/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern a() w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS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, 1 ms] (11) QDP (12) NonTerminationLoopProof [COMPLETE, 0 ms] (13) NO (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [EQUIVALENT, 0 ms] (18) QDP (19) PrologToPiTRSProof [SOUND, 0 ms] (20) PiTRS (21) DependencyPairsProof [EQUIVALENT, 0 ms] (22) PiDP (23) DependencyGraphProof [EQUIVALENT, 0 ms] (24) AND (25) PiDP (26) UsableRulesProof [EQUIVALENT, 0 ms] (27) PiDP (28) PiDPToQDPProof [EQUIVALENT, 0 ms] (29) QDP (30) NonTerminationLoopProof [COMPLETE, 0 ms] (31) NO (32) PiDP (33) UsableRulesProof [EQUIVALENT, 0 ms] (34) PiDP (35) PiDPToQDPProof [EQUIVALENT, 0 ms] (36) QDP (37) PrologToDTProblemTransformerProof [SOUND, 0 ms] (38) TRIPLES (39) TriplesToPiDPProof [SOUND, 0 ms] (40) PiDP (41) DependencyGraphProof [EQUIVALENT, 0 ms] (42) AND (43) PiDP (44) PiDPToQDPProof [EQUIVALENT, 0 ms] (45) QDP (46) NonTerminationLoopProof [COMPLETE, 0 ms] (47) NO (48) PiDP (49) PiDPToQDPProof [EQUIVALENT, 0 ms] (50) QDP (51) PrologToTRSTransformerProof [SOUND, 0 ms] (52) QTRS (53) QTRSRRRProof [EQUIVALENT, 56 ms] (54) QTRS (55) QTRSRRRProof [EQUIVALENT, 0 ms] (56) QTRS (57) QTRSRRRProof [EQUIVALENT, 3 ms] (58) QTRS (59) Overlay + Local Confluence [EQUIVALENT, 0 ms] (60) QTRS (61) DependencyPairsProof [EQUIVALENT, 0 ms] (62) QDP (63) DependencyGraphProof [EQUIVALENT, 0 ms] (64) AND (65) QDP (66) UsableRulesProof [EQUIVALENT, 0 ms] (67) QDP (68) QReductionProof [EQUIVALENT, 0 ms] (69) QDP (70) QDP (71) UsableRulesProof [EQUIVALENT, 0 ms] (72) QDP (73) QReductionProof [EQUIVALENT, 0 ms] (74) QDP (75) PrologToIRSwTTransformerProof [SOUND, 0 ms] (76) AND (77) IRSwT (78) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (79) IRSwT (80) IntTRSCompressionProof [EQUIVALENT, 20 ms] (81) IRSwT (82) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (83) IRSwT (84) IRSwTTerminationDigraphProof [EQUIVALENT, 7 ms] (85) IRSwT (86) FilterProof [EQUIVALENT, 0 ms] (87) IntTRS (88) IntTRSPeriodicNontermProof [COMPLETE, 0 ms] (89) NO (90) IRSwT (91) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (92) IRSwT (93) IntTRSCompressionProof [EQUIVALENT, 0 ms] (94) IRSwT (95) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (96) IRSwT (97) IRSwTTerminationDigraphProof [EQUIVALENT, 1 ms] (98) IRSwT (99) FilterProof [EQUIVALENT, 0 ms] (100) IntTRS (101) IntTRSPeriodicNontermProof [COMPLETE, 4 ms] (102) 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) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = F_IN_ evaluates to t =F_IN_ Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence F_IN_ -> G_IN_ with rule F_IN_ -> G_IN_ at position [] and matcher [ ] G_IN_ -> E_IN_ with rule G_IN_ -> E_IN_ at position [] and matcher [ ] E_IN_ -> F_IN_ with rule E_IN_ -> F_IN_ Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (13) NO ---------------------------------------- (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_ 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 ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) 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 ---------------------------------------- (17) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: B_IN_ -> C_IN_ C_IN_ -> D_IN_ D_IN_ -> B_IN_ R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) 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 ---------------------------------------- (20) 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. ---------------------------------------- (21) 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 ---------------------------------------- (22) 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 ---------------------------------------- (23) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 10 less nodes. ---------------------------------------- (24) Complex Obligation (AND) ---------------------------------------- (25) 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 ---------------------------------------- (26) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (27) 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 ---------------------------------------- (28) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (29) 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. ---------------------------------------- (30) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = F_IN_ evaluates to t =F_IN_ Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence F_IN_ -> G_IN_ with rule F_IN_ -> G_IN_ at position [] and matcher [ ] G_IN_ -> E_IN_ with rule G_IN_ -> E_IN_ at position [] and matcher [ ] E_IN_ -> F_IN_ with rule E_IN_ -> F_IN_ Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (31) NO ---------------------------------------- (32) 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 ---------------------------------------- (33) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (34) 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 ---------------------------------------- (35) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (36) Obligation: Q 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. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (37) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(a)", "(b)" ], [ "(a)", "(e)" ], [ "(b)", "(c)" ], [ "(c)", "(d)" ], [ "(d)", "(b)" ], [ "(e)", "(f)" ], [ "(f)", "(g)" ], [ "(g)", "(e)" ] ] }, "graph": { "nodes": { "160": { "goal": [{ "clause": 6, "scope": 7, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "150": { "goal": [{ "clause": 3, "scope": 3, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "161": { "goal": [{ "clause": -1, "scope": -1, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "151": { "goal": [{ "clause": -1, "scope": -1, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "162": { "goal": [{ "clause": 7, "scope": 8, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "152": { "goal": [{ "clause": 4, "scope": 4, "term": "(d)" }], "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": [] } }, "153": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "154": { "goal": [{ "clause": 2, "scope": 5, "term": "(b)" }], "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": [] } }, "155": { "goal": [{ "clause": -1, "scope": -1, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "145": { "goal": [ { "clause": -1, "scope": -1, "term": "(b)" }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "156": { "goal": [{ "clause": 1, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "146": { "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": [] } }, "157": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "4": { "goal": [ { "clause": 0, "scope": 1, "term": "(a)" }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "147": { "goal": [{ "clause": 2, "scope": 2, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "158": { "goal": [{ "clause": 5, "scope": 6, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "148": { "goal": [ { "clause": -1, "scope": 2, "term": null }, { "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": [] } }, "149": { "goal": [{ "clause": -1, "scope": -1, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 4, "label": "CASE" }, { "from": 4, "to": 145, "label": "ONLY EVAL with clause\na :- b.\nand substitution" }, { "from": 145, "to": 146, "label": "CASE" }, { "from": 146, "to": 147, "label": "PARALLEL" }, { "from": 146, "to": 148, "label": "PARALLEL" }, { "from": 147, "to": 149, "label": "ONLY EVAL with clause\nb :- c.\nand substitution" }, { "from": 148, "to": 156, "label": "FAILURE" }, { "from": 149, "to": 150, "label": "CASE" }, { "from": 150, "to": 151, "label": "ONLY EVAL with clause\nc :- d.\nand substitution" }, { "from": 151, "to": 152, "label": "CASE" }, { "from": 152, "to": 153, "label": "ONLY EVAL with clause\nd :- b.\nand substitution" }, { "from": 153, "to": 154, "label": "CASE" }, { "from": 154, "to": 155, "label": "ONLY EVAL with clause\nb :- c.\nand substitution" }, { "from": 155, "to": 149, "label": "INSTANCE" }, { "from": 156, "to": 157, "label": "ONLY EVAL with clause\na :- e.\nand substitution" }, { "from": 157, "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": 157, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (38) Obligation: Triples: cA :- cA. eB :- eB. aC :- cA. aC :- eB. Clauses: ccA :- ccA. ecB :- ecB. Afs: aC = aC ---------------------------------------- (39) 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 ---------------------------------------- (40) 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 ---------------------------------------- (41) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes. ---------------------------------------- (42) Complex Obligation (AND) ---------------------------------------- (43) 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 ---------------------------------------- (44) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (45) 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. ---------------------------------------- (46) 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 = EB_IN_ evaluates to t =EB_IN_ Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from EB_IN_ to EB_IN_. ---------------------------------------- (47) NO ---------------------------------------- (48) 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 ---------------------------------------- (49) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: CA_IN_ -> CA_IN_ R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (51) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 