/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern desc(g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2)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 [EQUIVALENT, 0 ms] (10) QDP (11) TransformationProof [EQUIVALENT, 0 ms] (12) QDP (13) TransformationProof [EQUIVALENT, 0 ms] (14) QDP (15) NonTerminationLoopProof [COMPLETE, 0 ms] (16) NO (17) PrologToPiTRSProof [SOUND, 0 ms] (18) PiTRS (19) DependencyPairsProof [EQUIVALENT, 0 ms] (20) PiDP (21) DependencyGraphProof [EQUIVALENT, 0 ms] (22) PiDP (23) UsableRulesProof [EQUIVALENT, 0 ms] (24) PiDP (25) PiDPToQDPProof [EQUIVALENT, 1 ms] (26) QDP (27) TransformationProof [EQUIVALENT, 0 ms] (28) QDP (29) TransformationProof [EQUIVALENT, 0 ms] (30) QDP (31) NonTerminationLoopProof [COMPLETE, 0 ms] (32) NO (33) PrologToDTProblemTransformerProof [SOUND, 0 ms] (34) TRIPLES (35) TriplesToPiDPProof [SOUND, 0 ms] (36) PiDP (37) DependencyGraphProof [EQUIVALENT, 0 ms] (38) PiDP (39) PiDPToQDPProof [EQUIVALENT, 1 ms] (40) QDP (41) TransformationProof [EQUIVALENT, 0 ms] (42) QDP (43) TransformationProof [EQUIVALENT, 0 ms] (44) QDP (45) NonTerminationLoopProof [COMPLETE, 0 ms] (46) NO (47) PrologToTRSTransformerProof [SOUND, 0 ms] (48) QTRS (49) Overlay + Local Confluence [EQUIVALENT, 0 ms] (50) QTRS (51) DependencyPairsProof [EQUIVALENT, 0 ms] (52) QDP (53) DependencyGraphProof [EQUIVALENT, 0 ms] (54) QDP (55) UsableRulesProof [EQUIVALENT, 0 ms] (56) QDP (57) QReductionProof [EQUIVALENT, 0 ms] (58) QDP (59) TransformationProof [EQUIVALENT, 0 ms] (60) QDP (61) TransformationProof [EQUIVALENT, 0 ms] (62) QDP (63) NonTerminationLoopProof [COMPLETE, 1 ms] (64) NO (65) PrologToIRSwTTransformerProof [SOUND, 0 ms] (66) IRSwT (67) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (68) IRSwT (69) IntTRSCompressionProof [EQUIVALENT, 29 ms] (70) IRSwT (71) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (72) IRSwT (73) IRSwTTerminationDigraphProof [EQUIVALENT, 10 ms] (74) IRSwT (75) IntTRSCompressionProof [EQUIVALENT, 0 ms] (76) IRSwT (77) IRSwTToIntTRSProof [SOUND, 0 ms] (78) IRSwT ---------------------------------------- (0) Obligation: Clauses: desc(X) :- desc(-(X, 1)). Query: desc(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: desc_in_1: (b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: desc_in_g(X) -> U1_g(X, desc_in_g(-(X, 1))) U1_g(X, desc_out_g(-(X, 1))) -> desc_out_g(X) The argument filtering Pi contains the following mapping: desc_in_g(x1) = desc_in_g(x1) U1_g(x1, x2) = U1_g(x2) desc_out_g(x1) = desc_out_g -(x1, x2) = -(x1, x2) 1 = 1 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: desc_in_g(X) -> U1_g(X, desc_in_g(-(X, 1))) U1_g(X, desc_out_g(-(X, 1))) -> desc_out_g(X) The argument filtering Pi contains the following mapping: desc_in_g(x1) = desc_in_g(x1) U1_g(x1, x2) = U1_g(x2) desc_out_g(x1) = desc_out_g -(x1, x2) = -(x1, x2) 1 = 1 ---------------------------------------- (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: DESC_IN_G(X) -> U1_G(X, desc_in_g(-(X, 1))) DESC_IN_G(X) -> DESC_IN_G(-(X, 1)) The TRS R consists of the following rules: desc_in_g(X) -> U1_g(X, desc_in_g(-(X, 1))) U1_g(X, desc_out_g(-(X, 1))) -> desc_out_g(X) The argument filtering Pi contains the following mapping: desc_in_g(x1) = desc_in_g(x1) U1_g(x1, x2) = U1_g(x2) desc_out_g(x1) = desc_out_g -(x1, x2) = -(x1, x2) 1 = 1 DESC_IN_G(x1) = DESC_IN_G(x1) U1_G(x1, x2) = U1_G(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: DESC_IN_G(X) -> U1_G(X, desc_in_g(-(X, 1))) DESC_IN_G(X) -> DESC_IN_G(-(X, 1)) The TRS R consists of the following rules: desc_in_g(X) -> U1_g(X, desc_in_g(-(X, 1))) U1_g(X, desc_out_g(-(X, 1))) -> desc_out_g(X) The argument filtering Pi contains the following mapping: desc_in_g(x1) = desc_in_g(x1) U1_g(x1, x2) = U1_g(x2) desc_out_g(x1) = desc_out_g -(x1, x2) = -(x1, x2) 1 = 1 DESC_IN_G(x1) = DESC_IN_G(x1) U1_G(x1, x2) = U1_G(x2) 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: DESC_IN_G(X) -> DESC_IN_G(-(X, 1)) The TRS R consists of the following rules: desc_in_g(X) -> U1_g(X, desc_in_g(-(X, 1))) U1_g(X, desc_out_g(-(X, 1))) -> desc_out_g(X) The argument filtering Pi contains the following mapping: desc_in_g(x1) = desc_in_g(x1) U1_g(x1, x2) = U1_g(x2) desc_out_g(x1) = desc_out_g -(x1, x2) = -(x1, x2) 1 = 1 DESC_IN_G(x1) = DESC_IN_G(x1) 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: DESC_IN_G(X) -> DESC_IN_G(-(X, 1)) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) PiDPToQDPProof (EQUIVALENT) 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: DESC_IN_G(X) -> DESC_IN_G(-(X, 1)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (11) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule DESC_IN_G(X) -> DESC_IN_G(-(X, 1)) we obtained the following new rules [LPAR04]: (DESC_IN_G(-(z0, 1)) -> DESC_IN_G(-(-(z0, 1), 1)),DESC_IN_G(-(z0, 1)) -> DESC_IN_G(-(-(z0, 1), 1))) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: DESC_IN_G(-(z0, 1)) -> DESC_IN_G(-(-(z0, 1), 1)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (13) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule DESC_IN_G(-(z0, 1)) -> DESC_IN_G(-(-(z0, 1), 1)) we obtained the following new rules [LPAR04]: (DESC_IN_G(-(-(z0, 1), 1)) -> DESC_IN_G(-(-(-(z0, 1), 1), 1)),DESC_IN_G(-(-(z0, 1), 1)) -> DESC_IN_G(-(-(-(z0, 1), 1), 1))) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: DESC_IN_G(-(-(z0, 1), 1)) -> DESC_IN_G(-(-(-(z0, 1), 1), 1)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (15) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = DESC_IN_G(-(-(z0, 1), 1)) evaluates to t =DESC_IN_G(-(-(-(z0, 1), 1), 1)) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [z0 / -(z0, 1)] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from DESC_IN_G(-(-(z0, 1), 1)) to DESC_IN_G(-(-(-(z0, 1), 1), 1)). ---------------------------------------- (16) NO ---------------------------------------- (17) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: desc_in_1: (b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: desc_in_g(X) -> U1_g(X, desc_in_g(-(X, 1))) U1_g(X, desc_out_g(-(X, 1))) -> desc_out_g(X) Pi is empty. Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (18) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: desc_in_g(X) -> U1_g(X, desc_in_g(-(X, 1))) U1_g(X, desc_out_g(-(X, 1))) -> desc_out_g(X) Pi is empty. ---------------------------------------- (19) 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: DESC_IN_G(X) -> U1_G(X, desc_in_g(-(X, 1))) DESC_IN_G(X) -> DESC_IN_G(-(X, 1)) The TRS R consists of the following rules: desc_in_g(X) -> U1_g(X, desc_in_g(-(X, 1))) U1_g(X, desc_out_g(-(X, 1))) -> desc_out_g(X) Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (20) Obligation: Pi DP problem: The TRS P consists of the following rules: DESC_IN_G(X) -> U1_G(X, desc_in_g(-(X, 1))) DESC_IN_G(X) -> DESC_IN_G(-(X, 1)) The TRS R consists of the following rules: desc_in_g(X) -> U1_g(X, desc_in_g(-(X, 1))) U1_g(X, desc_out_g(-(X, 1))) -> desc_out_g(X) Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (21) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (22) Obligation: Pi DP problem: The TRS P consists of the following rules: DESC_IN_G(X) -> DESC_IN_G(-(X, 1)) The TRS R consists of the following rules: desc_in_g(X) -> U1_g(X, desc_in_g(-(X, 1))) U1_g(X, desc_out_g(-(X, 1))) -> desc_out_g(X) Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (23) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (24) Obligation: Pi DP problem: The TRS P consists of the following rules: DESC_IN_G(X) -> DESC_IN_G(-(X, 