/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern perm(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 4 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 1 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) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [SOUND, 0 ms] (25) QDP (26) MRRProof [EQUIVALENT, 2 ms] (27) QDP (28) PisEmptyProof [EQUIVALENT, 0 ms] (29) YES ---------------------------------------- (0) Obligation: Clauses: perm([], []). perm(L, .(H, T)) :- ','(append2(V, .(H, U), L), ','(append1(V, U, W), perm(W, T))). append1([], L, L). append1(.(H, L1), L2, .(H, L3)) :- append1(L1, L2, L3). append2([], L, L). append2(.(H, L1), L2, .(H, L3)) :- append2(L1, L2, L3). Query: perm(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: perm_in_2: (b,f) append2_in_3: (f,f,b) append1_in_3: (b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) append2_in_aag([], L, L) -> append2_out_aag([], L, L) append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) append1_in_gga([], L, L) -> append1_out_gga([], L, L) append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) append2_in_aag(x1, x2, x3) = append2_in_aag(x3) .(x1, x2) = .(x2) append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) U2_ga(x1, x2, x3, x4) = U2_ga(x4) append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) append1_out_gga(x1, x2, x3) = append1_out_gga(x3) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) U3_ga(x1, x2, x3, x4) = U3_ga(x4) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) append2_in_aag([], L, L) -> append2_out_aag([], L, L) append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) append1_in_gga([], L, L) -> append1_out_gga([], L, L) append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) append2_in_aag(x1, x2, x3) = append2_in_aag(x3) .(x1, x2) = .(x2) append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) U2_ga(x1, x2, x3, x4) = U2_ga(x4) append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) append1_out_gga(x1, x2, x3) = append1_out_gga(x3) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) U3_ga(x1, x2, x3, x4) = U3_ga(x4) ---------------------------------------- (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: PERM_IN_GA(L, .(H, T)) -> U1_GA(L, H, T, append2_in_aag(V, .(H, U), L)) PERM_IN_GA(L, .(H, T)) -> APPEND2_IN_AAG(V, .(H, U), L) APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) -> U5_AAG(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND2_IN_AAG(L1, L2, L3) U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_GA(L, H, T, append1_in_gga(V, U, W)) U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) -> APPEND1_IN_GGA(V, U, W) APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) -> U4_GGA(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND1_IN_GGA(L1, L2, L3) U2_GA(L, H, T, append1_out_gga(V, U, W)) -> U3_GA(L, H, T, perm_in_ga(W, T)) U2_GA(L, H, T, append1_out_gga(V, U, W)) -> PERM_IN_GA(W, T) The TRS R consists of the following rules: perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) append2_in_aag([], L, L) -> append2_out_aag([], L, L) append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) append1_in_gga([], L, L) -> append1_out_gga([], L, L) append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) append2_in_aag(x1, x2, x3) = append2_in_aag(x3) .(x1, x2) = .(x2) append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) U2_ga(x1, x2, x3, x4) = U2_ga(x4) append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) append1_out_gga(x1, x2, x3) = append1_out_gga(x3) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) U3_ga(x1, x2, x3, x4) = U3_ga(x4) PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x4) APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3) U5_AAG(x1, x2, x3, x4, x5) = U5_AAG(x5) U2_GA(x1, x2, x3, x4) = U2_GA(x4) APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2) U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x5) U3_GA(x1, x2, x3, x4) = U3_GA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: PERM_IN_GA(L, .(H, T)) -> U1_GA(L, H, T, append2_in_aag(V, .(H, U), L)) PERM_IN_GA(L, .(H, T)) -> APPEND2_IN_AAG(V, .