3.82/1.80 YES 3.82/1.83 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 3.82/1.83 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.82/1.83 3.82/1.83 3.82/1.83 Left Termination of the query pattern 3.82/1.83 3.82/1.83 sublist(g,g) 3.82/1.83 3.82/1.83 w.r.t. the given Prolog program could successfully be proven: 3.82/1.83 3.82/1.83 (0) Prolog 3.82/1.83 (1) PrologToPiTRSProof [SOUND, 0 ms] 3.82/1.83 (2) PiTRS 3.82/1.83 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 3.82/1.83 (4) PiDP 3.82/1.83 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 3.82/1.83 (6) AND 3.82/1.83 (7) PiDP 3.82/1.83 (8) UsableRulesProof [EQUIVALENT, 0 ms] 3.82/1.83 (9) PiDP 3.82/1.83 (10) PiDPToQDPProof [SOUND, 0 ms] 3.82/1.83 (11) QDP 3.82/1.83 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 3.82/1.83 (13) YES 3.82/1.83 (14) PiDP 3.82/1.83 (15) UsableRulesProof [EQUIVALENT, 0 ms] 3.82/1.83 (16) PiDP 3.82/1.83 (17) PiDPToQDPProof [SOUND, 0 ms] 3.82/1.83 (18) QDP 3.82/1.83 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 3.82/1.83 (20) YES 3.82/1.83 3.82/1.83 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (0) 3.82/1.83 Obligation: 3.82/1.83 Clauses: 3.82/1.83 3.82/1.83 append([], Ys, Ys). 3.82/1.83 append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs). 3.82/1.83 sublist(X, Y) :- ','(append(P, X1, Y), append(X2, X, P)). 3.82/1.83 3.82/1.83 3.82/1.83 Query: sublist(g,g) 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (1) PrologToPiTRSProof (SOUND) 3.82/1.83 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 3.82/1.83 3.82/1.83 sublist_in_2: (b,b) 3.82/1.83 3.82/1.83 append_in_3: (f,f,b) (f,b,b) 3.82/1.83 3.82/1.83 Transforming Prolog into the following Term Rewriting System: 3.82/1.83 3.82/1.83 Pi-finite rewrite system: 3.82/1.83 The TRS R consists of the following rules: 3.82/1.83 3.82/1.83 sublist_in_gg(X, Y) -> U2_gg(X, Y, append_in_aag(P, X1, Y)) 3.82/1.83 append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys) 3.82/1.83 append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs)) 3.82/1.83 U1_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs)) 3.82/1.83 U2_gg(X, Y, append_out_aag(P, X1, Y)) -> U3_gg(X, Y, append_in_agg(X2, X, P)) 3.82/1.83 append_in_agg([], Ys, Ys) -> append_out_agg([], Ys, Ys) 3.82/1.83 append_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U1_agg(X, Xs, Ys, Zs, append_in_agg(Xs, Ys, Zs)) 3.82/1.83 U1_agg(X, Xs, Ys, Zs, append_out_agg(Xs, Ys, Zs)) -> append_out_agg(.(X, Xs), Ys, .(X, Zs)) 3.82/1.83 U3_gg(X, Y, append_out_agg(X2, X, P)) -> sublist_out_gg(X, Y) 3.82/1.83 3.82/1.83 The argument filtering Pi contains the following mapping: 3.82/1.83 sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) 3.82/1.83 3.82/1.83 U2_gg(x1, x2, x3) = U2_gg(x1, x3) 3.82/1.83 3.82/1.83 append_in_aag(x1, x2, x3) = append_in_aag(x3) 3.82/1.83 3.82/1.83 append_out_aag(x1, x2, x3) = append_out_aag(x1, x2) 3.82/1.83 3.82/1.83 .(x1, x2) = .