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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern a() w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS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) 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, 62 ms] (54) QTRS (55) QTRSRRRProof [EQUIVALENT, 4 ms] (56) QTRS (57) Overlay + Local Confluence [EQUIVALENT, 0 ms] (58) QTRS (59) DependencyPairsProof [EQUIVALENT, 0 ms] (60) QDP (61) DependencyGraphProof [EQUIVALENT, 0 ms] (62) AND (63) QDP (64) UsableRulesProof [EQUIVALENT, 0 ms] (65) QDP (66) QReductionProof [EQUIVALENT, 0 ms] (67) QDP (68) QDP (69) UsableRulesProof [EQUIVALENT, 0 ms] (70) QDP (71) QReductionProof [EQUIVALENT, 0 ms] (72) QDP (73) QTRSRRRProof [EQUIVALENT, 42 ms] (74) QTRS (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) IntTRSNonPeriodicNontermProof [COMPLETE, 8 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": 4, "program": { "directives": [], "clauses": [ [ "(a)", "(b)" ], [ "(a)", "(e)" ], [ "(b)", "(c)" ], [ "(c)", "(d)" ], [ "(d)", "(b)" ], [ "(e)", "(f)" ], [ "(f)", "(g)" ], [ "(g)", "(e)" ] ] }, "graph": { "nodes": { "77": { "goal": [{ "clause": 5, "scope": 6, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "45": { "goal": [{ "clause": 2, "scope": 2, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "78": { "goal": [{ "clause": -1, "scope": -1, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "46": { "goal": [ { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "68": { "goal": [{ "clause": -1, "scope": -1, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "79": { "goal": [{ "clause": 6, "scope": 7, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "69": { "goal": [{ "clause": 3, "scope": 3, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "4": { "goal": [{ "clause": -1, "scope": -1, "term": "(a)" }], "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": [] } }, "90": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "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": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "81": { "goal": [{ "clause": 7, "scope": 8, "term": "(g)" }], "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": [] } }, "72": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "73": { "goal": [{ "clause": 2, "scope": 5, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "41": { "goal": [ { "clause": -1, "scope": -1, "term": "(b)" }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "74": { "goal": [{ "clause": -1, "scope": -1, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "42": { "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": [] } }, "75": { "goal": [{ "clause": 1, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "76": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 4, "to": 5, "label": "CASE" }, { "from": 5, "to": 41, "label": "ONLY EVAL with clause\na :- b.\nand substitution" }, { "from": 41, "to": 42, "label": "CASE" }, { "from": 42, "to": 45, "label": "PARALLEL" }, { "from": 42, "to": 46, "label": "PARALLEL" }, { "from": 45, "to": 68, "label": "ONLY EVAL with clause\nb :- c.\nand substitution" }, { "from": 46, "to": 75, "label": "FAILURE" }, { "from": 68, "to": 69, "label": "CASE" }, { "from": 69, "to": 70, "label": "ONLY EVAL with clause\nc :- d.\nand substitution" }, { "from": 70, "to": 71, "label": "CASE" }, { "from": 71, "to": 72, "label": "ONLY EVAL with clause\nd :- b.\nand substitution" }, { "from": 72, "to": 73, "label": "CASE" }, { "from": 73, "to": 74, "label": "ONLY EVAL with clause\nb :- c.\nand substitution" }, { "from": 74, "to": 68, "label": "INSTANCE" }, { "from": 75, "to": 76, "label": "ONLY EVAL with clause\na :- e.\nand substitution" }, { "from": 76, "to": 77, "label": "CASE" }, { "from": 77, "to": 78, "label": "ONLY EVAL with clause\ne :- f.\nand substitution" }, { "from": 78, "to": 79, "label": "CASE" }, { "from": 79, "to": 80, "label": "ONLY EVAL with clause\nf :- g.\nand substitution" }, { "from": 80, "to": 81, "label": "CASE" }, { "from": 81, "to": 90, "label": "ONLY EVAL with clause\ng :- e.