5, "program": { "directives": [], "clauses": [ [ "(a)", "(b)" ], [ "(a)", "(e)" ], [ "(b)", "(c)" ], [ "(c)", "(d)" ], [ "(d)", "(b)" ], [ "(e)", "(f)" ], [ "(f)", "(g)" ], [ "(g)", "(e)" ] ] }, "graph": { "nodes": { "99": { "goal": [{ "clause": -1, "scope": -1, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "78": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "142": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "100": { "goal": [{ "clause": 4, "scope": 4, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "101": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "102": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "103": { "goal": [{ "clause": 5, "scope": 5, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "136": { "goal": [{ "clause": 7, "scope": 7, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5": { "goal": [{ "clause": -1, "scope": -1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "104": { "goal": [{ "clause": -1, "scope": -1, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "6": { "goal": [ { "clause": 0, "scope": 1, "term": "(a)" }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "90": { "goal": [{ "clause": -1, "scope": -1, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "128": { "goal": [{ "clause": -1, "scope": -1, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "118": { "goal": [{ "clause": 6, "scope": 6, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "83": { "goal": [{ "clause": 2, "scope": 2, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "73": { "goal": [{ "clause": 0, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "75": { "goal": [{ "clause": 1, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "97": { "goal": [{ "clause": 3, "scope": 3, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 5, "to": 6, "label": "CASE" }, { "from": 6, "to": 73, "label": "PARALLEL" }, { "from": 6, "to": 75, "label": "PARALLEL" }, { "from": 73, "to": 78, "label": "ONLY EVAL with clause\na :- b.\nand substitution" }, { "from": 75, "to": 102, "label": "ONLY EVAL with clause\na :- e.\nand substitution" }, { "from": 78, "to": 83, "label": "CASE" }, { "from": 83, "to": 90, "label": "ONLY EVAL with clause\nb :- c.\nand substitution" }, { "from": 90, "to": 97, "label": "CASE" }, { "from": 97, "to": 99, "label": "ONLY EVAL with clause\nc :- d.\nand substitution" }, { "from": 99, "to": 100, "label": "CASE" }, { "from": 100, "to": 101, "label": "ONLY EVAL with clause\nd :- b.\nand substitution" }, { "from": 101, "to": 78, "label": "INSTANCE" }, { "from": 102, "to": 103, "label": "CASE" }, { "from": 103, "to": 104, "label": "ONLY EVAL with clause\ne :- f.\nand substitution" }, { "from": 104, "to": 118, "label": "CASE" }, { "from": 118, "to": 128, "label": "ONLY EVAL with clause\nf :- g.\nand substitution" }, { "from": 128, "to": 136, "label": "CASE" }, { "from": 136, "to": 142, "label": "ONLY EVAL with clause\ng :- e.\nand substitution" }, { "from": 142, "to": 102, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (52) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f5_in -> U1(f78_in) U1(f78_out1) -> f5_out1 f5_in -> U2(f102_in) U2(f102_out1) -> f5_out1 f78_in -> U3(f78_in) U3(f78_out1) -> f78_out1 f102_in -> U4(f102_in) U4(f102_out1) -> f102_out1 Q is empty. ---------------------------------------- (53) QTRSRRRProof (EQUIVALENT) Used ordering: f5_in/0) U1/1)YES( f78_in/0) f78_out1/0) f5_out1/0) U2/1)YES( f102_in/0) f102_out1/0) U3/1)YES( U4/1)YES( Quasi precedence: [f5_in, f78_in, f102_in] f78_out1 > f5_out1 f102_out1 > f5_out1 Status: f5_in: multiset status f78_in: multiset status f78_out1: multiset status f5_out1: multiset status f102_in: multiset status f102_out1: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f78_out1) -> f5_out1 U2(f102_out1) -> f5_out1 ---------------------------------------- (54) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f5_in -> U1(f78_in) f5_in -> U2(f102_in) f78_in -> U3(f78_in) U3(f78_out1) -> f78_out1 f102_in -> U4(f102_in) U4(f102_out1) -> f102_out1 Q is empty. ---------------------------------------- (55) QTRSRRRProof (EQUIVALENT) Used ordering: f5_in/0) U1/1(YES) f78_in/0) U2/1(YES) f102_in/0) U3/1)YES( f78_out1/0) U4/1)YES( f102_out1/0) Quasi precedence: f5_in > U1_1 f5_in > f78_in f5_in > U2_1 f5_in > f102_in Status: f5_in: multiset status U1_1: multiset status f78_in: multiset status U2_1: multiset status f102_in: multiset status f78_out1: multiset status f102_out1: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f5_in -> U1(f78_in) f5_in -> U2(f102_in) ---------------------------------------- (56) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f78_in -> U3(f78_in) U3(f78_out1) -> f78_out1 f102_in -> U4(f102_in) U4(f102_out1) -> f102_out1 