(x1, x2) 3.82/1.83 3.82/1.83 U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5) 3.82/1.83 3.82/1.83 U3_gg(x1, x2, x3) = U3_gg(x3) 3.82/1.83 3.82/1.83 append_in_agg(x1, x2, x3) = append_in_agg(x2, x3) 3.82/1.83 3.82/1.83 append_out_agg(x1, x2, x3) = append_out_agg(x1) 3.82/1.83 3.82/1.83 U1_agg(x1, x2, x3, x4, x5) = U1_agg(x1, x5) 3.82/1.83 3.82/1.83 sublist_out_gg(x1, x2) = sublist_out_gg 3.82/1.83 3.82/1.83 3.82/1.83 3.82/1.83 3.82/1.83 3.82/1.83 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 3.82/1.83 3.82/1.83 3.82/1.83 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (2) 3.82/1.83 Obligation: 3.82/1.83 Pi-finite rewrite system: 3.82/1.83 The TRS R consists of the following rules: 3.82/1.83 3.82/1.83 sublist_in_gg(X, Y) -> U2_gg(X, Y, append_in_aag(P, X1, Y)) 3.82/1.83 append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys) 3.82/1.83 append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs)) 3.82/1.83 U1_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs)) 3.82/1.83 U2_gg(X, Y, append_out_aag(P, X1, Y)) -> U3_gg(X, Y, append_in_agg(X2, X, P)) 3.82/1.83 append_in_agg([], Ys, Ys) -> append_out_agg([], Ys, Ys) 3.82/1.83 append_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U1_agg(X, Xs, Ys, Zs, append_in_agg(Xs, Ys, Zs)) 3.82/1.83 U1_agg(X, Xs, Ys, Zs, append_out_agg(Xs, Ys, Zs)) -> append_out_agg(.(X, Xs), Ys, .(X, Zs)) 3.82/1.83 U3_gg(X, Y, append_out_agg(X2, X, P)) -> sublist_out_gg(X, Y) 3.82/1.83 3.82/1.83 The argument filtering Pi contains the following mapping: 3.82/1.83 sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) 3.82/1.83 3.82/1.83 U2_gg(x1, x2, x3) = U2_gg(x1, x3) 3.82/1.83 3.82/1.83 append_in_aag(x1, x2, x3) = append_in_aag(x3) 3.82/1.83 3.82/1.83 append_out_aag(x1, x2, x3) = append_out_aag(x1, x2) 3.82/1.83 3.82/1.83 .(x1, x2) = .(x1, x2) 3.82/1.83 3.82/1.83 U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5) 3.82/1.83 3.82/1.83 U3_gg(x1, x2, x3) = U3_gg(x3) 3.82/1.83 3.82/1.83 append_in_agg(x1, x2, x3) = append_in_agg(x2, x3) 3.82/1.83 3.82/1.83 append_out_agg(x1, x2, x3) = append_out_agg(x1) 3.82/1.83 3.82/1.83 U1_agg(x1, x2, x3, x4, x5) = U1_agg(x1, x5) 3.82/1.83 3.82/1.83 sublist_out_gg(x1, x2) = sublist_out_gg 3.82/1.83 3.82/1.83 3.82/1.83 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (3) DependencyPairsProof (EQUIVALENT) 3.82/1.83 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 3.82/1.83 Pi DP problem: 3.82/1.83 The TRS P consists of the following rules: 3.82/1.83 3.82/1.83 SUBLIST_IN_GG(X, Y) -> U2_GG(X, Y, append_in_aag(P, X1, Y)) 3.82/1.83 SUBLIST_IN_GG(X, Y) -> APPEND_IN_AAG(P, X1, Y) 3.82/1.83 APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U1_AAG(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs)) 3.82/1.83 APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAG(Xs, Ys, Zs) 3.82/1.83 U2_GG(X, Y, append_out_aag(P, X1, Y)) -> U3_GG(X, Y, append_in_agg(X2, X, P)) 3.82/1.83 U2_GG(X, Y, append_out_aag(P, X1, Y)) -> APPEND_IN_AGG(X2, X, P) 3.82/1.83 APPEND_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> U1_AGG(X, Xs, Ys, Zs, append_in_agg(Xs, Ys, Zs)) 3.82/1.83 APPEND_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGG(Xs, Ys, Zs) 