MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern suffix(g,a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) PiDP (7) UsableRulesProof [EQUIVALENT, 0 ms] (8) PiDP (9) PiDPToQDPProof [SOUND, 12 ms] (10) QDP (11) PrologToPiTRSProof [SOUND, 0 ms] (12) PiTRS (13) DependencyPairsProof [EQUIVALENT, 0 ms] (14) PiDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) PiDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) PiDP (19) PiDPToQDPProof [SOUND, 0 ms] (20) QDP (21) PrologToDTProblemTransformerProof [SOUND, 0 ms] (22) TRIPLES (23) TriplesToPiDPProof [SOUND, 10 ms] (24) PiDP (25) DependencyGraphProof [EQUIVALENT, 0 ms] (26) PiDP (27) PiDPToQDPProof [SOUND, 0 ms] (28) QDP (29) PrologToTRSTransformerProof [SOUND, 0 ms] (30) QTRS (31) DependencyPairsProof [EQUIVALENT, 0 ms] (32) QDP (33) DependencyGraphProof [EQUIVALENT, 0 ms] (34) QDP (35) MNOCProof [EQUIVALENT, 0 ms] (36) QDP (37) UsableRulesProof [EQUIVALENT, 0 ms] (38) QDP (39) QReductionProof [EQUIVALENT, 0 ms] (40) QDP (41) PrologToIRSwTTransformerProof [SOUND, 0 ms] (42) IRSwT (43) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (44) IRSwT (45) IntTRSCompressionProof [EQUIVALENT, 7 ms] (46) IRSwT (47) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (48) IRSwT (49) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (50) IRSwT (51) FilterProof [EQUIVALENT, 3 ms] (52) IntTRS (53) IntTRSPeriodicNontermProof [COMPLETE, 0 ms] (54) NO ---------------------------------------- (0) Obligation: Clauses: suffix(Xs, Ys) :- app(X1, Xs, Ys). app([], X, X). app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). Query: suffix(g,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: suffix_in_2: (b,f) app_in_3: (f,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys)) app_in_aga([], X, X) -> app_out_aga([], X, X) app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys) The argument filtering Pi contains the following mapping: suffix_in_ga(x1, x2) = suffix_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x1, x3) app_in_aga(x1, x2, x3) = app_in_aga(x2) app_out_aga(x1, x2, x3) = app_out_aga(x1, x2, x3) U2_aga(x1, x2, x3, x4, x5) = U2_aga(x3, x5) .(x1, x2) = .(x2) suffix_out_ga(x1, x2) = suffix_out_ga(x1, x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys)) app_in_aga([], X, X) -> app_out_aga([], X, X) app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys) The argument filtering Pi contains the following mapping: suffix_in_ga(x1, x2) = suffix_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x1, x3) app_in_aga(x1, x2, x3) = app_in_aga(x2) app_out_aga(x1, x2, x3) = app_out_aga(x1, x2, x3) U2_aga(x1, x2, x3, x4, x5) = U2_aga(x3, x5) .(x1, x2) = .(x2) suffix_out_ga(x1, x2) = suffix_out_ga(x1, x2) ---------------------------------------- (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: SUFFIX_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, app_in_aga(X1, Xs, Ys)) SUFFIX_IN_GA(Xs, Ys) -> APP_IN_AGA(X1, Xs, Ys) APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U2_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs) The TRS R consists of the following rules: suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys)) app_in_aga([], X, X) -> app_out_aga([], X, X) app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys) The argument filtering Pi contains the following mapping: suffix_in_ga(x1, x2) = suffix_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x1, x3) app_in_aga(x1, x2, x3) = app_in_aga(x2) app_out_aga(x1, x2, x3) = app_out_aga(x1, x2, x3) U2_aga(x1, x2, x3, x4, x5) = U2_aga(x3, x5) .(x1, x2) = .