/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern fl(a,a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2)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 [SOUND, 0 ms] (11) QDP (12) NonTerminationLoopProof [COMPLETE, 0 ms] (13) NO (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PrologToPiTRSProof [SOUND, 0 ms] (22) PiTRS (23) DependencyPairsProof [EQUIVALENT, 0 ms] (24) PiDP (25) DependencyGraphProof [EQUIVALENT, 0 ms] (26) AND (27) PiDP (28) UsableRulesProof [EQUIVALENT, 0 ms] (29) PiDP (30) PiDPToQDPProof [SOUND, 0 ms] (31) QDP (32) NonTerminationLoopProof [COMPLETE, 0 ms] (33) NO (34) PiDP (35) UsableRulesProof [EQUIVALENT, 0 ms] (36) PiDP (37) PiDPToQDPProof [SOUND, 0 ms] (38) QDP (39) QDPSizeChangeProof [EQUIVALENT, 0 ms] (40) YES (41) PrologToTRSTransformerProof [SOUND, 0 ms] (42) QTRS (43) QTRSRRRProof [EQUIVALENT, 71 ms] (44) QTRS (45) QTRSRRRProof [EQUIVALENT, 4 ms] (46) QTRS (47) QTRSRRRProof [EQUIVALENT, 2 ms] (48) QTRS (49) Overlay + Local Confluence [EQUIVALENT, 0 ms] (50) QTRS (51) DependencyPairsProof [EQUIVALENT, 0 ms] (52) QDP (53) UsableRulesProof [EQUIVALENT, 2 ms] (54) QDP (55) QReductionProof [EQUIVALENT, 0 ms] (56) QDP (57) NonTerminationLoopProof [COMPLETE, 0 ms] (58) NO (59) PrologToDTProblemTransformerProof [SOUND, 0 ms] (60) TRIPLES (61) TriplesToPiDPProof [SOUND, 0 ms] (62) PiDP (63) DependencyGraphProof [EQUIVALENT, 0 ms] (64) PiDP (65) PiDPToQDPProof [SOUND, 0 ms] (66) QDP (67) UsableRulesReductionPairsProof [EQUIVALENT, 5 ms] (68) QDP (69) DependencyGraphProof [EQUIVALENT, 0 ms] (70) QDP (71) NonTerminationLoopProof [COMPLETE, 0 ms] (72) NO (73) PrologToIRSwTTransformerProof [SOUND, 0 ms] (74) AND (75) IRSwT (76) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (77) IRSwT (78) IntTRSCompressionProof [EQUIVALENT, 40 ms] (79) IRSwT (80) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (81) IRSwT (82) IRSwTTerminationDigraphProof [EQUIVALENT, 7 ms] (83) IRSwT (84) FilterProof [EQUIVALENT, 0 ms] (85) IntTRS (86) IntTRSPeriodicNontermProof [COMPLETE, 0 ms] (87) NO (88) IRSwT (89) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (90) IRSwT (91) IntTRSCompressionProof [EQUIVALENT, 5 ms] (92) IRSwT (93) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (94) IRSwT (95) IRSwTTerminationDigraphProof [EQUIVALENT, 11 ms] (96) IRSwT (97) FilterProof [EQUIVALENT, 0 ms] (98) IntTRS ---------------------------------------- (0) Obligation: Clauses: fl([], [], 0). fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z)). append([], X, X). append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs). Query: fl(a,a,g) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: fl_in_3: (f,f,b) append_in_3: (f,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag(x3) s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag(x3) s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5) ---------------------------------------- (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: FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) FL_IN_AAG(.(E, X), R, s(Z)) -> APPEND_IN_AAA(E, Y, R) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U3_AAA(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_AAG(E, X, R, Z, fl_in_aag(X, Y, Z)) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag(x3) s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(x5) U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) FL_IN_AAG(.(E, X), R, s(Z)) -> APPEND_IN_AAA(E, Y, R) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U3_AAA(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_AAG(E, X, R, Z, fl_in_aag(X, Y, Z)) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag(x3) s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(x5) U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag(x3) s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA 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: APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) 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: APPEND_IN_AAA -> APPEND_IN_AAA 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 semiunifying a rule from P directly. s = APPEND_IN_AAA evaluates to t =APPEND_IN_AAA 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 APPEND_IN_AAA to APPEND_IN_AAA. ---------------------------------------- (13) NO ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag(x3) s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) 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: U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) The TRS R consists of the following rules: append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) 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: U1_AAG(Z, append_out_aaa) -> FL_IN_AAG(Z) FL_IN_AAG(s(Z)) -> U1_AAG(Z, append_in_aaa) The TRS R consists of the following rules: append_in_aaa -> append_out_aaa append_in_aaa -> U3_aaa(append_in_aaa) U3_aaa(append_out_aaa) -> append_out_aaa The set Q consists of the following terms: append_in_aaa U3_aaa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *FL_IN_AAG(s(Z)) -> U1_AAG(Z, append_in_aaa) The graph contains the following edges 1 > 1 *U1_AAG(Z, append_out_aaa) -> FL_IN_AAG(Z) The graph contains the following edges 1 >= 1 ---------------------------------------- (20) YES ---------------------------------------- (21) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: fl_in_3: (f,f,b) append_in_3: (f,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (22) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) ---------------------------------------- (23) 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: FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) FL_IN_AAG(.(E, X), R, s(Z)) -> APPEND_IN_AAA(E, Y, R) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U3_AAA(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_AAG(E, X, R, Z, fl_in_aag(X, Y, Z)) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(x5) U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) Obligation: Pi DP problem: The TRS P consists of the following rules: FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) FL_IN_AAG(.(E, X), R, s(Z)) -> APPEND_IN_AAA(E, Y, R) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U3_AAA(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_AAG(E, X, R, Z, fl_in_aag(X, Y, Z)) U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(x5) U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (25) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (26) Complex Obligation (AND) ---------------------------------------- (27) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (28) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (29) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (30) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: APPEND_IN_AAA -> APPEND_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (32) 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 = APPEND_IN_AAA evaluates to t =APPEND_IN_AAA 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 APPEND_IN_AAA to APPEND_IN_AAA. ---------------------------------------- (33) NO ---------------------------------------- (34) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) The TRS R consists of the following rules: fl_in_aag([], [], 0) -> fl_out_aag([], [], 0) fl_in_aag(.(E, X), R, s(Z)) -> U1_aag(E, X, R, Z, append_in_aaa(E, Y, R)) append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) -> U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z)) U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) -> fl_out_aag(.(E, X), R, s(Z)) The argument filtering Pi contains the following mapping: fl_in_aag(x1, x2, x3) = fl_in_aag(x3) 0 = 0 fl_out_aag(x1, x2, x3) = fl_out_aag s(x1) = s(x1) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (35) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (36) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) -> FL_IN_AAG(X, Y, Z) FL_IN_AAG(.(E, X), R, s(Z)) -> U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R)) The TRS R consists of the following rules: append_in_aaa([], X, X) -> append_out_aaa([], X, X) append_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs)) U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(X, Xs), Ys, .