3.82/1.83 3.82/1.83 The TRS R consists of the following rules: 3.82/1.83 3.82/1.83 sublist_in_gg(X, Y) -> U2_gg(X, Y, append_in_aag(P, X1, Y)) 3.82/1.83 append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys) 3.82/1.83 append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs)) 3.82/1.83 U1_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs)) 3.82/1.83 U2_gg(X, Y, append_out_aag(P, X1, Y)) -> U3_gg(X, Y, append_in_agg(X2, X, P)) 3.82/1.83 append_in_agg([], Ys, Ys) -> append_out_agg([], Ys, Ys) 3.82/1.83 append_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U1_agg(X, Xs, Ys, Zs, append_in_agg(Xs, Ys, Zs)) 3.82/1.83 U1_agg(X, Xs, Ys, Zs, append_out_agg(Xs, Ys, Zs)) -> append_out_agg(.(X, Xs), Ys, .(X, Zs)) 3.82/1.83 U3_gg(X, Y, append_out_agg(X2, X, P)) -> sublist_out_gg(X, Y) 3.82/1.83 3.82/1.83 The argument filtering Pi contains the following mapping: 3.82/1.83 sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) 3.82/1.83 3.82/1.83 U2_gg(x1, x2, x3) = U2_gg(x1, x3) 3.82/1.83 3.82/1.83 append_in_aag(x1, x2, x3) = append_in_aag(x3) 3.82/1.83 3.82/1.83 append_out_aag(x1, x2, x3) = append_out_aag(x1, x2) 3.82/1.83 3.82/1.83 .(x1, x2) = .(x1, x2) 3.82/1.83 3.82/1.83 U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5) 3.82/1.83 3.82/1.83 U3_gg(x1, x2, x3) = U3_gg(x3) 3.82/1.83 3.82/1.83 append_in_agg(x1, x2, x3) = append_in_agg(x2, x3) 3.82/1.83 3.82/1.83 append_out_agg(x1, x2, x3) = append_out_agg(x1) 3.82/1.83 3.82/1.83 U1_agg(x1, x2, x3, x4, x5) = U1_agg(x1, x5) 3.82/1.83 3.82/1.83 sublist_out_gg(x1, x2) = sublist_out_gg 3.82/1.83 3.82/1.83 SUBLIST_IN_GG(x1, x2) = SUBLIST_IN_GG(x1, x2) 3.82/1.83 3.82/1.83 U2_GG(x1, x2, x3) = U2_GG(x1, x3) 3.82/1.83 3.82/1.83 APPEND_IN_AAG(x1, x2, x3) = APPEND_IN_AAG(x3) 3.82/1.83 3.82/1.83 U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x1, x5) 3.82/1.83 3.82/1.83 U3_GG(x1, x2, x3) = U3_GG(x3) 3.82/1.83 3.82/1.83 APPEND_IN_AGG(x1, x2, x3) = APPEND_IN_AGG(x2, x3) 3.82/1.83 3.82/1.83 U1_AGG(x1, x2, x3, x4, x5) = U1_AGG(x1, x5) 3.82/1.83 3.82/1.83 3.82/1.83 We have to consider all (P,R,Pi)-chains 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (4) 3.82/1.83 Obligation: 3.82/1.83 Pi DP problem: 3.82/1.83 The TRS P consists of the following rules: 3.82/1.83 3.82/1.83 SUBLIST_IN_GG(X, Y) -> U2_GG(X, Y, append_in_aag(P, X1, Y)) 3.82/1.83 SUBLIST_IN_GG(X, Y) -> APPEND_IN_AAG(P, X1, Y) 3.82/1.83 APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U1_AAG(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs)) 3.82/1.83 APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAG(Xs, Ys, Zs) 3.82/1.83 U2_GG(X, Y, append_out_aag(P, X1, Y)) -> U3_GG(X, Y, append_in_agg(X2, X, P)) 3.82/1.83 U2_GG(X, Y, append_out_aag(P, X1, Y)) -> APPEND_IN_AGG(X2, X, P) 3.82/1.83 APPEND_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> U1_AGG(X, Xs, Ys, Zs, append_in_agg(Xs, Ys, Zs)) 3.82/1.83 APPEND_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGG(Xs, Ys, Zs) 3.82/1.83 3.82/1.83 The TRS R consists of the following rules: 3.82/1.83 3.82/1.83 