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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern sublist(g,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms]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, 7 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 16 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, 2 ms] (22) PiDP (23) DependencyGraphProof [EQUIVALENT, 0 ms] (24) AND (25) PiDP (26) UsableRulesProof [EQUIVALENT, 0 ms] (27) PiDP (28) PiDPToQDPProof [SOUND, 2 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) PrologToIRSwTTransformerProof [SOUND, 36 ms] (52) AND (53) IRSwT (54) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (55) TRUE (56) IRSwT (57) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (58) IRSwT (59) IntTRSCompressionProof [EQUIVALENT, 21 ms] (60) IRSwT (61) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (62) IRSwT (63) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (64) IRSwT (65) FilterProof [EQUIVALENT, 0 ms] (66) IntTRS (67) IntTRSPeriodicNontermProof [COMPLETE, 6 ms] (68) NO (69) PrologToDTProblemTransformerProof [SOUND, 41 ms] (70) TRIPLES (71) TriplesToPiDPProof [SOUND, 0 ms] (72) PiDP (73) DependencyGraphProof [EQUIVALENT, 0 ms] (74) AND (75) PiDP (76) UsableRulesProof [EQUIVALENT, 0 ms] (77) PiDP (78) PiDPToQDPProof [SOUND, 0 ms] (79) QDP (80) QDPSizeChangeProof [EQUIVALENT, 0 ms] (81) YES (82) PiDP (83) UsableRulesProof [EQUIVALENT, 0 ms] (84) PiDP (85) PiDPToQDPProof [SOUND, 0 ms] (86) QDP (87) PiDP (88) UsableRulesProof [EQUIVALENT, 0 ms] (89) PiDP ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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) ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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": 2, "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": { "33": { "goal": [{ "clause": 1, "scope": 2, "term": "(append1 X13 T9 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } }, "34": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "35": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "36": { "goal": [{ "clause": -1, "scope": -1, "term": "(append1 X45 T24 X46)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T24"], "free": [ "X45", "X46" ], "exprvars": [] } }, "28": { "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": [] } }, "29": { "goal": [{ "clause": -1, "scope": -1, "term": "(append1 X13 T9 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } }, "type": "Nodes", "197": { "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": [] } }, "198": { "goal": [{ "clause": 2, "scope": 3, "term": "(append2 T14 X15 T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "199": { "goal": [{ "clause": 3, "scope": 3, "term": "(append2 T14 X15 T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "200": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "201": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "202": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5": { "goal": [{ "clause": 4, "scope": 1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "203": { "goal": [{ "clause": -1, "scope": -1, "term": "(append2 T42 X76 T41)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T41"], "free": ["X76"], "exprvars": [] } }, "204": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "30": { "goal": [{ "clause": -1, "scope": -1, "term": "(append2 T14 X15 T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "31": { "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": [] } }, "32": { "goal": [{ "clause": 0, "scope": 2, "term": "(append1 X13 T9 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 5, "label": "CASE" }, { "from": 5, "to": 28, "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": 28, "to": 29, "label": "SPLIT 1" }, { "from": 28, "to": 30, "label": "SPLIT 2\nnew knowledge:\nT9 is ground\nreplacements:X13 -> T13,\nX14 -> T14" }, { "from": 29, "to": 31, "label": "CASE" }, { "from": 30, "to": 197, "label": "CASE" }, { "from": 31, "to": 32, "label": "PARALLEL" }, { "from": 31, "to": 33, "label": "PARALLEL" }, { "from": 32, "to": 34, "label": "ONLY EVAL with clause\nappend1([], X24, X24).\nand substitutionX13 -> [],\nT9 -> T20,\nX24 -> T20,\nX14 -> T20" }, { "from": 33, "to": 36, "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": 34, "to": 35, "label": "SUCCESS" }, { "from": 36, "to": 29, "label": "INSTANCE with matching:\nX13 -> X45\nT9 -> T24\nX14 -> X46" }, { "from": 197, "to": 198, "label": "PARALLEL" }, { "from": 197, "to": 199, "label": "PARALLEL" }, { "from": 198, "to": 200, "label": "EVAL with clause\nappend2([], X60, X60).\nand substitutionT14 -> [],\nX15 -> T32,\nX60 -> T32,\nT10 -> T32,\nX61 -> T32" }, { "from": 198, "to": 201, "label": "EVAL-BACKTRACK" }, { "from": 199, "to": 203, "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": 199, "to": 204, "label": "EVAL-BACKTRACK" }, { "from": 200, "to": 202, "label": "SUCCESS" }, { "from": 203, "to": 30, "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: f2_in(T9, T10) -> U1(f28_in(T9, T10), T9, T10) U1(f28_out1(X14, X15), T9, T10) -> f2_out1 f29_in(T20) -> f29_out1 f29_in(T24) -> U2(f29_in(T24), T24) U2(f29_out1, T24) -> f29_out1 f30_in(T32) -> f30_out1([], T32) f30_in(.(T39, T41)) -> U3(f30_in(T41), .(T39, T41)) U3(f30_out1(T42, X76), .(T39, T41)) -> f30_out1(.(T39, T42), X76) f28_in(T9, T10) -> U4(f29_in(T9), T9, T10) U4(f29_out1, T9, T10) -> U5(f30_in(T10), T9, T10) U5(f30_out1(T14, X15), T9, T10) -> f28_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: F2_IN(T9, T10) -> U1^1(f28_in(T9, T10), T9, T10) F2_IN(T9, T10) -> F28_IN(T9, T10) F29_IN(T24) -> U2^1(f29_in(T24), T24) F29_IN(T24) -> F29_IN(T24) F30_IN(.(T39, T41)) -> U3^1(f30_in(T41), .(T39, T41)) F30_IN(.(T39, T41)) -> F30_IN(T41) F28_IN(T9, T10) -> U4^1(f29_in(T9), T9, T10) F28_IN(T9, T10) -> F29_IN(T9) U4^1(f29_out1, T9, T10) -> U5^1(f30_in(T10), T9, T10) U4^1(f29_out1, T9, T10) -> F30_IN(T10) The TRS R consists of the following rules: f2_in(T9, T10) -> U1(f28_in(T9, T10), T9, T10) U1(f28_out1(X14, X15), T9, T10) -> f2_out1 f29_in(T20) -> f29_out1 f29_in(T24) -> U2(f29_in(T24), T24) U2(f29_out1, T24) -> f29_out1 f30_in(T32) -> f30_out1([], T32) f30_in(.(T39, T41)) -> U3(f30_in(T41), .(T39, T41)) U3(f30_out1(T42, X76), .(T39, T41)) -> f30_out1(.(T39, T42), X76) f28_in(T9, T10) -> U4(f29_in(T9), T9, T10) U4(f29_out1, T9, T10) -> U5(f30_in(T10), T9, T10) U5(f30_out1(T14, X15), T9, T10) -> f28_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: F30_IN(.(T39, T41)) -> F30_IN(T41) The TRS R consists of the following rules: f2_in(T9, T10) -> U1(f28_in(T9, T10), T9, T10) U1(f28_out1(X14, X15), T9, T10) -> f2_out1 f29_in(T20) -> f29_out1 f29_in(T24) -> U2(f29_in(T24), T24) U2(f29_out1, T24) -> f29_out1 f30_in(T32) -> f30_out1([], T32) f30_in(.(T39, T41)) -> U3(f30_in(T41), .(T39, T41)) U3(f30_out1(T42, X76), .(T39, T41)) -> f30_out1(.(T39, T42), X76) f28_in(T9, T10) -> U4(f29_in(T9), T9, T10) U4(f29_out1, T9, T10) -> U5(f30_in(T10), T9, T10) U5(f30_out1(T14, X15), T9, T10) -> f28_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: F30_IN(.(T39, T41)) -> F30_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: *F30_IN(.(T39, T41)) -> F30_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: F29_IN(T24) -> F29_IN(T24) The TRS R consists of the following rules: f2_in(T9, T10) -> U1(f28_in(T9, T10), T9, T10) U1(f28_out1(X14, X15), T9, T10) -> f2_out1 f29_in(T20) -> f29_out1 f29_in(T24) -> U2(f29_in(T24), T24) U2(f29_out1, T24) -> f29_out1 f30_in(T32) -> f30_out1([], T32) f30_in(.(T39, T41)) -> U3(f30_in(T41), .(T39, T41)) U3(f30_out1(T42, X76), .