/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern perm(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0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, 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) 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, 0 ms] (27) QDP (28) PisEmptyProof [EQUIVALENT, 0 ms] (29) YES ---------------------------------------- (0) Obligation: Clauses: app1(.(X, Xs), Ys, .(X, Zs)) :- app1(Xs, Ys, Zs). app1([], Ys, Ys). app2(.(X, Xs), Ys, .(X, Zs)) :- app2(Xs, Ys, Zs). app2([], Ys, Ys). perm(Xs, .(X, Ys)) :- ','(app2(X1s, .(X, X2s), Xs), ','(app1(X1s, X2s, Zs), perm(Zs, Ys))). perm([], []). 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) app2_in_3: (f,f,b) app1_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(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x4) app2_in_aag(x1, x2, x3) = app2_in_aag(x3) .(x1, x2) = .(x2) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) [] = [] app1_out_gga(x1, x2, x3) = app1_out_gga(x3) U5_ga(x1, x2, x3, x4) = U5_ga(x4) perm_out_ga(x1, x2) = perm_out_ga(x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x4) app2_in_aag(x1, x2, x3) = app2_in_aag(x3) .(x1, x2) = .(x2) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) [] = [] app1_out_gga(x1, x2, x3) = app1_out_gga(x3) U5_ga(x1, x2, x3, x4) = U5_ga(x4) perm_out_ga(x1, x2) = perm_out_ga(x2) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: PERM_IN_GA(Xs, .(X, Ys)) -> U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) PERM_IN_GA(Xs, .(X, Ys)) -> APP2_IN_AAG(X1s, .(X, X2s), Xs) APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U2_AAG(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP2_IN_AAG(Xs, Ys, Zs) U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> APP1_IN_GGA(X1s, X2s, Zs) APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U1_GGA(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP1_IN_GGA(Xs, Ys, Zs) U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_GA(Xs, X, Ys, perm_in_ga(Zs, Ys)) U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) The TRS R consists of the following rules: perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x4) app2_in_aag(x1, x2, x3) = app2_in_aag(x3) .(x1, x2) = .(x2) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) [] = [] app1_out_gga(x1, x2, x3) = app1_out_gga(x3) U5_ga(x1, x2, x3, x4) = U5_ga(x4) perm_out_ga(x1, x2) = perm_out_ga(x2) PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x4) APP2_IN_AAG(x1, x2, x3) = APP2_IN_AAG(x3) U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x5) U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6) APP1_IN_GGA(x1, x2, x3) = APP1_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x5) U5_GA(x1, x2, x3, x4) = U5_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(Xs, .(X, Ys)) -> U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) PERM_IN_GA(Xs, .(X, Ys)) -> APP2_IN_AAG(X1s, .(X, X2s), Xs) APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U2_AAG(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP2_IN_AAG(Xs, Ys, Zs) U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> APP1_IN_GGA(X1s, X2s, Zs) APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U1_GGA(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP1_IN_GGA(Xs, Ys, Zs) U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_GA(Xs, X, Ys, perm_in_ga(Zs, Ys)) U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) The TRS R consists of the following rules: perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x4) app2_in_aag(x1, x2, x3) = app2_in_aag(x3) .(x1, x2) = .(x2) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) [] = [] app1_out_gga(x1, x2, x3) = app1_out_gga(x3) U5_ga(x1, x2, x3, x4) = U5_ga(x4) perm_out_ga(x1, x2) = perm_out_ga(x2) PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x4) APP2_IN_AAG(x1, x2, x3) = APP2_IN_AAG(x3) U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x5) U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6) APP1_IN_GGA(x1, x2, x3) = APP1_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x5) U5_GA(x1, x2, x3, x4) = U5_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: APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP1_IN_GGA(Xs, Ys, Zs) The TRS R consists of the following rules: perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x4) app2_in_aag(x1, x2, x3) = app2_in_aag(x3) .(x1, x2) = .(x2) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) [] = [] app1_out_gga(x1, x2, x3) = app1_out_gga(x3) U5_ga(x1, x2, x3, x4) = U5_ga(x4) perm_out_ga(x1, x2) = perm_out_ga(x2) APP1_IN_GGA(x1, x2, x3) = APP1_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: APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP1_IN_GGA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APP1_IN_GGA(x1, x2, x3) = APP1_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: APP1_IN_GGA(.