sublist_in_gg(X, Y) -> U2_gg(X, Y, append_in_aag(P, X1, Y)) 3.82/1.83 append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys) 3.82/1.83 append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs)) 3.82/1.83 U1_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs)) 3.82/1.83 U2_gg(X, Y, append_out_aag(P, X1, Y)) -> U3_gg(X, Y, append_in_agg(X2, X, P)) 3.82/1.83 append_in_agg([], Ys, Ys) -> append_out_agg([], Ys, Ys) 3.82/1.83 append_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U1_agg(X, Xs, Ys, Zs, append_in_agg(Xs, Ys, Zs)) 3.82/1.83 U1_agg(X, Xs, Ys, Zs, append_out_agg(Xs, Ys, Zs)) -> append_out_agg(.(X, Xs), Ys, .(X, Zs)) 3.82/1.83 U3_gg(X, Y, append_out_agg(X2, X, P)) -> sublist_out_gg(X, Y) 3.82/1.83 3.82/1.83 The argument filtering Pi contains the following mapping: 3.82/1.83 sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) 3.82/1.83 3.82/1.83 U2_gg(x1, x2, x3) = U2_gg(x1, x3) 3.82/1.83 3.82/1.83 append_in_aag(x1, x2, x3) = append_in_aag(x3) 3.82/1.83 3.82/1.83 append_out_aag(x1, x2, x3) = append_out_aag(x1, x2) 3.82/1.83 3.82/1.83 .(x1, x2) = .(x1, x2) 3.82/1.83 3.82/1.83 U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5) 3.82/1.83 3.82/1.83 U3_gg(x1, x2, x3) = U3_gg(x3) 3.82/1.83 3.82/1.83 append_in_agg(x1, x2, x3) = append_in_agg(x2, x3) 3.82/1.83 3.82/1.83 append_out_agg(x1, x2, x3) = append_out_agg(x1) 3.82/1.83 3.82/1.83 U1_agg(x1, x2, x3, x4, x5) = U1_agg(x1, x5) 3.82/1.83 3.82/1.83 sublist_out_gg(x1, x2) = sublist_out_gg 3.82/1.83 3.82/1.83 SUBLIST_IN_GG(x1, x2) = SUBLIST_IN_GG(x1, x2) 3.82/1.83 3.82/1.83 U2_GG(x1, x2, x3) = U2_GG(x1, x3) 3.82/1.83 3.82/1.83 APPEND_IN_AAG(x1, x2, x3) = APPEND_IN_AAG(x3) 3.82/1.83 3.82/1.83 U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x1, x5) 3.82/1.83 3.82/1.83 U3_GG(x1, x2, x3) = U3_GG(x3) 3.82/1.83 3.82/1.83 APPEND_IN_AGG(x1, x2, x3) = APPEND_IN_AGG(x2, x3) 3.82/1.83 3.82/1.83 U1_AGG(x1, x2, x3, x4, x5) = U1_AGG(x1, x5) 3.82/1.83 3.82/1.83 3.82/1.83 We have to consider all (P,R,Pi)-chains 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (5) DependencyGraphProof (EQUIVALENT) 3.82/1.83 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes. 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (6) 3.82/1.83 Complex Obligation (AND) 3.82/1.83 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (7) 3.82/1.83 Obligation: 3.82/1.83 Pi DP problem: 3.82/1.83 The TRS P consists of the following rules: 3.82/1.83 3.82/1.83 APPEND_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGG(Xs, Ys, Zs) 3.82/1.83 3.82/1.83 The TRS R consists of the following rules: 3.82/1.83 3.82/1.83 sublist_in_gg(X, Y) -> U2_gg(X, Y, append_in_aag(P, X1, Y)) 3.82/1.83 append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys) 3.82/1.83 append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs)) 3.82/1.83 U1_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs)) 3.82/1.83 U2_gg(X, Y, append_out_aag(P, X1, Y)) -> U3_gg(X, Y, append_in_agg(X2, X, P)) 3.82/1.83 append_in_agg([], Ys, Ys) -> append_out_agg([], Ys, Ys) 3.82/1.83 append_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U1_agg(X, Xs, Ys, Zs, append_in_agg(Xs, Ys, Zs)) 3.82/1.83 U1_agg(X, Xs, Ys, Zs, append_out_agg(Xs, Ys, Zs)) -> append_out_agg(.