Q is empty. ---------------------------------------- (57) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U3(x_1)) = 2*x_1 POL(U4(x_1)) = 2*x_1 POL(f102_in) = 0 POL(f102_out1) = 1 POL(f78_in) = 0 POL(f78_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U3(f78_out1) -> f78_out1 U4(f102_out1) -> f102_out1 ---------------------------------------- (58) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f78_in -> U3(f78_in) f102_in -> U4(f102_in) Q is empty. ---------------------------------------- (59) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (60) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f78_in -> U3(f78_in) f102_in -> U4(f102_in) The set Q consists of the following terms: f78_in f102_in ---------------------------------------- (61) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: F78_IN -> F78_IN F102_IN -> F102_IN The TRS R consists of the following rules: f78_in -> U3(f78_in) f102_in -> U4(f102_in) The set Q consists of the following terms: f78_in f102_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (64) Complex Obligation (AND) ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: F102_IN -> F102_IN The TRS R consists of the following rules: f78_in -> U3(f78_in) f102_in -> U4(f102_in) The set Q consists of the following terms: f78_in f102_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (66) 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. ---------------------------------------- (67) Obligation: Q DP problem: The TRS P consists of the following rules: F102_IN -> F102_IN R is empty. The set Q consists of the following terms: f78_in f102_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (68) 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]. f78_in f102_in ---------------------------------------- (69) Obligation: Q DP problem: The TRS P consists of the following rules: F102_IN -> F102_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: F78_IN -> F78_IN The TRS R consists of the following rules: f78_in -> U3(f78_in) f102_in -> U4(f102_in) The set Q consists of the following terms: f78_in f102_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) 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. ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: F78_IN -> F78_IN R is empty. The set Q consists of the following terms: f78_in f102_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) 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]. f78_in f102_in ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: F78_IN -> F78_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (75) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 2, "program": { "directives": [], "clauses": [ [ "(a)", "(b)" ], [ "(a)", "(e)" ], [ "(b)", "(c)" ], [ "(c)", "(d)" ], [ "(d)", "(b)" ], [ "(e)", "(f)" ], [ "(f)", "(g)" ], [ "(g)", "(e)" ] ] }, "graph": { "nodes": { "44": { "goal": [{ "clause": 2, "scope": 2, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "67": { "goal": [{ "clause": -1, "scope": -1, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "24": { "goal": [{ "clause": 0, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "68": { "goal": [{ "clause": 4, "scope": 4, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "25": { "goal": [{ "clause": 1, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "47": { "goal": [{ "clause": -1, "scope": -1, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "69": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "26": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "3": { "goal": [ { "clause": 0, "scope": 1, "term": "(a)" }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "80": { "goal": [{ "clause": -1, "scope": -1, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "70": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "71": { "goal": [{ "clause": 5, "scope": 5, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "93": { "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": [] } }, "72": { "goal": [{ "clause": -1, "scope": -1, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "86": { "goal": [{ "clause": 7, "scope": 7, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "76": { "goal": [{ "clause": 6, "scope": 6, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 3, "label": "CASE" }, { "from": 3, "to": 24, "label": "PARALLEL" }, { "from": 3, "to": 25, "label": "PARALLEL" }, { "from": 24, "to": 26, "label": "ONLY EVAL with clause\na :- b.\nand substitution" }, { "from": 25, "to": 70, "label": "ONLY EVAL with clause\na :- e.\nand substitution" }, { "from": 26, "to": 44, "label": "CASE" }, { "from": 44, "to": 47, "label": "ONLY EVAL with clause\nb :- c.\nand substitution" }, { "from": 47, "to": 61, "label": "CASE" }, { "from": 61, "to": 67, "label": "ONLY EVAL with clause\nc :- d.