1)) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (25) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: DESC_IN_G(X) -> DESC_IN_G(-(X, 1)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (27) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule DESC_IN_G(X) -> DESC_IN_G(-(X, 1)) we obtained the following new rules [LPAR04]: (DESC_IN_G(-(z0, 1)) -> DESC_IN_G(-(-(z0, 1), 1)),DESC_IN_G(-(z0, 1)) -> DESC_IN_G(-(-(z0, 1), 1))) ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: DESC_IN_G(-(z0, 1)) -> DESC_IN_G(-(-(z0, 1), 1)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (29) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule DESC_IN_G(-(z0, 1)) -> DESC_IN_G(-(-(z0, 1), 1)) we obtained the following new rules [LPAR04]: (DESC_IN_G(-(-(z0, 1), 1)) -> DESC_IN_G(-(-(-(z0, 1), 1), 1)),DESC_IN_G(-(-(z0, 1), 1)) -> DESC_IN_G(-(-(-(z0, 1), 1), 1))) ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: DESC_IN_G(-(-(z0, 1), 1)) -> DESC_IN_G(-(-(-(z0, 1), 1), 1)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (31) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = DESC_IN_G(-(-(z0, 1), 1)) evaluates to t =DESC_IN_G(-(-(-(z0, 1), 1), 1)) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [z0 / -(z0, 1)] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from DESC_IN_G(-(-(z0, 1), 1)) to DESC_IN_G(-(-(-(z0, 1), 1), 1)). ---------------------------------------- (32) NO ---------------------------------------- (33) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 4, "program": { "directives": [], "clauses": [[ "(desc X)", "(desc (- X (1)))" ]] }, "graph": { "nodes": { "34": { "goal": [{ "clause": 0, "scope": 2, "term": "(desc (- T3 (1)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "47": { "goal": [{ "clause": -1, "scope": -1, "term": "(desc (- (- T7 (1)) (1)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": [], "exprvars": [] } }, "4": { "goal": [{ "clause": -1, "scope": -1, "term": "(desc T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "5": { "goal": [{ "clause": 0, "scope": 1, "term": "(desc T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "28": { "goal": [{ "clause": -1, "scope": -1, "term": "(desc (- T3 (1)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "type": "Nodes" }, "edges": [ { "from": 4, "to": 5, "label": "CASE" }, { "from": 5, "to": 28, "label": "ONLY EVAL with clause\ndesc(X2) :- desc(-(X2, 1)).\nand substitutionT1 -> T3,\nX2 -> T3" }, { "from": 28, "to": 34, "label": "CASE" }, { "from": 34, "to": 47, "label": "ONLY EVAL with clause\ndesc(X6) :- desc(-(X6, 1)).\nand substitutionT3 -> T7,\nX6 -> -(T7, 1)" }, { "from": 47, "to": 4, "label": "INSTANCE with matching:\nT1 -> -(-(T7, 1), 1)" } ], "type": "Graph" } } ---------------------------------------- (34) Obligation: Triples: descA(X1) :- descA(-(-(X1, 1), 1)). Clauses: desccA(X1) :- desccA(-(-(X1, 1), 1)). Afs: descA(x1) = descA(x1) ---------------------------------------- (35) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: descA_in_1: (b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: DESCA_IN_G(X1) -> U1_G(X1, descA_in_g(-(-(X1, 1), 1))) DESCA_IN_G(X1) -> DESCA_IN_G(-(-(X1, 1), 1)) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (36) Obligation: Pi DP problem: The TRS P consists of the following rules: DESCA_IN_G(X1) -> U1_G(X1, descA_in_g(-(-(X1, 1), 1))) DESCA_IN_G(X1) -> DESCA_IN_G(-(-(X1, 1), 1)) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (37) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (38) Obligation: Pi DP problem: The TRS P consists of the following rules: DESCA_IN_G(X1) -> DESCA_IN_G(-(-(X1, 1), 1)) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (39) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: DESCA_IN_G(X1) -> DESCA_IN_G(-(-(X1, 