(H, U), L) APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) -> U5_AAG(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND2_IN_AAG(L1, L2, L3) U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_GA(L, H, T, append1_in_gga(V, U, W)) U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) -> APPEND1_IN_GGA(V, U, W) APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) -> U4_GGA(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND1_IN_GGA(L1, L2, L3) U2_GA(L, H, T, append1_out_gga(V, U, W)) -> U3_GA(L, H, T, perm_in_ga(W, T)) U2_GA(L, H, T, append1_out_gga(V, U, W)) -> PERM_IN_GA(W, T) The TRS R consists of the following rules: perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) append2_in_aag([], L, L) -> append2_out_aag([], L, L) append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) append1_in_gga([], L, L) -> append1_out_gga([], L, L) append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) append2_in_aag(x1, x2, x3) = append2_in_aag(x3) .(x1, x2) = .(x2) append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) U2_ga(x1, x2, x3, x4) = U2_ga(x4) append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) append1_out_gga(x1, x2, x3) = append1_out_gga(x3) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) U3_ga(x1, x2, x3, x4) = U3_ga(x4) PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x4) APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3) U5_AAG(x1, x2, x3, x4, x5) = U5_AAG(x5) U2_GA(x1, x2, x3, x4) = U2_GA(x4) APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2) U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x5) U3_GA(x1, x2, x3, x4) = U3_GA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND1_IN_GGA(L1, L2, L3) The TRS R consists of the following rules: perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) append2_in_aag([], L, L) -> append2_out_aag([], L, L) append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) append1_in_gga([], L, L) -> append1_out_gga([], L, L) append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) append2_in_aag(x1, x2, x3) = append2_in_aag(x3) .(x1, x2) = .(x2) append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) U2_ga(x1, x2, x3, x4) = U2_ga(x4) append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) append1_out_gga(x1, x2, x3) = append1_out_gga(x3) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) U3_ga(x1, x2, x3, x4) = U3_ga(x4) APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2) 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: APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND1_IN_GGA(L1, L2, L3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2) 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: APPEND1_IN_GGA(.(L1), L2) -> APPEND1_IN_GGA(L1, L2) 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: *APPEND1_IN_GGA(.(L1), L2) -> APPEND1_IN_GGA(L1, L2) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND2_IN_AAG(L1, L2, L3) The TRS R consists of the following rules: perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) append2_in_aag([], L, L) -> append2_out_aag([], L, L) append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) append1_in_gga([], L, L) -> append1_out_gga([], L, L) append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) append2_in_aag(x1, x2, x3) = append2_in_aag(x3) .(x1, x2) = .(x2) append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) U2_ga(x1, x2, x3, x4) = U2_ga(x4) append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) append1_out_gga(x1, x2, x3) = append1_out_gga(x3) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) U3_ga(x1, x2, x3, x4) = U3_ga(x4) APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3) 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: APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND2_IN_AAG(L1, L2, L3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3) 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: APPEND2_IN_AAG(.(L3)) -> APPEND2_IN_AAG(L3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *APPEND2_IN_AAG(.(L3)) -> APPEND2_IN_AAG(L3) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_GA(L, H, T, append1_in_gga(V, U, W)) U2_GA(L, H, T, append1_out_gga(V, U, W)) -> PERM_IN_GA(W, T) PERM_IN_GA(L, .(H, T)) -> U1_GA(L, H, T, append2_in_aag(V, .