\nand substitution" }, { "from": 67, "to": 68, "label": "CASE" }, { "from": 68, "to": 69, "label": "ONLY EVAL with clause\nd :- b.\nand substitution" }, { "from": 69, "to": 26, "label": "INSTANCE" }, { "from": 70, "to": 71, "label": "CASE" }, { "from": 71, "to": 72, "label": "ONLY EVAL with clause\ne :- f.\nand substitution" }, { "from": 72, "to": 76, "label": "CASE" }, { "from": 76, "to": 80, "label": "ONLY EVAL with clause\nf :- g.\nand substitution" }, { "from": 80, "to": 86, "label": "CASE" }, { "from": 86, "to": 93, "label": "ONLY EVAL with clause\ng :- e.\nand substitution" }, { "from": 93, "to": 70, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (76) Complex Obligation (AND) ---------------------------------------- (77) Obligation: Rules: f86_in -> f93_in :|: TRUE f93_out -> f86_out :|: TRUE f72_in -> f76_in :|: TRUE f76_out -> f72_out :|: TRUE f80_out -> f76_out :|: TRUE f76_in -> f80_in :|: TRUE f71_out -> f70_out :|: TRUE f70_in -> f71_in :|: TRUE f72_out -> f71_out :|: TRUE f71_in -> f72_in :|: TRUE f70_out -> f93_out :|: TRUE f93_in -> f70_in :|: TRUE f80_in -> f86_in :|: TRUE f86_out -> f80_out :|: TRUE f2_in -> f3_in :|: TRUE f3_out -> f2_out :|: TRUE f3_in -> f24_in :|: TRUE f24_out -> f3_out :|: TRUE f3_in -> f25_in :|: TRUE f25_out -> f3_out :|: TRUE f70_out -> f25_out :|: TRUE f25_in -> f70_in :|: TRUE Start term: f2_in ---------------------------------------- (78) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f86_in -> f93_in :|: TRUE f72_in -> f76_in :|: TRUE f76_in -> f80_in :|: TRUE f70_in -> f71_in :|: TRUE f71_in -> f72_in :|: TRUE f93_in -> f70_in :|: TRUE f80_in -> f86_in :|: TRUE ---------------------------------------- (79) Obligation: Rules: f86_in -> f93_in :|: TRUE f72_in -> f76_in :|: TRUE f76_in -> f80_in :|: TRUE f70_in -> f71_in :|: TRUE f71_in -> f72_in :|: TRUE f93_in -> f70_in :|: TRUE f80_in -> f86_in :|: TRUE ---------------------------------------- (80) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (81) Obligation: Rules: f86_in -> f86_in :|: TRUE ---------------------------------------- (82) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (83) Obligation: Rules: f86_in -> f86_in :|: TRUE ---------------------------------------- (84) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f86_in -> f86_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (85) Obligation: Termination digraph: Nodes: (1) f86_in -> f86_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (86) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f86_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (87) Obligation: Rules: f86_in -> f86_in :|: TRUE ---------------------------------------- (88) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (89) NO ---------------------------------------- (90) Obligation: Rules: f26_in -> f44_in :|: TRUE f44_out -> f26_out :|: TRUE f61_in -> f67_in :|: TRUE f67_out -> f61_out :|: TRUE f61_out -> f47_out :|: TRUE f47_in -> f61_in :|: TRUE f69_out -> f68_out :|: TRUE f68_in -> f69_in :|: TRUE f69_in -> f26_in :|: TRUE f26_out -> f69_out :|: TRUE f67_in -> f68_in :|: TRUE f68_out -> f67_out :|: TRUE f47_out -> f44_out :|: TRUE f44_in -> f47_in :|: TRUE f2_in -> f3_in :|: TRUE f3_out -> f2_out :|: TRUE f3_in -> f24_in :|: TRUE f24_out -> f3_out :|: TRUE f3_in -> f25_in :|: TRUE f25_out -> f3_out :|: TRUE f24_in -> f26_in :|: TRUE f26_out -> f24_out :|: TRUE Start term: f2_in ---------------------------------------- (91) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f26_in -> f44_in :|: TRUE f61_in -> f67_in :|: TRUE f47_in -> f61_in :|: TRUE f68_in -> f69_in :|: TRUE f69_in -> f26_in :|: TRUE f67_in -> f68_in :|: TRUE f44_in -> f47_in :|: TRUE ---------------------------------------- (92) Obligation: Rules: f26_in -> f44_in :|: TRUE f61_in -> f67_in :|: TRUE f47_in -> f61_in :|: TRUE f68_in -> f69_in :|: TRUE f69_in -> f26_in :|: TRUE f67_in -> f68_in :|: TRUE f44_in -> f47_in :|: TRUE ---------------------------------------- (93) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (94) Obligation: Rules: f47_in -> f47_in :|: TRUE ---------------------------------------- (95) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (96) Obligation: Rules: f47_in -> f47_in :|: TRUE ---------------------------------------- (97) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f47_in -> f47_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (98) Obligation: Termination digraph: Nodes: (1) f47_in -> f47_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (99) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f47_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (100) Obligation: Rules: f47_in -> f47_in :|: TRUE ---------------------------------------- (101) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (102) NO