(X, Zs)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5) FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (37) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: U1_AAG(Z, append_out_aaa) -> FL_IN_AAG(Z) FL_IN_AAG(s(Z)) -> U1_AAG(Z, append_in_aaa) The TRS R consists of the following rules: append_in_aaa -> append_out_aaa append_in_aaa -> U3_aaa(append_in_aaa) U3_aaa(append_out_aaa) -> append_out_aaa The set Q consists of the following terms: append_in_aaa U3_aaa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (39) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *FL_IN_AAG(s(Z)) -> U1_AAG(Z, append_in_aaa) The graph contains the following edges 1 > 1 *U1_AAG(Z, append_out_aaa) -> FL_IN_AAG(Z) The graph contains the following edges 1 >= 1 ---------------------------------------- (40) YES ---------------------------------------- (41) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 18, "program": { "directives": [], "clauses": [ [ "(fl ([]) ([]) (0))", null ], [ "(fl (. E X) R (s Z))", "(',' (append E Y R) (fl X Y Z))" ], [ "(append ([]) X X)", null ], [ "(append (. X Xs) Ys (. X Zs))", "(append Xs Ys Zs)" ] ] }, "graph": { "nodes": { "55": { "goal": [{ "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "18": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "19": { "goal": [ { "clause": 0, "scope": 1, "term": "(fl T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "type": "Nodes", "131": { "goal": [{ "clause": 3, "scope": 2, "term": "(append T16 X14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "100": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "111": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "122": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T16 X14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "123": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T22 T21 T15)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": [], "exprvars": [] } }, "211": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "113": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T16 X14 T17) (fl T18 X14 T15))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": ["X14"], "exprvars": [] } }, "212": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "213": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "115": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "214": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T39 X47 T40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X47"], "exprvars": [] } }, "127": { "goal": [ { "clause": 2, "scope": 2, "term": "(append T16 X14 T17)" }, { "clause": 3, "scope": 2, "term": "(append T16 X14 T17)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "215": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "129": { "goal": [{ "clause": 2, "scope": 2, "term": "(append T16 X14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "95": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "54": { "goal": [{ "clause": 0, "scope": 1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 18, "to": 19, "label": "CASE" }, { "from": 19, "to": 54, "label": "PARALLEL" }, { "from": 19, "to": 55, "label": "PARALLEL" }, { "from": 54, "to": 95, "label": "EVAL with clause\nfl([], [], 0).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> 0" }, { "from": 54, "to": 100, "label": "EVAL-BACKTRACK" }, { "from": 55, "to": 113, "label": "EVAL with clause\nfl(.(X10, X11), X12, s(X13)) :- ','(append(X10, X14, X12), fl(X11, X14, X13)).\nand substitutionX10 -> T16,\nX11 -> T18,\nT1 -> .(T16, T18),\nT2 -> T17,\nX12 -> T17,\nX13 -> T15,\nT3 -> s(T15),\nT12 -> T16,\nT14 -> T17,\nT13 -> T18" }, { "from": 55, "to": 115, "label": "EVAL-BACKTRACK" }, { "from": 95, "to": 111, "label": "SUCCESS" }, { "from": 113, "to": 122, "label": "SPLIT 1" }, { "from": 113, "to": 123, "label": "SPLIT 2\nreplacements:X14 -> T21,\nT18 -> T22" }, { "from": 122, "to": 127, "label": "CASE" }, { "from": 123, "to": 18, "label": "INSTANCE with matching:\nT1 -> T22\nT2 -> T21\nT3 -> T15" }, { "from": 127, "to": 129, "label": "PARALLEL" }, { "from": 127, "to": 131, "label": "PARALLEL" }, { "from": 129, "to": 211, "label": "EVAL with clause\nappend([], X31, X31).\nand substitutionT16 -> [],\nX14 -> T29,\nX31 -> T29,\nT17 -> T29,\nX32 -> T29" }, { "from": 129, "to": 212, "label": "EVAL-BACKTRACK" }, { "from": 131, "to": 214, "label": "EVAL with clause\nappend(.(X43, X44), X45, .(X43, X46)) :- append(X44, X45, X46).\nand substitutionX43 -> T36,\nX44 -> T39,\nT16 -> .(T36, T39),\nX14 -> X47,\nX45 -> X47,\nX46 -> T40,\nT17 -> .