(x2) suffix_out_ga(x1, x2) = suffix_out_ga(x1, x2) SUFFIX_IN_GA(x1, x2) = SUFFIX_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x1, x3) APP_IN_AGA(x1, x2, x3) = APP_IN_AGA(x2) U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: SUFFIX_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, app_in_aga(X1, Xs, Ys)) SUFFIX_IN_GA(Xs, Ys) -> APP_IN_AGA(X1, Xs, Ys) APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U2_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs) The TRS R consists of the following rules: suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys)) app_in_aga([], X, X) -> app_out_aga([], X, X) app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys) The argument filtering Pi contains the following mapping: suffix_in_ga(x1, x2) = suffix_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x1, x3) app_in_aga(x1, x2, x3) = app_in_aga(x2) app_out_aga(x1, x2, x3) = app_out_aga(x1, x2, x3) U2_aga(x1, x2, x3, x4, x5) = U2_aga(x3, x5) .(x1, x2) = .(x2) suffix_out_ga(x1, x2) = suffix_out_ga(x1, x2) SUFFIX_IN_GA(x1, x2) = SUFFIX_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA'(x1, x3) APP_IN_AGA(x1, x2, x3) = APP_IN_AGA(x2) U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs) The TRS R consists of the following rules: suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys)) app_in_aga([], X, X) -> app_out_aga([], X, X) app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys) The argument filtering Pi contains the following mapping: suffix_in_ga(x1, x2) = suffix_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x1, x3) app_in_aga(x1, x2, x3) = app_in_aga(x2) app_out_aga(x1, x2, x3) = app_out_aga(x1, x2, x3) U2_aga(x1, x2, x3, x4, x5) = U2_aga(x3, x5) .(x1, x2) = .(x2) suffix_out_ga(x1, x2) = suffix_out_ga(x1, x2) APP_IN_AGA(x1, x2, x3) = APP_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APP_IN_AGA(x1, x2, x3) = APP_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: APP_IN_AGA(Ys) -> APP_IN_AGA(Ys) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (11) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: suffix_in_2: (b,f) app_in_3: (f,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys)) app_in_aga([], X, X) -> app_out_aga([], X, X) app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys) The argument filtering Pi contains the following mapping: suffix_in_ga(x1, x2) = suffix_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) app_in_aga(x1, x2, x3) = app_in_aga(x2) app_out_aga(x1, x2, x3) = app_out_aga(x1, x3) U2_aga(x1, x2, x3, x4, x5) = U2_aga(x5) .(x1, x2) = .(x2) suffix_out_ga(x1, x2) = suffix_out_ga(x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (12) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys)) app_in_aga([], X, X) -> app_out_aga([], X, X) app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys) The argument filtering Pi contains the following mapping: suffix_in_ga(x1, x2) = suffix_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) app_in_aga(x1, x2, x3) = app_in_aga(x2) app_out_aga(x1, x2, x3) = app_out_aga(x1, x3) U2_aga(x1, x2, x3, x4, x5) = U2_aga(x5) .(x1, x2) = .(x2) suffix_out_ga(x1, x2) = suffix_out_ga(x2) ---------------------------------------- (13) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: SUFFIX_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, app_in_aga(X1, Xs, Ys)) SUFFIX_IN_GA(Xs, Ys) -> APP_IN_AGA(X1, Xs, Ys) APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U2_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs) The TRS R consists of the following rules: suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys)) app_in_aga([], X, X) -> app_out_aga([], X, X) app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys) The argument filtering Pi contains the following mapping: suffix_in_ga(x1, x2) = suffix_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) app_in_aga(x1, x2, x3) = app_in_aga(x2) app_out_aga(x1, x2, x3) = app_out_aga(x1, x3) U2_aga(x1, x2, x3, x4, x5) = U2_aga(x5) .(x1, x2) = .