/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern sublist(g,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms]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, 5 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) PrologToPiTRSProof [SOUND, 0 ms] (20) PiTRS (21) DependencyPairsProof [EQUIVALENT, 0 ms] (22) PiDP (23) DependencyGraphProof [EQUIVALENT, 2 ms] (24) AND (25) PiDP (26) UsableRulesProof [EQUIVALENT, 0 ms] (27) PiDP (28) PiDPToQDPProof [SOUND, 0 ms] (29) QDP (30) QDPSizeChangeProof [EQUIVALENT, 0 ms] (31) YES (32) PiDP (33) UsableRulesProof [EQUIVALENT, 0 ms] (34) PiDP (35) PiDPToQDPProof [SOUND, 0 ms] (36) QDP (37) PrologToTRSTransformerProof [SOUND, 0 ms] (38) QTRS (39) DependencyPairsProof [EQUIVALENT, 0 ms] (40) QDP (41) DependencyGraphProof [EQUIVALENT, 0 ms] (42) AND (43) QDP (44) UsableRulesProof [EQUIVALENT, 0 ms] (45) QDP (46) QDPSizeChangeProof [EQUIVALENT, 0 ms] (47) YES (48) QDP (49) UsableRulesProof [EQUIVALENT, 0 ms] (50) QDP (51) PrologToDTProblemTransformerProof [SOUND, 47 ms] (52) TRIPLES (53) TriplesToPiDPProof [SOUND, 0 ms] (54) PiDP (55) DependencyGraphProof [EQUIVALENT, 0 ms] (56) AND (57) PiDP (58) UsableRulesProof [EQUIVALENT, 0 ms] (59) PiDP (60) PiDPToQDPProof [SOUND, 0 ms] (61) QDP (62) QDPSizeChangeProof [EQUIVALENT, 0 ms] (63) YES (64) PiDP (65) UsableRulesProof [EQUIVALENT, 0 ms] (66) PiDP (67) PiDPToQDPProof [SOUND, 0 ms] (68) QDP (69) PiDP (70) UsableRulesProof [EQUIVALENT, 0 ms] (71) PiDP (72) PrologToIRSwTTransformerProof [SOUND, 28 ms] (73) AND (74) IRSwT (75) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (76) TRUE (77) IRSwT (78) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (79) IRSwT (80) IntTRSCompressionProof [EQUIVALENT, 20 ms] (81) IRSwT (82) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (83) IRSwT (84) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (85) IRSwT (86) FilterProof [EQUIVALENT, 0 ms] (87) IntTRS (88) IntTRSPeriodicNontermProof [COMPLETE, 6 ms] (89) NO ---------------------------------------- (0) Obligation: Clauses: append1([], Ys, Ys). append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs). append2([], Ys, Ys). append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs). sublist(X, Y) :- ','(append1(U, X, V), append2(V, W, Y)). Query: sublist(g,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: sublist_in_2: (b,b) append1_in_3: (f,b,f) append2_in_3: (b,f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V)) append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys) append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs)) U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y)) append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys) append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs)) U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y) The argument filtering Pi contains the following mapping: sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) U3_gg(x1, x2, x3) = U3_gg(x1, x2, x3) append1_in_aga(x1, x2, x3) = append1_in_aga(x2) append1_out_aga(x1, x2, x3) = append1_out_aga(x1, x2, x3) U1_aga(x1, x2, x3, x4, x5) = U1_aga(x3, x5) .(x1, x2) = .(x2) U4_gg(x1, x2, x3) = U4_gg(x1, x2, x3) append2_in_gag(x1, x2, x3) = append2_in_gag(x1, x3) [] = [] append2_out_gag(x1, x2, x3) = append2_out_gag(x1, x2, x3) U2_gag(x1, x2, x3, x4, x5) = U2_gag(x2, x4, x5) sublist_out_gg(x1, x2) = sublist_out_gg(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: sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V)) append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys) append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs)) U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y)) append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys) append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs)) U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y) The argument filtering Pi contains the following mapping: sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) U3_gg(x1, x2, x3) = U3_gg(x1, x2, x3) append1_in_aga(x1, x2, x3) = append1_in_aga(x2) append1_out_aga(x1, x2, x3) = append1_out_aga(x1, x2, x3) U1_aga(x1, x2, x3, x4, x5) = U1_aga(x3, x5) .(x1, x2) = .(x2) U4_gg(x1, x2, x3) = U4_gg(x1, x2, x3) append2_in_gag(x1, x2, x3) = append2_in_gag(x1, x3) [] = [] append2_out_gag(x1, x2, x3) = append2_out_gag(x1, x2, x3) U2_gag(x1, x2, x3, x4, x5) = U2_gag(x2, x4, x5) sublist_out_gg(x1, x2) = sublist_out_gg(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: SUBLIST_IN_GG(X, Y) -> U3_GG(X, Y, append1_in_aga(U, X, V)) SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AGA(U, X, V) APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs) U3_GG(X, Y, append1_out_aga(U, X, V)) -> U4_GG(X, Y, append2_in_gag(V, W, Y)) U3_GG(X, Y, append1_out_aga(U, X, V)) -> APPEND2_IN_GAG(V, W, Y) APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V)) append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys) append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs)) U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y)) append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys) append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs)) U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y) The argument filtering Pi contains the following mapping: sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) U3_gg(x1, x2, x3) = U3_gg(x1, x2, x3) append1_in_aga(x1, x2, x3) = append1_in_aga(x2) append1_out_aga(x1, x2, x3) = append1_out_aga(x1, x2, x3) U1_aga(x1, x2, x3, x4, x5) = U1_aga(x3, x5) .(x1, x2) = .(x2) U4_gg(x1, x2, x3) = U4_gg(x1, x2, x3) append2_in_gag(x1, x2, x3) = append2_in_gag(x1, x3) [] = [] append2_out_gag(x1, x2, x3) = append2_out_gag(x1, x2, x3) U2_gag(x1, x2, x3, x4, x5) = U2_gag(x2, x4, x5) sublist_out_gg(x1, x2) = sublist_out_gg(x1, x2) SUBLIST_IN_GG(x1, x2) = SUBLIST_IN_GG(x1, x2) U3_GG(x1, x2, x3) = U3_GG(x1, x2, x3) APPEND1_IN_AGA(x1, x2, x3) = APPEND1_IN_AGA(x2) U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x3, x5) U4_GG(x1, x2, x3) = U4_GG(x1, x2, x3) APPEND2_IN_GAG(x1, x2, x3) = APPEND2_IN_GAG(x1, x3) U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x2, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBLIST_IN_GG(X, Y) -> U3_GG(X, Y, append1_in_aga(U, X, V)) SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AGA(U, X, V) APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs) U3_GG(X, Y, append1_out_aga(U, X, V)) -> U4_GG(X, Y, append2_in_gag(V, W, Y)) U3_GG(X, Y, append1_out_aga(U, X, V)) -> APPEND2_IN_GAG(V, W, Y) APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V)) append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys) append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs)) U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y)) append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys) append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs)) U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y) The argument filtering Pi contains the following mapping: sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) U3_gg(x1, x2, x3) = U3_gg(x1, x2, x3) append1_in_aga(x1, x2, x3) = append1_in_aga(x2) append1_out_aga(x1, x2, x3) = append1_out_aga(x1, x2, x3) U1_aga(x1, x2, x3, x4, x5) = U1_aga(x3, x5) .