(T36, T40),\nT37 -> T39,\nT38 -> T40" }, { "from": 131, "to": 215, "label": "EVAL-BACKTRACK" }, { "from": 211, "to": 213, "label": "SUCCESS" }, { "from": 214, "to": 122, "label": "INSTANCE with matching:\nT16 -> T39\nX14 -> X47\nT17 -> T40" } ], "type": "Graph" } } ---------------------------------------- (42) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f18_in(0) -> f18_out1 f18_in(s(T15)) -> U1(f113_in(T15), s(T15)) U1(f113_out1, s(T15)) -> f18_out1 f122_in -> f122_out1 f122_in -> U2(f122_in) U2(f122_out1) -> f122_out1 f113_in(T15) -> U3(f122_in, T15) U3(f122_out1, T15) -> U4(f18_in(T15), T15) U4(f18_out1, T15) -> f113_out1 Q is empty. ---------------------------------------- (43) QTRSRRRProof (EQUIVALENT) Used ordering: f18_in/1(YES) 0/0) f18_out1/0) s/1(YES) U1/2(YES,YES) f113_in/1(YES) f113_out1/0) f122_in/0) f122_out1/0) U2/1)YES( U3/2(YES,YES) U4/2(YES,YES) Quasi precedence: 0 > f18_out1 > f113_out1 > [U1_2, U4_2] s_1 > f18_out1 > f113_out1 > [U1_2, U4_2] s_1 > f113_in_1 > [f122_in, f122_out1] > [f18_in_1, U3_2] > [U1_2, U4_2] Status: f18_in_1: multiset status 0: multiset status f18_out1: multiset status s_1: [1] U1_2: multiset status f113_in_1: multiset status f113_out1: multiset status f122_in: multiset status f122_out1: multiset status U3_2: multiset status U4_2: [2,1] With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f18_in(0) -> f18_out1 f18_in(s(T15)) -> U1(f113_in(T15), s(T15)) U1(f113_out1, s(T15)) -> f18_out1 f113_in(T15) -> U3(f122_in, T15) U3(f122_out1, T15) -> U4(f18_in(T15), T15) U4(f18_out1, T15) -> f113_out1 ---------------------------------------- (44) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f122_in -> f122_out1 f122_in -> U2(f122_in) U2(f122_out1) -> f122_out1 Q is empty. ---------------------------------------- (45) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = x_1 POL(f122_in) = 2 POL(f122_out1) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f122_in -> f122_out1 ---------------------------------------- (46) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f122_in -> U2(f122_in) U2(f122_out1) -> f122_out1 Q is empty. ---------------------------------------- (47) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = 2*x_1 POL(f122_in) = 0 POL(f122_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U2(f122_out1) -> f122_out1 ---------------------------------------- (48) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f122_in -> U2(f122_in) Q is empty. ---------------------------------------- (49) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (50) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f122_in -> U2(f122_in) The set Q consists of the following terms: f122_in ---------------------------------------- (51) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: F122_IN -> F122_IN The TRS R consists of the following rules: f122_in -> U2(f122_in) The set Q consists of the following terms: f122_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) 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. ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: F122_IN -> F122_IN R is empty. The set Q consists of the following terms: f122_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) 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]. f122_in ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: F122_IN -> F122_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) 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 = F122_IN evaluates to t =F122_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 F122_IN to F122_IN. ---------------------------------------- (58) NO ---------------------------------------- (59) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 50, "program": { "directives": [], "clauses": [ [ "(fl ([]) ([]) (0))", null ], [ "(fl (. E X) R (s Z))", "(',' (append E Y R) (fl X Y Z))" ], [ "(append ([]) X X)", null ], [ "(append (. X Xs) Ys (. X Zs))", "(append Xs Ys Zs)" ] ] }, "graph": { "nodes": { "type": "Nodes", "140": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "141": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T15 X13 T16) (fl T17 X13 T14))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": ["X13"], "exprvars": [] } }, "142": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "143": { "goal": [ { "clause": 2, "scope": 2, "term": "(',' (append T15 X13 T16) (fl T17 X13 T14))" }, { "clause": 3, "scope": 2, "term": "(',' (append T15 X13 T16) (fl T17 X13 T14))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": ["X13"], "exprvars": [] } }, "145": { "goal": [{ "clause": 2, "scope": 2, "term": "(',' (append T15 X13 T16) (fl T17 X13 T14))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": ["X13"], "exprvars": [] } }, "135": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(fl T1 T2 (0))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "146": { "goal": [{ "clause": 3, "scope": 2, "term": "(',' (append T15 X13 T16) (fl T17 X13 T14))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": ["X13"], "exprvars": [] } }, "137": { "goal": [{ "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [[ "(fl T1 T2 T3)", "(fl ([]) ([]) (0))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "148": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T23 T24 T14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "149": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "237": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T34 X38 T35) (fl T36 X38 T14))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": ["X38"], "exprvars": [] } }, "139": { "goal": [{ "clause": 1, "scope": 1, "term": "(fl T1 T2 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "238": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "50": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "51": { "goal": [ { "clause": 0, "scope": 1, "term": "(fl T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 50, "to": 51, "label": "CASE" }, { "from": 51, "to": 135, "label": "EVAL with clause\nfl([], [], 0).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> 0" }, { "from": 51, "to": 137, "label": "EVAL-BACKTRACK" }, { "from": 135, "to": 139, "label": "SUCCESS" }, { "from": 137, "to": 141, "label": "EVAL with clause\nfl(.(X9, X10), X11, s(X12)) :- ','(append(X9, X13, X11), fl(X10, X13, X12)).\nand substitutionX9 -> T15,\nX10 -> T17,\nT1 -> .(T15, T17),\nT2 -> T16,\nX11 -> T16,\nX12 -> T14,\nT3 -> s(T14),\nT11 -> T15,\nT13 -> T16,\nT12 -> T17" }, { "from": 137, "to": 142, "label": "EVAL-BACKTRACK" }, { "from": 139, "to": 140, "label": "BACKTRACK\nfor clause: fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z))because of non-unification" }, { "from": 141, "to": 143, "label": "CASE" }, { "from": 143, "to": 145, "label": "PARALLEL" }, { "from": 143, "to": 146, "label": "PARALLEL" }, { "from": 145, "to": 148, "label": "EVAL with clause\nappend([], X22, X22).\nand substitutionT15 -> [],\nX13 -> T24,\nX22 -> T24,\nT16 -> T24,\nX23 -> T24,\nT17 -> T23,\nT22 -> T24" }, { "from": 145, "to": 149, "label": "EVAL-BACKTRACK" }, { "from": 146, "to": 237, "label": "EVAL with clause\nappend(.(X34, X35), X36, .(X34, X37)) :- append(X35, X36, X37).\nand substitutionX34 -> T31,\nX35 -> T34,\nT15 -> .(T31, T34),\nX13 -> X38,\nX36 -> X38,\nX37 -> T35,\nT16 -> .(T31, T35),\nT32 -> T34,\nT33 -> T35,\nT17 -> T36" }, { "from": 146, "to": 238, "label": "EVAL-BACKTRACK" }, { "from": 148, "to": 50, "label": "INSTANCE with matching:\nT1 -> T23\nT2 -> T24\nT3 -> T14" }, { "from": 237, "to": 141, "label": "INSTANCE with matching:\nT15 -> T34\nX13 -> X38\nT16 -> T35\nT17 -> T36" } ], "type": "Graph" } } ---------------------------------------- (60) Obligation: Triples: pB([], X1, X1, X2, X3) :- flA(X2, X1, X3). pB(.(X1, X2), X3, .(X1, X4), X5, X6) :- pB(X2, X3, X4, X5, X6). flA(.(X1, X2), X3, s(X4)) :- pB(X1, X5, X3, X2, X4). Clauses: flcA([], [], 0). flcA(.(X1, X2), X3, s(X4)) :- qcB(X1, X5, X3, X2, X4). qcB([], X1, X1, X2, X3) :- flcA(X2, X1, X3). qcB(.(X1, X2), X3, .(X1, X4), X5, X6) :- qcB(X2, X3, X4, X5, X6). Afs: flA(x1, x2, x3) = flA(x3) ---------------------------------------- (61) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: flA_in_3: (f,f,b) pB_in_5: (f,f,f,f,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: FLA_IN_AAG(.(X1, X2), X3, s(X4)) -> U3_AAG(X1, X2, X3, X4, pB_in_aaaag(X1, X5, X3, X2, X4)) FLA_IN_AAG(.(X1, X2), X3, s(X4)) -> PB_IN_AAAAG(X1, X5, X3, X2, X4) PB_IN_AAAAG([], X1, X1, X2, X3) -> U1_AAAAG(X1, X2, X3, flA_in_aag(X2, X1, X3)) PB_IN_AAAAG([], X1, X1, X2, X3) -> FLA_IN_AAG(X2, X1, X3) PB_IN_AAAAG(.(X1, X2), X3, .(X1, X4), X5, X6) -> U2_AAAAG(X1, X2, X3, X4, X5, X6, pB_in_aaaag(X2, X3, X4, X5, X6)) PB_IN_AAAAG(.(X1, X2), X3, .