/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern a() w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS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) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [EQUIVALENT, 0 ms] (11) QDP (12) PiDP (13) UsableRulesProof [EQUIVALENT, 0 ms] (14) PiDP (15) PrologToPiTRSProof [SOUND, 0 ms] (16) PiTRS (17) DependencyPairsProof [EQUIVALENT, 0 ms] (18) PiDP (19) DependencyGraphProof [EQUIVALENT, 0 ms] (20) AND (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [EQUIVALENT, 0 ms] (25) QDP (26) PiDP (27) UsableRulesProof [EQUIVALENT, 0 ms] (28) PiDP (29) PrologToDTProblemTransformerProof [SOUND, 0 ms] (30) TRIPLES (31) TriplesToPiDPProof [SOUND, 0 ms] (32) PiDP (33) DependencyGraphProof [EQUIVALENT, 0 ms] (34) AND (35) PiDP (36) PiDPToQDPProof [EQUIVALENT, 0 ms] (37) QDP (38) PiDP (39) PrologToTRSTransformerProof [SOUND, 0 ms] (40) QTRS (41) QTRSRRRProof [EQUIVALENT, 45 ms] (42) QTRS (43) QTRSRRRProof [EQUIVALENT, 0 ms] (44) QTRS (45) Overlay + Local Confluence [EQUIVALENT, 1 ms] (46) QTRS (47) DependencyPairsProof [EQUIVALENT, 0 ms] (48) QDP (49) DependencyGraphProof [EQUIVALENT, 0 ms] (50) AND (51) QDP (52) UsableRulesProof [EQUIVALENT, 0 ms] (53) QDP (54) QReductionProof [EQUIVALENT, 0 ms] (55) QDP (56) QDP (57) UsableRulesProof [EQUIVALENT, 0 ms] (58) QDP (59) QReductionProof [EQUIVALENT, 0 ms] (60) QDP (61) PrologToIRSwTTransformerProof [SOUND, 0 ms] (62) AND (63) IRSwT (64) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (65) IRSwT (66) IntTRSCompressionProof [EQUIVALENT, 4 ms] (67) IRSwT (68) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (69) IRSwT (70) IRSwTTerminationDigraphProof [EQUIVALENT, 4 ms] (71) IRSwT (72) FilterProof [EQUIVALENT, 0 ms] (73) IntTRS (74) IntTRSPeriodicNontermProof [COMPLETE, 0 ms] (75) NO (76) IRSwT (77) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (78) IRSwT (79) IntTRSCompressionProof [EQUIVALENT, 0 ms] (80) IRSwT (81) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (82) IRSwT (83) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (84) IRSwT (85) FilterProof [EQUIVALENT, 0 ms] (86) IntTRS (87) IntTRSPeriodicNontermProof [COMPLETE, 0 ms] (88) NO ---------------------------------------- (0) Obligation: Clauses: a :- b. a :- e. b :- c. c :- d. d :- b. e :- f. f :- g. g :- e. Query: 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: Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: a_in_ -> U1_(b_in_) b_in_ -> U3_(c_in_) c_in_ -> U4_(d_in_) d_in_ -> U5_(b_in_) U5_(b_out_) -> d_out_ U4_(d_out_) -> c_out_ U3_(c_out_) -> b_out_ U1_(b_out_) -> a_out_ a_in_ -> U2_(e_in_) e_in_ -> U6_(f_in_) f_in_ -> U7_(g_in_) g_in_ -> U8_(e_in_) U8_(e_out_) -> g_out_ U7_(g_out_) -> f_out_ U6_(f_out_) -> e_out_ U2_(e_out_) -> a_out_ Pi is empty. Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: a_in_ -> U1_(b_in_) b_in_ -> U3_(c_in_) c_in_ -> U4_(d_in_) d_in_ -> U5_(b_in_) U5_(b_out_) -> d_out_ U4_(d_out_) -> c_out_ U3_(c_out_) -> b_out_ U1_(b_out_) -> a_out_ a_in_ -> U2_(e_in_) e_in_ -> U6_(f_in_) f_in_ -> U7_(g_in_) g_in_ -> U8_(e_in_) U8_(e_out_) -> g_out_ U7_(g_out_) -> f_out_ U6_(f_out_) -> e_out_ U2_(e_out_) -> a_out_ Pi is empty. ---------------------------------------- (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: A_IN_ -> U1_^1(b_in_) A_IN_ -> B_IN_ B_IN_ -> U3_^1(c_in_) B_IN_ -> C_IN_ C_IN_ -> U4_^1(d_in_) C_IN_ -> D_IN_ D_IN_ -> U5_^1(b_in_) D_IN_ -> B_IN_ A_IN_ -> U2_^1(e_in_) A_IN_ -> E_IN_ E_IN_ -> U6_^1(f_in_) E_IN_ -> F_IN_ F_IN_ -> U7_^1(g_in_) F_IN_ -> G_IN_ G_IN_ -> U8_^1(e_in_) G_IN_ -> E_IN_ The TRS R consists of the following rules: a_in_ -> U1_(b_in_) b_in_ -> U3_(c_in_) c_in_ -> U4_(d_in_) d_in_ -> U5_(b_in_) U5_(b_out_) -> d_out_ U4_(d_out_) -> c_out_ U3_(c_out_) -> b_out_ U1_(b_out_) -> a_out_ a_in_ -> U2_(e_in_) e_in_ -> U6_(f_in_) f_in_ -> U7_(g_in_) g_in_ -> U8_(e_in_) U8_(e_out_) -> g_out_ U7_(g_out_) -> f_out_ U6_(f_out_) -> e_out_ U2_(e_out_) -> a_out_ Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: A_IN_ -> U1_^1(b_in_) A_IN_ -> B_IN_ B_IN_ -> U3_^1(c_in_) B_IN_ -> C_IN_ C_IN_ -> U4_^1(d_in_) C_IN_ -> D_IN_ D_IN_ -> U5_^1(b_in_) D_IN_ -> B_IN_ A_IN_ -> U2_^1(e_in_) A_IN_ -> E_IN_ E_IN_ -> U6_^1(f_in_) E_IN_ -> F_IN_ F_IN_ -> U7_^1(g_in_) F_IN_ -> G_IN_ G_IN_ -> U8_^1(e_in_) G_IN_ -> E_IN_ The TRS R consists of the following rules: a_in_ -> U1_(b_in_) b_in_ -> U3_(c_in_) c_in_ -> U4_(d_in_) d_in_ -> U5_(b_in_) U5_(b_out_) -> d_out_ U4_(d_out_) -> c_out_ U3_(c_out_) -> b_out_ U1_(b_out_) -> a_out_ a_in_ -> U2_(e_in_) e_in_ -> U6_(f_in_) f_in_ -> U7_(g_in_) g_in_ -> U8_(e_in_) U8_(e_out_) -> g_out_ U7_(g_out_) -> f_out_ U6_(f_out_) -> e_out_ U2_(e_out_) -> a_out_ Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 10 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: E_IN_ -> F_IN_ F_IN_ -> G_IN_ G_IN_ -> E_IN_ The TRS R consists of the following rules: a_in_ -> U1_(b_in_) b_in_ -> U3_(c_in_) c_in_ -> U4_(d_in_) d_in_ -> U5_(b_in_) U5_(b_out_) -> d_out_ U4_(d_out_) -> c_out_ U3_(c_out_) -> b_out_ U1_(b_out_) -> a_out_ a_in_ -> U2_(e_in_) e_in_ -> U6_(f_in_) f_in_ -> U7_(g_in_) g_in_ -> U8_(e_in_) U8_(e_out_) -> g_out_ U7_(g_out_) -> f_out_ U6_(f_out_) -> e_out_ U2_(e_out_) -> a_out_ Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: E_IN_ -> F_IN_ F_IN_ -> G_IN_ G_IN_ -> E_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: E_IN_ -> F_IN_ F_IN_ -> G_IN_ G_IN_ -> E_IN_ R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) Obligation: Pi DP problem: The TRS P consists of the following rules: B_IN_ -> C_IN_ C_IN_ -> D_IN_ D_IN_ -> B_IN_ The TRS R consists of the following rules: a_in_ -> U1_(b_in_) b_in_ -> U3_(c_in_) c_in_ -> U4_(d_in_) d_in_ -> U5_(b_in_) U5_(b_out_) -> d_out_ U4_(d_out_) -> c_out_ U3_(c_out_) -> b_out_ U1_(b_out_) -> a_out_ a_in_ -> U2_(e_in_) e_in_ -> U6_(f_in_) f_in_ -> U7_(g_in_) g_in_ -> U8_(e_in_) U8_(e_out_) -> g_out_ U7_(g_out_) -> f_out_ U6_(f_out_) -> e_out_ U2_(e_out_) -> a_out_ Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (13) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: B_IN_ -> C_IN_ C_IN_ -> D_IN_ D_IN_ -> B_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: a_in_ -> U1_(b_in_) b_in_ -> U3_(c_in_) c_in_ -> U4_(d_in_) d_in_ -> U5_(b_in_) U5_(b_out_) -> d_out_ U4_(d_out_) -> c_out_ U3_(c_out_) -> b_out_ U1_(b_out_) -> a_out_ a_in_ -> U2_(e_in_) e_in_ -> U6_(f_in_) f_in_ -> U7_(g_in_) g_in_ -> U8_(e_in_) U8_(e_out_) -> g_out_ U7_(g_out_) -> f_out_ U6_(f_out_) -> e_out_ U2_(e_out_) -> a_out_ Pi is empty. Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (16) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: a_in_ -> U1_(b_in_) b_in_ -> U3_(c_in_) c_in_ -> U4_(d_in_) d_in_ -> U5_(b_in_) U5_(b_out_) -> d_out_ U4_(d_out_) -> c_out_ U3_(c_out_) -> b_out_ U1_(b_out_) -> a_out_ a_in_ -> U2_(e_in_) e_in_ -> U6_(f_in_) f_in_ -> U7_(g_in_) g_in_ -> U8_(e_in_) U8_(e_out_) -> g_out_ U7_(g_out_) -> f_out_ U6_(f_out_) -> e_out_ U2_(e_out_) -> a_out_ Pi is empty. ---------------------------------------- (17) 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: A_IN_ -> U1_^1(b_in_) A_IN_ -> B_IN_ B_IN_ -> U3_^1(c_in_) B_IN_ -> C_IN_ C_IN_ -> U4_^1(d_in_) C_IN_ -> D_IN_ D_IN_ -> U5_^1(b_in_) D_IN_ -> B_IN_ A_IN_ -> U2_^1(e_in_) A_IN_ -> E_IN_ E_IN_ -> U6_^1(f_in_) E_IN_ -> F_IN_ F_IN_ -> U7_^1(g_in_) F_IN_ -> G_IN_ G_IN_ -> U8_^1(e_in_) G_IN_ -> E_IN_ The TRS R consists of the following rules: a_in_ -> U1_(b_in_) b_in_ -> U3_(c_in_) c_in_ -> U4_(d_in_) d_in_ -> U5_(b_in_) U5_(b_out_) -> d_out_ U4_(d_out_) -> c_out_ U3_(c_out_) -> b_out_ U1_(b_out_) -> a_out_ a_in_ -> U2_(e_in_) e_in_ -> U6_(f_in_) f_in_ -> U7_(g_in_) g_in_ -> U8_(e_in_) U8_(e_out_) -> g_out_ U7_(g_out_) -> f_out_ U6_(f_out_) -> e_out_ U2_(e_out_) -> a_out_ Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: A_IN_ -> U1_^1(b_in_) A_IN_ -> B_IN_ B_IN_ -> U3_^1(c_in_) B_IN_ -> C_IN_ C_IN_ -> U4_^1(d_in_) C_IN_ -> D_IN_ D_IN_ -> U5_^1(b_in_) D_IN_ -> B_IN_ A_IN_ -> U2_^1(e_in_) A_IN_ -> E_IN_ E_IN_ -> U6_^1(f_in_) E_IN_ -> F_IN_ F_IN_ -> U7_^1(g_in_) F_IN_ -> G_IN_ G_IN_ -> U8_^1(e_in_) G_IN_ -> E_IN_ The TRS R consists of the following rules: a_in_ -> U1_(b_in_) b_in_ -> U3_(c_in_) c_in_ -> U4_(d_in_) d_in_ -> U5_(b_in_) U5_(b_out_) -> d_out_ U4_(d_out_) -> c_out_ U3_(c_out_) -> b_out_ U1_(b_out_) -> a_out_ a_in_ -> U2_(e_in_) e_in_ -> U6_(f_in_) f_in_ -> U7_(g_in_) g_in_ -> U8_(e_in_) U8_(e_out_) -> g_out_ U7_(g_out_) -> f_out_ U6_(f_out_) -> e_out_ U2_(e_out_) -> a_out_ Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 10 less nodes. ---------------------------------------- (20) Complex Obligation (AND) ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: E_IN_ -> F_IN_ F_IN_ -> G_IN_ G_IN_ -> E_IN_ The TRS R consists of the following rules: a_in_ -> U1_(b_in_) b_in_ -> U3_(c_in_) c_in_ -> U4_(d_in_) d_in_ -> U5_(b_in_) U5_(b_out_) -> d_out_ U4_(d_out_) -> c_out_ U3_(c_out_) -> b_out_ U1_(b_out_) -> a_out_ a_in_ -> U2_(e_in_) e_in_ -> U6_(f_in_) f_in_ -> U7_(g_in_) g_in_ -> U8_(e_in_) U8_(e_out_) -> g_out_ U7_(g_out_) -> f_out_ U6_(f_out_) -> e_out_ U2_(e_out_) -> a_out_ Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: E_IN_ -> F_IN_ F_IN_ -> G_IN_ G_IN_ -> E_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: E_IN_ -> F_IN_ F_IN_ -> G_IN_ G_IN_ -> E_IN_ R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) Obligation: Pi DP problem: The TRS P consists of the following rules: B_IN_ -> C_IN_ C_IN_ -> D_IN_ D_IN_ -> B_IN_ The TRS R consists of the following rules: a_in_ -> U1_(b_in_) b_in_ -> U3_(c_in_) c_in_ -> U4_(d_in_) d_in_ -> U5_(b_in_) U5_(b_out_) -> d_out_ U4_(d_out_) -> c_out_ U3_(c_out_) -> b_out_ U1_(b_out_) -> a_out_ a_in_ -> U2_(e_in_) e_in_ -> U6_(f_in_) f_in_ -> U7_(g_in_) g_in_ -> U8_(e_in_) U8_(e_out_) -> g_out_ U7_(g_out_) -> f_out_ U6_(f_out_) -> e_out_ U2_(e_out_) -> a_out_ Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (27) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: B_IN_ -> C_IN_ C_IN_ -> D_IN_ D_IN_ -> B_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(a)", "(b)" ], [ "(a)", "(e)" ], [ "(b)", "(c)" ], [ "(c)", "(d)" ], [ "(d)", "(b)" ], [ "(e)", "(f)" ], [ "(f)", "(g)" ], [ "(g)", "(e)" ] ] }, "graph": { "nodes": { "33": { "goal": [{ "clause": 2, "scope": 2, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "34": { "goal": [ { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "35": { "goal": [{ "clause": -1, "scope": -1, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "27": { "goal": [ { "clause": 0, "scope": 1, "term": "(a)" }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "150": { "goal": [{ "clause": 1, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "151": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "152": { "goal": [{ "clause": 5, "scope": 6, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "153": { "goal": [{ "clause": -1, "scope": -1, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "154": { "goal": [{ "clause": 6, "scope": 7, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "144": { "goal": [{ "clause": 3, "scope": 3, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "155": { "goal": [{ "clause": -1, "scope": -1, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "145": { "goal": [{ "clause": -1, "scope": -1, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "156": { "goal": [{ "clause": 7, "scope": 8, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "146": { "goal": [{ "clause": 4, "scope": 4, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "157": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "147": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "148": { "goal": [{ "clause": 2, "scope": 5, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "149": { "goal": [{ "clause": -1, "scope": -1, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "31": { "goal": [ { "clause": -1, "scope": -1, "term": "(b)" }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "32": { "goal": [ { "clause": 2, "scope": 2, "term": "(b)" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 27, "label": "CASE" }, { "from": 27, "to": 31, "label": "ONLY EVAL with clause\na :- b.\nand substitution" }, { "from": 31, "to": 32, "label": "CASE" }, { "from": 32, "to": 33, "label": "PARALLEL" }, { "from": 32, "to": 34, "label": "PARALLEL" }, { "from": 33, "to": 35, "label": "ONLY EVAL with clause\nb :- c.\nand substitution" }, { "from": 34, "to": 150, "label": "FAILURE" }, { "from": 35, "to": 144, "label": "CASE" }, { "from": 144, "to": 145, "label": "ONLY EVAL with clause\nc :- d.\nand substitution" }, { "from": 145, "to": 146, "label": "CASE" }, { "from": 146, "to": 147, "label": "ONLY EVAL with clause\nd :- b.\nand substitution" }, { "from": 147, "to": 148, "label": "CASE" }, { "from": 148, "to": 149, "label": "ONLY EVAL with clause\nb :- c.\nand substitution" }, { "from": 149, "to": 35, "label": "INSTANCE" }, { "from": 150, "to": 151, "label": "ONLY EVAL with clause\na :- e.\nand substitution" }, { "from": 151, "to": 152, "label": "CASE" }, { "from": 152, "to": 153, "label": "ONLY EVAL with clause\ne :- f.\nand substitution" }, { "from": 153, "to": 154, "label": "CASE" }, { "from": 154, "to": 155, "label": "ONLY EVAL with clause\nf :- g.\nand substitution" }, { "from": 155, "to": 156, "label": "CASE" }, { "from": 156, "to": 157, "label": "ONLY EVAL with clause\ng :- e.\nand substitution" }, { "from": 157, "to": 151, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (30) Obligation: Triples: cA :- cA. eB :- eB. aC :- cA. aC :- eB. Clauses: ccA :- ccA. ecB :- ecB. Afs: aC = aC ---------------------------------------- (31) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: AC_IN_ -> U3_^1(cA_in_) AC_IN_ -> CA_IN_ CA_IN_ -> U1_^1(cA_in_) CA_IN_ -> CA_IN_ AC_IN_ -> U4_^1(eB_in_) AC_IN_ -> EB_IN_ EB_IN_ -> U2_^1(eB_in_) EB_IN_ -> EB_IN_ 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 ---------------------------------------- (32) Obligation: Pi DP problem: The TRS P consists of the following rules: AC_IN_ -> U3_^1(cA_in_) AC_IN_ -> CA_IN_ CA_IN_ -> U1_^1(cA_in_) CA_IN_ -> CA_IN_ AC_IN_ -> U4_^1(eB_in_) AC_IN_ -> EB_IN_ EB_IN_ -> U2_^1(eB_in_) EB_IN_ -> EB_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (33) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes. ---------------------------------------- (34) Complex Obligation (AND) ---------------------------------------- (35) Obligation: Pi DP problem: The TRS P consists of the following rules: EB_IN_ -> EB_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (36) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: EB_IN_ -> EB_IN_ R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (38) Obligation: Pi DP problem: The TRS P consists of the following rules: CA_IN_ -> CA_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (39) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 2, "program": { "directives": [], "clauses": [ [ "(a)", "(b)" ], [ "(a)", "(e)" ], [ "(b)", "(c)" ], [ "(c)", "(d)" ], [ "(d)", "(b)" ], [ "(e)", "(f)" ], [ "(f)", "(g)" ], [ "(g)", "(e)" ] ] }, "graph": { "nodes": { "77": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "59": { "goal": [{ "clause": 2, "scope": 2, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "28": { "goal": [{ "clause": 0, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "29": { "goal": [{ "clause": 1, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "160": { "goal": [{ "clause": 6, "scope": 6, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "161": { "goal": [{ "clause": -1, "scope": -1, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "162": { "goal": [{ "clause": 7, "scope": 7, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "163": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "158": { "goal": [{ "clause": 5, "scope": 5, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5": { "goal": [ { "clause": 0, "scope": 1, "term": "(a)" }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "159": { "goal": [{ "clause": -1, "scope": -1, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "60": { "goal": [{ "clause": -1, "scope": -1, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "71": { "goal": [{ "clause": 4, "scope": 4, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "82": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "61": { "goal": [{ "clause": 3, "scope": 3, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "62": { "goal": [{ "clause": -1, "scope": -1, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "30": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 5, "label": "CASE" }, { "from": 5, "to": 28, "label": "PARALLEL" }, { "from": 5, "to": 29, "label": "PARALLEL" }, { "from": 28, "to": 30, "label": "ONLY EVAL with clause\na :- b.\nand substitution" }, { "from": 29, "to": 82, "label": "ONLY EVAL with clause\na :- e.\nand substitution" }, { "from": 30, "to": 59, "label": "CASE" }, { "from": 59, "to": 60, "label": "ONLY EVAL with clause\nb :- c.\nand substitution" }, { "from": 60, "to": 61, "label": "CASE" }, { "from": 61, "to": 62, "label": "ONLY EVAL with clause\nc :- d.\nand substitution" }, { "from": 62, "to": 71, "label": "CASE" }, { "from": 71, "to": 77, "label": "ONLY EVAL with clause\nd :- b.\nand substitution" }, { "from": 77, "to": 30, "label": "INSTANCE" }, { "from": 82, "to": 158, "label": "CASE" }, { "from": 158, "to": 159, "label": "ONLY EVAL with clause\ne :- f.\nand substitution" }, { "from": 159, "to": 160, "label": "CASE" }, { "from": 160, "to": 161, "label": "ONLY EVAL with clause\nf :- g.\nand substitution" }, { "from": 161, "to": 162, "label": "CASE" }, { "from": 162, "to": 163, "label": "ONLY EVAL with clause\ng :- e.