(H, U), L)) The TRS R consists of the following rules: perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(L, .(H, T)) -> U1_ga(L, H, T, append2_in_aag(V, .(H, U), L)) append2_in_aag([], L, L) -> append2_out_aag([], L, L) append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_ga(L, H, T, append1_in_gga(V, U, W)) append1_in_gga([], L, L) -> append1_out_gga([], L, L) append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) U2_ga(L, H, T, append1_out_gga(V, U, W)) -> U3_ga(L, H, T, perm_in_ga(W, T)) U3_ga(L, H, T, perm_out_ga(W, T)) -> perm_out_ga(L, .(H, T)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x4) append2_in_aag(x1, x2, x3) = append2_in_aag(x3) .(x1, x2) = .(x2) append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) U2_ga(x1, x2, x3, x4) = U2_ga(x4) append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) append1_out_gga(x1, x2, x3) = append1_out_gga(x3) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) U3_ga(x1, x2, x3, x4) = U3_ga(x4) PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x4) U2_GA(x1, x2, x3, x4) = U2_GA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) -> U2_GA(L, H, T, append1_in_gga(V, U, W)) U2_GA(L, H, T, append1_out_gga(V, U, W)) -> PERM_IN_GA(W, T) PERM_IN_GA(L, .(H, T)) -> U1_GA(L, H, T, append2_in_aag(V, .(H, U), L)) The TRS R consists of the following rules: append1_in_gga([], L, L) -> append1_out_gga([], L, L) append1_in_gga(.(H, L1), L2, .(H, L3)) -> U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3)) append2_in_aag([], L, L) -> append2_out_aag([], L, L) append2_in_aag(.(H, L1), L2, .(H, L3)) -> U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3)) U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) -> append1_out_gga(.(H, L1), L2, .(H, L3)) U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) -> append2_out_aag(.(H, L1), L2, .(H, L3)) The argument filtering Pi contains the following mapping: [] = [] append2_in_aag(x1, x2, x3) = append2_in_aag(x3) .(x1, x2) = .(x2) append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2) U5_aag(x1, x2, x3, x4, x5) = U5_aag(x5) append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2) append1_out_gga(x1, x2, x3) = append1_out_gga(x3) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x5) PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x4) U2_GA(x1, x2, x3, x4) = U2_GA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GA(append2_out_aag(V, .(U))) -> U2_GA(append1_in_gga(V, U)) U2_GA(append1_out_gga(W)) -> PERM_IN_GA(W) PERM_IN_GA(L) -> U1_GA(append2_in_aag(L)) The TRS R consists of the following rules: append1_in_gga([], L) -> append1_out_gga(L) append1_in_gga(.(L1), L2) -> U4_gga(append1_in_gga(L1, L2)) append2_in_aag(L) -> append2_out_aag([], L) append2_in_aag(.(L3)) -> U5_aag(append2_in_aag(L3)) U4_gga(append1_out_gga(L3)) -> append1_out_gga(.(L3)) U5_aag(append2_out_aag(L1, L2)) -> append2_out_aag(.(L1), L2) The set Q consists of the following terms: append1_in_gga(x0, x1) append2_in_aag(x0) U4_gga(x0) U5_aag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: U1_GA(append2_out_aag(V, .(U))) -> U2_GA(append1_in_gga(V, U)) U2_GA(append1_out_gga(W)) -> PERM_IN_GA(W) PERM_IN_GA(L) -> U1_GA(append2_in_aag(L)) Strictly oriented rules of the TRS R: append1_in_gga([], L) -> append1_out_gga(L) append1_in_gga(.(L1), L2) -> U4_gga(append1_in_gga(L1, L2)) append2_in_aag(L) -> append2_out_aag([], L) append2_in_aag(.(L3)) -> U5_aag(append2_in_aag(L3)) U4_gga(append1_out_gga(L3)) -> append1_out_gga(.(L3)) U5_aag(append2_out_aag(L1, L2)) -> append2_out_aag(.(L1), L2) Used ordering: Knuth-Bendix order [KBO] with precedence:append1_in_gga_2 > [] > PERM_IN_GA_1 > ._1 > U2_GA_1 > append2_in_aag_1 > U1_GA_1 > U5_aag_1 > append2_out_aag_2 > U4_gga_1 > append1_out_gga_1 and weight map: []=2 append1_out_gga_1=6 ._1=7 U4_gga_1=7 append2_in_aag_1=2 U5_aag_1=7 U1_GA_1=3 U2_GA_1=1 PERM_IN_GA_1=6 append1_in_gga_2=8 append2_out_aag_2=0 The variable weight is 1 ---------------------------------------- (27) Obligation: Q DP problem: P is empty. R is empty. The set Q consists of the following terms: append1_in_gga(x0, x1) append2_in_aag(x0) U4_gga(x0) U5_aag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (28) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (29) YES