(X1, X4), X5, X6) -> PB_IN_AAAAG(X2, X3, X4, X5, X6) R is empty. The argument filtering Pi contains the following mapping: flA_in_aag(x1, x2, x3) = flA_in_aag(x3) s(x1) = s(x1) pB_in_aaaag(x1, x2, x3, x4, x5) = pB_in_aaaag(x5) FLA_IN_AAG(x1, x2, x3) = FLA_IN_AAG(x3) U3_AAG(x1, x2, x3, x4, x5) = U3_AAG(x4, x5) PB_IN_AAAAG(x1, x2, x3, x4, x5) = PB_IN_AAAAG(x5) U1_AAAAG(x1, x2, x3, x4) = U1_AAAAG(x3, x4) U2_AAAAG(x1, x2, x3, x4, x5, x6, x7) = U2_AAAAG(x6, x7) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (62) Obligation: Pi DP problem: The TRS P consists of the following rules: FLA_IN_AAG(.(X1, X2), X3, s(X4)) -> U3_AAG(X1, X2, X3, X4, pB_in_aaaag(X1, X5, X3, X2, X4)) FLA_IN_AAG(.(X1, X2), X3, s(X4)) -> PB_IN_AAAAG(X1, X5, X3, X2, X4) PB_IN_AAAAG([], X1, X1, X2, X3) -> U1_AAAAG(X1, X2, X3, flA_in_aag(X2, X1, X3)) PB_IN_AAAAG([], X1, X1, X2, X3) -> FLA_IN_AAG(X2, X1, X3) PB_IN_AAAAG(.(X1, X2), X3, .(X1, X4), X5, X6) -> U2_AAAAG(X1, X2, X3, X4, X5, X6, pB_in_aaaag(X2, X3, X4, X5, X6)) PB_IN_AAAAG(.(X1, X2), X3, .(X1, X4), X5, X6) -> PB_IN_AAAAG(X2, X3, X4, X5, X6) R is empty. The argument filtering Pi contains the following mapping: flA_in_aag(x1, x2, x3) = flA_in_aag(x3) s(x1) = s(x1) pB_in_aaaag(x1, x2, x3, x4, x5) = pB_in_aaaag(x5) FLA_IN_AAG(x1, x2, x3) = FLA_IN_AAG(x3) U3_AAG(x1, x2, x3, x4, x5) = U3_AAG(x4, x5) PB_IN_AAAAG(x1, x2, x3, x4, x5) = PB_IN_AAAAG(x5) U1_AAAAG(x1, x2, x3, x4) = U1_AAAAG(x3, x4) U2_AAAAG(x1, x2, x3, x4, x5, x6, x7) = U2_AAAAG(x6, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (63) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (64) Obligation: Pi DP problem: The TRS P consists of the following rules: FLA_IN_AAG(.(X1, X2), X3, s(X4)) -> PB_IN_AAAAG(X1, X5, X3, X2, X4) PB_IN_AAAAG([], X1, X1, X2, X3) -> FLA_IN_AAG(X2, X1, X3) PB_IN_AAAAG(.(X1, X2), X3, .(X1, X4), X5, X6) -> PB_IN_AAAAG(X2, X3, X4, X5, X6) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) FLA_IN_AAG(x1, x2, x3) = FLA_IN_AAG(x3) PB_IN_AAAAG(x1, x2, x3, x4, x5) = PB_IN_AAAAG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (65) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: FLA_IN_AAG(s(X4)) -> PB_IN_AAAAG(X4) PB_IN_AAAAG(X3) -> FLA_IN_AAG(X3) PB_IN_AAAAG(X6) -> PB_IN_AAAAG(X6) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (67) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. The following dependency pairs can be deleted: FLA_IN_AAG(s(X4)) -> PB_IN_AAAAG(X4) No rules are removed from R. Used ordering: POLO with Polynomial interpretation [POLO]: POL(FLA_IN_AAG(x_1)) = 2 + x_1 POL(PB_IN_AAAAG(x_1)) = 2 + x_1 POL(s(x_1)) = 2 + 2*x_1 ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: PB_IN_AAAAG(X3) -> FLA_IN_AAG(X3) PB_IN_AAAAG(X6) -> PB_IN_AAAAG(X6) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (69) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: PB_IN_AAAAG(X6) -> PB_IN_AAAAG(X6) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (71) 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 = PB_IN_AAAAG(X6) evaluates to t =PB_IN_AAAAG(X6) 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 PB_IN_AAAAG(X6) to PB_IN_AAAAG(X6). ---------------------------------------- (72) NO ---------------------------------------- (73) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 17, "program": { "directives": [], "clauses": [ [ "(fl ([]) ([]) (0))", null ], [ "(fl (. E X) R (s Z))", "(',' (append E Y R) (fl X Y Z))" ], [ "(append ([]) X X)", null ], [ "(append (. X Xs) Ys (. X Zs))", "(append Xs Ys Zs)" ] ] }, "graph": { "nodes": { "17": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "type": "Nodes", "130": { "goal": [{ "clause": 3, "scope": 2, "term": "(append T16 X14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "120": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T16 X14 T17) (fl T18 X14 T15))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": ["X14"], "exprvars": [] } }, "121": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "132": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "133": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "134": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "124": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T16 X14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "125": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T22 T21 T15)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": [], "exprvars": [] } }, "136": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T39 X47 T40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X47"], "exprvars": [] } }, "126": { "goal": [ { "clause": 2, "scope": 2, "term": "(append T16 X14 T17)" }, { "clause": 3, "scope": 2, "term": "(append T16 X14 T17)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "138": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "117": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "128": { "goal": [{ "clause": 2, "scope": 2, "term": "(append T16 X14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "118": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "119": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "52": { "goal": [{ "clause": 0, "scope": 1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "20": { "goal": [ { "clause": 0, "scope": 1, "term": "(fl T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "53": { "goal": [{ "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 17, "to": 20, "label": "CASE" }, { "from": 20, "to": 52, "label": "PARALLEL" }, { "from": 20, "to": 53, "label": "PARALLEL" }, { "from": 52, "to": 117, "label": "EVAL with clause\nfl([], [], 0).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> 0" }, { "from": 52, "to": 118, "label": "EVAL-BACKTRACK" }, { "from": 53, "to": 120, "label": "EVAL with clause\nfl(.(X10, X11), X12, s(X13)) :- ','(append(X10, X14, X12), fl(X11, X14, X13)).\nand substitutionX10 -> T16,\nX11 -> T18,\nT1 -> .(T16, T18),\nT2 -> T17,\nX12 -> T17,\nX13 -> T15,\nT3 -> s(T15),\nT12 -> T16,\nT14 -> T17,\nT13 -> T18" }, { "from": 53, "to": 121, "label": "EVAL-BACKTRACK" }, { "from": 117, "to": 119, "label": "SUCCESS" }, { "from": 120, "to": 124, "label": "SPLIT 1" }, { "from": 120, "to": 125, "label": "SPLIT 2\nreplacements:X14 -> T21,\nT18 -> T22" }, { "from": 124, "to": 126, "label": "CASE" }, { "from": 125, "to": 17, "label": "INSTANCE with matching:\nT1 -> T22\nT2 -> T21\nT3 -> T15" }, { "from": 126, "to": 128, "label": "PARALLEL" }, { "from": 126, "to": 130, "label": "PARALLEL" }, { "from": 128, "to": 132, "label": "EVAL with clause\nappend([], X31, X31).\nand substitutionT16 -> [],\nX14 -> T29,\nX31 -> T29,\nT17 -> T29,\nX32 -> T29" }, { "from": 128, "to": 133, "label": "EVAL-BACKTRACK" }, { "from": 130, "to": 136, "label": "EVAL with clause\nappend(.(X43, X44), X45, .(X43, X46)) :- append(X44, X45, X46).\nand substitutionX43 -> T36,\nX44 -> T39,\nT16 -> .(T36, T39),\nX14 -> X47,\nX45 -> X47,\nX46 -> T40,\nT17 -> .(T36, T40),\nT37 -> T39,\nT38 -> T40" }, { "from": 130, "to": 138, "label": "EVAL-BACKTRACK" }, { "from": 132, "to": 134, "label": "SUCCESS" }, { "from": 136, "to": 124, "label": "INSTANCE with matching:\nT16 -> T39\nX14 -> X47\nT17 -> T40" } ], "type": "Graph" } } ---------------------------------------- (74) Complex Obligation (AND) ---------------------------------------- (75) Obligation: Rules: f136_out -> f130_out :|: TRUE f130_in -> f138_in :|: TRUE f138_out -> f130_out :|: TRUE f130_in -> f136_in :|: TRUE f136_in -> f124_in :|: TRUE f124_out -> f136_out :|: TRUE f130_out -> f126_out :|: TRUE f126_in -> f128_in :|: TRUE f126_in -> f130_in :|: TRUE f128_out -> f126_out :|: TRUE f124_in -> f126_in :|: TRUE f126_out -> f124_out :|: TRUE f20_out(T3) -> f17_out(T3) :|: TRUE f17_in(x) -> f20_in(x) :|: TRUE f20_in(x1) -> f53_in(x1) :|: TRUE f53_out(x2) -> f20_out(x2) :|: TRUE f20_in(x3) -> f52_in(x3) :|: TRUE f52_out(x4) -> f20_out(x4) :|: TRUE f120_out(T15) -> f53_out(s(T15)) :|: TRUE f121_out -> f53_out(x5) :|: TRUE f53_in(s(x6)) -> f120_in(x6) :|: TRUE f53_in(x7) -> f121_in :|: TRUE f124_out -> f125_in(x8) :|: TRUE f125_out(x9) -> f120_out(x9) :|: TRUE f120_in(x10) -> f124_in :|: TRUE Start term: f17_in(T3) ---------------------------------------- (76) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f130_in -> f136_in :|: TRUE f136_in -> f124_in :|: TRUE f126_in -> f130_in :|: TRUE f124_in -> f126_in :|: TRUE ---------------------------------------- (77) Obligation: Rules: f130_in -> f136_in :|: TRUE f136_in -> f124_in :|: TRUE f126_in -> f130_in :|: TRUE f124_in -> f126_in :|: TRUE ---------------------------------------- (78) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (79) Obligation: Rules: f126_in -> f126_in :|: TRUE ---------------------------------------- (80) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (81) Obligation: Rules: f126_in -> f126_in :|: TRUE ---------------------------------------- (82) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f126_in -> f126_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (83) Obligation: Termination digraph: Nodes: (1) f126_in -> f126_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (84) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f126_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (85) Obligation: Rules: f126_in -> f126_in :|: TRUE ---------------------------------------- (86) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (87) NO ---------------------------------------- (88) Obligation: Rules: f136_out -> f130_out :|: TRUE f130_in -> f138_in :|: TRUE f138_out -> f130_out :|: TRUE f130_in -> f136_in :|: TRUE f132_in -> f132_out :|: TRUE f124_in -> f126_in :|: TRUE f126_out -> f124_out :|: TRUE f132_out -> f128_out :|: TRUE f128_in -> f133_in :|: TRUE f128_in -> f132_in :|: TRUE f133_out -> f128_out :|: TRUE f20_out(T3) -> f17_out(T3) :|: TRUE f17_in(x) -> f20_in(x) :|: TRUE f20_in(x1) -> f53_in(x1) :|: TRUE f53_out(x2) -> f20_out(x2) :|: TRUE f20_in(x3) -> f52_in(x3) :|: TRUE f52_out(x4) -> f20_out(x4) :|: TRUE f136_in -> f124_in :|: TRUE f124_out -> f136_out :|: TRUE f125_in(T15) -> f17_in(T15) :|: TRUE f17_out(x5) -> f125_out(x5) :|: TRUE f130_out -> f126_out :|: TRUE f126_in -> f128_in :|: TRUE f126_in -> f130_in :|: TRUE f128_out -> f126_out :|: TRUE f120_out(x6) -> f53_out(s(x6)) :|: TRUE f121_out -> f53_out(x7) :|: TRUE f53_in(s(x8)) -> f120_in(x8) :|: TRUE f53_in(x9) -> f121_in :|: TRUE f124_out -> f125_in(x10) :|: TRUE f125_out(x11) -> f120_out(x11) :|: TRUE f120_in(x12) -> f124_in :|: TRUE Start term: f17_in(T3) ---------------------------------------- (89) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f136_out -> f130_out :|: TRUE f130_in -> f136_in :|: TRUE f132_in -> f132_out :|: TRUE f124_in -> f126_in :|: TRUE f126_out -> f124_out :|: TRUE f132_out -> f128_out :|: TRUE f128_in -> f132_in :|: TRUE f17_in(x) -> f20_in(x) :|: TRUE f20_in(x1) -> f53_in(x1) :|: TRUE f136_in -> f124_in :|: TRUE f124_out -> f136_out :|: TRUE f125_in(T15) -> f17_in(T15) :|: TRUE f130_out -> f126_out :|: TRUE f126_in -> f128_in :|: TRUE f126_in -> f130_in :|: TRUE f128_out -> f126_out :|: TRUE f53_in(s(x8)) -> f120_in(x8) :|: TRUE f124_out -> f125_in(x10) :|: TRUE f120_in(x12) -> f124_in :|: TRUE ---------------------------------------- (90) Obligation: Rules: f136_out -> f130_out :|: TRUE f130_in -> f136_in :|: TRUE f132_in -> f132_out :|: TRUE f124_in -> f126_in :|: TRUE f126_out -> f124_out :|: TRUE f132_out -> f128_out :|: TRUE f128_in -> f132_in :|: TRUE f17_in(x) -> f20_in(x) :|: TRUE f20_in(x1) -> f53_in(x1) :|: TRUE f136_in -> f124_in :|: TRUE f124_out -> f136_out :|: TRUE f125_in(T15) -> f17_in(T15) :|: TRUE f130_out -> f126_out :|: TRUE f126_in -> f128_in :|: TRUE f126_in -> f130_in :|: TRUE f128_out -> f126_out :|: TRUE f53_in(s(x8)) -> f120_in(x8) :|: TRUE f124_out -> f125_in(x10) :|: TRUE f120_in(x12) -> f124_in :|: TRUE ---------------------------------------- (91) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (92) Obligation: Rules: f126_out -> f124_in :|: TRUE f126_out -> f126_out :|: TRUE f124_in -> f124_in :|: TRUE f124_in -> f126_out :|: TRUE ---------------------------------------- (93) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (94) Obligation: Rules: f126_out -> f124_in :|: TRUE f126_out -> f126_out :|: TRUE f124_in -> f124_in :|: TRUE f124_in -> f126_out :|: TRUE ---------------------------------------- (95) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f126_out -> f124_in :|: TRUE (2) f126_out -> f126_out :|: TRUE (3) f124_in -> f124_in :|: TRUE (4) f124_in -> f126_out :|: TRUE Arcs: (1) -> (3), (4) (2) -> (1), (2) (3) -> (3), (4) (4) -> (1), (2) This digraph is fully evaluated! ---------------------------------------- (96) Obligation: Termination digraph: Nodes: (1) f126_out -> f124_in :|: TRUE (2) f126_out -> f126_out :|: TRUE (3) f124_in -> f126_out :|: TRUE (4) f124_in -> f124_in :|: TRUE Arcs: (1) -> (3), (4) (2) -> (1), (2) (3) -> (1), (2) (4) -> (3), (4) This digraph is fully evaluated! ---------------------------------------- (97) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f126_out() f124_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (98) Obligation: Rules: f126_out -> f124_in :|: TRUE f126_out -> f126_out :|: TRUE f124_in -> f126_out :|: TRUE f124_in -> f124_in :|: TRUE