/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern p2(a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS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, 5 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, 3 ms] (20) QDP (21) PrologToTRSTransformerProof [SOUND, 0 ms] (22) QTRS (23) QTRSRRRProof [EQUIVALENT, 4 ms] (24) QTRS (25) Overlay + Local Confluence [EQUIVALENT, 0 ms] (26) QTRS (27) DependencyPairsProof [EQUIVALENT, 0 ms] (28) QDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) QDP (31) QReductionProof [EQUIVALENT, 0 ms] (32) QDP (33) PrologToDTProblemTransformerProof [SOUND, 0 ms] (34) TRIPLES (35) TriplesToPiDPProof [SOUND, 0 ms] (36) PiDP (37) DependencyGraphProof [EQUIVALENT, 0 ms] (38) PiDP (39) PiDPToQDPProof [SOUND, 1 ms] (40) QDP (41) PrologToIRSwTTransformerProof [SOUND, 0 ms] (42) IRSwT (43) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (44) IRSwT (45) IntTRSCompressionProof [EQUIVALENT, 20 ms] (46) IRSwT (47) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (48) IRSwT (49) IRSwTTerminationDigraphProof [EQUIVALENT, 7 ms] (50) IRSwT (51) FilterProof [EQUIVALENT, 3 ms] (52) IntTRS (53) IntTRSPeriodicNontermProof [COMPLETE, 8 ms] (54) NO ---------------------------------------- (0) Obligation: Clauses: p1(f(X)) :- p1(X). p2(f(X)) :- p2(X). Query: p2(a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: p2_in_1: (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p2_in_a(f(X)) -> U2_a(X, p2_in_a(X)) U2_a(X, p2_out_a(X)) -> p2_out_a(f(X)) The argument filtering Pi contains the following mapping: p2_in_a(x1) = p2_in_a U2_a(x1, x2) = U2_a(x2) p2_out_a(x1) = p2_out_a(x1) f(x1) = f(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: p2_in_a(f(X)) -> U2_a(X, p2_in_a(X)) U2_a(X, p2_out_a(X)) -> p2_out_a(f(X)) The argument filtering Pi contains the following mapping: p2_in_a(x1) = p2_in_a U2_a(x1, x2) = U2_a(x2) p2_out_a(x1) = p2_out_a(x1) f(x1) = f(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: P2_IN_A(f(X)) -> U2_A(X, p2_in_a(X)) P2_IN_A(f(X)) -> P2_IN_A(X) The TRS R consists of the following rules: p2_in_a(f(X)) -> U2_a(X, p2_in_a(X)) U2_a(X, p2_out_a(X)) -> p2_out_a(f(X)) The argument filtering Pi contains the following mapping: p2_in_a(x1) = p2_in_a U2_a(x1, x2) = U2_a(x2) p2_out_a(x1) = p2_out_a(x1) f(x1) = f(x1) P2_IN_A(x1) = P2_IN_A U2_A(x1, x2) = U2_A(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: P2_IN_A(f(X)) -> U2_A(X, p2_in_a(X)) P2_IN_A(f(X)) -> P2_IN_A(X) The TRS R consists of the following rules: p2_in_a(f(X)) -> U2_a(X, p2_in_a(X)) U2_a(X, p2_out_a(X)) -> p2_out_a(f(X)) The argument filtering Pi contains the following mapping: p2_in_a(x1) = p2_in_a U2_a(x1, x2) = U2_a(x2) p2_out_a(x1) = p2_out_a(x1) f(x1) = f(x1) P2_IN_A(x1) = P2_IN_A U2_A(x1, x2) = U2_A(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: P2_IN_A(f(X)) -> P2_IN_A(X) The TRS R consists of the following rules: p2_in_a(f(X)) -> U2_a(X, p2_in_a(X)) U2_a(X, p2_out_a(X)) -> p2_out_a(f(X)) The argument filtering Pi contains the following mapping: p2_in_a(x1) = p2_in_a U2_a(x1, x2) = U2_a(x2) p2_out_a(x1) = p2_out_a(x1) f(x1) = f(x1) P2_IN_A(x1) = P2_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: P2_IN_A(f(X)) -> P2_IN_A(X) R is empty. The argument filtering Pi contains the following mapping: f(x1) = f(x1) P2_IN_A(x1) = P2_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: P2_IN_A -> P2_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: p2_in_1: (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p2_in_a(f(X)) -> U2_a(X, p2_in_a(X)) U2_a(X, p2_out_a(X)) -> p2_out_a(f(X)) The argument filtering Pi contains the following mapping: p2_in_a(x1) = p2_in_a U2_a(x1, x2) = U2_a(x2) p2_out_a(x1) = p2_out_a(x1) f(x1) = f(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: p2_in_a(f(X)) -> U2_a(X, p2_in_a(X)) U2_a(X, p2_out_a(X)) -> p2_out_a(f(X)) The argument filtering Pi contains the following mapping: p2_in_a(x1) = p2_in_a U2_a(x1, x2) = U2_a(x2) p2_out_a(x1) = p2_out_a(x1) f(x1) = f(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: P2_IN_A(f(X)) -> U2_A(X, p2_in_a(X)) P2_IN_A(f(X)) -> P2_IN_A(X) The TRS R consists of the following rules: p2_in_a(f(X)) -> U2_a(X, p2_in_a(X)) U2_a(X, p2_out_a(X)) -> p2_out_a(f(X)) The