(X, Xs), Ys, .(X, Zs)) 3.82/1.83 U3_gg(X, Y, append_out_agg(X2, X, P)) -> sublist_out_gg(X, Y) 3.82/1.83 3.82/1.83 The argument filtering Pi contains the following mapping: 3.82/1.83 sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) 3.82/1.83 3.82/1.83 U2_gg(x1, x2, x3) = U2_gg(x1, x3) 3.82/1.83 3.82/1.83 append_in_aag(x1, x2, x3) = append_in_aag(x3) 3.82/1.83 3.82/1.83 append_out_aag(x1, x2, x3) = append_out_aag(x1, x2) 3.82/1.83 3.82/1.83 .(x1, x2) = .(x1, x2) 3.82/1.83 3.82/1.83 U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5) 3.82/1.83 3.82/1.83 U3_gg(x1, x2, x3) = U3_gg(x3) 3.82/1.83 3.82/1.83 append_in_agg(x1, x2, x3) = append_in_agg(x2, x3) 3.82/1.83 3.82/1.83 append_out_agg(x1, x2, x3) = append_out_agg(x1) 3.82/1.83 3.82/1.83 U1_agg(x1, x2, x3, x4, x5) = U1_agg(x1, x5) 3.82/1.83 3.82/1.83 sublist_out_gg(x1, x2) = sublist_out_gg 3.82/1.83 3.82/1.83 APPEND_IN_AGG(x1, x2, x3) = APPEND_IN_AGG(x2, x3) 3.82/1.83 3.82/1.83 3.82/1.83 We have to consider all (P,R,Pi)-chains 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (8) UsableRulesProof (EQUIVALENT) 3.82/1.83 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (9) 3.82/1.83 Obligation: 3.82/1.83 Pi DP problem: 3.82/1.83 The TRS P consists of the following rules: 3.82/1.83 3.82/1.83 APPEND_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGG(Xs, Ys, Zs) 3.82/1.83 3.82/1.83 R is empty. 3.82/1.83 The argument filtering Pi contains the following mapping: 3.82/1.83 .(x1, x2) = .(x1, x2) 3.82/1.83 3.82/1.83 APPEND_IN_AGG(x1, x2, x3) = APPEND_IN_AGG(x2, x3) 3.82/1.83 3.82/1.83 3.82/1.83 We have to consider all (P,R,Pi)-chains 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (10) PiDPToQDPProof (SOUND) 3.82/1.83 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (11) 3.82/1.83 Obligation: 3.82/1.83 Q DP problem: 3.82/1.83 The TRS P consists of the following rules: 3.82/1.83 3.82/1.83 APPEND_IN_AGG(Ys, .(X, Zs)) -> APPEND_IN_AGG(Ys, Zs) 3.82/1.83 3.82/1.83 R is empty. 3.82/1.83 Q is empty. 3.82/1.83 We have to consider all (P,Q,R)-chains. 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (12) QDPSizeChangeProof (EQUIVALENT) 3.82/1.83 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. 3.82/1.83 3.82/1.83 From the DPs we obtained the following set of size-change graphs: 3.82/1.83 *APPEND_IN_AGG(Ys, .(X, Zs)) -> APPEND_IN_AGG(Ys, Zs) 3.82/1.83 The graph contains the following edges 1 >= 1, 2 > 2 3.82/1.83 3.82/1.83 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (13) 3.82/1.83 YES 3.82/1.83 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (14) 3.82/1.83 Obligation: 3.82/1.83 Pi DP problem: 3.82/1.83 The TRS P consists of the following rules: 3.82/1.83 3.82/1.83 APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAG(Xs, Ys, Zs) 3.82/1.83 3.82/1.83 The TRS R consists of the following rules: 3.82/1.83 3.82/1.83 sublist_in_gg(X, Y) -> U2_gg(X, Y, append_in_aag(P, X1, Y)) 