1), 1)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (41) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule DESCA_IN_G(X1) -> DESCA_IN_G(-(-(X1, 1), 1)) we obtained the following new rules [LPAR04]: (DESCA_IN_G(-(-(z0, 1), 1)) -> DESCA_IN_G(-(-(-(-(z0, 1), 1), 1), 1)),DESCA_IN_G(-(-(z0, 1), 1)) -> DESCA_IN_G(-(-(-(-(z0, 1), 1), 1), 1))) ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: DESCA_IN_G(-(-(z0, 1), 1)) -> DESCA_IN_G(-(-(-(-(z0, 1), 1), 1), 1)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (43) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule DESCA_IN_G(-(-(z0, 1), 1)) -> DESCA_IN_G(-(-(-(-(z0, 1), 1), 1), 1)) we obtained the following new rules [LPAR04]: (DESCA_IN_G(-(-(-(-(z0, 1), 1), 1), 1)) -> DESCA_IN_G(-(-(-(-(-(-(z0, 1), 1), 1), 1), 1), 1)),DESCA_IN_G(-(-(-(-(z0, 1), 1), 1), 1)) -> DESCA_IN_G(-(-(-(-(-(-(z0, 1), 1), 1), 1), 1), 1))) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: DESCA_IN_G(-(-(-(-(z0, 1), 1), 1), 1)) -> DESCA_IN_G(-(-(-(-(-(-(z0, 1), 1), 1), 1), 1), 1)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (45) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = DESCA_IN_G(-(-(-(-(z0, 1), 1), 1), 1)) evaluates to t =DESCA_IN_G(-(-(-(-(-(-(z0, 1), 1), 1), 1), 1), 1)) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [z0 / -(-(z0, 1), 1)] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from DESCA_IN_G(-(-(-(-(z0, 1), 1), 1), 1)) to DESCA_IN_G(-(-(-(-(-(-(z0, 1), 1), 1), 1), 1), 1)). ---------------------------------------- (46) NO ---------------------------------------- (47) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 6, "program": { "directives": [], "clauses": [[ "(desc X)", "(desc (- X (1)))" ]] }, "graph": { "nodes": { "6": { "goal": [{ "clause": -1, "scope": -1, "term": "(desc T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "7": { "goal": [{ "clause": 0, "scope": 1, "term": "(desc T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "type": "Nodes", "30": { "goal": [{ "clause": -1, "scope": -1, "term": "(desc (- T5 (1)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 6, "to": 7, "label": "CASE" }, { "from": 7, "to": 30, "label": "ONLY EVAL with clause\ndesc(X4) :- desc(-(X4, 1)).\nand substitutionT1 -> T5,\nX4 -> T5" }, { "from": 30, "to": 6, "label": "INSTANCE with matching:\nT1 -> -(T5, 1)" } ], "type": "Graph" } } ---------------------------------------- (48) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f6_in(T5) -> U1(f6_in(-(T5, 1)), T5) U1(f6_out1, T5) -> f6_out1 Q is empty. ---------------------------------------- (49) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (50) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f6_in(T5) -> U1(f6_in(-(T5, 1)), T5) U1(f6_out1, T5) -> f6_out1 The set Q consists of the following terms: f6_in(x0) U1(f6_out1, x0) ---------------------------------------- (51) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: F6_IN(T5) -> U1^1(f6_in(-(T5, 1)), T5) F6_IN(T5) -> F6_IN(-(T5, 1)) The TRS R consists of the following rules: f6_in(T5) -> U1(f6_in(-(T5, 1)), T5) U1(f6_out1, T5) -> f6_out1 The set Q consists of the following terms: f6_in(x0) U1(f6_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: F6_IN(T5) -> F6_IN(-(T5, 1)) The TRS R consists of the following rules: f6_in(T5) -> U1(f6_in(-(T5, 1)), T5) U1(f6_out1, T5) -> f6_out1 The set Q consists of the following terms: f6_in(x0) U1(f6_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) 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. ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: F6_IN(T5) -> F6_IN(-(T5, 1)) R is empty. The set Q consists of the following terms: f6_in(x0) U1(f6_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) 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]. f6_in(x0) U1(f6_out1, x0) ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: F6_IN(T5) -> F6_IN(-(T5, 1)) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule F6_IN(T5) -> F6_IN(-(T5, 1)) we obtained the following new rules [LPAR04]: (F6_IN(-(z0, 1)) -> F6_IN(-(-(z0, 1), 1)),F6_IN(-(z0, 1)) -> F6_IN(-(-(z0, 1), 1))) ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: F6_IN(-(z0, 1)) -> F6_IN(-(-(z0, 1), 1)) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule F6_IN(-(z0, 1)) -> F6_IN(-(-(z0, 1), 1)) we obtained the following new rules [LPAR04]: (F6_IN(-(-(z0, 1), 1)) -> F6_IN(-(-(-(z0, 1), 1), 1)),F6_IN(-(-(z0, 1), 1)) -> F6_IN(-(-(-(z0, 1), 1), 1))) ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: F6_IN(-(-(z0, 1), 1)) -> F6_IN(-(-(-(z0, 1), 1), 1)) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = F6_IN(-(-(z0, 1), 1)) evaluates to t =F6_IN(-(-(-(z0, 1), 1), 1)) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [z0 / -(z0, 1)] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from F6_IN(-(-(z0, 1), 1)) to F6_IN(-(-(-(z0, 1), 1), 1)). ---------------------------------------- (64) NO ---------------------------------------- (65) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 11, "program": { "directives": [], "clauses": [[ "(desc X)", "(desc (- X (1)))" ]] }, "graph": { "nodes": { "11": { "goal": [{ "clause": -1, "scope": -1, "term": "(desc T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "12": { "goal": [{ "clause": 0, "scope": 1, "term": "(desc T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "46": { "goal": [{ "clause": -1, "scope": -1, "term": "(desc (- T5 (1)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "type": "Nodes" }, "edges": [ { "from": 11, "to": 12, "label": "CASE" }, { "from": 12, "to": 46, "label": "ONLY EVAL with clause\ndesc(X4) :- desc(-(X4, 1)).\nand substitutionT1 -> T5,\nX4 -> T5" }, { "from": 46, "to": 11, "label": "INSTANCE with matching:\nT1 -> -(T5, 1)" } ], "type": "Graph" } } ---------------------------------------- (66) Obligation: Rules: f46_in(T5) -> f11_in(T5 - 1) :|: TRUE f11_out(x - 1) -> f46_out(x) :|: TRUE f46_out(x1) -> f12_out(x1) :|: TRUE f12_in(x2) -> f46_in(x2) :|: TRUE f12_out(T1) -> f11_out(T1) :|: TRUE f11_in(x3) -> f12_in(x3) :|: TRUE Start term: f11_in(T1) ---------------------------------------- (67) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f46_in(T5) -> f11_in(T5 - 1) :|: TRUE f12_in(x2) -> f46_in(x2) :|: TRUE f11_in(x3) -> f12_in(x3) :|: TRUE ---------------------------------------- (68) Obligation: Rules: f46_in(T5) -> f11_in(T5 - 1) :|: TRUE f12_in(x2) -> f46_in(x2) :|: TRUE f11_in(x3) -> f12_in(x3) :|: TRUE ---------------------------------------- (69) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (70) Obligation: Rules: f12_in(x2:0) -> f12_in(x2:0 - 1) :|: TRUE ---------------------------------------- (71) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (72) Obligation: Rules: f12_in(x2:0) -> f12_in(arith) :|: TRUE && arith = x2:0 - 1 ---------------------------------------- (73) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f12_in(x2:0) -> f12_in(arith) :|: TRUE && arith = x2:0 - 1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (74) Obligation: Termination digraph: Nodes: (1) f12_in(x2:0) -> f12_in(arith) :|: TRUE && arith = x2:0 - 1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (75) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (76) Obligation: Rules: f12_in(x2:0:0) -> f12_in(x2:0:0 - 1) :|: TRUE ---------------------------------------- (77) IRSwTToIntTRSProof (SOUND) Applied path-length measure to transform intTRS with terms to intTRS. ---------------------------------------- (78) Obligation: Rules: f12_in(x) -> f12_in(x - 1) :|: TRUE