argument filtering Pi contains the following mapping: p2_in_a(x1) = p2_in_a U2_a(x1, x2) = U2_a(x2) p2_out_a(x1) = p2_out_a(x1) f(x1) = f(x1) P2_IN_A(x1) = P2_IN_A U2_A(x1, x2) = U2_A(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: P2_IN_A(f(X)) -> U2_A(X, p2_in_a(X)) P2_IN_A(f(X)) -> P2_IN_A(X) The TRS R consists of the following rules: p2_in_a(f(X)) -> U2_a(X, p2_in_a(X)) U2_a(X, p2_out_a(X)) -> p2_out_a(f(X)) The argument filtering Pi contains the following mapping: p2_in_a(x1) = p2_in_a U2_a(x1, x2) = U2_a(x2) p2_out_a(x1) = p2_out_a(x1) f(x1) = f(x1) P2_IN_A(x1) = P2_IN_A U2_A(x1, x2) = U2_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 1 less node. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: P2_IN_A(f(X)) -> P2_IN_A(X) The TRS R consists of the following rules: p2_in_a(f(X)) -> U2_a(X, p2_in_a(X)) U2_a(X, p2_out_a(X)) -> p2_out_a(f(X)) The argument filtering Pi contains the following mapping: p2_in_a(x1) = p2_in_a U2_a(x1, x2) = U2_a(x2) p2_out_a(x1) = p2_out_a(x1) f(x1) = f(x1) P2_IN_A(x1) = P2_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: P2_IN_A(f(X)) -> P2_IN_A(X) R is empty. The argument filtering Pi contains the following mapping: f(x1) = f(x1) P2_IN_A(x1) = P2_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: P2_IN_A -> P2_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": 8, "program": { "directives": [], "clauses": [ [ "(p1 (f X))", "(p1 X)" ], [ "(p2 (f X))", "(p2 X)" ] ] }, "graph": { "nodes": { "59": { "goal": [{ "clause": -1, "scope": -1, "term": "(p2 T5)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "8": { "goal": [{ "clause": -1, "scope": -1, "term": "(p2 T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "9": { "goal": [{ "clause": 1, "scope": 1, "term": "(p2 T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "60": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes" }, "edges": [ { "from": 8, "to": 9, "label": "CASE" }, { "from": 9, "to": 59, "label": "EVAL with clause\np2(f(X3)) :- p2(X3).\nand substitutionX3 -> T5,\nT1 -> f(T5),\nT4 -> T5" }, { "from": 9, "to": 60, "label": "EVAL-BACKTRACK" }, { "from": 59, "to": 8, "label": "INSTANCE with matching:\nT1 -> T5" } ], "type": "Graph" } } ---------------------------------------- (22) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f8_in -> U1(f8_in) U1(f8_out1(T5)) -> f8_out1(f(T5)) Q is empty. ---------------------------------------- (23) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1)) = 2*x_1 POL(f(x_1)) = x_1 POL(f8_in) = 0 POL(f8_out1(x_1)) = 2 + x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f8_out1(T5)) -> f8_out1(f(T5)) ---------------------------------------- (24) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f8_in -> U1(f8_in) Q is empty. ---------------------------------------- (25) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (26) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f8_in -> U1(f8_in) The set Q consists of the following terms: f8_in ---------------------------------------- (27) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: F8_IN -> F8_IN The TRS R consists of the following rules: f8_in -> U1(f8_in) The set Q consists of the following terms: f8_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: F8_IN -> F8_IN R is empty. The set Q consists of the following terms: f8_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. f8_in ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: F8_IN -> F8_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 10, "program": { "directives": [], "clauses": [ [ "(p1 (f X))", "(p1 X)" ], [ "(p2 (f X))", "(p2 X)" ] ] }, "graph": { "nodes": { "11": { "goal": [{ "clause": 1, "scope": 1, "term": "(p2 T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "55": { "goal": [{ "clause": -1, "scope": -1, "term": "(p2 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "56": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "61": { "goal": [{ "clause": 1, "scope": 2, "term": "(p2 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "64": { "goal": [{ "clause": -1, "scope": -1, "term": "(p2 T8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "10": { "goal": [{ "clause": -1, "scope": -1, "term": "(p2 T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "65": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 10, "to": 11, "label": "CASE" }, { "from": 11, "to": 55, "label": "EVAL with clause\np2(f(X2)) :- p2(X2).