\nand substitution" }, { "from": 163, "to": 82, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (40) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f2_in -> U1(f30_in) U1(f30_out1) -> f2_out1 f2_in -> U2(f82_in) U2(f82_out1) -> f2_out1 f30_in -> U3(f30_in) U3(f30_out1) -> f30_out1 f82_in -> U4(f82_in) U4(f82_out1) -> f82_out1 Q is empty. ---------------------------------------- (41) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1)) = 2 + 2*x_1 POL(U2(x_1)) = 2*x_1 POL(U3(x_1)) = 2*x_1 POL(U4(x_1)) = 2*x_1 POL(f2_in) = 2 POL(f2_out1) = 1 POL(f30_in) = 0 POL(f30_out1) = 1 POL(f82_in) = 0 POL(f82_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f30_out1) -> f2_out1 f2_in -> U2(f82_in) U2(f82_out1) -> f2_out1 U3(f30_out1) -> f30_out1 U4(f82_out1) -> f82_out1 ---------------------------------------- (42) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f2_in -> U1(f30_in) f30_in -> U3(f30_in) f82_in -> U4(f82_in) Q is empty. ---------------------------------------- (43) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1)) = 1 + 2*x_1 POL(U3(x_1)) = 2*x_1 POL(U4(x_1)) = x_1 POL(f2_in) = 2 POL(f30_in) = 0 POL(f82_in) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f2_in -> U1(f30_in) ---------------------------------------- (44) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f30_in -> U3(f30_in) f82_in -> U4(f82_in) Q is empty. ---------------------------------------- (45) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (46) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f30_in -> U3(f30_in) f82_in -> U4(f82_in) The set Q consists of the following terms: f30_in f82_in ---------------------------------------- (47) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: F30_IN -> F30_IN F82_IN -> F82_IN The TRS R consists of the following rules: f30_in -> U3(f30_in) f82_in -> U4(f82_in) The set Q consists of the following terms: f30_in f82_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (50) Complex Obligation (AND) ---------------------------------------- (51) Obligation: Q DP problem: The TRS P consists of the following rules: F82_IN -> F82_IN The TRS R consists of the following rules: f30_in -> U3(f30_in) f82_in -> U4(f82_in) The set Q consists of the following terms: f30_in f82_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (52) 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. ---------------------------------------- (53) Obligation: Q DP problem: The TRS P consists of the following rules: F82_IN -> F82_IN R is empty. The set Q consists of the following terms: f30_in f82_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (54) 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]. f30_in f82_in ---------------------------------------- (55) Obligation: Q DP problem: The TRS P consists of the following rules: F82_IN -> F82_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: F30_IN -> F30_IN The TRS R consists of the following rules: f30_in -> U3(f30_in) f82_in -> U4(f82_in) The set Q consists of the following terms: f30_in f82_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) 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. ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: F30_IN -> F30_IN R is empty. The set Q consists of the following terms: f30_in f82_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) 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]. f30_in f82_in ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: F30_IN -> F30_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 1, "program": { "directives": [], "clauses": [ [ "(a)", "(b)" ], [ "(a)", "(e)" ], [ "(b)", "(c)" ], [ "(c)", "(d)" ], [ "(d)", "(b)" ], [ "(e)", "(f)" ], [ "(f)", "(g)" ], [ "(g)", "(e)" ] ] }, "graph": { "nodes": { "58": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "48": { "goal": [{ "clause": 0, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "140": { "goal": [{ "clause": 6, "scope": 6, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "141": { "goal": [{ "clause": -1, "scope": -1, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "142": { "goal": [{ "clause": 7, "scope": 7, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "143": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "123": { "goal": [{ "clause": 4, "scope": 4, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "113": { "goal": [{ "clause": 3, "scope": 3, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "135": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "126": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "138": { "goal": [{ "clause": 5, "scope": 5, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "139": { "goal": [{ "clause": -1, "scope": -1, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "107": { "goal": [{ "clause": -1, "scope": -1, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "9": { "goal": [ { "clause": 0, "scope": 1, "term": "(a)" }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "119": { "goal": [{ "clause": -1, "scope": -1, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "50": { "goal": [{ "clause": 1, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "96": { "goal": [{ "clause": 2, "scope": 2, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 9, "label": "CASE" }, { "from": 9, "to": 48, "label": "PARALLEL" }, { "from": 9, "to": 50, "label": "PARALLEL" }, { "from": 48, "to": 58, "label": "ONLY EVAL with clause\na :- b.\nand substitution" }, { "from": 50, "to": 135, "label": "ONLY EVAL with clause\na :- e.\nand substitution" }, { "from": 58, "to": 96, "label": "CASE" }, { "from": 96, "to": 107, "label": "ONLY EVAL with clause\nb :- c.\nand substitution" }, { "from": 107, "to": 113, "label": "CASE" }, { "from": 113, "to": 119, "label": "ONLY EVAL with clause\nc :- d.\nand substitution" }, { "from": 119, "to": 123, "label": "CASE" }, { "from": 123, "to": 126, "label": "ONLY EVAL with clause\nd :- b.\nand substitution" }, { "from": 126, "to": 58, "label": "INSTANCE" }, { "from": 135, "to": 138, "label": "CASE" }, { "from": 138, "to": 139, "label": "ONLY EVAL with clause\ne :- f.\nand substitution" }, { "from": 139, "to": 140, "label": "CASE" }, { "from": 140, "to": 141, "label": "ONLY EVAL with clause\nf :- g.\nand substitution" }, { "from": 141, "to": 142, "label": "CASE" }, { "from": 142, "to": 143, "label": "ONLY EVAL with clause\ng :- e.\nand substitution" }, { "from": 143, "to": 135, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (62) Complex Obligation (AND) ---------------------------------------- (63) Obligation: Rules: f139_in -> f140_in :|: TRUE f140_out -> f139_out :|: TRUE f141_out -> f140_out :|: TRUE f140_in -> f141_in :|: TRUE f135_in -> f138_in :|: TRUE f138_out -> f135_out :|: TRUE f138_in -> f139_in :|: TRUE f139_out -> f138_out :|: TRUE f141_in -> f142_in :|: TRUE f142_out -> f141_out :|: TRUE f135_out -> f143_out :|: TRUE f143_in -> f135_in :|: TRUE f142_in -> f143_in :|: TRUE f143_out -> f142_out :|: TRUE f1_in -> f9_in :|: TRUE f9_out -> f1_out :|: TRUE f48_out -> f9_out :|: TRUE f50_out -> f9_out :|: TRUE f9_in -> f50_in :|: TRUE f9_in -> f48_in :|: TRUE f50_in -> f135_in :|: TRUE f135_out -> f50_out :|: TRUE Start term: f1_in ---------------------------------------- (64) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f139_in -> f140_in :|: TRUE f140_in -> f141_in :|: TRUE f135_in -> f138_in :|: TRUE f138_in -> f139_in :|: TRUE f141_in -> f142_in :|: TRUE f143_in -> f135_in :|: TRUE f142_in -> f143_in :|: TRUE ---------------------------------------- (65) Obligation: Rules: f139_in -> f140_in :|: TRUE f140_in -> f141_in :|: TRUE f135_in -> f138_in :|: TRUE f138_in -> f139_in :|: TRUE f141_in -> f142_in :|: TRUE f143_in -> f135_in :|: TRUE f142_in -> f143_in :|: TRUE ---------------------------------------- (66) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (67) Obligation: Rules: f143_in -> f143_in :|: TRUE ---------------------------------------- (68) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (69) Obligation: Rules: f143_in -> f143_in :|: TRUE ---------------------------------------- (70) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f143_in -> f143_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (71) Obligation: Termination digraph: Nodes: (1) f143_in -> f143_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (72) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f143_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (73) Obligation: Rules: f143_in -> f143_in :|: TRUE ---------------------------------------- (74) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (75) NO ---------------------------------------- (76) Obligation: Rules: f107_out -> f96_out :|: TRUE f96_in -> f107_in :|: TRUE f126_out -> f123_out :|: TRUE f123_in -> f126_in :|: TRUE f113_out -> f107_out :|: TRUE f107_in -> f113_in :|: TRUE f113_in -> f119_in :|: TRUE f119_out -> f113_out :|: TRUE f119_in -> f123_in :|: TRUE f123_out -> f119_out :|: TRUE f58_out -> f126_out :|: TRUE f126_in -> f58_in :|: TRUE f96_out -> f58_out :|: TRUE f58_in -> f96_in :|: TRUE f1_in -> f9_in :|: TRUE f9_out -> f1_out :|: TRUE f48_out -> f9_out :|: TRUE f50_out -> f9_out :|: TRUE f9_in -> f50_in :|: TRUE f9_in -> f48_in :|: TRUE f58_out -> f48_out :|: TRUE f48_in -> f58_in :|: TRUE Start term: f1_in ---------------------------------------- (77) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f96_in -> f107_in :|: TRUE f123_in -> f126_in :|: TRUE f107_in -> f113_in :|: TRUE f113_in -> f119_in :|: TRUE f119_in -> f123_in :|: TRUE f126_in -> f58_in :|: TRUE f58_in -> f96_in :|: TRUE ---------------------------------------- (78) Obligation: Rules: f96_in -> f107_in :|: TRUE f123_in -> f126_in :|: TRUE f107_in -> f113_in :|: TRUE f113_in -> f119_in :|: TRUE f119_in -> f123_in :|: TRUE f126_in -> f58_in :|: TRUE f58_in -> f96_in :|: TRUE ---------------------------------------- (79) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (80) Obligation: Rules: f96_in -> f96_in :|: TRUE ---------------------------------------- (81) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (82) Obligation: Rules: f96_in -> f96_in :|: TRUE ---------------------------------------- (83) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f96_in -> f96_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (84) Obligation: Termination digraph: Nodes: (1) f96_in -> f96_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (85) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f96_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (86) Obligation: Rules: f96_in -> f96_in :|: TRUE ---------------------------------------- (87) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (88) NO