(T39, T41)) -> f30_out1(.(T39, T42), X76) f28_in(T9, T10) -> U4(f29_in(T9), T9, T10) U4(f29_out1, T9, T10) -> U5(f30_in(T10), T9, T10) U5(f30_out1(T14, X15), T9, T10) -> f28_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: F29_IN(T24) -> F29_IN(T24) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) 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": { "17": { "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": [] } }, "192": { "goal": [{ "clause": -1, "scope": -1, "term": "(append1 X13 T9 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } }, "193": { "goal": [{ "clause": -1, "scope": -1, "term": "(append2 T14 X15 T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "type": "Nodes", "194": { "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": [] } }, "195": { "goal": [{ "clause": 0, "scope": 2, "term": "(append1 X13 T9 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } }, "196": { "goal": [{ "clause": 1, "scope": 2, "term": "(append1 X13 T9 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X13", "X14" ], "exprvars": [] } }, "241": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "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": [] } }, "232": { "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": [] } }, "233": { "goal": [{ "clause": 2, "scope": 3, "term": "(append2 T14 X15 T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "244": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "245": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "235": { "goal": [{ "clause": 3, "scope": 3, "term": "(append2 T14 X15 T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X15"], "exprvars": [] } }, "6": { "goal": [{ "clause": 4, "scope": 1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "205": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "227": { "goal": [{ "clause": -1, "scope": -1, "term": "(append1 X45 T24 X46)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T24"], "free": [ "X45", "X46" ], "exprvars": [] } }, "206": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 6, "label": "CASE" }, { "from": 6, "to": 17, "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": 17, "to": 192, "label": "SPLIT 1" }, { "from": 17, "to": 193, "label": "SPLIT 2\nnew knowledge:\nT9 is ground\nreplacements:X13 -> T13,\nX14 -> T14" }, { "from": 192, "to": 194, "label": "CASE" }, { "from": 193, "to": 232, "label": "CASE" }, { "from": 194, "to": 195, "label": "PARALLEL" }, { "from": 194, "to": 196, "label": "PARALLEL" }, { "from": 195, "to": 205, "label": "ONLY EVAL with clause\nappend1([], X24, X24).\nand substitutionX13 -> [],\nT9 -> T20,\nX24 -> T20,\nX14 -> T20" }, { "from": 196, "to": 227, "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": 205, "to": 206, "label": "SUCCESS" }, { "from": 227, "to": 192, "label": "INSTANCE with matching:\nX13 -> X45\nT9 -> T24\nX14 -> X46" }, { "from": 232, "to": 233, "label": "PARALLEL" }, { "from": 232, "to": 235, "label": "PARALLEL" }, { "from": 233, "to": 241, "label": "EVAL with clause\nappend2([], X60, X60).\nand substitutionT14 -> [],\nX15 -> T32,\nX60 -> T32,\nT10 -> T32,\nX61 -> T32" }, { "from": 233, "to": 244, "label": "EVAL-BACKTRACK" }, { "from": 235, "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": 235, "to": 253, "label": "EVAL-BACKTRACK" }, { "from": 241, "to": 245, "label": "SUCCESS" }, { "from": 252, "to": 193, "label": "INSTANCE with matching:\nT14 -> T42\nX15 -> X76\nT10 -> T41" } ], "type": "Graph" } } ---------------------------------------- (52) Complex Obligation (AND) ---------------------------------------- (53) Obligation: Rules: f235_out(T10) -> f232_out(T10) :|: TRUE f232_in(x) -> f235_in(x) :|: TRUE f232_in(x1) -> f233_in(x1) :|: TRUE f233_out(x2) -> f232_out(x2) :|: TRUE f252_in(T41) -> f193_in(T41) :|: TRUE f193_out(x3) -> f252_out(x3) :|: TRUE f235_in(x4) -> f253_in :|: TRUE f252_out(x5) -> f235_out(.(x6, x5)) :|: TRUE f253_out -> f235_out(x7) :|: TRUE f235_in(.(x8, x9)) -> f252_in(x9) :|: TRUE f193_in(x10) -> f232_in(x10) :|: TRUE f232_out(x11) -> f193_out(x11) :|: TRUE f6_out(T1, T2) -> f3_out(T1, T2) :|: TRUE f3_in(x12, x13) -> f6_in(x12, x13) :|: TRUE f17_out(x14, x15) -> f6_out(x14, x15) :|: TRUE f6_in(x16, x17) -> f17_in(x16, x17) :|: TRUE f193_out(x18) -> f17_out(x19, x18) :|: TRUE f17_in(x20, x21) -> f192_in(x20) :|: TRUE f192_out(x22) -> f193_in(x23) :|: TRUE Start term: f3_in(T1, T2) ---------------------------------------- (54) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (55) TRUE ---------------------------------------- (56) Obligation: Rules: f227_out(T24) -> f196_out(T24) :|: TRUE f196_in(x) -> f227_in(x) :|: TRUE f192_in(T9) -> f194_in(T9) :|: TRUE f194_out(x1) -> f192_out(x1) :|: TRUE f195_out(x2) -> f194_out(x2) :|: TRUE f194_in(x3) -> f195_in(x3) :|: TRUE f196_out(x4) -> f194_out(x4) :|: TRUE f194_in(x5) -> f196_in(x5) :|: TRUE f192_out(x6) -> f227_out(x6) :|: TRUE f227_in(x7) -> f192_in(x7) :|: TRUE f6_out(T1, T2) -> f3_out(T1, T2) :|: TRUE f3_in(x8, x9) -> f6_in(x8, x9) :|: TRUE f17_out(x10, x11) -> f6_out(x10, x11) :|: TRUE f6_in(x12, x13) -> f17_in(x12, x13) :|: TRUE f193_out(x14) -> f17_out(x15, x14) :|: TRUE f17_in(x16, x17) -> f192_in(x16) :|: TRUE f192_out(x18) -> f193_in(x19) :|: TRUE Start term: f3_in(T1, T2) ---------------------------------------- (57) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f196_in(x) -> f227_in(x) :|: TRUE f192_in(T9) -> f194_in(T9) :|: TRUE f194_in(x5) -> f196_in(x5) :|: TRUE f227_in(x7) -> f192_in(x7) :|: TRUE ---------------------------------------- (58) Obligation: Rules: f196_in(x) -> f227_in(x) :|: TRUE f192_in(T9) -> f194_in(T9) :|: TRUE f194_in(x5) -> f196_in(x5) :|: TRUE f227_in(x7) -> f192_in(x7) :|: TRUE ---------------------------------------- (59) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (60) Obligation: Rules: f192_in(T9:0) -> f192_in(T9:0) :|: TRUE ---------------------------------------- (61) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (62) Obligation: Rules: f192_in(T9:0) -> f192_in(T9:0) :|: TRUE ---------------------------------------- (63) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f192_in(T9:0) -> f192_in(T9:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (64) Obligation: Termination digraph: Nodes: (1) f192_in(T9:0) -> f192_in(T9:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (65) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f192_in(VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (66) Obligation: Rules: f192_in(T9:0) -> f192_in(T9:0) :|: TRUE ---------------------------------------- (67) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, T9:0) -> f(1, T9:0) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1, -8) ---------------------------------------- (68) NO ---------------------------------------- (69) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(append1 ([]) Ys Ys)", null ], [ "(append1 (. 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:- ','(append1(X5, X3, X6), append2(X6, X7, X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T6,\nX4 -> T6" }, { "from": 108, "to": 109, "label": "CASE" }, { "from": 109, "to": 110, "label": "PARALLEL" }, { "from": 109, "to": 111, "label": "PARALLEL" }, { "from": 110, "to": 112, "label": "ONLY EVAL with clause\nappend1([], X20, X20).\nand substitutionX5 -> [],\nT5 -> T15,\nX20 -> T15,\nX6 -> T15" }, { "from": 111, "to": 210, "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": 112, "to": 113, "label": "CASE" }, { "from": 113, "to": 114, "label": "PARALLEL" }, { "from": 113, "to": 115, "label": "PARALLEL" }, { "from": 114, "to": 116, "label": "EVAL with clause\nappend2([], X33, X33).