\nand substitutionX2 -> T4,\nT1 -> f(T4),\nT3 -> T4" }, { "from": 11, "to": 56, "label": "EVAL-BACKTRACK" }, { "from": 55, "to": 61, "label": "CASE" }, { "from": 61, "to": 64, "label": "EVAL with clause\np2(f(X5)) :- p2(X5).\nand substitutionX5 -> T8,\nT4 -> f(T8),\nT7 -> T8" }, { "from": 61, "to": 65, "label": "EVAL-BACKTRACK" }, { "from": 64, "to": 10, "label": "INSTANCE with matching:\nT1 -> T8" } ], "type": "Graph" } } ---------------------------------------- (34) Obligation: Triples: p2A(f(f(X1))) :- p2A(X1). Clauses: p2cA(f(f(X1))) :- p2cA(X1). Afs: p2A(x1) = p2A ---------------------------------------- (35) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: p2A_in_1: (f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: P2A_IN_A(f(f(X1))) -> U1_A(X1, p2A_in_a(X1)) P2A_IN_A(f(f(X1))) -> P2A_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: p2A_in_a(x1) = p2A_in_a f(x1) = f(x1) P2A_IN_A(x1) = P2A_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 ---------------------------------------- (36) Obligation: Pi DP problem: The TRS P consists of the following rules: P2A_IN_A(f(f(X1))) -> U1_A(X1, p2A_in_a(X1)) P2A_IN_A(f(f(X1))) -> P2A_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: p2A_in_a(x1) = p2A_in_a f(x1) = f(x1) P2A_IN_A(x1) = P2A_IN_A U1_A(x1, x2) = U1_A(x2) 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: P2A_IN_A(f(f(X1))) -> P2A_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: f(x1) = f(x1) P2A_IN_A(x1) = P2A_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (39) PiDPToQDPProof (SOUND) 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: P2A_IN_A -> P2A_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (41) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 7, "program": { "directives": [], "clauses": [ [ "(p1 (f X))", "(p1 X)" ], [ "(p2 (f X))", "(p2 X)" ] ] }, "graph": { "nodes": { "12": { "goal": [{ "clause": 1, "scope": 1, "term": "(p2 T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "7": { "goal": [{ "clause": -1, "scope": -1, "term": "(p2 T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "62": { "goal": [{ "clause": -1, "scope": -1, "term": "(p2 T5)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "63": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 7, "to": 12, "label": "CASE" }, { "from": 12, "to": 62, "label": "EVAL with clause\np2(f(X3)) :- p2(X3).\nand substitutionX3 -> T5,\nT1 -> f(T5),\nT4 -> T5" }, { "from": 12, "to": 63, "label": "EVAL-BACKTRACK" }, { "from": 62, "to": 7, "label": "INSTANCE with matching:\nT1 -> T5" } ], "type": "Graph" } } ---------------------------------------- (42) Obligation: Rules: f12_in -> f62_in :|: TRUE f12_in -> f63_in :|: TRUE f62_out -> f12_out :|: TRUE f63_out -> f12_out :|: TRUE f7_out -> f62_out :|: TRUE f62_in -> f7_in :|: TRUE f7_in -> f12_in :|: TRUE f12_out -> f7_out :|: TRUE Start term: f7_in ---------------------------------------- (43) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f12_in -> f62_in :|: TRUE f62_in -> f7_in :|: TRUE f7_in -> f12_in :|: TRUE ---------------------------------------- (44) Obligation: Rules: f12_in -> f62_in :|: TRUE f62_in -> f7_in :|: TRUE f7_in -> f12_in :|: TRUE ---------------------------------------- (45) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (46) Obligation: Rules: f12_in -> f12_in :|: TRUE ---------------------------------------- (47) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (48) Obligation: Rules: f12_in -> f12_in :|: TRUE ---------------------------------------- (49) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f12_in -> f12_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (50) Obligation: Termination digraph: Nodes: (1) f12_in -> f12_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (51) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f12_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (52) Obligation: Rules: f12_in -> f12_in :|: TRUE ---------------------------------------- (53) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (54) NO