(x1, x2) = .(x2) U4_gg(x1, x2, x3) = U4_gg(x1, x2, x3) append2_in_gag(x1, x2, x3) = append2_in_gag(x1, x3) [] = [] append2_out_gag(x1, x2, x3) = append2_out_gag(x1, x2, x3) U2_gag(x1, x2, x3, x4, x5) = U2_gag(x2, x4, x5) sublist_out_gg(x1, x2) = sublist_out_gg(x1, x2) SUBLIST_IN_GG(x1, x2) = SUBLIST_IN_GG(x1, x2) U3_GG(x1, x2, x3) = U3_GG(x1, x2, x3) APPEND1_IN_AGA(x1, x2, x3) = APPEND1_IN_AGA(x2) U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x3, x5) U4_GG(x1, x2, x3) = U4_GG(x1, x2, x3) APPEND2_IN_GAG(x1, x2, x3) = APPEND2_IN_GAG(x1, x3) U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x2, 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 6 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V)) append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys) append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs)) U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y)) append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys) append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs)) U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y) The argument filtering Pi contains the following mapping: sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) U3_gg(x1, x2, x3) = U3_gg(x1, x2, x3) append1_in_aga(x1, x2, x3) = append1_in_aga(x2) append1_out_aga(x1, x2, x3) = append1_out_aga(x1, x2, x3) U1_aga(x1, x2, x3, x4, x5) = U1_aga(x3, x5) .(x1, x2) = .(x2) U4_gg(x1, x2, x3) = U4_gg(x1, x2, x3) append2_in_gag(x1, x2, x3) = append2_in_gag(x1, x3) [] = [] append2_out_gag(x1, x2, x3) = append2_out_gag(x1, x2, x3) U2_gag(x1, x2, x3, x4, x5) = U2_gag(x2, x4, x5) sublist_out_gg(x1, x2) = sublist_out_gg(x1, x2) APPEND2_IN_GAG(x1, x2, x3) = APPEND2_IN_GAG(x1, x3) 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: APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND2_IN_GAG(x1, x2, x3) = APPEND2_IN_GAG(x1, x3) 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: APPEND2_IN_GAG(.(Xs), .(Zs)) -> APPEND2_IN_GAG(Xs, Zs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) 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: *APPEND2_IN_GAG(.(Xs), .(Zs)) -> APPEND2_IN_GAG(Xs, Zs) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V)) append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys) append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs)) U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y)) append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys) append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs)) U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y) The argument filtering Pi contains the following mapping: sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) U3_gg(x1, x2, x3) = U3_gg(x1, x2, x3) append1_in_aga(x1, x2, x3) = append1_in_aga(x2) append1_out_aga(x1, x2, x3) = append1_out_aga(x1, x2, x3) U1_aga(x1, x2, x3, x4, x5) = U1_aga(x3, x5) .(x1, x2) = .(x2) U4_gg(x1, x2, x3) = U4_gg(x1, x2, x3) append2_in_gag(x1, x2, x3) = append2_in_gag(x1, x3) [] = [] append2_out_gag(x1, x2, x3) = append2_out_gag(x1, x2, x3) U2_gag(x1, x2, x3, x4, x5) = U2_gag(x2, x4, x5) sublist_out_gg(x1, x2) = sublist_out_gg(x1, x2) APPEND1_IN_AGA(x1, x2, x3) = APPEND1_IN_AGA(x2) 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: APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND1_IN_AGA(x1, x2, x3) = APPEND1_IN_AGA(x2) 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: APPEND1_IN_AGA(Ys) -> APPEND1_IN_AGA(Ys) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: sublist_in_2: (b,b) append1_in_3: (f,b,f) append2_in_3: (b,f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V)) append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys) append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs)) U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y)) append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys) append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs)) U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y) The argument filtering Pi contains the following mapping: sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) U3_gg(x1, x2, x3) = U3_gg(x2, x3) append1_in_aga(x1, x2, x3) = append1_in_aga(x2) append1_out_aga(x1, x2, x3) = append1_out_aga(x1, x3) U1_aga(x1, x2, x3, x4, x5) = U1_aga(x5) .(x1, x2) = .(x2) U4_gg(x1, x2, x3) = U4_gg(x3) append2_in_gag(x1, x2, x3) = append2_in_gag(x1, x3) [] = [] append2_out_gag(x1, x2, x3) = append2_out_gag(x2) U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5) sublist_out_gg(x1, x2) = sublist_out_gg Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (20) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V)) append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys) append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs)) U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y)) append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys) append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs)) U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y) The argument filtering Pi contains the following mapping: sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) U3_gg(x1, x2, x3) = U3_gg(x2, x3) append1_in_aga(x1, x2, x3) = append1_in_aga(x2) append1_out_aga(x1, x2, x3) = append1_out_aga(x1, x3) U1_aga(x1, x2, x3, x4, x5) = U1_aga(x5) .(x1, x2) = .(x2) U4_gg(x1, x2, x3) = U4_gg(x3) append2_in_gag(x1, x2, x3) = append2_in_gag(x1, x3) [] = [] append2_out_gag(x1, x2, x3) = append2_out_gag(x2) U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5) sublist_out_gg(x1, x2) = sublist_out_gg ---------------------------------------- (21) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: SUBLIST_IN_GG(X, Y) -> U3_GG(X, Y, append1_in_aga(U, X, V)) SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AGA(U, X, V) APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs) U3_GG(X, Y, append1_out_aga(U, X, V)) -> U4_GG(X, Y, append2_in_gag(V, W, Y)) U3_GG(X, Y, append1_out_aga(U, X, V)) -> APPEND2_IN_GAG(V, W, Y) APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V)) append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys) append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs)) U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y)) append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys) append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs)) U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y) The argument filtering Pi contains the following mapping: sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) U3_gg(x1, x2, x3) = U3_gg(x2, x3) append1_in_aga(x1, x2, x3) = append1_in_aga(x2) append1_out_aga(x1, x2, x3) = append1_out_aga(x1, x3) U1_aga(x1, x2, x3, x4, x5) = U1_aga(x5) .(x1, x2) = .(x2) U4_gg(x1, x2, x3) = U4_gg(x3) append2_in_gag(x1, x2, x3) = append2_in_gag(x1, x3) [] = [] append2_out_gag(x1, x2, x3) = append2_out_gag(x2) U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5) sublist_out_gg(x1, x2) = sublist_out_gg SUBLIST_IN_GG(x1, x2) = SUBLIST_IN_GG(x1, x2) U3_GG(x1, x2, x3) = U3_GG(x2, x3) APPEND1_IN_AGA(x1, x2, x3) = APPEND1_IN_AGA(x2) U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x5) U4_GG(x1, x2, x3) = U4_GG(x3) APPEND2_IN_GAG(x1, x2, x3) = APPEND2_IN_GAG(x1, x3) U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBLIST_IN_GG(X, Y) -> U3_GG(X, Y, append1_in_aga(U, X, V)) SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AGA(U, X, V) APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs) U3_GG(X, Y, append1_out_aga(U, X, V)) -> U4_GG(X, Y, append2_in_gag(V, W, Y)) U3_GG(X, Y, append1_out_aga(U, X, V)) -> APPEND2_IN_GAG(V, W, Y) APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V)) append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys) append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs)) U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y)) append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys) append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs)) U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y) The argument filtering Pi contains the following mapping: sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) U3_gg(x1, x2, x3) = U3_gg(x2, x3) append1_in_aga(x1, x2, x3) = append1_in_aga(x2) append1_out_aga(x1, x2, x3) = append1_out_aga(x1, x3) U1_aga(x1, x2, x3, x4, x5) = U1_aga(x5) .(x1, x2) = .(x2) U4_gg(x1, x2, x3) = U4_gg(x3) append2_in_gag(x1, x2, x3) = append2_in_gag(x1, x3) [] = [] append2_out_gag(x1, x2, x3) = append2_out_gag(x2) U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5) sublist_out_gg(x1, x2) = sublist_out_gg SUBLIST_IN_GG(x1, x2) = SUBLIST_IN_GG(x1, x2) U3_GG(x1, x2, x3) = U3_GG(x2, x3) APPEND1_IN_AGA(x1, x2, x3) = APPEND1_IN_AGA'(x2) U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x5) U4_GG(x1, x2, x3) = U4_GG(x3) APPEND2_IN_GAG(x1, x2, x3) = APPEND2_IN_GAG(x1, x3) U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (23) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes. ---------------------------------------- (24) Complex Obligation (AND) ---------------------------------------- (25) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V)) append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys) append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs)) U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y)) append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys) append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs)) U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y) The argument filtering Pi contains the following mapping: sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) U3_gg(x1, x2, x3) = U3_gg(x2, x3) append1_in_aga(x1, x2, x3) = append1_in_aga(x2) append1_out_aga(x1, x2, x3) = append1_out_aga(x1, x3) U1_aga(x1, x2, x3, x4, x5) = U1_aga(x5) .(x1, x2) = .(x2) U4_gg(x1, x2, x3) = U4_gg(x3) append2_in_gag(x1, x2, x3) = append2_in_gag(x1, x3) [] = [] append2_out_gag(x1, x2, x3) = append2_out_gag(x2) U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5) sublist_out_gg(x1, x2) = sublist_out_gg APPEND2_IN_GAG(x1, x2, x3) = APPEND2_IN_GAG(x1, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (26) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (27) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND2_IN_GAG(x1, x2, x3) = APPEND2_IN_GAG(x1, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (28) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: APPEND2_IN_GAG(.(Xs), .(Zs)) -> APPEND2_IN_GAG(Xs, Zs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (30) 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: *APPEND2_IN_GAG(.(Xs), .(Zs)) -> APPEND2_IN_GAG(Xs, Zs) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (31) YES ---------------------------------------- (32) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V)) append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys) append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs)) U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs)) U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y)) append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys) append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs)) U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs)) U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y) The argument filtering Pi contains the following mapping: sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) U3_gg(x1, x2, x3) = U3_gg(x2, x3) append1_in_aga(x1, x2, x3) = append1_in_aga(x2) append1_out_aga(x1, x2, x3) = append1_out_aga(x1, x3) U1_aga(x1, x2, x3, x4, x5) = U1_aga(x5) .(x1, x2) = .(x2) U4_gg(x1, x2, x3) = U4_gg(x3) append2_in_gag(x1, x2, x3) = append2_in_gag(x1, x3) [] = [] append2_out_gag(x1, x2, x3) = append2_out_gag(x2) U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5) sublist_out_gg(x1, x2) = sublist_out_gg APPEND1_IN_AGA(x1, x2, x3) = APPEND1_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (33) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (34) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND1_IN_AGA(x1, x2, x3) = APPEND1_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (35) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: APPEND1_IN_AGA(Ys) -> APPEND1_IN_AGA(Ys) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (37) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 12, "program": { "directives": [], "clauses": [ [ "(append1 ([]) Ys Ys)", null ], [ "(append1 (. X Xs) Ys (. X Zs))", "(append1 Xs Ys Zs)" ], [ "(append2 ([]) Ys Ys)", null ], [ "(append2 (. X Xs) Ys (. X Zs))", "(append2 Xs Ys Zs)" ], [ "(sublist X Y)", "(',' (append1 U X V) (append2 V W Y))" ] ] }, "graph": { "nodes": { "12": { "goal": [{ "clause": -1, "scope": -1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "171": { "goal": [{ "clause": 4, "scope": 1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "type": "Nodes", "172": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append1 X13 T9 X14) (append2 X14 X15 