\nand substitution" }, { "from": 90, "to": 76, "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": 1, "program": { "directives": [], "clauses": [ [ "(a)", "(b)" ], [ "(a)", "(e)" ], [ "(b)", "(c)" ], [ "(c)", "(d)" ], [ "(d)", "(b)" ], [ "(e)", "(f)" ], [ "(f)", "(g)" ], [ "(g)", "(e)" ] ] }, "graph": { "nodes": { "55": { "goal": [{ "clause": 3, "scope": 3, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "66": { "goal": [{ "clause": 6, "scope": 6, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "67": { "goal": [{ "clause": -1, "scope": -1, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "57": { "goal": [{ "clause": -1, "scope": -1, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "47": { "goal": [{ "clause": 0, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "48": { "goal": [{ "clause": 1, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "59": { "goal": [{ "clause": 4, "scope": 4, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "160": { "goal": [{ "clause": 7, "scope": 7, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "161": { "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": [] } }, "9": { "goal": [ { "clause": 0, "scope": 1, "term": "(a)" }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "50": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "61": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "51": { "goal": [{ "clause": 2, "scope": 2, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "63": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "53": { "goal": [{ "clause": -1, "scope": -1, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "64": { "goal": [{ "clause": 5, "scope": 5, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "65": { "goal": [{ "clause": -1, "scope": -1, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 9, "label": "CASE" }, { "from": 9, "to": 47, "label": "PARALLEL" }, { "from": 9, "to": 48, "label": "PARALLEL" }, { "from": 47, "to": 50, "label": "ONLY EVAL with clause\na :- b.\nand substitution" }, { "from": 48, "to": 63, "label": "ONLY EVAL with clause\na :- e.\nand substitution" }, { "from": 50, "to": 51, "label": "CASE" }, { "from": 51, "to": 53, "label": "ONLY EVAL with clause\nb :- c.\nand substitution" }, { "from": 53, "to": 55, "label": "CASE" }, { "from": 55, "to": 57, "label": "ONLY EVAL with clause\nc :- d.\nand substitution" }, { "from": 57, "to": 59, "label": "CASE" }, { "from": 59, "to": 61, "label": "ONLY EVAL with clause\nd :- b.\nand substitution" }, { "from": 61, "to": 50, "label": "INSTANCE" }, { "from": 63, "to": 64, "label": "CASE" }, { "from": 64, "to": 65, "label": "ONLY EVAL with clause\ne :- f.\nand substitution" }, { "from": 65, "to": 66, "label": "CASE" }, { "from": 66, "to": 67, "label": "ONLY EVAL with clause\nf :- g.\nand substitution" }, { "from": 67, "to": 160, "label": "CASE" }, { "from": 160, "to": 161, "label": "ONLY EVAL with clause\ng :- e.\nand substitution" }, { "from": 161, "to": 63, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (52) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f1_in -> U1(f50_in) U1(f50_out1) -> f1_out1 f1_in -> U2(f63_in) U2(f63_out1) -> f1_out1 f50_in -> U3(f50_in) U3(f50_out1) -> f50_out1 f63_in -> U4(f63_in) U4(f63_out1) -> f63_out1 Q is empty. ---------------------------------------- (53) 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(f1_in) = 2 POL(f1_out1) = 1 POL(f50_in) = 0 POL(f50_out1) = 1 POL(f63_in) = 0 POL(f63_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f50_out1) -> f1_out1 f1_in -> U2(f63_in) U2(f63_out1) -> f1_out1 U3(f50_out1) -> f50_out1 U4(f63_out1) -> f63_out1 ---------------------------------------- (54) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f1_in -> U1(f50_in) f50_in -> U3(f50_in) f63_in -> U4(f63_in) Q is empty. ---------------------------------------- (55) 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(f1_in) = 2 POL(f50_in) = 0 POL(f63_in) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f1_in -> U1(f50_in) ---------------------------------------- (56) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f50_in -> U3(f50_in) f63_in -> U4(f63_in) Q is empty. ---------------------------------------- (57) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (58) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f50_in -> U3(f50_in) f63_in -> U4(f63_in) The set Q consists of the following terms: f50_in f63_in ---------------------------------------- (59) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: F50_IN -> F50_IN F63_IN -> F63_IN The TRS R consists of the following rules: f50_in -> U3(f50_in) f63_in -> U4(f63_in) The set Q consists of the following terms: f50_in f63_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (62) Complex Obligation (AND) ---------------------------------------- (63) Obligation: Q DP problem: The TRS P consists of the following rules: F63_IN -> F63_IN The TRS R consists of the following rules: f50_in -> U3(f50_in) f63_in -> U4(f63_in) The set Q consists of the following terms: f50_in f63_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (64) 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. ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: F63_IN -> F63_IN R is empty. The set Q consists of the following terms: f50_in f63_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (66) 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]. f50_in f63_in ---------------------------------------- (67) Obligation: Q DP problem: The TRS P consists of the following rules: F63_IN -> F63_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: F50_IN -> F50_IN The TRS R consists of the following rules: f50_in -> U3(f50_in) f63_in -> U4(f63_in) The set Q consists of the following terms: f50_in f63_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) 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. ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: F50_IN -> F50_IN R is empty. The set Q consists of the following terms: f50_in f63_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) 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]. f50_in f63_in ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: F50_IN -> F50_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) QTRSRRRProof (EQUIVALENT) Used ordering: f1_in/0) U1/1)YES( f50_in/0) f50_out1/0) f1_out1/0) U2/1)YES( f63_in/0) f63_out1/0) U3/1)YES( U4/1)YES( Quasi precedence: [f1_in, f50_in, f63_in] f50_out1 > f1_out1 f63_out1 > f1_out1 Status: f1_in: multiset status f50_in: multiset status f50_out1: multiset status f1_out1: multiset status f63_in: multiset status f63_out1: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f50_out1) -> f1_out1 U2(f63_out1) -> f1_out1 ---------------------------------------- (74) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f1_in -> U1(f50_in) f1_in -> U2(f63_in) f50_in -> U3(f50_in) U3(f50_out1) -> f50_out1 f63_in -> U4(f63_in) U4(f63_out1) -> f63_out1 Q is empty. ---------------------------------------- (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": 1, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "56": { "goal": [{ "clause": 3, "scope": 3, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "58": { "goal": [{ "clause": -1, "scope": -1, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "49": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "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": [] } }, "3": { "goal": [ { "clause": 0, "scope": 1, "term": "(a)" }, { "clause": 1, "scope": 1, "term": "(a)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "103": { "goal": [{ "clause": -1, "scope": -1, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "104": { "goal": [{ "clause": 5, "scope": 5, "term": "(e)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "105": { "goal": [{ "clause": -1, "scope": -1, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "106": { "goal": [{ "clause": 6, "scope": 6, "term": "(f)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "107": { "goal": [{ "clause": -1, "scope": -1, "term": "(g)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "60": { "goal": [{ "clause": 4, "scope": 4, "term": "(d)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "62": { "goal": [{ "clause": -1, "scope": -1, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "52": { "goal": [{ "clause": 2, "scope": 2, "term": "(b)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "43": { "goal": [{ "clause": 0, "scope": 1, "term": "(a)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "54": { "goal": [{ "clause": -1, "scope": -1, "term": "(c)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 3, "label": "CASE" }, { "from": 3, "to": 43, "label": "PARALLEL" }, { "from": 3, "to": 44, "label": "PARALLEL" }, { "from": 43, "to": 49, "label": "ONLY EVAL with clause\na :- b.\nand substitution" }, { "from": 44, "to": 103, "label": "ONLY EVAL with clause\na :- e.\nand substitution" }, { "from": 49, "to": 52, "label": "CASE" }, { "from": 52, "to": 54, "label": "ONLY EVAL with clause\nb :- c.\nand substitution" }, { "from": 54, "to": 56, "label": "CASE" }, { "from": 56, "to": 58, "label": "ONLY EVAL with clause\nc :- d.\nand substitution" }, { "from": 58, "to": 60, "label": "CASE" }, { "from": 60, "to": 62, "label": "ONLY EVAL with clause\nd :- b.\nand substitution" }, { "from": 62, "to": 49, "label": "INSTANCE" }, { "from": 103, "to": 104, "label": "CASE" }, { "from": 104, "to": 105, "label": "ONLY EVAL with clause\ne :- f.\nand substitution" }, { "from": 105, "to": 106, "label": "CASE" }, { "from": 106, "to": 107, "label": "ONLY EVAL with clause\nf :- g.\nand substitution" }, { "from": 107, "to": 162, "label": "CASE" }, { "from": 162, "to": 163, "label": "ONLY EVAL with clause\ng :- e.\nand substitution" }, { "from": 163, "to": 103, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (76) Complex Obligation (AND) ---------------------------------------- (77) Obligation: Rules: f162_out -> f107_out :|: TRUE f107_in -> f162_in :|: TRUE f163_out -> f162_out :|: TRUE f162_in -> f163_in :|: TRUE f103_out -> f163_out :|: TRUE f163_in -> f103_in :|: TRUE f105_in -> f106_in :|: TRUE f106_out -> f105_out :|: TRUE f103_in -> f104_in :|: TRUE f104_out -> f103_out :|: TRUE f104_in -> f105_in :|: TRUE f105_out -> f104_out :|: TRUE f107_out -> f106_out :|: TRUE f106_in -> f107_in :|: TRUE f2_in -> f3_in :|: TRUE f3_out -> f2_out :|: TRUE f3_in -> f43_in :|: TRUE f44_out -> f3_out :|: TRUE f43_out -> f3_out :|: TRUE f3_in -> f44_in :|: TRUE f44_in -> f103_in :|: TRUE f103_out -> f44_out :|: TRUE Start term: f2_in ---------------------------------------- (78) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f107_in -> f162_in :|: TRUE f162_in -> f163_in :|: TRUE f163_in -> f103_in :|: TRUE f105_in -> f106_in :|: TRUE f103_in -> f104_in :|: TRUE f104_in -> f105_in :|: TRUE f106_in -> f107_in :|: TRUE ---------------------------------------- (79) Obligation: Rules: f107_in -> f162_in :|: TRUE f162_in -> f163_in :|: TRUE f163_in -> f103_in :|: TRUE f105_in -> f106_in :|: TRUE f103_in -> f104_in :|: TRUE f104_in -> f105_in :|: TRUE f106_in -> f107_in :|: TRUE ---------------------------------------- (80) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (81) Obligation: Rules: f107_in -> f107_in :|: TRUE ---------------------------------------- (82) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (83) Obligation: Rules: f107_in -> f107_in :|: TRUE ---------------------------------------- (84) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f107_in -> f107_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (85) Obligation: Termination digraph: Nodes: (1) f107_in -> f107_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (86) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f107_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (87) Obligation: Rules: f107_in -> f107_in :|: TRUE ---------------------------------------- (88) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Proved unsatisfiability of the following formula, indicating that the system is never left after entering: ((run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T)) Proved satisfiability of the following formula, indicating that the system is entered at least once: (run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) ---------------------------------------- (89) NO ---------------------------------------- (90) Obligation: Rules: f58_in -> f60_in :|: TRUE f60_out -> f58_out :|: TRUE f54_in -> f56_in :|: TRUE f56_out -> f54_out :|: TRUE f58_out -> f56_out :|: TRUE f56_in -> f58_in :|: TRUE f49_out -> f62_out :|: TRUE f62_in -> f49_in :|: TRUE f62_out -> f60_out :|: TRUE f60_in -> f62_in :|: TRUE f52_out -> f49_out :|: TRUE f49_in -> f52_in :|: TRUE f54_out -> f52_out :|: TRUE f52_in -> f54_in :|: TRUE f2_in -> f3_in :|: TRUE f3_out -> f2_out :|: TRUE f3_in -> f43_in :|: TRUE f44_out -> f3_out :|: TRUE f43_out -> f3_out :|: TRUE f3_in -> f44_in :|: TRUE f43_in -> f49_in :|: TRUE f49_out -> f43_out :|: TRUE Start term: f2_in ---------------------------------------- (91) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f58_in -> f60_in :|: TRUE f54_in -> f56_in :|: TRUE f56_in -> f58_in :|: TRUE f62_in -> f49_in :|: TRUE f60_in -> f62_in :|: TRUE f49_in -> f52_in :|: TRUE f52_in -> f54_in :|: TRUE ---------------------------------------- (92) Obligation: Rules: f58_in -> f60_in :|: TRUE f54_in -> f56_in :|: TRUE f56_in -> f58_in :|: TRUE f62_in -> f49_in :|: TRUE f60_in -> f62_in :|: TRUE f49_in -> f52_in :|: TRUE f52_in -> f54_in :|: TRUE ---------------------------------------- (93) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (94) Obligation: Rules: f54_in -> f54_in :|: TRUE ---------------------------------------- (95) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (96) Obligation: Rules: f54_in -> f54_in :|: TRUE ---------------------------------------- (97) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f54_in -> f54_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (98) Obligation: Termination digraph: Nodes: (1) f54_in -> f54_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (99) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f54_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (100) Obligation: Rules: f54_in -> f54_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