ "clause": 3, "scope": 1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "116": { "goal": [{ "clause": 1, "scope": 2, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "149": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "139": { "goal": [{ "clause": 0, "scope": 3, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "8": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T4) (q T4))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "129": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "9": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "119": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "10": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 6, "label": "CASE" }, { "from": 6, "to": 8, "label": "ONLY EVAL with clause\ngoal(X3) :- ','(p(X3), q(X3)).\nand substitutionT1 -> T4,\nX3 -> T4" }, { "from": 8, "to": 9, "label": "SPLIT 1" }, { "from": 8, "to": 10, "label": "SPLIT 2\nnew knowledge:\nT4 is ground" }, { "from": 9, "to": 114, "label": "CASE" }, { "from": 10, "to": 161, "label": "CASE" }, { "from": 114, "to": 115, "label": "PARALLEL" }, { "from": 114, "to": 116, "label": "PARALLEL" }, { "from": 115, "to": 119, "label": "EVAL with clause\np(a).\nand substitutionT4 -> a" }, { "from": 115, "to": 120, "label": "EVAL-BACKTRACK" }, { "from": 116, "to": 129, "label": "ONLY EVAL with clause\np(X7) :- p(X8).\nand substitutionT4 -> T7,\nX7 -> T7" }, { "from": 119, "to": 122, "label": "SUCCESS" }, { "from": 129, "to": 136, "label": "CASE" }, { "from": 136, "to": 139, "label": "PARALLEL" }, { "from": 136, "to": 140, "label": "PARALLEL" }, { "from": 139, "to": 147, "label": "ONLY EVAL with clause\np(a).\nand substitutionX8 -> a" }, { "from": 140, "to": 158, "label": "ONLY EVAL with clause\np(X14) :- p(X15).\nand substitutionX8 -> X16,\nX14 -> X16" }, { "from": 147, "to": 149, "label": "SUCCESS" }, { "from": 158, "to": 129, "label": "INSTANCE with matching:\nX8 -> X15" }, { "from": 161, "to": 162, "label": "EVAL with clause\nq(b).\nand substitutionT4 -> b" }, { "from": 161, "to": 163, "label": "EVAL-BACKTRACK" }, { "from": 162, "to": 164, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (20) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f3_in(T4) -> U1(f8_in(T4), T4) U1(f8_out1, T4) -> f3_out1 f129_in -> f129_out1 f129_in -> U2(f129_in) U2(f129_out1) -> f129_out1 f9_in(a) -> f9_out1 f9_in(T7) -> U3(f129_in, T7) U3(f129_out1, T7) -> f9_out1 f10_in(b) -> f10_out1 f8_in(T4) -> U4(f9_in(T4), T4) U4(f9_out1, T4) -> U5(f10_in(T4), T4) U5(f10_out1, T4) -> f8_out1 Q is empty. ---------------------------------------- (21) QTRSRRRProof (EQUIVALENT) Used ordering: f3_in/1(YES) U1/2(YES,YES) f8_in/1(YES) f8_out1/0) f3_out1/0) f129_in/0) f129_out1/0) U2/1)YES( f9_in/1(YES) a/0) f9_out1/0) U3/2(YES,YES) f10_in/1(YES) b/0) f10_out1/0) U4/2(YES,YES) U5/2(YES,YES) Quasi precedence: f3_in_1 > U1_2 > [f3_out1, a, f9_out1, f10_in_1] f3_in_1 > f8_in_1 > f9_in_1 > [f129_in, f129_out1] > [f3_out1, a, f9_out1, f10_in_1] f3_in_1 > f8_in_1 > f9_in_1 > U3_2 > [f3_out1, a, f9_out1, f10_in_1] f3_in_1 > f8_in_1 > U4_2 > U5_2 > [f3_out1, a, f9_out1, f10_in_1] b > f10_out1 > f8_out1 > [f3_out1, a, f9_out1, f10_in_1] Status: f3_in_1: multiset status U1_2: multiset status f8_in_1: [1] f8_out1: multiset status f3_out1: multiset status f129_in: multiset status f129_out1: multiset status f9_in_1: multiset status a: multiset status f9_out1: multiset status U3_2: multiset status f10_in_1: multiset status b: multiset status f10_out1: multiset status U4_2: multiset status U5_2: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f3_in(T4) -> U1(f8_in(T4), T4) U1(f8_out1, T4) -> f3_out1 f9_in(a) -> f9_out1 f9_in(T7) -> U3(f129_in, T7) U3(f129_out1, T7) -> f9_out1 f10_in(b) -> f10_out1 f8_in(T4) -> U4(f9_in(T4), T4) U4(f9_out1, T4) -> U5(f10_in(T4), T4) U5(f10_out1, T4) -> f8_out1 ---------------------------------------- (22) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f129_in -> f129_out1 f129_in -> U2(f129_in) U2(f129_out1) -> f129_out1 Q is empty. ---------------------------------------- (23) QTRSRRRProof (EQUIVALENT) Used ordering: f129_in/0) f129_out1/0) U2/1)YES( Quasi precedence: f129_in > f129_out1 Status: f129_in: multiset status f129_out1: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f129_in -> f129_out1 ---------------------------------------- (24) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f129_in -> U2(f129_in) U2(f129_out1) -> f129_out1 Q is empty. ---------------------------------------- (25) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = 2*x_1 POL(f129_in) = 0 POL(f129_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U2(f129_out1) -> f129_out1 ---------------------------------------- (26) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f129_in -> U2(f129_in) Q is empty. ---------------------------------------- (27) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (28) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f129_in -> U2(f129_in) The set Q consists of the following terms: f129_in ---------------------------------------- (29) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: F129_IN -> F129_IN The TRS R consists of the following rules: f129_in -> U2(f129_in) The set Q consists of the following terms: f129_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) 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. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: F129_IN -> F129_IN R is empty. The set Q consists of the following terms: f129_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) 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]. f129_in ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: F129_IN -> F129_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 1, "program": { "directives": [], "clauses": [ [ "(p (a))", null ], [ "(p X)", "(p Y)" ], [ "(q (b))", null ], [ "(goal X)", "(',' (p X) (q X))" ] ] }, "graph": { "nodes": { "44": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "45": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "46": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "47": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "48": { "goal": [ { "clause": 0, "scope": 3, "term": "(p X8)" }, { "clause": 1, "scope": 3, "term": "(p X8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "49": { "goal": [{ "clause": 0, "scope": 3, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "39": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "type": "Nodes", "165": { "goal": [{ "clause": 2, "scope": 4, "term": "(q T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "166": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "167": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "168": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "4": { "goal": [{ "clause": 3, "scope": 1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "7": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T4) (q T4))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "50": { "goal": [{ "clause": 1, "scope": 3, "term": "(p X8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X8"], "exprvars": [] } }, "40": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "51": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "41": { "goal": [ { "clause": 0, "scope": 2, "term": "(p T4)" }, { "clause": 1, "scope": 2, "term": "(p T4)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "52": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "42": { "goal": [{ "clause": 0, "scope": 2, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "53": { "goal": [{ "clause": -1, "scope": -1, "term": "(p X15)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "43": { "goal": [{ "clause": 1, "scope": 2, "term": "(p T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 4, "label": "CASE" }, { "from": 4, "to": 7, "label": "ONLY EVAL with clause\ngoal(X3) :- ','(p(X3), q(X3)).\nand substitutionT1 -> T4,\nX3 -> T4" }, { "from": 7, "to": 39, "label": "SPLIT 1" }, { "from": 7, "to": 40, "label": "SPLIT 2\nnew knowledge:\nT4 is ground" }, { "from": 39, "to": 41, "label": "CASE" }, { "from": 40, "to": 165, "label": "CASE" }, { "from": 41, "to": 42, "label": "PARALLEL" }, { "from": 41, "to": 43, "label": "PARALLEL" }, { "from": 42, "to": 44, "label": "EVAL with clause\np(a).\nand substitutionT4 -> a" }, { "from": 42, "to": 45, "label": "EVAL-BACKTRACK" }, { "from": 43, "to": 47, "label": "ONLY EVAL with clause\np(X7) :- p(X8).\nand substitutionT4 -> T7,\nX7 -> T7" }, { "from": 44, "to": 46, "label": "SUCCESS" }, { "from": 47, "to": 48, "label": "CASE" }, { "from": 48, "to": 49, "label": "PARALLEL" }, { "from": 48, "to": 50, "label": "PARALLEL" }, { "from": 49, "to": 51, "label": "ONLY EVAL with clause\np(a).\nand substitutionX8 -> a" }, { "from": 50, "to": 53, "label": "ONLY EVAL with clause\np(X14) :- p(X15).\nand substitutionX8 -> X16,\nX14 -> X16" }, { "from": 51, "to": 52, "label": "SUCCESS" }, { "from": 53, "to": 47, "label": "INSTANCE with matching:\nX8 -> X15" }, { "from": 165, "to": 166, "label": "EVAL with clause\nq(b).