T10))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T10" ], "free": [ "X13", "X14", "X15" ], "exprvars": [] } }, "184": { "goal": [ { "clause": 0, "scope": 2, "term": "(append1 X13 T9 X14)" }, { "clause": 1, "scope": 2, "term": "(append1 X13 T9 X14)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } }, "250": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "251": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "252": { "goal": [{ "clause": -1, "scope": -1, "term": "(append2 T42 X76 T41)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T41"], "free": ["X76"], "exprvars": [] } }, "253": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "177": { "goal": [{ "clause": -1, "scope": -1, "term": "(append1 X13 T9 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } }, "210": { "goal": [{ "clause": 0, "scope": 2, "term": "(append1 X13 T9 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } }, "178": { "goal": [{ "clause": -1, "scope": -1, "term": "(append2 T14 X15 T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "212": { "goal": [{ "clause": 1, "scope": 2, "term": "(append1 X13 T9 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } }, "246": { "goal": [ { "clause": 2, "scope": 3, "term": "(append2 T14 X15 T10)" }, { "clause": 3, "scope": 3, "term": "(append2 T14 X15 T10)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "225": { "goal": [{ "clause": -1, "scope": -1, "term": "(append1 X45 T24 X46)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T24"], "free": [ "X45", "X46" ], "exprvars": [] } }, "247": { "goal": [{ "clause": 2, "scope": 3, "term": "(append2 T14 X15 T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "215": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "248": { "goal": [{ "clause": 3, "scope": 3, "term": "(append2 T14 X15 T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "216": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "249": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 12, "to": 171, "label": "CASE" }, { "from": 171, "to": 172, "label": "ONLY EVAL with clause\nsublist(X11, X12) :- ','(append1(X13, X11, X14), append2(X14, X15, X12)).\nand substitutionT1 -> T9,\nX11 -> T9,\nT2 -> T10,\nX12 -> T10" }, { "from": 172, "to": 177, "label": "SPLIT 1" }, { "from": 172, "to": 178, "label": "SPLIT 2\nnew knowledge:\nT9 is ground\nreplacements:X13 -> T13,\nX14 -> T14" }, { "from": 177, "to": 184, "label": "CASE" }, { "from": 178, "to": 246, "label": "CASE" }, { "from": 184, "to": 210, "label": "PARALLEL" }, { "from": 184, "to": 212, "label": "PARALLEL" }, { "from": 210, "to": 215, "label": "ONLY EVAL with clause\nappend1([], X24, X24).\nand substitutionX13 -> [],\nT9 -> T20,\nX24 -> T20,\nX14 -> T20" }, { "from": 212, "to": 225, "label": "ONLY EVAL with clause\nappend1(.(X40, X41), X42, .(X40, X43)) :- append1(X41, X42, X43).\nand substitutionX40 -> X44,\nX41 -> X45,\nX13 -> .(X44, X45),\nT9 -> T24,\nX42 -> T24,\nX43 -> X46,\nX14 -> .(X44, X46)" }, { "from": 215, "to": 216, "label": "SUCCESS" }, { "from": 225, "to": 177, "label": "INSTANCE with matching:\nX13 -> X45\nT9 -> T24\nX14 -> X46" }, { "from": 246, "to": 247, "label": "PARALLEL" }, { "from": 246, "to": 248, "label": "PARALLEL" }, { "from": 247, "to": 249, "label": "EVAL with clause\nappend2([], X60, X60).\nand substitutionT14 -> [],\nX15 -> T32,\nX60 -> T32,\nT10 -> T32,\nX61 -> T32" }, { "from": 247, "to": 250, "label": "EVAL-BACKTRACK" }, { "from": 248, "to": 252, "label": "EVAL with clause\nappend2(.(X72, X73), X74, .(X72, X75)) :- append2(X73, X74, X75).\nand substitutionX72 -> T39,\nX73 -> T42,\nT14 -> .(T39, T42),\nX15 -> X76,\nX74 -> X76,\nX75 -> T41,\nT10 -> .(T39, T41),\nT40 -> T42" }, { "from": 248, "to": 253, "label": "EVAL-BACKTRACK" }, { "from": 249, "to": 251, "label": "SUCCESS" }, { "from": 252, "to": 178, "label": "INSTANCE with matching:\nT14 -> T42\nX15 -> X76\nT10 -> T41" } ], "type": "Graph" } } ---------------------------------------- (38) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f12_in(T9, T10) -> U1(f172_in(T9, T10), T9, T10) U1(f172_out1(X14, X15), T9, T10) -> f12_out1 f177_in(T20) -> f177_out1 f177_in(T24) -> U2(f177_in(T24), T24) U2(f177_out1, T24) -> f177_out1 f178_in(T32) -> f178_out1([], T32) f178_in(.(T39, T41)) -> U3(f178_in(T41), .(T39, T41)) U3(f178_out1(T42, X76), .(T39, T41)) -> f178_out1(.(T39, T42), X76) f172_in(T9, T10) -> U4(f177_in(T9), T9, T10) U4(f177_out1, T9, T10) -> U5(f178_in(T10), T9, T10) U5(f178_out1(T14, X15), T9, T10) -> f172_out1(T14, X15) Q is empty. ---------------------------------------- (39) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: F12_IN(T9, T10) -> U1^1(f172_in(T9, T10), T9, T10) F12_IN(T9, T10) -> F172_IN(T9, T10) F177_IN(T24) -> U2^1(f177_in(T24), T24) F177_IN(T24) -> F177_IN(T24) F178_IN(.(T39, T41)) -> U3^1(f178_in(T41), .(T39, T41)) F178_IN(.(T39, T41)) -> F178_IN(T41) F172_IN(T9, T10) -> U4^1(f177_in(T9), T9, T10) F172_IN(T9, T10) -> F177_IN(T9) U4^1(f177_out1, T9, T10) -> U5^1(f178_in(T10), T9, T10) U4^1(f177_out1, T9, T10) -> F178_IN(T10) The TRS R consists of the following rules: f12_in(T9, T10) -> U1(f172_in(T9, T10), T9, T10) U1(f172_out1(X14, X15), T9, T10) -> f12_out1 f177_in(T20) -> f177_out1 f177_in(T24) -> U2(f177_in(T24), T24) U2(f177_out1, T24) -> f177_out1 f178_in(T32) -> f178_out1([], T32) f178_in(.(T39, T41)) -> U3(f178_in(T41), .(T39, T41)) U3(f178_out1(T42, X76), .(T39, T41)) -> f178_out1(.(T39, T42), X76) f172_in(T9, T10) -> U4(f177_in(T9), T9, T10) U4(f177_out1, T9, T10) -> U5(f178_in(T10), T9, T10) U5(f178_out1(T14, X15), T9, T10) -> f172_out1(T14, X15) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes. ---------------------------------------- (42) Complex Obligation (AND) ---------------------------------------- (43) Obligation: Q DP problem: The TRS P consists of the following rules: F178_IN(.(T39, T41)) -> F178_IN(T41) The TRS R consists of the following rules: f12_in(T9, T10) -> U1(f172_in(T9, T10), T9, T10) U1(f172_out1(X14, X15), T9, T10) -> f12_out1 f177_in(T20) -> f177_out1 f177_in(T24) -> U2(f177_in(T24), T24) U2(f177_out1, T24) -> f177_out1 f178_in(T32) -> f178_out1([], T32) f178_in(.(T39, T41)) -> U3(f178_in(T41), .(T39, T41)) U3(f178_out1(T42, X76), .(T39, T41)) -> f178_out1(.(T39, T42), X76) f172_in(T9, T10) -> U4(f177_in(T9), T9, T10) U4(f177_out1, T9, T10) -> U5(f178_in(T10), T9, T10) U5(f178_out1(T14, X15), T9, T10) -> f172_out1(T14, X15) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (44) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: F178_IN(.(T39, T41)) -> F178_IN(T41) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (46) 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: *F178_IN(.(T39, T41)) -> F178_IN(T41) The graph contains the following edges 1 > 1 ---------------------------------------- (47) YES ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: F177_IN(T24) -> F177_IN(T24) The TRS R consists of the following rules: f12_in(T9, T10) -> U1(f172_in(T9, T10), T9, T10) U1(f172_out1(X14, X15), T9, T10) -> f12_out1 f177_in(T20) -> f177_out1 f177_in(T24) -> U2(f177_in(T24), T24) U2(f177_out1, T24) -> f177_out1 f178_in(T32) -> f178_out1([], T32) f178_in(.(T39, T41)) -> U3(f178_in(T41), .(T39, T41)) U3(f178_out1(T42, X76), .(T39, T41)) -> f178_out1(.(T39, T42), X76) f172_in(T9, T10) -> U4(f177_in(T9), T9, T10) U4(f177_out1, T9, T10) -> U5(f178_in(T10), T9, T10) U5(f178_out1(T14, X15), T9, T10) -> f172_out1(T14, X15) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: F177_IN(T24) -> F177_IN(T24) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 4, "program": { "directives": [], "clauses": [ [ "(append1 ([]) Ys Ys)", null ], [ "(append1 (. X Xs) Ys (. X Zs))", "(append1 Xs Ys Zs)" ], [ "(append2 ([]) Ys Ys)", null ], [ "(append2 (. X Xs) Ys (. X Zs))", "(append2 Xs Ys Zs)" ], [ "(sublist X Y)", "(',' (append1 U X V) (append2 V W Y))" ] ] }, "graph": { "nodes": { "44": { "goal": [ { "clause": 0, "scope": 2, "term": "(',' (append1 X5 T5 X6) (append2 X6 X7 T6))" }, { "clause": 1, "scope": 2, "term": "(',' (append1 X5 T5 X6) (append2 X6 X7 T6))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T5", "T6" ], "free": [ "X5", "X6", "X7" ], "exprvars": [] } }, "27": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append1 X5 T5 X6) (append2 X6 X7 T6))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T5", "T6" ], "free": [ "X5", "X6", "X7" ], "exprvars": [] } }, "193": { "goal": [{ "clause": -1, "scope": -1, "term": "(append2 T15 X7 T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T6", "T15" ], "free": ["X7"], "exprvars": [] } }, "270": { "goal": [{ "clause": -1, "scope": -1, "term": "(append2 T61 X130 T60)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T60"], "free": ["X130"], "exprvars": [] } }, "type": "Nodes", "194": { "goal": [ { "clause": 2, "scope": 3, "term": "(append2 T15 X7 T6)" }, { "clause": 3, "scope": 3, "term": "(append2 T15 X7 T6)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T6", "T15" ], "free": ["X7"], "exprvars": [] } }, "271": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "195": { "goal": [{ "clause": 2, "scope": 3, "term": "(append2 T15 X7 T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T6", "T15" ], "free": ["X7"], "exprvars": [] } }, "272": { "goal": [ { "clause": 2, "scope": 6, "term": "(append2 T61 X130 T60)" }, { "clause": 3, "scope": 6, "term": "(append2 T61 X130 T60)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T60"], "free": ["X130"], "exprvars": [] } }, "196": { "goal": [{ "clause": 3, "scope": 3, "term": "(append2 T15 X7 T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T6", "T15" ], "free": ["X7"], "exprvars": [] } }, "273": { "goal": [{ "clause": 2, "scope": 6, "term": "(append2 T61 X130 T60)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T60"], "free": ["X130"], "exprvars": [] } }, "274": { "goal": [{ "clause": 3, "scope": 6, "term": "(append2 T61 X130 T60)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T60"], "free": ["X130"], "exprvars": [] } }, "275": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "254": { "goal": [ { "clause": 0, "scope": 4, "term": "(append1 X76 T37 X77)" }, { "clause": 1, "scope": 4, "term": "(append1 X76 T37 X77)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T37"], "free": [ "X76", "X77" ], "exprvars": [] } }, "276": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "255": { "goal": [{ "clause": 0, "scope": 4, "term": "(append1 X76 T37 X77)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T37"], "free": [ "X76", "X77" ], "exprvars": [] } }, "277": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "256": { "goal": [{ "clause": 1, "scope": 4, "term": "(append1 X76 T37 X77)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T37"], "free": [ "X76", "X77" ], "exprvars": [] } }, "278": { "goal": [{ "clause": -1, "scope": -1, "term": "(append2 T78 X159 T77)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": ["X159"], "exprvars": [] } }, "235": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append1 X76 T37 X77) (append2 (. X75 X77) X7 T6))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T6", "T37" ], "free": [ "X7", "X75", "X76", "X77" ], "exprvars": [] } }, "257": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "279": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "258": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "237": { "goal": [{ "clause": -1, "scope": -1, "term": "(append1 X76 T37 X77)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T37"], "free": [ "X76", "X77" ], "exprvars": [] } }, "238": { "goal": [{ "clause": -1, "scope": -1, "term": "(append2 (. X75 T40) X7 T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [ "X7", "X75" ], "exprvars": [] } }, "14": { "goal": [{ "clause": 4, "scope": 1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "186": { "goal": [{ "clause": 0, "scope": 2, "term": "(',' (append1 X5 T5 X6) (append2 X6 X7 T6))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T5", "T6" ], "free": [ "X5", "X6", "X7" ], "exprvars": [] } }, "187": { "goal": [{ "clause": 1, "scope": 2, "term": "(',' (append1 X5 T5 X6) (append2 X6 X7 T6))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T5", "T6" ], "free": [ "X5", "X6", "X7" ], "exprvars": [] } }, "222": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "223": { "goal": [{ "clause": -1, "scope": -1, "term": "(append2 T30 X49 T31)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T30", "T31" ], "free": ["X49"], "exprvars": [] } }, "267": { "goal": [{ "clause": -1, "scope": -1, "term": "(append1 X107 T50 X108)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T50"], "free": [ "X107", "X108" ], "exprvars": [] } }, "4": { "goal": [{ "clause": -1, "scope": -1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "202": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "224": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "268": { "goal": [ { "clause": 2, "scope": 5, "term": "(append2 (. X75 T40) X7 T6)" }, { "clause": 3, "scope": 5, "term": "(append2 (. X75 T40) X7 T6)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [ "X7", "X75" ], "exprvars": [] } }, "269": { "goal": [{ "clause": 3, "scope": 5, "term": "(append2 (. X75 T40) X7 T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [ "X7", "X75" ], "exprvars": [] } }, "206": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 4, "to": 14, "label": "CASE" }, { "from": 14, "to": 27, "label": "ONLY EVAL with clause\nsublist(X3, X4) :- ','(append1(X5, X3, X6), append2(X6, X7, X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T6,\nX4 -> T6" }, { "from": 27, "to": 44, "label": "CASE" }, { "from": 44, "to": 186, "label": "PARALLEL" }, { "from": 44, "to": 187, "label": "PARALLEL" }, { "from": 186, "to": 193, "label": "ONLY EVAL with clause\nappend1([], X20, X20).\nand substitutionX5 -> [],\nT5 -> T15,\nX20 -> T15,\nX6 -> T15" }, { "from": 187, "to": 235, "label": "ONLY EVAL with clause\nappend1(.(X71, X72), X73, .(X71, X74)) :- append1(X72, X73, X74).\nand substitutionX71 -> X75,\nX72 -> X76,\nX5 -> .(X75, X76),\nT5 -> T37,\nX73 -> T37,\nX74 -> X77,\nX6 -> .(X75, X77)" }, { "from": 193, "to": 194, "label": "CASE" }, { "from": 194, "to": 195, "label": "PARALLEL" }, { "from": 194, "to": 196, "label": "PARALLEL" }, { "from": 195, "to": 202, "label": "EVAL with clause\nappend2([], X33, X33).\nand substitutionT15 -> [],\nX7 -> T22,\nX33 -> T22,\nT6 -> T22,\nX34 -> T22" }, { "from": 195, "to": 206, "label": "EVAL-BACKTRACK" }, { "from": 196, "to": 223, "label": "EVAL with clause\nappend2(.(X45, X46), X47, .(X45, X48)) :- append2(X46, X47, X48).\nand substitutionX45 -> T29,\nX46 -> T30,\nT15 -> .(T29, T30),\nX7 -> X49,\nX47 -> X49,\nX48 -> T31,\nT6 -> .(T29, T31)" }, { "from": 196, "to": 224, "label": "EVAL-BACKTRACK" }, { "from": 202, "to": 222, "label": "SUCCESS" }, { "from": 223, "to": 193, "label": "INSTANCE with matching:\nT15 -> T30\nX7 -> X49\nT6 -> T31" }, { "from": 235, "to": 237, "label": "SPLIT 1" }, { "from": 235, "to": 238, "label": "SPLIT 2\nnew knowledge:\nT37 is ground\nreplacements:X76 -> T39,\nX77 -> T40" }, { "from": 237, "to": 254, "label": "CASE" }, { "from": 238, "to": 268, "label": "CASE" }, { "from": 254, "to": 255, "label": "PARALLEL" }, { "from": 254, "to": 256, "label": "PARALLEL" }, { "from": 255, "to": 257, "label": "ONLY EVAL with clause\nappend1([], X86, X86).\nand substitutionX76 -> [],\nT37 -> T46,\nX86 -> T46,\nX77 -> T46" }, { "from": 256, "to": 267, "label": "ONLY EVAL with clause\nappend1(.(X102, X103), X104, .(X102, X105)) :- append1(X103, X104, X105).\nand substitutionX102 -> X106,\nX103 -> X107,\nX76 -> .(X106, X107),\nT37 -> T50,\nX104 -> T50,\nX105 -> X108,\nX77 -> .(X106, X108)" }, { "from": 257, "to": 258, "label": "SUCCESS" }, { "from": 267, "to": 237, "label": "INSTANCE with matching:\nX76 -> X107\nT37 -> T50\nX77 -> X108" }, { "from": 268, "to": 269, "label": "BACKTRACK\nfor clause: append2([], Ys, Ys)because of non-unification" }, { "from": 269, "to": 270, "label": "EVAL with clause\nappend2(.(X125, X126), X127, .(X125, X128)) :- append2(X126, X127, X128).\nand substitutionX75 -> T59,\nX125 -> T59,\nT40 -> T61,\nX126 -> T61,\nX7 -> X130,\nX127 -> X130,\nX129 -> T59,\nX128 -> T60,\nT6 -> .(T59, T60),\nT58 -> T61" }, { "from": 269, "to": 271, "label": "EVAL-BACKTRACK" }, { "from": 270, "to": 272, "label": "CASE" }, { "from": 272, "to": 273, "label": "PARALLEL" }, { "from": 272, "to": 274, "label": "PARALLEL" }, { "from": 273, "to": 275, "label": "EVAL with clause\nappend2([], X143, X143).\nand substitutionT61 -> [],\nX130 -> T68,\nX143 -> T68,\nT60 -> T68,\nX144 -> T68" }, { "from": 273, "to": 276, "label": "EVAL-BACKTRACK" }, { "from": 274, "to": 278, "label": "EVAL with clause\nappend2(.(X155, X156), X157, .(X155, X158)) :- append2(X156, X157, X158).\nand substitutionX155 -> T75,\nX156 -> T78,\nT61 -> .(T75, T78),\nX130 -> X159,\nX157 -> X159,\nX158 -> T77,\nT60 -> .(T75, T77),\nT76 -> T78" }, { "from": 274, "to": 279, "label": "EVAL-BACKTRACK" }, { "from": 275, "to": 277, "label": "SUCCESS" }, { "from": 278, "to": 270, "label": "INSTANCE with matching:\nT61 -> T78\nX130 -> X159\nT60 -> T77" } ], "type": "Graph" } } ---------------------------------------- (52) Obligation: Triples: append2A(.(X1, X2), X3, .(X1, X4)) :- append2A(X2, X3, X4). append1B(.(X1, X2), X3, .(X1, X4)) :- append1B(X2, X3, X4). append2C(.(X1, X2), X3, .(X1, X4)) :- append2C(X2, X3, X4). sublistD(X1, X2) :- append2A(X1, X3, X2). sublistD(X1, X2) :- append1B(X3, X1, X4). sublistD(X1, .(X2, X3)) :- ','(append1cB(X4, X1, X5), append2C(X5, X6, X3)). Clauses: append2cA([], X1, X1). append2cA(.(X1, X2), X3, .(X1, X4)) :- append2cA(X2, X3, X4). append1cB([], X1, X1). append1cB(.(X1, X2), X3, .(X1, X4)) :- append1cB(X2, X3, X4). append2cC([], X1, X1). append2cC(.(X1, X2), X3, .(X1, X4)) :- append2cC(X2, X3, X4). Afs: sublistD(x1, x2) = sublistD(x1, x2) ---------------------------------------- (53) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: sublistD_in_2: (b,b) append2A_in_3: (b,f,b) append1B_in_3: (f,b,f) append1cB_in_3: (f,b,f) append2C_in_3: (b,f,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: SUBLISTD_IN_GG(X1, X2) -> U4_GG(X1, X2, append2A_in_gag(X1, X3, X2)) SUBLISTD_IN_GG(X1, X2) -> APPEND2A_IN_GAG(X1, X3, X2) APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U1_GAG(X1, X2, X3, X4, append2A_in_gag(X2, X3, X4)) APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2A_IN_GAG(X2, X3, X4) SUBLISTD_IN_GG(X1, X2) -> U5_GG(X1, X2, append1B_in_aga(X3, X1, X4)) SUBLISTD_IN_GG(X1, X2) -> APPEND1B_IN_AGA(X3, X1, X4) APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U2_AGA(X1, X2, X3, X4, append1B_in_aga(X2, X3, X4)) APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPEND1B_IN_AGA(X2, X3, X4) SUBLISTD_IN_GG(X1, .(X2, X3)) -> U6_GG(X1, X2, X3, append1cB_in_aga(X4, X1, X5)) U6_GG(X1, X2, X3, append1cB_out_aga(X4, X1, X5)) -> U7_GG(X1, X2, X3, append2C_in_gag(X5, X6, X3)) U6_GG(X1, X2, X3, append1cB_out_aga(X4, X1, X5)) -> APPEND2C_IN_GAG(X5, X6, X3) APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U3_GAG(X1, X2, X3, X4, append2C_in_gag(X2, X3, X4)) APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2C_IN_GAG(X2, X3, X4) The TRS R consists of the following rules: append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1) append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4)) U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: append2A_in_gag(x1, x2, x3) = append2A_in_gag(x1, x3) .(x1, x2) = .