(x2) suffix_out_ga(x1, x2) = suffix_out_ga(x2) SUFFIX_IN_GA(x1, x2) = SUFFIX_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) APP_IN_AGA(x1, x2, x3) = APP_IN_AGA(x2) U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: SUFFIX_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, app_in_aga(X1, Xs, Ys)) SUFFIX_IN_GA(Xs, Ys) -> APP_IN_AGA(X1, Xs, Ys) APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U2_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs) The TRS R consists of the following rules: suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys)) app_in_aga([], X, X) -> app_out_aga([], X, X) app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys) The argument filtering Pi contains the following mapping: suffix_in_ga(x1, x2) = suffix_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) app_in_aga(x1, x2, x3) = app_in_aga(x2) app_out_aga(x1, x2, x3) = app_out_aga(x1, x3) U2_aga(x1, x2, x3, x4, x5) = U2_aga(x5) .(x1, x2) = .(x2) suffix_out_ga(x1, x2) = suffix_out_ga(x2) SUFFIX_IN_GA(x1, x2) = SUFFIX_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) APP_IN_AGA(x1, x2, x3) = APP_IN_AGA(x2) U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs) The TRS R consists of the following rules: suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys)) app_in_aga([], X, X) -> app_out_aga([], X, X) app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs)) U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys) The argument filtering Pi contains the following mapping: suffix_in_ga(x1, x2) = suffix_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) app_in_aga(x1, x2, x3) = app_in_aga(x2) app_out_aga(x1, x2, x3) = app_out_aga(x1, x3) U2_aga(x1, x2, x3, x4, x5) = U2_aga(x5) .(x1, x2) = .(x2) suffix_out_ga(x1, x2) = suffix_out_ga(x2) APP_IN_AGA(x1, x2, x3) = APP_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APP_IN_AGA(x1, x2, x3) = APP_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: APP_IN_AGA(Ys) -> APP_IN_AGA(Ys) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(suffix Xs Ys)", "(app X1 Xs Ys)" ], [ "(app ([]) X X)", null ], [ "(app (. X Xs) Ys (. X Zs))", "(app Xs Ys Zs)" ] ] }, "graph": { "nodes": { "66": { "goal": [{ "clause": -1, "scope": -1, "term": "(app X6 T5 T7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": ["X6"], "exprvars": [] } }, "67": { "goal": [ { "clause": 1, "scope": 2, "term": "(app X6 T5 T7)" }, { "clause": 2, "scope": 2, "term": "(app X6 T5 T7)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": ["X6"], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(suffix T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "69": { "goal": [{ "clause": 1, "scope": 2, "term": "(app X6 T5 T7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": ["X6"], "exprvars": [] } }, "8": { "goal": [{ "clause": 0, "scope": 1, "term": "(suffix T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "70": { "goal": [{ "clause": 2, "scope": 2, "term": "(app X6 T5 T7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": ["X6"], "exprvars": [] } }, "71": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "72": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "75": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "86": { "goal": [{ "clause": -1, "scope": -1, "term": "(app X29 T19 T22)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": ["X29"], "exprvars": [] } }, "87": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 8, "label": "CASE" }, { "from": 8, "to": 66, "label": "ONLY EVAL with clause\nsuffix(X4, X5) :- app(X6, X4, X5).\nand substitutionT1 -> T5,\nX4 -> T5,\nT2 -> T7,\nX5 -> T7,\nT6 -> T7" }, { "from": 66, "to": 67, "label": "CASE" }, { "from": 67, "to": 69, "label": "PARALLEL" }, { "from": 67, "to": 70, "label": "PARALLEL" }, { "from": 69, "to": 71, "label": "EVAL with clause\napp([], X11, X11).\nand substitutionX6 -> [],\nT5 -> T12,\nX11 -> T12,\nT7 -> T12" }, { "from": 69, "to": 72, "label": "EVAL-BACKTRACK" }, { "from": 70, "to": 86, "label": "EVAL with clause\napp(.(X24, X25), X26, .(X24, X27)) :- app(X25, X26, X27).\nand substitutionX24 -> T20,\nX25 -> X29,\nX6 -> .(T20, X29),\nT5 -> T19,\nX26 -> T19,\nX28 -> T20,\nX27 -> T22,\nT7 -> .(T20, T22),\nT21 -> T22" }, { "from": 70, "to": 87, "label": "EVAL-BACKTRACK" }, { "from": 71, "to": 75, "label": "SUCCESS" }, { "from": 86, "to": 66, "label": "INSTANCE with matching:\nX6 -> X29\nT5 -> T19\nT7 -> T22" } ], "type": "Graph" } } ---------------------------------------- (22) Obligation: Triples: appA(.(X1, X2), X3, .(X1, X4)) :- appA(X2, X3, X4). suffixB(X1, X2) :- appA(X3, X1, X2). Clauses: appcA([], X1, X1). appcA(.(X1, X2), X3, .(X1, X4)) :- appcA(X2, X3, X4). Afs: suffixB(x1, x2) = suffixB(x1) ---------------------------------------- (23) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: suffixB_in_2: (b,f) appA_in_3: (f,b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: SUFFIXB_IN_GA(X1, X2) -> U2_GA(X1, X2, appA_in_aga(X3, X1, X2)) SUFFIXB_IN_GA(X1, X2) -> APPA_IN_AGA(X3, X1, X2) APPA_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U1_AGA(X1, X2, X3, X4, appA_in_aga(X2, X3, X4)) APPA_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_AGA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: appA_in_aga(x1, x2, x3) = appA_in_aga(x2) .(x1, x2) = .(x2) SUFFIXB_IN_GA(x1, x2) = SUFFIXB_IN_GA(x1) U2_GA(x1, x2, x3) = U2_GA(x1, x3) APPA_IN_AGA(x1, x2, x3) = APPA_IN_AGA(x2) U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x3, x5) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (24) Obligation: Pi DP problem: The TRS P consists of the following rules: SUFFIXB_IN_GA(X1, X2) -> U2_GA(X1, X2, appA_in_aga(X3, X1, X2)) SUFFIXB_IN_GA(X1, X2) -> APPA_IN_AGA(X3, X1, X2) APPA_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U1_AGA(X1, X2, X3, X4, appA_in_aga(X2, X3, X4)) APPA_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_AGA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: appA_in_aga(x1, x2, x3) = appA_in_aga(x2) .(x1, x2) = .(x2) SUFFIXB_IN_GA(x1, x2) = SUFFIXB_IN_GA'(x1) U2_GA(x1, x2, x3) = U2_GA(x1, x3) APPA_IN_AGA(x1, x2, x3) = APPA_IN_AGA(x2) U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (25) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (26) Obligation: Pi DP problem: The TRS P consists of the following rules: APPA_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_AGA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPA_IN_AGA(x1, x2, x3) = APPA_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (27) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: APPA_IN_AGA(X3) -> APPA_IN_AGA(X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (29) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 1, "program": { "directives": [], "clauses": [ [ "(suffix Xs Ys)", "(app X1 Xs Ys)" ], [ "(app ([]) X X)", null ], [ "(app (. X Xs) Ys (. X Zs))", "(app Xs Ys Zs)" ] ] }, "graph": { "nodes": { "77": { "goal": [ { "clause": 1, "scope": 2, "term": "(app X11 T10 T12)" }, { "clause": 2, "scope": 2, "term": "(app X11 T10 T12)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X11"], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(suffix T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "137": { "goal": [{ "clause": -1, "scope": -1, "term": "(app X36 T26 T29)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T26"], "free": ["X36"], "exprvars": [] } }, "6": { "goal": [{ "clause": 0, "scope": 1, "term": "(suffix T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "138": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "80": { "goal": [{ "clause": 1, "scope": 2, "term": "(app X11 T10 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X11"], "exprvars": [] } }, "82": { "goal": [{ "clause": 2, "scope": 2, "term": "(app X11 T10 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X11"], "exprvars": [] } }, "83": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "84": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "85": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "64": { "goal": [{ "clause": -1, "scope": -1, "term": "(app X11 T10 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X11"], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 6, "label": "CASE" }, { "from": 6, "to": 64, "label": "ONLY EVAL with clause\nsuffix(X9, X10) :- app(X11, X9, X10).\nand substitutionT1 -> T10,\nX9 -> T10,\nT2 -> T12,\nX10 -> T12,\nT11 -> T12" }, { "from": 64, "to": 77, "label": "CASE" }, { "from": 77, "to": 80, "label": "PARALLEL" }, { "from": 77, "to": 82, "label": "PARALLEL" }, { "from": 80, "to": 83, "label": "EVAL with clause\napp([], X18, X18).\nand substitutionX11 -> [],\nT10 -> T19,\nX18 -> T19,\nT12 -> T19" }, { "from": 80, "to": 84, "label": "EVAL-BACKTRACK" }, { "from": 82, "to": 137, "label": "EVAL with clause\napp(.(X31, X32), X33, .(X31, X34)) :- app(X32, X33, X34).\nand substitutionX31 -> T27,\nX32 -> X36,\nX11 -> .(T27, X36),\nT10 -> T26,\nX33 -> T26,\nX35 -> T27,\nX34 -> T29,\nT12 -> .(T27, T29),\nT28 -> T29" }, { "from": 82, "to": 138, "label": "EVAL-BACKTRACK" }, { "from": 83, "to": 85, "label": "SUCCESS" }, { "from": 137, "to": 64, "label": "INSTANCE with matching:\nX11 -> X36\nT10 -> T26\nT12 -> T29" } ], "type": "Graph" } } ---------------------------------------- (30) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f1_in(T10) -> U1(f64_in(T10), T10) U1(f64_out1, T10) -> f1_out1 f64_in(T19) -> f64_out1 f64_in(T26) -> U2(f64_in(T26), T26) U2(f64_out1, T26) -> f64_out1 Q is empty. ---------------------------------------- (31) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: F1_IN(T10) -> U1^1(f64_in(T10), T10) F1_IN(T10) -> F64_IN(T10) F64_IN(T26) -> U2^1(f64_in(T26), T26) F64_IN(T26) -> F64_IN(T26) The TRS R consists of the following rules: f1_in(T10) -> U1(f64_in(T10), T10) U1(f64_out1, T10) -> f1_out1 f64_in(T19) -> f64_out1 f64_in(T26) -> U2(f64_in(T26), T26) U2(f64_out1, T26) -> f64_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: F64_IN(T26) -> F64_IN(T26) The TRS R consists of the following rules: f1_in(T10) -> U1(f64_in(T10), T10) U1(f64_out1, T10) -> f1_out1 f64_in(T19) -> f64_out1 f64_in(T26) -> U2(f64_in(T26), T26) U2(f64_out1, T26) -> f64_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) MNOCProof (EQUIVALENT) We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: F64_IN(T26) -> F64_IN(T26) The TRS R consists of the following rules: f1_in(T10) -> U1(f64_in(T10), T10) U1(f64_out1, T10) -> f1_out1 f64_in(T19) -> f64_out1 f64_in(T26) -> U2(f64_in(T26), T26) U2(f64_out1, T26) -> f64_out1 The set Q consists of the following terms: f1_in(x0) U1(f64_out1, x0) f64_in(x0) U2(f64_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) 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. ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: F64_IN(T26) -> F64_IN(T26) R is empty. The set Q consists of the following terms: f1_in(x0) U1(f64_out1, x0) f64_in(x0) U2(f64_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) 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]. f1_in(x0) U1(f64_out1, x0) f64_in(x0) U2(f64_out1, x0) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: F64_IN(T26) -> F64_IN(T26) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 2, "program": { "directives": [], "clauses": [ [ "(suffix Xs Ys)", "(app X1 Xs Ys)" ], [ "(app ([]) X X)", null ], [ "(app (. X Xs) Ys (. X Zs))", "(app Xs Ys Zs)" ] ] }, "graph": { "nodes": { "88": { "goal": [{ "clause": -1, "scope": -1, "term": "(app X36 T26 T29)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T26"], "free": ["X36"], "exprvars": [] } }, "78": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(suffix T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "68": { "goal": [{ "clause": -1, "scope": -1, "term": "(app X11 T10 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X11"], "exprvars": [] } }, "79": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "7": { "goal": [{ "clause": 0, "scope": 1, "term": "(suffix T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "81": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "73": { "goal": [ { "clause": 1, "scope": 2, "term": "(app X11 T10 T12)" }, { "clause": 2, "scope": 2, "term": "(app X11 T10 T12)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X11"], "exprvars": [] } }, "95": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "74": { "goal": [{ "clause": 1, "scope": 2, "term": "(app X11 T10 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X11"], "exprvars": [] } }, "76": { "goal": [{ "clause": 2, "scope": 2, "term": "(app X11 T10 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X11"], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 7, "label": "CASE" }, { "from": 7, "to": 68, "label": "ONLY EVAL with clause\nsuffix(X9, X10) :- app(X11, X9, X10).\nand substitutionT1 -> T10,\nX9 -> T10,\nT2 -> T12,\nX10 -> T12,\nT11 -> T12" }, { "from": 68, "to": 73, "label": "CASE" }, { "from": 73, "to": 74, "label": "PARALLEL" }, { "from": 73, "to": 76, "label": "PARALLEL" }, { "from": 74, "to": 78, "label": "EVAL with clause\napp([], X18, X18).\nand substitutionX11 -> [],\nT10 -> T19,\nX18 -> T19,\nT12 -> T19" }, { "from": 74, "to": 79, "label": "EVAL-BACKTRACK" }, { "from": 76, "to": 88, "label": "EVAL with clause\napp(.(X31, X32), X33, .(X31, X34)) :- app(X32, X33, X34).\nand substitutionX31 -> T27,\nX32 -> X36,\nX11 -> .(T27, X36),\nT10 -> T26,\nX33 -> T26,\nX35 -> T27,\nX34 -> T29,\nT12 -> .(T27, T29),\nT28 -> T29" }, { "from": 76, "to": 95, "label": "EVAL-BACKTRACK" }, { "from": 78, "to": 81, "label": "SUCCESS" }, { "from": 88, "to": 68, "label": "INSTANCE with matching:\nX11 -> X36\nT10 -> T26\nT12 -> T29" } ], "type": "Graph" } } ---------------------------------------- (42) Obligation: Rules: f68_out(T26) -> f88_out(T26) :|: TRUE f88_in(x) -> f68_in(x) :|: TRUE f95_out -> f76_out(T10) :|: TRUE f88_out(x1) -> f76_out(x1) :|: TRUE f76_in(x2) -> f88_in(x2) :|: TRUE f76_in(x3) -> f95_in :|: TRUE f73_in(x4) -> f74_in(x4) :|: TRUE f74_out(x5) -> f73_out(x5) :|: TRUE f76_out(x6) -> f73_out(x6) :|: TRUE f73_in(x7) -> f76_in(x7) :|: TRUE f73_out(x8) -> f68_out(x8) :|: TRUE f68_in(x9) -> f73_in(x9) :|: TRUE f7_out(T1) -> f2_out(T1) :|: TRUE f2_in(x10) -> f7_in(x10) :|: TRUE f7_in(x11) -> f68_in(x11) :|: TRUE f68_out(x12) -> f7_out(x12) :|: TRUE Start term: f2_in(T1) ---------------------------------------- (43) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f88_in(x) -> f68_in(x) :|: TRUE f76_in(x2) -> f88_in(x2) :|: TRUE f73_in(x7) -> f76_in(x7) :|: TRUE f68_in(x9) -> f73_in(x9) :|: TRUE ---------------------------------------- (44) Obligation: Rules: f88_in(x) -> f68_in(x) :|: TRUE f76_in(x2) -> f88_in(x2) :|: TRUE f73_in(x7) -> f76_in(x7) :|: TRUE f68_in(x9) -> f73_in(x9) :|: TRUE ---------------------------------------- (45) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (46) Obligation: Rules: f73_in(x7:0) -> f73_in(x7:0) :|: TRUE ---------------------------------------- (47) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (48) Obligation: Rules: f73_in(x7:0) -> f73_in(x7:0) :|: TRUE ---------------------------------------- (49) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f73_in(x7:0) -> f73_in(x7:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (50) Obligation: Termination digraph: Nodes: (1) f73_in(x7:0) -> f73_in(x7:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (51) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f73_in(VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (52) Obligation: Rules: f73_in(x7:0) -> f73_in(x7:0) :|: TRUE ---------------------------------------- (53) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, x7:0) -> f(1, x7:0) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1, -8) ---------------------------------------- (54) NO