\nand substitutionT4 -> b" }, { "from": 165, "to": 167, "label": "EVAL-BACKTRACK" }, { "from": 166, "to": 168, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (36) Obligation: Rules: f53_in -> f47_in :|: TRUE f47_out -> f53_out :|: TRUE f53_out -> f50_out :|: TRUE f50_in -> f53_in :|: TRUE f48_in -> f49_in :|: TRUE f50_out -> f48_out :|: TRUE f49_out -> f48_out :|: TRUE f48_in -> f50_in :|: TRUE f48_out -> f47_out :|: TRUE f47_in -> f48_in :|: TRUE f4_out(T1) -> f1_out(T1) :|: TRUE f1_in(x) -> f4_in(x) :|: TRUE f7_out(T4) -> f4_out(T4) :|: TRUE f4_in(x1) -> f7_in(x1) :|: TRUE f40_out(x2) -> f7_out(x2) :|: TRUE f7_in(x3) -> f39_in(x3) :|: TRUE f39_out(x4) -> f40_in(x4) :|: TRUE f39_in(x5) -> f41_in(x5) :|: TRUE f41_out(x6) -> f39_out(x6) :|: TRUE f42_out(x7) -> f41_out(x7) :|: TRUE f41_in(x8) -> f43_in(x8) :|: TRUE f41_in(x9) -> f42_in(x9) :|: TRUE f43_out(x10) -> f41_out(x10) :|: TRUE f43_in(T7) -> f47_in :|: TRUE f47_out -> f43_out(x11) :|: TRUE Start term: f1_in(T1) ---------------------------------------- (37) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f53_in -> f47_in :|: TRUE f50_in -> f53_in :|: TRUE f48_in -> f50_in :|: TRUE f47_in -> f48_in :|: TRUE ---------------------------------------- (38) Obligation: Rules: f53_in -> f47_in :|: TRUE f50_in -> f53_in :|: TRUE f48_in -> f50_in :|: TRUE f47_in -> f48_in :|: TRUE ---------------------------------------- (39) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (40) Obligation: Rules: f48_in -> f48_in :|: TRUE ---------------------------------------- (41) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (42) Obligation: Rules: f48_in -> f48_in :|: TRUE ---------------------------------------- (43) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f48_in -> f48_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (44) Obligation: Termination digraph: Nodes: (1) f48_in -> f48_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (45) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f48_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (46) Obligation: Rules: f48_in -> f48_in :|: TRUE ---------------------------------------- (47) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Proved unsatisfiability of the following formula, indicating that the system is never left after entering: ((run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T)) Proved satisfiability of the following formula, indicating that the system is entered at least once: (run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) ---------------------------------------- (48) NO ---------------------------------------- (49) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: goal_in_1: (b) p_in_1: (b) (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g goal_out_g(x1) = goal_out_g Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (50) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g goal_out_g(x1) = goal_out_g ---------------------------------------- (51) 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: GOAL_IN_G(X) -> U2_G(X, p_in_g(X)) GOAL_IN_G(X) -> P_IN_G(X) P_IN_G(X) -> U1_G(X, p_in_a(Y)) P_IN_G(X) -> P_IN_A(Y) P_IN_A(X) -> U1_A(X, p_in_a(Y)) P_IN_A(X) -> P_IN_A(Y) U2_G(X, p_out_g(X)) -> U3_G(X, q_in_g(X)) U2_G(X, p_out_g(X)) -> Q_IN_G(X) The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g goal_out_g(x1) = goal_out_g GOAL_IN_G(x1) = GOAL_IN_G(x1) U2_G(x1, x2) = U2_G(x1, x2) P_IN_G(x1) = P_IN_G(x1) U1_G(x1, x2) = U1_G(x2) P_IN_A(x1) = P_IN_A U1_A(x1, x2) = U1_A(x2) U3_G(x1, x2) = U3_G(x2) Q_IN_G(x1) = Q_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (52) Obligation: Pi DP problem: The TRS P consists of the following rules: GOAL_IN_G(X) -> U2_G(X, p_in_g(X)) GOAL_IN_G(X) -> P_IN_G(X) P_IN_G(X) -> U1_G(X, p_in_a(Y)) P_IN_G(X) -> P_IN_A(Y) P_IN_A(X) -> U1_A(X, p_in_a(Y)) P_IN_A(X) -> P_IN_A(Y) U2_G(X, p_out_g(X)) -> U3_G(X, q_in_g(X)) U2_G(X, p_out_g(X)) -> Q_IN_G(X) The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g goal_out_g(x1) = goal_out_g GOAL_IN_G(x1) = GOAL_IN_G(x1) U2_G(x1, x2) = U2_G(x1, x2) P_IN_G(x1) = P_IN_G(x1) U1_G(x1, x2) = U1_G(x2) P_IN_A(x1) = P_IN_A U1_A(x1, x2) = U1_A(x2) U3_G(x1, x2) = U3_G(x2) Q_IN_G(x1) = Q_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (53) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes. ---------------------------------------- (54) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_A(X) -> P_IN_A(Y) The TRS R consists of the following rules: goal_in_g(X) -> U2_g(X, p_in_g(X)) p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_a(Y)) p_in_a(a) -> p_out_a(a) p_in_a(X) -> U1_a(X, p_in_a(Y)) U1_a(X, p_out_a(Y)) -> p_out_a(X) U1_g(X, p_out_a(Y)) -> p_out_g(X) U2_g(X, p_out_g(X)) -> U3_g(X, q_in_g(X)) q_in_g(b) -> q_out_g(b) U3_g(X, q_out_g(X)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U2_g(x1, x2) = U2_g(x1, x2) p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a U1_a(x1, x2) = U1_a(x2) U3_g(x1, x2) = U3_g(x2) q_in_g(x1) = q_in_g(x1) b = b q_out_g(x1) = q_out_g goal_out_g(x1) = goal_out_g P_IN_A(x1) = P_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (55) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (56) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_A(X) -> P_IN_A(Y) R is empty. The argument filtering Pi contains the following mapping: P_IN_A(x1) = P_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (57) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: P_IN_A -> P_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains.