MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern sublist(g,a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) 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, 0 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) MNOCProof [EQUIVALENT, 0 ms] (45) QDP (46) UsableRulesProof [EQUIVALENT, 0 ms] (47) QDP (48) QReductionProof [EQUIVALENT, 0 ms] (49) QDP (50) QDP (51) MNOCProof [EQUIVALENT, 0 ms] (52) QDP (53) UsableRulesProof [EQUIVALENT, 0 ms] (54) QDP (55) QReductionProof [EQUIVALENT, 0 ms] (56) QDP (57) PrologToIRSwTTransformerProof [SOUND, 0 ms] (58) AND (59) IRSwT (60) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (61) TRUE (62) IRSwT (63) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (64) IRSwT (65) IntTRSCompressionProof [EQUIVALENT, 23 ms] (66) IRSwT (67) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (68) IRSwT (69) IRSwTTerminationDigraphProof [EQUIVALENT, 7 ms] (70) IRSwT (71) FilterProof [EQUIVALENT, 0 ms] (72) IntTRS (73) IntTRSPeriodicNontermProof [COMPLETE, 7 ms] (74) NO (75) PrologToDTProblemTransformerProof [SOUND, 0 ms] (76) TRIPLES (77) TriplesToPiDPProof [SOUND, 2 ms] (78) PiDP (79) DependencyGraphProof [EQUIVALENT, 0 ms] (80) AND (81) PiDP (82) UsableRulesProof [EQUIVALENT, 0 ms] (83) PiDP (84) PiDPToQDPProof [SOUND, 0 ms] (85) QDP (86) QDPSizeChangeProof [EQUIVALENT, 0 ms] (87) YES (88) PiDP (89) UsableRulesProof [EQUIVALENT, 0 ms] (90) PiDP (91) PiDPToQDPProof [SOUND, 0 ms] (92) QDP (93) PiDP (94) UsableRulesProof [EQUIVALENT, 0 ms] (95) PiDP ---------------------------------------- (0) Obligation: Clauses: sublist(X, Y) :- ','(append(U, X, V), append(V, W, Y)). append([], Ys, Ys). append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs). Query: sublist(g,a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: sublist_in_2: (b,f) append_in_3: (f,b,f) (b,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V)) append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys) append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y)) append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys) append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs)) U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y) The argument filtering Pi contains the following mapping: sublist_in_ga(x1, x2) = sublist_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) append_in_aga(x1, x2, x3) = append_in_aga(x2) append_out_aga(x1, x2, x3) = append_out_aga(x1, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x5) .(x1, x2) = .(x2) U2_ga(x1, x2, x3) = U2_ga(x3) append_in_gaa(x1, x2, x3) = append_in_gaa(x1) [] = [] append_out_gaa(x1, x2, x3) = append_out_gaa U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) sublist_out_ga(x1, x2) = sublist_out_ga 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_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V)) append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys) append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y)) append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys) append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs)) U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y) The argument filtering Pi contains the following mapping: sublist_in_ga(x1, x2) = sublist_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) append_in_aga(x1, x2, x3) = append_in_aga(x2) append_out_aga(x1, x2, x3) = append_out_aga(x1, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x5) .(x1, x2) = .(x2) U2_ga(x1, x2, x3) = U2_ga(x3) append_in_gaa(x1, x2, x3) = append_in_gaa(x1) [] = [] append_out_gaa(x1, x2, x3) = append_out_gaa U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) sublist_out_ga(x1, x2) = sublist_out_ga ---------------------------------------- (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_GA(X, Y) -> U1_GA(X, Y, append_in_aga(U, X, V)) SUBLIST_IN_GA(X, Y) -> APPEND_IN_AGA(U, X, V) APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U3_AGA(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs) U1_GA(X, Y, append_out_aga(U, X, V)) -> U2_GA(X, Y, append_in_gaa(V, W, Y)) U1_GA(X, Y, append_out_aga(U, X, V)) -> APPEND_IN_GAA(V, W, Y) APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U3_GAA(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V)) append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys) append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y)) append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys) append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs)) U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y) The argument filtering Pi contains the following mapping: sublist_in_ga(x1, x2) = sublist_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) append_in_aga(x1, x2, x3) = append_in_aga(x2) append_out_aga(x1, x2, x3) = append_out_aga(x1, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x5) .(x1, x2) = .(x2) U2_ga(x1, x2, x3) = U2_ga(x3) append_in_gaa(x1, x2, x3) = append_in_gaa(x1) [] = [] append_out_gaa(x1, x2, x3) = append_out_gaa U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) sublist_out_ga(x1, x2) = sublist_out_ga SUBLIST_IN_GA(x1, x2) = SUBLIST_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) APPEND_IN_AGA(x1, x2, x3) = APPEND_IN_AGA(x2) U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x5) U2_GA(x1, x2, x3) = U2_GA(x3) APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1) U3_GAA(x1, x2, x3, x4, x5) = U3_GAA(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_GA(X, Y) -> U1_GA(X, Y, append_in_aga(U, X, V)) SUBLIST_IN_GA(X, Y) -> APPEND_IN_AGA(U, X, V) APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U3_AGA(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs) U1_GA(X, Y, append_out_aga(U, X, V)) -> U2_GA(X, Y, append_in_gaa(V, W, Y)) U1_GA(X, Y, append_out_aga(U, X, V)) -> APPEND_IN_GAA(V, W, Y) APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U3_GAA(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V)) append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys) append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y)) append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys) append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs)) U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y) The argument filtering Pi contains the following mapping: sublist_in_ga(x1, x2) = sublist_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) append_in_aga(x1, x2, x3) = append_in_aga(x2) append_out_aga(x1, x2, x3) = append_out_aga(x1, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x5) .(x1, x2) = .(x2) U2_ga(x1, x2, x3) = U2_ga(x3) append_in_gaa(x1, x2, x3) = append_in_gaa(x1) [] = [] append_out_gaa(x1, x2, x3) = append_out_gaa U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) sublist_out_ga(x1, x2) = sublist_out_ga SUBLIST_IN_GA(x1, x2) = SUBLIST_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) APPEND_IN_AGA(x1, x2, x3) = APPEND_IN_AGA(x2) U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x5) U2_GA(x1, x2, x3) = U2_GA(x3) APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1) U3_GAA(x1, x2, x3, x4, x5) = U3_GAA(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: APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V)) append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys) append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y)) append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys) append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs)) U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y) The argument filtering Pi contains the following mapping: sublist_in_ga(x1, x2) = sublist_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) append_in_aga(x1, x2, x3) = append_in_aga(x2) append_out_aga(x1, x2, x3) = append_out_aga(x1, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x5) .(x1, x2) = .(x2) U2_ga(x1, x2, x3) = U2_ga(x3) append_in_gaa(x1, x2, x3) = append_in_gaa(x1) [] = [] append_out_gaa(x1, x2, x3) = append_out_gaa U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) sublist_out_ga(x1, x2) = sublist_out_ga APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: APPEND_IN_GAA(.(Xs)) -> APPEND_IN_GAA(Xs) 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: *APPEND_IN_GAA(.(Xs)) -> APPEND_IN_GAA(Xs) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V)) append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys) append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y)) append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys) append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs)) U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y) The argument filtering Pi contains the following mapping: sublist_in_ga(x1, x2) = sublist_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) append_in_aga(x1, x2, x3) = append_in_aga(x2) append_out_aga(x1, x2, x3) = append_out_aga(x1, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x5) .(x1, x2) = .(x2) U2_ga(x1, x2, x3) = U2_ga(x3) append_in_gaa(x1, x2, x3) = append_in_gaa(x1) [] = [] append_out_gaa(x1, x2, x3) = append_out_gaa U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5) sublist_out_ga(x1, x2) = sublist_out_ga APPEND_IN_AGA(x1, x2, x3) = APPEND_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: APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND_IN_AGA(x1, x2, x3) = APPEND_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: APPEND_IN_AGA(Ys) -> APPEND_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,f) append_in_3: (f,b,f) (b,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V)) append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys) append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y)) append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys) append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs)) U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y) The argument filtering Pi contains the following mapping: sublist_in_ga(x1, x2) = sublist_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x1, x3) append_in_aga(x1, x2, x3) = append_in_aga(x2) append_out_aga(x1, x2, x3) = append_out_aga(x1, x2, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x3, x5) .(x1, x2) = .(x2) U2_ga(x1, x2, x3) = U2_ga(x1, x3) append_in_gaa(x1, x2, x3) = append_in_gaa(x1) [] = [] append_out_gaa(x1, x2, x3) = append_out_gaa(x1) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x2, x5) sublist_out_ga(x1, x2) = sublist_out_ga(x1) 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_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V)) append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys) append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y)) append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys) append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs)) U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y) The argument filtering Pi contains the following mapping: sublist_in_ga(x1, x2) = sublist_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x1, x3) append_in_aga(x1, x2, x3) = append_in_aga(x2) append_out_aga(x1, x2, x3) = append_out_aga(x1, x2, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x3, x5) .(x1, x2) = .(x2) U2_ga(x1, x2, x3) = U2_ga(x1, x3) append_in_gaa(x1, x2, x3) = append_in_gaa(x1) [] = [] append_out_gaa(x1, x2, x3) = append_out_gaa(x1) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x2, x5) sublist_out_ga(x1, x2) = sublist_out_ga(x1) ---------------------------------------- (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_GA(X, Y) -> U1_GA(X, Y, append_in_aga(U, X, V)) SUBLIST_IN_GA(X, Y) -> APPEND_IN_AGA(U, X, V) APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U3_AGA(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs) U1_GA(X, Y, append_out_aga(U, X, V)) -> U2_GA(X, Y, append_in_gaa(V, W, Y)) U1_GA(X, Y, append_out_aga(U, X, V)) -> APPEND_IN_GAA(V, W, Y) APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U3_GAA(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V)) append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys) append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y)) append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys) append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs)) U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y) The argument filtering Pi contains the following mapping: sublist_in_ga(x1, x2) = sublist_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x1, x3) append_in_aga(x1, x2, x3) = append_in_aga(x2) append_out_aga(x1, x2, x3) = append_out_aga(x1, x2, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x3, x5) .(x1, x2) = .