\nand substitutionT15 -> [],\nX7 -> T22,\nX33 -> T22,\nT6 -> T22,\nX34 -> T22" }, { "from": 114, "to": 117, "label": "EVAL-BACKTRACK" }, { "from": 115, "to": 167, "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": 115, "to": 175, "label": "EVAL-BACKTRACK" }, { "from": 116, "to": 118, "label": "SUCCESS" }, { "from": 167, "to": 112, "label": "INSTANCE with matching:\nT15 -> T30\nX7 -> X49\nT6 -> T31" }, { "from": 210, "to": 212, "label": "SPLIT 1" }, { "from": 210, "to": 213, "label": "SPLIT 2\nnew knowledge:\nT37 is ground\nreplacements:X76 -> T39,\nX77 -> T40" }, { "from": 212, "to": 219, "label": "CASE" }, { "from": 213, "to": 225, "label": "CASE" }, { "from": 219, "to": 220, "label": "PARALLEL" }, { "from": 219, "to": 221, "label": "PARALLEL" }, { "from": 220, "to": 222, "label": "ONLY EVAL with clause\nappend1([], X86, X86).\nand substitutionX76 -> [],\nT37 -> T46,\nX86 -> T46,\nX77 -> T46" }, { "from": 221, "to": 224, "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": 222, "to": 223, "label": "SUCCESS" }, { "from": 224, "to": 212, "label": "INSTANCE with matching:\nX76 -> X107\nT37 -> T50\nX77 -> X108" }, { "from": 225, "to": 226, "label": "BACKTRACK\nfor clause: append2([], Ys, Ys)because of non-unification" }, { "from": 226, "to": 254, "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": 226, "to": 255, "label": "EVAL-BACKTRACK" }, { "from": 254, "to": 264, "label": "CASE" }, { "from": 264, "to": 265, "label": "PARALLEL" }, { "from": 264, "to": 266, "label": "PARALLEL" }, { "from": 265, "to": 267, "label": "EVAL with clause\nappend2([], X143, X143).\nand substitutionT61 -> [],\nX130 -> T68,\nX143 -> T68,\nT60 -> T68,\nX144 -> T68" }, { "from": 265, "to": 268, "label": "EVAL-BACKTRACK" }, { "from": 266, "to": 270, "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": 266, "to": 271, "label": "EVAL-BACKTRACK" }, { "from": 267, "to": 269, "label": "SUCCESS" }, { "from": 270, "to": 254, "label": "INSTANCE with matching:\nT61 -> T78\nX130 -> X159\nT60 -> T77" } ], "type": "Graph" } } ---------------------------------------- (70) 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) ---------------------------------------- (71) 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 ---------------------------------------- (72) 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 ---------------------------------------- (73) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes. ---------------------------------------- (74) Complex Obligation (AND) ---------------------------------------- (75) 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 ---------------------------------------- (76) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (77) 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 ---------------------------------------- (78) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (79) 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. ---------------------------------------- (80) 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 ---------------------------------------- (81) YES ---------------------------------------- (82) 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 ---------------------------------------- (83) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (84) 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 ---------------------------------------- (85) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (86) 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. ---------------------------------------- (87) 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 ---------------------------------------- (88) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (89) 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