3.82/1.83 append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys) 3.82/1.83 append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs)) 3.82/1.83 U1_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs)) 3.82/1.83 U2_gg(X, Y, append_out_aag(P, X1, Y)) -> U3_gg(X, Y, append_in_agg(X2, X, P)) 3.82/1.83 append_in_agg([], Ys, Ys) -> append_out_agg([], Ys, Ys) 3.82/1.83 append_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U1_agg(X, Xs, Ys, Zs, append_in_agg(Xs, Ys, Zs)) 3.82/1.83 U1_agg(X, Xs, Ys, Zs, append_out_agg(Xs, Ys, Zs)) -> append_out_agg(.(X, Xs), Ys, .(X, Zs)) 3.82/1.83 U3_gg(X, Y, append_out_agg(X2, X, P)) -> sublist_out_gg(X, Y) 3.82/1.83 3.82/1.83 The argument filtering Pi contains the following mapping: 3.82/1.83 sublist_in_gg(x1, x2) = sublist_in_gg(x1, x2) 3.82/1.83 3.82/1.83 U2_gg(x1, x2, x3) = U2_gg(x1, x3) 3.82/1.83 3.82/1.83 append_in_aag(x1, x2, x3) = append_in_aag(x3) 3.82/1.83 3.82/1.83 append_out_aag(x1, x2, x3) = append_out_aag(x1, x2) 3.82/1.83 3.82/1.83 .(x1, x2) = .(x1, x2) 3.82/1.83 3.82/1.83 U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5) 3.82/1.83 3.82/1.83 U3_gg(x1, x2, x3) = U3_gg(x3) 3.82/1.83 3.82/1.83 append_in_agg(x1, x2, x3) = append_in_agg(x2, x3) 3.82/1.83 3.82/1.83 append_out_agg(x1, x2, x3) = append_out_agg(x1) 3.82/1.83 3.82/1.83 U1_agg(x1, x2, x3, x4, x5) = U1_agg(x1, x5) 3.82/1.83 3.82/1.83 sublist_out_gg(x1, x2) = sublist_out_gg 3.82/1.83 3.82/1.83 APPEND_IN_AAG(x1, x2, x3) = APPEND_IN_AAG(x3) 3.82/1.83 3.82/1.83 3.82/1.83 We have to consider all (P,R,Pi)-chains 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (15) UsableRulesProof (EQUIVALENT) 3.82/1.83 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (16) 3.82/1.83 Obligation: 3.82/1.83 Pi DP problem: 3.82/1.83 The TRS P consists of the following rules: 3.82/1.83 3.82/1.83 APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAG(Xs, Ys, Zs) 3.82/1.83 3.82/1.83 R is empty. 3.82/1.83 The argument filtering Pi contains the following mapping: 3.82/1.83 .(x1, x2) = .(x1, x2) 3.82/1.83 3.82/1.83 APPEND_IN_AAG(x1, x2, x3) = APPEND_IN_AAG(x3) 3.82/1.83 3.82/1.83 3.82/1.83 We have to consider all (P,R,Pi)-chains 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (17) PiDPToQDPProof (SOUND) 3.82/1.83 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (18) 3.82/1.83 Obligation: 3.82/1.83 Q DP problem: 3.82/1.83 The TRS P consists of the following rules: 3.82/1.83 3.82/1.83 APPEND_IN_AAG(.(X, Zs)) -> APPEND_IN_AAG(Zs) 3.82/1.83 3.82/1.83 R is empty. 3.82/1.83 Q is empty. 3.82/1.83 We have to consider all (P,Q,R)-chains. 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (19) QDPSizeChangeProof (EQUIVALENT) 3.82/1.83 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. 3.82/1.83 3.82/1.83 From the DPs we obtained the following set of size-change graphs: 3.82/1.83 *APPEND_IN_AAG(.(X, Zs)) -> APPEND_IN_AAG(Zs) 3.82/1.83 The graph contains the following edges 1 > 1 3.82/1.83 3.82/1.83 3.82/1.83 ---------------------------------------- 3.82/1.83 3.82/1.83 (20) 3.82/1.83 YES 3.82/1.85 EOF