(x2) U2_ga(x1, x2, x3) = U2_ga(x1, x3) append_in_gaa(x1, x2, x3) = append_in_gaa(x1) [] = [] append_out_gaa(x1, x2, x3) = append_out_gaa(x1) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x2, x5) sublist_out_ga(x1, x2) = sublist_out_ga(x1) SUBLIST_IN_GA(x1, x2) = SUBLIST_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x1, x3) APPEND_IN_AGA(x1, x2, x3) = APPEND_IN_AGA(x2) U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x3, x5) U2_GA(x1, x2, x3) = U2_GA(x1, x3) APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1) U3_GAA(x1, x2, x3, x4, x5) = U3_GAA(x2, 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_GA(X, Y) -> U1_GA(X, Y, append_in_aga(U, X, V)) SUBLIST_IN_GA(X, Y) -> APPEND_IN_AGA(U, X, V) APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U3_AGA(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs) U1_GA(X, Y, append_out_aga(U, X, V)) -> U2_GA(X, Y, append_in_gaa(V, W, Y)) U1_GA(X, Y, append_out_aga(U, X, V)) -> APPEND_IN_GAA(V, W, Y) APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U3_GAA(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V)) append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys) append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y)) append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys) append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs)) U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y) The argument filtering Pi contains the following mapping: sublist_in_ga(x1, x2) = sublist_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x1, x3) append_in_aga(x1, x2, x3) = append_in_aga(x2) append_out_aga(x1, x2, x3) = append_out_aga(x1, x2, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x3, x5) .(x1, x2) = .(x2) U2_ga(x1, x2, x3) = U2_ga(x1, x3) append_in_gaa(x1, x2, x3) = append_in_gaa(x1) [] = [] append_out_gaa(x1, x2, x3) = append_out_gaa(x1) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x2, x5) sublist_out_ga(x1, x2) = sublist_out_ga(x1) SUBLIST_IN_GA(x1, x2) = SUBLIST_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x1, x3) APPEND_IN_AGA(x1, x2, x3) = APPEND_IN_AGA(x2) U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x3, x5) U2_GA(x1, x2, x3) = U2_GA(x1, x3) APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1) U3_GAA(x1, x2, x3, x4, x5) = U3_GAA(x2, 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: APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V)) append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys) append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y)) append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys) append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs)) U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y) The argument filtering Pi contains the following mapping: sublist_in_ga(x1, x2) = sublist_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x1, x3) append_in_aga(x1, x2, x3) = append_in_aga(x2) append_out_aga(x1, x2, x3) = append_out_aga(x1, x2, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x3, x5) .(x1, x2) = .(x2) U2_ga(x1, x2, x3) = U2_ga(x1, x3) append_in_gaa(x1, x2, x3) = append_in_gaa(x1) [] = [] append_out_gaa(x1, x2, x3) = append_out_gaa(x1) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x2, x5) sublist_out_ga(x1, x2) = sublist_out_ga(x1) APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1) 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: APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1) 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: APPEND_IN_GAA(.(Xs)) -> APPEND_IN_GAA(Xs) 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: *APPEND_IN_GAA(.(Xs)) -> APPEND_IN_GAA(Xs) The graph contains the following edges 1 > 1 ---------------------------------------- (31) YES ---------------------------------------- (32) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs) The TRS R consists of the following rules: sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V)) append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys) append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs)) U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs)) U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y)) append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys) append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs)) U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs)) U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y) The argument filtering Pi contains the following mapping: sublist_in_ga(x1, x2) = sublist_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x1, x3) append_in_aga(x1, x2, x3) = append_in_aga(x2) append_out_aga(x1, x2, x3) = append_out_aga(x1, x2, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x3, x5) .(x1, x2) = .(x2) U2_ga(x1, x2, x3) = U2_ga(x1, x3) append_in_gaa(x1, x2, x3) = append_in_gaa(x1) [] = [] append_out_gaa(x1, x2, x3) = append_out_gaa(x1) U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x2, x5) sublist_out_ga(x1, x2) = sublist_out_ga(x1) APPEND_IN_AGA(x1, x2, x3) = APPEND_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: APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND_IN_AGA(x1, x2, x3) = APPEND_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: APPEND_IN_AGA(Ys) -> APPEND_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": 10, "program": { "directives": [], "clauses": [ [ "(sublist X Y)", "(',' (append U X V) (append V W Y))" ], [ "(append ([]) Ys Ys)", null ], [ "(append (. X Xs) Ys (. X Zs))", "(append Xs Ys Zs)" ] ] }, "graph": { "nodes": { "44": { "goal": [{ "clause": -1, "scope": -1, "term": "(append X13 T10 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [ "X13", "X14" ], "exprvars": [] } }, "45": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T16 X15 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "13": { "goal": [{ "clause": 0, "scope": 1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "46": { "goal": [ { "clause": 1, "scope": 2, "term": "(append X13 T10 X14)" }, { "clause": 2, "scope": 2, "term": "(append X13 T10 X14)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [ "X13", "X14" ], "exprvars": [] } }, "26": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append X13 T10 X14) (append X14 X15 T12))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [ "X13", "X14", "X15" ], "exprvars": [] } }, "type": "Nodes", "199": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "200": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "135": { "goal": [ { "clause": 1, "scope": 3, "term": "(append T16 X15 T12)" }, { "clause": 2, "scope": 3, "term": "(append T16 X15 T12)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "201": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "136": { "goal": [{ "clause": 1, "scope": 3, "term": "(append T16 X15 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "202": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T44 X76 T45)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X76"], "exprvars": [] } }, "137": { "goal": [{ "clause": 2, "scope": 3, "term": "(append T16 X15 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "203": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "118": { "goal": [{ "clause": -1, "scope": -1, "term": "(append X45 T26 X46)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T26"], "free": [ "X45", "X46" ], "exprvars": [] } }, "73": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "53": { "goal": [{ "clause": 1, "scope": 2, "term": "(append X13 T10 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [ "X13", "X14" ], "exprvars": [] } }, "75": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "10": { "goal": [{ "clause": -1, "scope": -1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "54": { "goal": [{ "clause": 2, "scope": 2, "term": "(append X13 T10 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [ "X13", "X14" ], "exprvars": [] } } }, "edges": [ { "from": 10, "to": 13, "label": "CASE" }, { "from": 13, "to": 26, "label": "ONLY EVAL with clause\nsublist(X11, X12) :- ','(append(X13, X11, X14), append(X14, X15, X12)).