(Xs), Ys) -> APP1_IN_GGA(Xs, Ys) 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: *APP1_IN_GGA(.(Xs), Ys) -> APP1_IN_GGA(Xs, Ys) 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: APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP2_IN_AAG(Xs, Ys, Zs) The TRS R consists of the following rules: perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x4) app2_in_aag(x1, x2, x3) = app2_in_aag(x3) .(x1, x2) = .(x2) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) [] = [] app1_out_gga(x1, x2, x3) = app1_out_gga(x3) U5_ga(x1, x2, x3, x4) = U5_ga(x4) perm_out_ga(x1, x2) = perm_out_ga(x2) APP2_IN_AAG(x1, x2, x3) = APP2_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: APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP2_IN_AAG(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APP2_IN_AAG(x1, x2, x3) = APP2_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: APP2_IN_AAG(.(Zs)) -> APP2_IN_AAG(Zs) 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: *APP2_IN_AAG(.(Zs)) -> APP2_IN_AAG(Zs) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) PERM_IN_GA(Xs, .(X, Ys)) -> U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) The TRS R consists of the following rules: perm_in_ga(Xs, .(X, Ys)) -> U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) -> perm_out_ga(Xs, .(X, Ys)) The argument filtering Pi contains the following mapping: perm_in_ga(x1, x2) = perm_in_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x4) app2_in_aag(x1, x2, x3) = app2_in_aag(x3) .(x1, x2) = .(x2) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) [] = [] app1_out_gga(x1, x2, x3) = app1_out_gga(x3) U5_ga(x1, x2, x3, x4) = U5_ga(x4) perm_out_ga(x1, x2) = perm_out_ga(x2) PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x4) U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6) 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: U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) -> U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs)) U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) -> PERM_IN_GA(Zs, Ys) PERM_IN_GA(Xs, .(X, Ys)) -> U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs)) The TRS R consists of the following rules: app1_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs)) app1_in_gga([], Ys, Ys) -> app1_out_gga([], Ys, Ys) app2_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs)) app2_in_aag([], Ys, Ys) -> app2_out_aag([], Ys, Ys) U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) -> app1_out_gga(.(X, Xs), Ys, .(X, Zs)) U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) -> app2_out_aag(.(X, Xs), Ys, .(X, Zs)) The argument filtering Pi contains the following mapping: app2_in_aag(x1, x2, x3) = app2_in_aag(x3) .(x1, x2) = .(x2) U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5) app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2) app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5) [] = [] app1_out_gga(x1, x2, x3) = app1_out_gga(x3) PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x4) U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6) 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: U3_GA(app2_out_aag(X1s, .(X2s))) -> U4_GA(app1_in_gga(X1s, X2s)) U4_GA(app1_out_gga(Zs)) -> PERM_IN_GA(Zs) PERM_IN_GA(Xs) -> U3_GA(app2_in_aag(Xs)) The TRS R consists of the following rules: app1_in_gga(.(Xs), Ys) -> U1_gga(app1_in_gga(Xs, Ys)) app1_in_gga([], Ys) -> app1_out_gga(Ys) app2_in_aag(.(Zs)) -> U2_aag(app2_in_aag(Zs)) app2_in_aag(Ys) -> app2_out_aag([], Ys) U1_gga(app1_out_gga(Zs)) -> app1_out_gga(.(Zs)) U2_aag(app2_out_aag(Xs, Ys)) -> app2_out_aag(.(Xs), Ys) The set Q consists of the following terms: app1_in_gga(x0, x1) app2_in_aag(x0) U1_gga(x0) U2_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: U3_GA(app2_out_aag(X1s, .(X2s))) -> U4_GA(app1_in_gga(X1s, X2s)) U4_GA(app1_out_gga(Zs)) -> PERM_IN_GA(Zs) PERM_IN_GA(Xs) -> U3_GA(app2_in_aag(Xs)) Strictly oriented rules of the TRS R: app1_in_gga(.(Xs), Ys) -> U1_gga(app1_in_gga(Xs, Ys)) app1_in_gga([], Ys) -> app1_out_gga(Ys) app2_in_aag(.(Zs)) -> U2_aag(app2_in_aag(Zs)) app2_in_aag(Ys) -> app2_out_aag([], Ys) U1_gga(app1_out_gga(Zs)) -> app1_out_gga(.(Zs)) U2_aag(app2_out_aag(Xs, Ys)) -> app2_out_aag(.(Xs), Ys) Used ordering: Knuth-Bendix order [KBO] with precedence:U3_GA_1 > app2_in_aag_1 > ._1 > app1_in_gga_2 > PERM_IN_GA_1 > U1_gga_1 > U2_aag_1 > U4_GA_1 > app2_out_aag_2 > app1_out_gga_1 > [] and weight map: []=2 ._1=3 U1_gga_1=3 app1_out_gga_1=7 app2_in_aag_1=4 U2_aag_1=3 U3_GA_1=2 U4_GA_1=1 PERM_IN_GA_1=7 app1_in_gga_2=5 app2_out_aag_2=1 The variable weight is 1 ---------------------------------------- (27) Obligation: Q DP problem: P is empty. R is empty. The set Q consists of the following terms: app1_in_gga(x0, x1) app2_in_aag(x0) U1_gga(x0) U2_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