(x2) append1B_in_aga(x1, x2, x3) = append1B_in_aga(x2) append1cB_in_aga(x1, x2, x3) = append1cB_in_aga(x2) append1cB_out_aga(x1, x2, x3) = append1cB_out_aga(x1, x2, x3) U10_aga(x1, x2, x3, x4, x5) = U10_aga(x3, x5) append2C_in_gag(x1, x2, x3) = append2C_in_gag(x1, x3) SUBLISTD_IN_GG(x1, x2) = SUBLISTD_IN_GG(x1, x2) U4_GG(x1, x2, x3) = U4_GG(x1, x2, x3) APPEND2A_IN_GAG(x1, x2, x3) = APPEND2A_IN_GAG(x1, x3) U1_GAG(x1, x2, x3, x4, x5) = U1_GAG(x2, x4, x5) U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3) APPEND1B_IN_AGA(x1, x2, x3) = APPEND1B_IN_AGA(x2) U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x3, x5) U6_GG(x1, x2, x3, x4) = U6_GG(x1, x3, x4) U7_GG(x1, x2, x3, x4) = U7_GG(x1, x3, x4) APPEND2C_IN_GAG(x1, x2, x3) = APPEND2C_IN_GAG(x1, x3) U3_GAG(x1, x2, x3, x4, x5) = U3_GAG(x2, x4, x5) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (54) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBLISTD_IN_GG(X1, X2) -> U4_GG(X1, X2, append2A_in_gag(X1, X3, X2)) SUBLISTD_IN_GG(X1, X2) -> APPEND2A_IN_GAG(X1, X3, X2) APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U1_GAG(X1, X2, X3, X4, append2A_in_gag(X2, X3, X4)) APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2A_IN_GAG(X2, X3, X4) SUBLISTD_IN_GG(X1, X2) -> U5_GG(X1, X2, append1B_in_aga(X3, X1, X4)) SUBLISTD_IN_GG(X1, X2) -> APPEND1B_IN_AGA(X3, X1, X4) APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U2_AGA(X1, X2, X3, X4, append1B_in_aga(X2, X3, X4)) APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPEND1B_IN_AGA(X2, X3, X4) SUBLISTD_IN_GG(X1, .(X2, X3)) -> U6_GG(X1, X2, X3, append1cB_in_aga(X4, X1, X5)) U6_GG(X1, X2, X3, append1cB_out_aga(X4, X1, X5)) -> U7_GG(X1, X2, X3, append2C_in_gag(X5, X6, X3)) U6_GG(X1, X2, X3, append1cB_out_aga(X4, X1, X5)) -> APPEND2C_IN_GAG(X5, X6, X3) APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U3_GAG(X1, X2, X3, X4, append2C_in_gag(X2, X3, X4)) APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2C_IN_GAG(X2, X3, X4) The TRS R consists of the following rules: append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1) append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4)) U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: append2A_in_gag(x1, x2, x3) = append2A_in_gag(x1, x3) .(x1, x2) = .(x2) append1B_in_aga(x1, x2, x3) = append1B_in_aga(x2) append1cB_in_aga(x1, x2, x3) = append1cB_in_aga(x2) append1cB_out_aga(x1, x2, x3) = append1cB_out_aga(x1, x2, x3) U10_aga(x1, x2, x3, x4, x5) = U10_aga(x3, x5) append2C_in_gag(x1, x2, x3) = append2C_in_gag(x1, x3) SUBLISTD_IN_GG(x1, x2) = SUBLISTD_IN_GG(x1, x2) U4_GG(x1, x2, x3) = U4_GG(x1, x2, x3) APPEND2A_IN_GAG(x1, x2, x3) = APPEND2A_IN_GAG(x1, x3) U1_GAG(x1, x2, x3, x4, x5) = U1_GAG(x2, x4, x5) U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3) APPEND1B_IN_AGA(x1, x2, x3) = APPEND1B_IN_AGA(x2) U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x3, x5) U6_GG(x1, x2, x3, x4) = U6_GG(x1, x3, x4) U7_GG(x1, x2, x3, x4) = U7_GG(x1, x3, x4) APPEND2C_IN_GAG(x1, x2, x3) = APPEND2C_IN_GAG(x1, x3) U3_GAG(x1, x2, x3, x4, x5) = U3_GAG(x2, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (55) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes. ---------------------------------------- (56) Complex Obligation (AND) ---------------------------------------- (57) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2C_IN_GAG(X2, X3, X4) The TRS R consists of the following rules: append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1) append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4)) U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) append1cB_in_aga(x1, x2, x3) = append1cB_in_aga(x2) append1cB_out_aga(x1, x2, x3) = append1cB_out_aga(x1, x2, x3) U10_aga(x1, x2, x3, x4, x5) = U10_aga(x3, x5) APPEND2C_IN_GAG(x1, x2, x3) = APPEND2C_IN_GAG(x1, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (58) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (59) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2C_IN_GAG(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND2C_IN_GAG(x1, x2, x3) = APPEND2C_IN_GAG(x1, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (60) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (61) Obligation: Q DP problem: The TRS P consists of the following rules: APPEND2C_IN_GAG(.(X2), .(X4)) -> APPEND2C_IN_GAG(X2, X4) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (62) 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: *APPEND2C_IN_GAG(.(X2), .(X4)) -> APPEND2C_IN_GAG(X2, X4) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (63) YES ---------------------------------------- (64) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPEND1B_IN_AGA(X2, X3, X4) The TRS R consists of the following rules: append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1) append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4)) U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) append1cB_in_aga(x1, x2, x3) = append1cB_in_aga(x2) append1cB_out_aga(x1, x2, x3) = append1cB_out_aga(x1, x2, x3) U10_aga(x1, x2, x3, x4, x5) = U10_aga(x3, x5) APPEND1B_IN_AGA(x1, x2, x3) = APPEND1B_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (65) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (66) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPEND1B_IN_AGA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND1B_IN_AGA(x1, x2, x3) = APPEND1B_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (67) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: APPEND1B_IN_AGA(X3) -> APPEND1B_IN_AGA(X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (69) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2A_IN_GAG(X2, X3, X4) The TRS R consists of the following rules: append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1) append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4)) U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) append1cB_in_aga(x1, x2, x3) = append1cB_in_aga(x2) append1cB_out_aga(x1, x2, x3) = append1cB_out_aga(x1, x2, x3) U10_aga(x1, x2, x3, x4, x5) = U10_aga(x3, x5) APPEND2A_IN_GAG(x1, x2, x3) = APPEND2A_IN_GAG(x1, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (70) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (71) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2A_IN_GAG(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND2A_IN_GAG(x1, x2, x3) = APPEND2A_IN_GAG(x1, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (72) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 3, "program": { "directives": [], "clauses": [ [ "(append1 ([]) Ys Ys)", null ], [ "(append1 (. X Xs) Ys (. X Zs))", "(append1 Xs Ys Zs)" ], [ "(append2 ([]) Ys Ys)", null ], [ "(append2 (. X Xs) Ys (. X Zs))", "(append2 Xs Ys Zs)" ], [ "(sublist X Y)", "(',' (append1 U X V) (append2 V W Y))" ] ] }, "graph": { "nodes": { "180": { "goal": [{ "clause": -1, "scope": -1, "term": "(append2 T14 X15 T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "170": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append1 X13 T9 X14) (append2 X14 X15 T10))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T10" ], "free": [ "X13", "X14", "X15" ], "exprvars": [] } }, "type": "Nodes", "240": { "goal": [{ "clause": 3, "scope": 3, "term": "(append2 T14 X15 T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "241": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "220": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "242": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "221": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "243": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "123": { "goal": [{ "clause": 4, "scope": 1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "244": { "goal": [{ "clause": -1, "scope": -1, "term": "(append2 T42 X76 T41)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T41"], "free": ["X76"], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "179": { "goal": [{ "clause": -1, "scope": -1, "term": "(append1 X13 T9 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } }, "245": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "236": { "goal": [ { "clause": 2, "scope": 3, "term": "(append2 T14 X15 T10)" }, { "clause": 3, "scope": 3, "term": "(append2 T14 X15 T10)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "226": { "goal": [{ "clause": -1, "scope": -1, "term": "(append1 X45 T24 X46)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T24"], "free": [ "X45", "X46" ], "exprvars": [] } }, "217": { "goal": [ { "clause": 0, "scope": 2, "term": "(append1 X13 T9 X14)" }, { "clause": 1, "scope": 2, "term": "(append1 X13 T9 X14)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } }, "239": { "goal": [{ "clause": 2, "scope": 3, "term": "(append2 T14 X15 T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "218": { "goal": [{ "clause": 0, "scope": 2, "term": "(append1 X13 T9 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } }, "219": { "goal": [{ "clause": 1, "scope": 2, "term": "(append1 X13 T9 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 123, "label": "CASE" }, { "from": 123, "to": 170, "label": "ONLY EVAL with clause\nsublist(X11, X12) :- ','(append1(X13, X11, X14), append2(X14, X15, X12)).\nand substitutionT1 -> T9,\nX11 -> T9,\nT2 -> T10,\nX12 -> T10" }, { "from": 170, "to": 179, "label": "SPLIT 1" }, { "from": 170, "to": 180, "label": "SPLIT 2\nnew knowledge:\nT9 is ground\nreplacements:X13 -> T13,\nX14 -> T14" }, { "from": 179, "to": 217, "label": "CASE" }, { "from": 180, "to": 236, "label": "CASE" }, { "from": 217, "to": 218, "label": "PARALLEL" }, { "from": 217, "to": 219, "label": "PARALLEL" }, { "from": 218, "to": 220, "label": "ONLY EVAL with clause\nappend1([], X24, X24).\nand substitutionX13 -> [],\nT9 -> T20,\nX24 -> T20,\nX14 -> T20" }, { "from": 219, "to": 226, "label": "ONLY EVAL with clause\nappend1(.(X40, X41), X42, .(X40, X43)) :- append1(X41, X42, X43).\nand substitutionX40 -> X44,\nX41 -> X45,\nX13 -> .(X44, X45),\nT9 -> T24,\nX42 -> T24,\nX43 -> X46,\nX14 -> .(X44, X46)" }, { "from": 220, "to": 221, "label": "SUCCESS" }, { "from": 226, "to": 179, "label": "INSTANCE with matching:\nX13 -> X45\nT9 -> T24\nX14 -> X46" }, { "from": 236, "to": 239, "label": "PARALLEL" }, { "from": 236, "to": 240, "label": "PARALLEL" }, { "from": 239, "to": 241, "label": "EVAL with clause\nappend2([], X60, X60).\nand substitutionT14 -> [],\nX15 -> T32,\nX60 -> T32,\nT10 -> T32,\nX61 -> T32" }, { "from": 239, "to": 242, "label": "EVAL-BACKTRACK" }, { "from": 240, "to": 244, "label": "EVAL with clause\nappend2(.(X72, X73), X74, .(X72, X75)) :- append2(X73, X74, X75).\nand substitutionX72 -> T39,\nX73 -> T42,\nT14 -> .(T39, T42),\nX15 -> X76,\nX74 -> X76,\nX75 -> T41,\nT10 -> .(T39, T41),\nT40 -> T42" }, { "from": 240, "to": 245, "label": "EVAL-BACKTRACK" }, { "from": 241, "to": 243, "label": "SUCCESS" }, { "from": 244, "to": 180, "label": "INSTANCE with matching:\nT14 -> T42\nX15 -> X76\nT10 -> T41" } ], "type": "Graph" } } ---------------------------------------- (73) Complex Obligation (AND) ---------------------------------------- (74) Obligation: Rules: f236_in(T10) -> f240_in(T10) :|: TRUE f236_in(x) -> f239_in(x) :|: TRUE f240_out(x1) -> f236_out(x1) :|: TRUE f239_out(x2) -> f236_out(x2) :|: TRUE f240_in(x3) -> f245_in :|: TRUE f240_in(.(T39, T41)) -> f244_in(T41) :|: TRUE f244_out(x4) -> f240_out(.(x5, x4)) :|: TRUE f245_out -> f240_out(x6) :|: TRUE f244_in(x7) -> f180_in(x7) :|: TRUE f180_out(x8) -> f244_out(x8) :|: TRUE f180_in(x9) -> f236_in(x9) :|: TRUE f236_out(x10) -> f180_out(x10) :|: TRUE f3_in(T1, T2) -> f123_in(T1, T2) :|: TRUE f123_out(x11, x12) -> f3_out(x11, x12) :|: TRUE f170_out(x13, x14) -> f123_out(x13, x14) :|: TRUE f123_in(x15, x16) -> f170_in(x15, x16) :|: TRUE f170_in(x17, x18) -> f179_in(x17) :|: TRUE f180_out(x19) -> f170_out(x20, x19) :|: TRUE f179_out(x21) -> f180_in(x22) :|: TRUE Start term: f3_in(T1, T2) ---------------------------------------- (75) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (76) TRUE ---------------------------------------- (77) Obligation: Rules: f219_in(T24) -> f226_in(T24) :|: TRUE f226_out(x) -> f219_out(x) :|: TRUE f226_in(x1) -> f179_in(x1) :|: TRUE f179_out(x2) -> f226_out(x2) :|: TRUE f179_in(T9) -> f217_in(T9) :|: TRUE f217_out(x3) -> f179_out(x3) :|: TRUE f217_in(x4) -> f219_in(x4) :|: TRUE f219_out(x5) -> f217_out(x5) :|: TRUE f217_in(x6) -> f218_in(x6) :|: TRUE f218_out(x7) -> f217_out(x7) :|: TRUE f3_in(T1, T2) -> f123_in(T1, T2) :|: TRUE f123_out(x8, x9) -> f3_out(x8, x9) :|: TRUE f170_out(x10, x11) -> f123_out(x10, x11) :|: TRUE f123_in(x12, x13) -> f170_in(x12, x13) :|: TRUE f170_in(x14, x15) -> f179_in(x14) :|: TRUE f180_out(x16) -> f170_out(x17, x16) :|: TRUE f179_out(x18) -> f180_in(x19) :|: TRUE Start term: f3_in(T1, T2) ---------------------------------------- (78) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f219_in(T24) -> f226_in(T24) :|: TRUE f226_in(x1) -> f179_in(x1) :|: TRUE f179_in(T9) -> f217_in(T9) :|: TRUE f217_in(x4) -> f219_in(x4) :|: TRUE ---------------------------------------- (79) Obligation: Rules: f219_in(T24) -> f226_in(T24) :|: TRUE f226_in(x1) -> f179_in(x1) :|: TRUE f179_in(T9) -> f217_in(T9) :|: TRUE f217_in(x4) -> f219_in(x4) :|: TRUE ---------------------------------------- (80) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (81) Obligation: Rules: f219_in(T24:0) -> f219_in(T24:0) :|: TRUE ---------------------------------------- (82) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (83) Obligation: Rules: f219_in(T24:0) -> f219_in(T24:0) :|: TRUE ---------------------------------------- (84) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f219_in(T24:0) -> f219_in(T24:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (85) Obligation: Termination digraph: Nodes: (1) f219_in(T24:0) -> f219_in(T24:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (86) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f219_in(VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (87) Obligation: Rules: f219_in(T24:0) -> f219_in(T24:0) :|: TRUE ---------------------------------------- (88) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, T24:0) -> f(1, T24:0) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1, -8) ---------------------------------------- (89) NO