\nand substitutionT1 -> T10,\nX11 -> T10,\nT2 -> T12,\nX12 -> T12,\nT11 -> T12" }, { "from": 26, "to": 44, "label": "SPLIT 1" }, { "from": 26, "to": 45, "label": "SPLIT 2\nnew knowledge:\nT10 is ground\nreplacements:X13 -> T15,\nX14 -> T16" }, { "from": 44, "to": 46, "label": "CASE" }, { "from": 45, "to": 135, "label": "CASE" }, { "from": 46, "to": 53, "label": "PARALLEL" }, { "from": 46, "to": 54, "label": "PARALLEL" }, { "from": 53, "to": 73, "label": "ONLY EVAL with clause\nappend([], X24, X24).\nand substitutionX13 -> [],\nT10 -> T22,\nX24 -> T22,\nX14 -> T22" }, { "from": 54, "to": 118, "label": "ONLY EVAL with clause\nappend(.(X40, X41), X42, .(X40, X43)) :- append(X41, X42, X43).\nand substitutionX40 -> X44,\nX41 -> X45,\nX13 -> .(X44, X45),\nT10 -> T26,\nX42 -> T26,\nX43 -> X46,\nX14 -> .(X44, X46)" }, { "from": 73, "to": 75, "label": "SUCCESS" }, { "from": 118, "to": 44, "label": "INSTANCE with matching:\nX13 -> X45\nT10 -> T26\nX14 -> X46" }, { "from": 135, "to": 136, "label": "PARALLEL" }, { "from": 135, "to": 137, "label": "PARALLEL" }, { "from": 136, "to": 199, "label": "EVAL with clause\nappend([], X60, X60).\nand substitutionT16 -> [],\nX15 -> T34,\nX60 -> T34,\nT12 -> T34,\nX61 -> T34" }, { "from": 136, "to": 200, "label": "EVAL-BACKTRACK" }, { "from": 137, "to": 202, "label": "EVAL with clause\nappend(.(X72, X73), X74, .(X72, X75)) :- append(X73, X74, X75).\nand substitutionX72 -> T41,\nX73 -> T44,\nT16 -> .(T41, T44),\nX15 -> X76,\nX74 -> X76,\nX75 -> T45,\nT12 -> .(T41, T45),\nT42 -> T44,\nT43 -> T45" }, { "from": 137, "to": 203, "label": "EVAL-BACKTRACK" }, { "from": 199, "to": 201, "label": "SUCCESS" }, { "from": 202, "to": 45, "label": "INSTANCE with matching:\nT16 -> T44\nX15 -> X76\nT12 -> T45" } ], "type": "Graph" } } ---------------------------------------- (38) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f10_in(T10) -> U1(f26_in(T10), T10) U1(f26_out1, T10) -> f10_out1 f44_in(T22) -> f44_out1 f44_in(T26) -> U2(f44_in(T26), T26) U2(f44_out1, T26) -> f44_out1 f45_in -> f45_out1 f45_in -> U3(f45_in) U3(f45_out1) -> f45_out1 f26_in(T10) -> U4(f44_in(T10), T10) U4(f44_out1, T10) -> U5(f45_in, T10) U5(f45_out1, T10) -> f26_out1 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: F10_IN(T10) -> U1^1(f26_in(T10), T10) F10_IN(T10) -> F26_IN(T10) F44_IN(T26) -> U2^1(f44_in(T26), T26) F44_IN(T26) -> F44_IN(T26) F45_IN -> U3^1(f45_in) F45_IN -> F45_IN F26_IN(T10) -> U4^1(f44_in(T10), T10) F26_IN(T10) -> F44_IN(T10) U4^1(f44_out1, T10) -> U5^1(f45_in, T10) U4^1(f44_out1, T10) -> F45_IN The TRS R consists of the following rules: f10_in(T10) -> U1(f26_in(T10), T10) U1(f26_out1, T10) -> f10_out1 f44_in(T22) -> f44_out1 f44_in(T26) -> U2(f44_in(T26), T26) U2(f44_out1, T26) -> f44_out1 f45_in -> f45_out1 f45_in -> U3(f45_in) U3(f45_out1) -> f45_out1 f26_in(T10) -> U4(f44_in(T10), T10) U4(f44_out1, T10) -> U5(f45_in, T10) U5(f45_out1, T10) -> f26_out1 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: F45_IN -> F45_IN The TRS R consists of the following rules: f10_in(T10) -> U1(f26_in(T10), T10) U1(f26_out1, T10) -> f10_out1 f44_in(T22) -> f44_out1 f44_in(T26) -> U2(f44_in(T26), T26) U2(f44_out1, T26) -> f44_out1 f45_in -> f45_out1 f45_in -> U3(f45_in) U3(f45_out1) -> f45_out1 f26_in(T10) -> U4(f44_in(T10), T10) U4(f44_out1, T10) -> U5(f45_in, T10) U5(f45_out1, T10) -> f26_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (44) MNOCProof (EQUIVALENT) We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: F45_IN -> F45_IN The TRS R consists of the following rules: f10_in(T10) -> U1(f26_in(T10), T10) U1(f26_out1, T10) -> f10_out1 f44_in(T22) -> f44_out1 f44_in(T26) -> U2(f44_in(T26), T26) U2(f44_out1, T26) -> f44_out1 f45_in -> f45_out1 f45_in -> U3(f45_in) U3(f45_out1) -> f45_out1 f26_in(T10) -> U4(f44_in(T10), T10) U4(f44_out1, T10) -> U5(f45_in, T10) U5(f45_out1, T10) -> f26_out1 The set Q consists of the following terms: f10_in(x0) U1(f26_out1, x0) f44_in(x0) U2(f44_out1, x0) f45_in U3(f45_out1) f26_in(x0) U4(f44_out1, x0) U5(f45_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (46) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: F45_IN -> F45_IN R is empty. The set Q consists of the following terms: f10_in(x0) U1(f26_out1, x0) f44_in(x0) U2(f44_out1, x0) f45_in U3(f45_out1) f26_in(x0) U4(f44_out1, x0) U5(f45_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. f10_in(x0) U1(f26_out1, x0) f44_in(x0) U2(f44_out1, x0) f45_in U3(f45_out1) f26_in(x0) U4(f44_out1, x0) U5(f45_out1, x0) ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: F45_IN -> F45_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: F44_IN(T26) -> F44_IN(T26) The TRS R consists of the following rules: f10_in(T10) -> U1(f26_in(T10), T10) U1(f26_out1, T10) -> f10_out1 f44_in(T22) -> f44_out1 f44_in(T26) -> U2(f44_in(T26), T26) U2(f44_out1, T26) -> f44_out1 f45_in -> f45_out1 f45_in -> U3(f45_in) U3(f45_out1) -> f45_out1 f26_in(T10) -> U4(f44_in(T10), T10) U4(f44_out1, T10) -> U5(f45_in, T10) U5(f45_out1, T10) -> f26_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) MNOCProof (EQUIVALENT) We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: F44_IN(T26) -> F44_IN(T26) The TRS R consists of the following rules: f10_in(T10) -> U1(f26_in(T10), T10) U1(f26_out1, T10) -> f10_out1 f44_in(T22) -> f44_out1 f44_in(T26) -> U2(f44_in(T26), T26) U2(f44_out1, T26) -> f44_out1 f45_in -> f45_out1 f45_in -> U3(f45_in) U3(f45_out1) -> f45_out1 f26_in(T10) -> U4(f44_in(T10), T10) U4(f44_out1, T10) -> U5(f45_in, T10) U5(f45_out1, T10) -> f26_out1 The set Q consists of the following terms: f10_in(x0) U1(f26_out1, x0) f44_in(x0) U2(f44_out1, x0) f45_in U3(f45_out1) f26_in(x0) U4(f44_out1, x0) U5(f45_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: F44_IN(T26) -> F44_IN(T26) R is empty. The set Q consists of the following terms: f10_in(x0) U1(f26_out1, x0) f44_in(x0) U2(f44_out1, x0) f45_in U3(f45_out1) f26_in(x0) U4(f44_out1, x0) U5(f45_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. f10_in(x0) U1(f26_out1, x0) f44_in(x0) U2(f44_out1, x0) f45_in U3(f45_out1) f26_in(x0) U4(f44_out1, x0) U5(f45_out1, x0) ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: F44_IN(T26) -> F44_IN(T26) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 9, "program": { "directives": [], "clauses": [ [ "(sublist X Y)", "(',' (append U X V) (append V W Y))" ], [ "(append ([]) Ys Ys)", null ], [ "(append (. X Xs) Ys (. X Zs))", "(append Xs Ys Zs)" ] ] }, "graph": { "nodes": { "33": { "goal": [{ "clause": -1, "scope": -1, "term": "(append X13 T10 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [ "X13", "X14" ], "exprvars": [] } }, "12": { "goal": [{ "clause": 0, "scope": 1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "34": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T16 X15 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "36": { "goal": [ { "clause": 1, "scope": 2, "term": "(append X13 T10 X14)" }, { "clause": 2, "scope": 2, "term": "(append X13 T10 X14)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [ "X13", "X14" ], "exprvars": [] } }, "37": { "goal": [{ "clause": 1, "scope": 2, "term": "(append X13 T10 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [ "X13", "X14" ], "exprvars": [] } }, "38": { "goal": [{ "clause": 2, "scope": 2, "term": "(append X13 T10 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [ "X13", "X14" ], "exprvars": [] } }, "28": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append X13 T10 X14) (append X14 X15 T12))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [ "X13", "X14", "X15" ], "exprvars": [] } }, "type": "Nodes", "100": { "goal": [ { "clause": 1, "scope": 3, "term": "(append T16 X15 T12)" }, { "clause": 2, "scope": 3, "term": "(append T16 X15 T12)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "101": { "goal": [{ "clause": 1, "scope": 3, "term": "(append T16 X15 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "102": { "goal": [{ "clause": 2, "scope": 3, "term": "(append T16 X15 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "113": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T44 X76 T45)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X76"], "exprvars": [] } }, "104": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "105": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "116": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "107": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "9": { "goal": [{ "clause": -1, "scope": -1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "40": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "41": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "43": { "goal": [{ "clause": -1, "scope": -1, "term": "(append X45 T26 X46)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T26"], "free": [ "X45", "X46" ], "exprvars": [] } } }, "edges": [ { "from": 9, "to": 12, "label": "CASE" }, { "from": 12, "to": 28, "label": "ONLY EVAL with clause\nsublist(X11, X12) :- ','(append(X13, X11, X14), append(X14, X15, X12)).\nand substitutionT1 -> T10,\nX11 -> T10,\nT2 -> T12,\nX12 -> T12,\nT11 -> T12" }, { "from": 28, "to": 33, "label": "SPLIT 1" }, { "from": 28, "to": 34, "label": "SPLIT 2\nnew knowledge:\nT10 is ground\nreplacements:X13 -> T15,\nX14 -> T16" }, { "from": 33, "to": 36, "label": "CASE" }, { "from": 34, "to": 100, "label": "CASE" }, { "from": 36, "to": 37, "label": "PARALLEL" }, { "from": 36, "to": 38, "label": "PARALLEL" }, { "from": 37, "to": 40, "label": "ONLY EVAL with clause\nappend([], X24, X24).\nand substitutionX13 -> [],\nT10 -> T22,\nX24 -> T22,\nX14 -> T22" }, { "from": 38, "to": 43, "label": "ONLY EVAL with clause\nappend(.(X40, X41), X42, .(X40, X43)) :- append(X41, X42, X43).\nand substitutionX40 -> X44,\nX41 -> X45,\nX13 -> .(X44, X45),\nT10 -> T26,\nX42 -> T26,\nX43 -> X46,\nX14 -> .(X44, X46)" }, { "from": 40, "to": 41, "label": "SUCCESS" }, { "from": 43, "to": 33, "label": "INSTANCE with matching:\nX13 -> X45\nT10 -> T26\nX14 -> X46" }, { "from": 100, "to": 101, "label": "PARALLEL" }, { "from": 100, "to": 102, "label": "PARALLEL" }, { "from": 101, "to": 104, "label": "EVAL with clause\nappend([], X60, X60).\nand substitutionT16 -> [],\nX15 -> T34,\nX60 -> T34,\nT12 -> T34,\nX61 -> T34" }, { "from": 101, "to": 105, "label": "EVAL-BACKTRACK" }, { "from": 102, "to": 113, "label": "EVAL with clause\nappend(.(X72, X73), X74, .(X72, X75)) :- append(X73, X74, X75).\nand substitutionX72 -> T41,\nX73 -> T44,\nT16 -> .(T41, T44),\nX15 -> X76,\nX74 -> X76,\nX75 -> T45,\nT12 -> .(T41, T45),\nT42 -> T44,\nT43 -> T45" }, { "from": 102, "to": 116, "label": "EVAL-BACKTRACK" }, { "from": 104, "to": 107, "label": "SUCCESS" }, { "from": 113, "to": 34, "label": "INSTANCE with matching:\nT16 -> T44\nX15 -> X76\nT12 -> T45" } ], "type": "Graph" } } ---------------------------------------- (58) Complex Obligation (AND) ---------------------------------------- (59) Obligation: Rules: f34_out -> f113_out :|: TRUE f113_in -> f34_in :|: TRUE f34_in -> f100_in :|: TRUE f100_out -> f34_out :|: TRUE f100_in -> f102_in :|: TRUE f101_out -> f100_out :|: TRUE f100_in -> f101_in :|: TRUE f102_out -> f100_out :|: TRUE f113_out -> f102_out :|: TRUE f102_in -> f113_in :|: TRUE f116_out -> f102_out :|: TRUE f102_in -> f116_in :|: TRUE f9_in(T1) -> f12_in(T1) :|: TRUE f12_out(x) -> f9_out(x) :|: TRUE f28_out(T10) -> f12_out(T10) :|: TRUE f12_in(x1) -> f28_in(x1) :|: TRUE f28_in(x2) -> f33_in(x2) :|: TRUE f34_out -> f28_out(x3) :|: TRUE f33_out(x4) -> f34_in :|: TRUE Start term: f9_in(T1) ---------------------------------------- (60) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (61) TRUE ---------------------------------------- (62) Obligation: Rules: f43_in(T26) -> f33_in(T26) :|: TRUE f33_out(x) -> f43_out(x) :|: TRUE f33_in(T10) -> f36_in(T10) :|: TRUE f36_out(x1) -> f33_out(x1) :|: TRUE f38_in(x2) -> f43_in(x2) :|: TRUE f43_out(x3) -> f38_out(x3) :|: TRUE f36_in(x4) -> f37_in(x4) :|: TRUE f37_out(x5) -> f36_out(x5) :|: TRUE f36_in(x6) -> f38_in(x6) :|: TRUE f38_out(x7) -> f36_out(x7) :|: TRUE f9_in(T1) -> f12_in(T1) :|: TRUE f12_out(x8) -> f9_out(x8) :|: TRUE f28_out(x9) -> f12_out(x9) :|: TRUE f12_in(x10) -> f28_in(x10) :|: TRUE f28_in(x11) -> f33_in(x11) :|: TRUE f34_out -> f28_out(x12) :|: TRUE f33_out(x13) -> f34_in :|: TRUE Start term: f9_in(T1) ---------------------------------------- (63) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f43_in(T26) -> f33_in(T26) :|: TRUE f33_in(T10) -> f36_in(T10) :|: TRUE f38_in(x2) -> f43_in(x2) :|: TRUE f36_in(x6) -> f38_in(x6) :|: TRUE ---------------------------------------- (64) Obligation: Rules: f43_in(T26) -> f33_in(T26) :|: TRUE f33_in(T10) -> f36_in(T10) :|: TRUE f38_in(x2) -> f43_in(x2) :|: TRUE f36_in(x6) -> f38_in(x6) :|: TRUE ---------------------------------------- (65) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (66) Obligation: Rules: f38_in(x2:0) -> f38_in(x2:0) :|: TRUE ---------------------------------------- (67) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (68) Obligation: Rules: f38_in(x2:0) -> f38_in(x2:0) :|: TRUE ---------------------------------------- (69) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f38_in(x2:0) -> f38_in(x2:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (70) Obligation: Termination digraph: Nodes: (1) f38_in(x2:0) -> f38_in(x2:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (71) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f38_in(VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (72) Obligation: Rules: f38_in(x2:0) -> f38_in(x2:0) :|: TRUE ---------------------------------------- (73) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, x2:0) -> f(1, x2:0) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1, -8) ---------------------------------------- (74) NO ---------------------------------------- (75) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 8, "program": { "directives": [], "clauses": [ [ "(sublist X Y)", "(',' (append U X V) (append V W Y))" ], [ "(append ([]) Ys Ys)", null ], [ "(append (. X Xs) Ys (. X Zs))", "(append Xs Ys Zs)" ] ] }, "graph": { "nodes": { "22": { "goal": [ { "clause": 1, "scope": 2, "term": "(',' (append X5 T5 X6) (append X6 X7 T7))" }, { "clause": 2, "scope": 2, "term": "(',' (append X5 T5 X6) (append X6 X7 T7))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [ "X5", "X6", "X7" ], "exprvars": [] } }, "23": { "goal": [{ "clause": 1, "scope": 2, "term": "(',' (append X5 T5 X6) (append X6 X7 T7))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [ "X5", "X6", "X7" ], "exprvars": [] } }, "24": { "goal": [{ "clause": 2, "scope": 2, "term": "(',' (append X5 T5 X6) (append X6 X7 T7))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [ "X5", "X6", "X7" ], "exprvars": [] } }, "25": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T16 X7 T7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T16"], "free": ["X7"], "exprvars": [] } }, "27": { "goal": [ { "clause": 1, "scope": 3, "term": "(append T16 X7 T7)" }, { "clause": 2, "scope": 3, "term": "(append T16 X7 T7)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T16"], "free": ["X7"], "exprvars": [] } }, "29": { "goal": [{ "clause": 1, "scope": 3, "term": "(append T16 X7 T7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T16"], "free": ["X7"], "exprvars": [] } }, "192": { "goal": [ { "clause": 1, "scope": 5, "term": "(append (. X75 T42) X7 T7)" }, { "clause": 2, "scope": 5, "term": "(append (. X75 T42) X7 T7)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X7", "X75" ], "exprvars": [] } }, "type": "Nodes", "195": { "goal": [{ "clause": 2, "scope": 5, "term": "(append (. X75 T42) X7 T7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X7", "X75" ], "exprvars": [] } }, "132": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "111": { "goal": [{ "clause": -1, "scope": -1, "term": "(append X76 T39 X77)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T39"], "free": [ "X76", "X77" ], "exprvars": [] } }, "112": { "goal": [{ "clause": -1, "scope": -1, "term": "(append (. X75 T42) X7 T7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X7", "X75" ], "exprvars": [] } }, "134": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "119": { "goal": [ { "clause": 1, "scope": 4, "term": "(append X76 T39 X77)" }, { "clause": 2, "scope": 4, "term": "(append X76 T39 X77)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T39"], "free": [ "X76", "X77" ], "exprvars": [] } }, "30": { "goal": [{ "clause": 2, "scope": 3, "term": "(append T16 X7 T7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T16"], "free": ["X7"], "exprvars": [] } }, "31": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "32": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "11": { "goal": [{ "clause": 0, "scope": 1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "35": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "39": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T31 X49 T33)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T31"], "free": ["X49"], "exprvars": [] } }, "185": { "goal": [{ "clause": -1, "scope": -1, "term": "(append X107 T52 X108)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T52"], "free": [ "X107", "X108" ], "exprvars": [] } }, "222": { "goal": [ { "clause": 1, "scope": 6, "term": "(append T63 X130 T64)" }, { "clause": 2, "scope": 6, "term": "(append T63 X130 T64)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X130"], "exprvars": [] } }, "223": { "goal": [{ "clause": 1, "scope": 6, "term": "(append T63 X130 T64)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X130"], "exprvars": [] } }, "125": { "goal": [{ "clause": 1, "scope": 4, "term": "(append X76 T39 X77)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T39"], "free": [ "X76", "X77" ], "exprvars": [] } }, "224": { "goal": [{ "clause": 2, "scope": 6, "term": "(append T63 X130 T64)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X130"], "exprvars": [] } }, "126": { "goal": [{ "clause": 2, "scope": 4, "term": "(append X76 T39 X77)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T39"], "free": [ "X76", "X77" ], "exprvars": [] } }, "225": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "204": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T63 X130 T64)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X130"], "exprvars": [] } }, "226": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "106": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append X76 T39 X77) (append (. X75 X77) X7 T7))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T39"], "free": [ "X7", "X75", "X76", "X77" ], "exprvars": [] } }, "205": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "227": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "8": { "goal": [{ "clause": -1, "scope": -1, "term": "(sublist T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "228": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T81 X159 T82)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X159"], "exprvars": [] } }, "229": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "42": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "21": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append X5 T5 X6) (append X6 X7 T7))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [ "X5", "X6", "X7" ], "exprvars": [] } } }, "edges": [ { "from": 8, "to": 11, "label": "CASE" }, { "from": 11, "to": 21, "label": "ONLY EVAL with clause\nsublist(X3, X4) :- ','(append(X5, X3, X6), append(X6, X7, X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7" }, { "from": 21, "to": 22, "label": "CASE" }, { "from": 22, "to": 23, "label": "PARALLEL" }, { "from": 22, "to": 24, "label": "PARALLEL" }, { "from": 23, "to": 25, "label": "ONLY EVAL with clause\nappend([], X20, X20).\nand substitutionX5 -> [],\nT5 -> T16,\nX20 -> T16,\nX6 -> T16" }, { "from": 24, "to": 106, "label": "ONLY EVAL with clause\nappend(.(X71, X72), X73, .(X71, X74)) :- append(X72, X73, X74).\nand substitutionX71 -> X75,\nX72 -> X76,\nX5 -> .(X75, X76),\nT5 -> T39,\nX73 -> T39,\nX74 -> X77,\nX6 -> .(X75, X77)" }, { "from": 25, "to": 27, "label": "CASE" }, { "from": 27, "to": 29, "label": "PARALLEL" }, { "from": 27, "to": 30, "label": "PARALLEL" }, { "from": 29, "to": 31, "label": "EVAL with clause\nappend([], X33, X33).\nand substitutionT16 -> [],\nX7 -> T23,\nX33 -> T23,\nT7 -> T23,\nX34 -> T23" }, { "from": 29, "to": 32, "label": "EVAL-BACKTRACK" }, { "from": 30, "to": 39, "label": "EVAL with clause\nappend(.(X45, X46), X47, .(X45, X48)) :- append(X46, X47, X48).\nand substitutionX45 -> T30,\nX46 -> T31,\nT16 -> .(T30, T31),\nX7 -> X49,\nX47 -> X49,\nX48 -> T33,\nT7 -> .(T30, T33),\nT32 -> T33" }, { "from": 30, "to": 42, "label": "EVAL-BACKTRACK" }, { "from": 31, "to": 35, "label": "SUCCESS" }, { "from": 39, "to": 25, "label": "INSTANCE with matching:\nT16 -> T31\nX7 -> X49\nT7 -> T33" }, { "from": 106, "to": 111, "label": "SPLIT 1" }, { "from": 106, "to": 112, "label": "SPLIT 2\nnew knowledge:\nT39 is ground\nreplacements:X76 -> T41,\nX77 -> T42" }, { "from": 111, "to": 119, "label": "CASE" }, { "from": 112, "to": 192, "label": "CASE" }, { "from": 119, "to": 125, "label": "PARALLEL" }, { "from": 119, "to": 126, "label": "PARALLEL" }, { "from": 125, "to": 132, "label": "ONLY EVAL with clause\nappend([], X86, X86).\nand substitutionX76 -> [],\nT39 -> T48,\nX86 -> T48,\nX77 -> T48" }, { "from": 126, "to": 185, "label": "ONLY EVAL with clause\nappend(.(X102, X103), X104, .(X102, X105)) :- append(X103, X104, X105).\nand substitutionX102 -> X106,\nX103 -> X107,\nX76 -> .(X106, X107),\nT39 -> T52,\nX104 -> T52,\nX105 -> X108,\nX77 -> .(X106, X108)" }, { "from": 132, "to": 134, "label": "SUCCESS" }, { "from": 185, "to": 111, "label": "INSTANCE with matching:\nX76 -> X107\nT39 -> T52\nX77 -> X108" }, { "from": 192, "to": 195, "label": "BACKTRACK\nfor clause: append([], Ys, Ys)because of non-unification" }, { "from": 195, "to": 204, "label": "EVAL with clause\nappend(.(X125, X126), X127, .(X125, X128)) :- append(X126, X127, X128).\nand substitutionX75 -> T61,\nX125 -> T61,\nT42 -> T63,\nX126 -> T63,\nX7 -> X130,\nX127 -> X130,\nX129 -> T61,\nX128 -> T64,\nT7 -> .(T61, T64),\nT60 -> T63,\nT62 -> T64" }, { "from": 195, "to": 205, "label": "EVAL-BACKTRACK" }, { "from": 204, "to": 222, "label": "CASE" }, { "from": 222, "to": 223, "label": "PARALLEL" }, { "from": 222, "to": 224, "label": "PARALLEL" }, { "from": 223, "to": 225, "label": "EVAL with clause\nappend([], X143, X143).\nand substitutionT63 -> [],\nX130 -> T71,\nX143 -> T71,\nT64 -> T71,\nX144 -> T71" }, { "from": 223, "to": 226, "label": "EVAL-BACKTRACK" }, { "from": 224, "to": 228, "label": "EVAL with clause\nappend(.(X155, X156), X157, .(X155, X158)) :- append(X156, X157, X158).\nand substitutionX155 -> T78,\nX156 -> T81,\nT63 -> .(T78, T81),\nX130 -> X159,\nX157 -> X159,\nX158 -> T82,\nT64 -> .(T78, T82),\nT79 -> T81,\nT80 -> T82" }, { "from": 224, "to": 229, "label": "EVAL-BACKTRACK" }, { "from": 225, "to": 227, "label": "SUCCESS" }, { "from": 228, "to": 204, "label": "INSTANCE with matching:\nT63 -> T81\nX130 -> X159\nT64 -> T82" } ], "type": "Graph" } } ---------------------------------------- (76) Obligation: Triples: appendA(.(X1, X2), X3, .(X1, X4)) :- appendA(X2, X3, X4). appendB(.(X1, X2), X3, .(X1, X4)) :- appendB(X2, X3, X4). appendC(.(X1, X2), X3, .(X1, X4)) :- appendC(X2, X3, X4). sublistD(X1, X2) :- appendA(X1, X3, X2). sublistD(X1, X2) :- appendB(X3, X1, X4). sublistD(X1, .(X2, X3)) :- ','(appendcB(X4, X1, X5), appendC(X5, X6, X3)). Clauses: appendcA([], X1, X1). appendcA(.(X1, X2), X3, .(X1, X4)) :- appendcA(X2, X3, X4). appendcB([], X1, X1). appendcB(.(X1, X2), X3, .(X1, X4)) :- appendcB(X2, X3, X4). appendcC([], X1, X1). appendcC(.(X1, X2), X3, .(X1, X4)) :- appendcC(X2, X3, X4). Afs: sublistD(x1, x2) = sublistD(x1) ---------------------------------------- (77) 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,f) appendA_in_3: (b,f,f) appendB_in_3: (f,b,f) appendcB_in_3: (f,b,f) appendC_in_3: (b,f,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: SUBLISTD_IN_GA(X1, X2) -> U4_GA(X1, X2, appendA_in_gaa(X1, X3, X2)) SUBLISTD_IN_GA(X1, X2) -> APPENDA_IN_GAA(X1, X3, X2) APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U1_GAA(X1, X2, X3, X4, appendA_in_gaa(X2, X3, X4)) APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDA_IN_GAA(X2, X3, X4) SUBLISTD_IN_GA(X1, X2) -> U5_GA(X1, X2, appendB_in_aga(X3, X1, X4)) SUBLISTD_IN_GA(X1, X2) -> APPENDB_IN_AGA(X3, X1, X4) APPENDB_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U2_AGA(X1, X2, X3, X4, appendB_in_aga(X2, X3, X4)) APPENDB_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPENDB_IN_AGA(X2, X3, X4) SUBLISTD_IN_GA(X1, .(X2, X3)) -> U6_GA(X1, X2, X3, appendcB_in_aga(X4, X1, X5)) U6_GA(X1, X2, X3, appendcB_out_aga(X4, X1, X5)) -> U7_GA(X1, X2, X3, appendC_in_gaa(X5, X6, X3)) U6_GA(X1, X2, X3, appendcB_out_aga(X4, X1, X5)) -> APPENDC_IN_GAA(X5, X6, X3) APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U3_GAA(X1, X2, X3, X4, appendC_in_gaa(X2, X3, X4)) APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDC_IN_GAA(X2, X3, X4) The TRS R consists of the following rules: appendcB_in_aga([], X1, X1) -> appendcB_out_aga([], X1, X1) appendcB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, appendcB_in_aga(X2, X3, X4)) U10_aga(X1, X2, X3, X4, appendcB_out_aga(X2, X3, X4)) -> appendcB_out_aga(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: appendA_in_gaa(x1, x2, x3) = appendA_in_gaa(x1) .(x1, x2) = .(x2) appendB_in_aga(x1, x2, x3) = appendB_in_aga(x2) appendcB_in_aga(x1, x2, x3) = appendcB_in_aga(x2) appendcB_out_aga(x1, x2, x3) = appendcB_out_aga(x1, x2, x3) U10_aga(x1, x2, x3, x4, x5) = U10_aga(x3, x5) appendC_in_gaa(x1, x2, x3) = appendC_in_gaa(x1) SUBLISTD_IN_GA(x1, x2) = SUBLISTD_IN_GA(x1) U4_GA(x1, x2, x3) = U4_GA(x1, x3) APPENDA_IN_GAA(x1, x2, x3) = APPENDA_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x2, x5) U5_GA(x1, x2, x3) = U5_GA(x1, x3) APPENDB_IN_AGA(x1, x2, x3) = APPENDB_IN_AGA(x2) U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x3, x5) U6_GA(x1, x2, x3, x4) = U6_GA(x1, x4) U7_GA(x1, x2, x3, x4) = U7_GA(x1, x4) APPENDC_IN_GAA(x1, x2, x3) = APPENDC_IN_GAA(x1) U3_GAA(x1, x2, x3, x4, x5) = U3_GAA(x2, x5) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (78) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBLISTD_IN_GA(X1, X2) -> U4_GA(X1, X2, appendA_in_gaa(X1, X3, X2)) SUBLISTD_IN_GA(X1, X2) -> APPENDA_IN_GAA(X1, X3, X2) APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U1_GAA(X1, X2, X3, X4, appendA_in_gaa(X2, X3, X4)) APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDA_IN_GAA(X2, X3, X4) SUBLISTD_IN_GA(X1, X2) -> U5_GA(X1, X2, appendB_in_aga(X3, X1, X4)) SUBLISTD_IN_GA(X1, X2) -> APPENDB_IN_AGA(X3, X1, X4) APPENDB_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U2_AGA(X1, X2, X3, X4, appendB_in_aga(X2, X3, X4)) APPENDB_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPENDB_IN_AGA(X2, X3, X4) SUBLISTD_IN_GA(X1, .(X2, X3)) -> U6_GA(X1, X2, X3, appendcB_in_aga(X4, X1, X5)) U6_GA(X1, X2, X3, appendcB_out_aga(X4, X1, X5)) -> U7_GA(X1, X2, X3, appendC_in_gaa(X5, X6, X3)) U6_GA(X1, X2, X3, appendcB_out_aga(X4, X1, X5)) -> APPENDC_IN_GAA(X5, X6, X3) APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U3_GAA(X1, X2, X3, X4, appendC_in_gaa(X2, X3, X4)) APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDC_IN_GAA(X2, X3, X4) The TRS R consists of the following rules: appendcB_in_aga([], X1, X1) -> appendcB_out_aga([], X1, X1) appendcB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, appendcB_in_aga(X2, X3, X4)) U10_aga(X1, X2, X3, X4, appendcB_out_aga(X2, X3, X4)) -> appendcB_out_aga(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: appendA_in_gaa(x1, x2, x3) = appendA_in_gaa(x1) .(x1, x2) = .(x2) appendB_in_aga(x1, x2, x3) = appendB_in_aga(x2) appendcB_in_aga(x1, x2, x3) = appendcB_in_aga(x2) appendcB_out_aga(x1, x2, x3) = appendcB_out_aga(x1, x2, x3) U10_aga(x1, x2, x3, x4, x5) = U10_aga(x3, x5) appendC_in_gaa(x1, x2, x3) = appendC_in_gaa(x1) SUBLISTD_IN_GA(x1, x2) = SUBLISTD_IN_GA(x1) U4_GA(x1, x2, x3) = U4_GA(x1, x3) APPENDA_IN_GAA(x1, x2, x3) = APPENDA_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x2, x5) U5_GA(x1, x2, x3) = U5_GA(x1, x3) APPENDB_IN_AGA(x1, x2, x3) = APPENDB_IN_AGA(x2) U2_AGA(x1, x2, x3, x4, x5) = U2_AGA(x3, x5) U6_GA(x1, x2, x3, x4) = U6_GA(x1, x4) U7_GA(x1, x2, x3, x4) = U7_GA(x1, x4) APPENDC_IN_GAA(x1, x2, x3) = APPENDC_IN_GAA(x1) U3_GAA(x1, x2, x3, x4, x5) = U3_GAA(x2, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (79) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes. ---------------------------------------- (80) Complex Obligation (AND) ---------------------------------------- (81) Obligation: Pi DP problem: The TRS P consists of the following rules: APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDC_IN_GAA(X2, X3, X4) The TRS R consists of the following rules: appendcB_in_aga([], X1, X1) -> appendcB_out_aga([], X1, X1) appendcB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, appendcB_in_aga(X2, X3, X4)) U10_aga(X1, X2, X3, X4, appendcB_out_aga(X2, X3, X4)) -> appendcB_out_aga(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) appendcB_in_aga(x1, x2, x3) = appendcB_in_aga(x2) appendcB_out_aga(x1, x2, x3) = appendcB_out_aga(x1, x2, x3) U10_aga(x1, x2, x3, x4, x5) = U10_aga(x3, x5) APPENDC_IN_GAA(x1, x2, x3) = APPENDC_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (82) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (83) Obligation: Pi DP problem: The TRS P consists of the following rules: APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDC_IN_GAA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPENDC_IN_GAA(x1, x2, x3) = APPENDC_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (84) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (85) Obligation: Q DP problem: The TRS P consists of the following rules: APPENDC_IN_GAA(.(X2)) -> APPENDC_IN_GAA(X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (86) 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: *APPENDC_IN_GAA(.(X2)) -> APPENDC_IN_GAA(X2) The graph contains the following edges 1 > 1 ---------------------------------------- (87) YES ---------------------------------------- (88) Obligation: Pi DP problem: The TRS P consists of the following rules: APPENDB_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPENDB_IN_AGA(X2, X3, X4) The TRS R consists of the following rules: appendcB_in_aga([], X1, X1) -> appendcB_out_aga([], X1, X1) appendcB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, appendcB_in_aga(X2, X3, X4)) U10_aga(X1, X2, X3, X4, appendcB_out_aga(X2, X3, X4)) -> appendcB_out_aga(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) appendcB_in_aga(x1, x2, x3) = appendcB_in_aga(x2) appendcB_out_aga(x1, x2, x3) = appendcB_out_aga(x1, x2, x3) U10_aga(x1, x2, x3, x4, x5) = U10_aga(x3, x5) APPENDB_IN_AGA(x1, x2, x3) = APPENDB_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (89) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (90) Obligation: Pi DP problem: The TRS P consists of the following rules: APPENDB_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPENDB_IN_AGA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPENDB_IN_AGA(x1, x2, x3) = APPENDB_IN_AGA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (91) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: APPENDB_IN_AGA(X3) -> APPENDB_IN_AGA(X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (93) Obligation: Pi DP problem: The TRS P consists of the following rules: APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDA_IN_GAA(X2, X3, X4) The TRS R consists of the following rules: appendcB_in_aga([], X1, X1) -> appendcB_out_aga([], X1, X1) appendcB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, appendcB_in_aga(X2, X3, X4)) U10_aga(X1, X2, X3, X4, appendcB_out_aga(X2, X3, X4)) -> appendcB_out_aga(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) appendcB_in_aga(x1, x2, x3) = appendcB_in_aga(x2) appendcB_out_aga(x1, x2, x3) = appendcB_out_aga(x1, x2, x3) U10_aga(x1, x2, x3, x4, x5) = U10_aga(x3, x5) APPENDA_IN_GAA(x1, x2, x3) = APPENDA_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (94) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (95) Obligation: Pi DP problem: The TRS P consists of the following rules: APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDA_IN_GAA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPENDA_IN_GAA(x1, x2, x3) = APPENDA_IN_GAA(x1) We have to consider all (P,R,Pi)-chains