/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern perm(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0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(.(X0, X), Y, .(X0, Z)) :- app1(X, Y, Z). app1([], Y, Y). app2(.(X0, X), Y, .(X0, Z)) :- app2(X, Y, Z). app2([], Y, Y). perm(X, .(X0, Y)) :- ','(app1(X1, .(X0, X2), X), ','(app2(X1, X2, Z), perm(Z, Y))). 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) app1_in_3: (f,f,b) app2_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(X, .(X0, Y)) -> U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X)) app1_in_aag(.(X0, X), Y, .(X0, Z)) -> U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z)) app1_in_aag([], Y, Y) -> app1_out_aag([], Y, Y) U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) -> app1_out_aag(.(X0, X), Y, .(X0, Z)) U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) -> U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z)) app2_in_gga(.(X0, X), Y, .(X0, Z)) -> U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z)) app2_in_gga([], Y, Y) -> app2_out_gga([], Y, Y) U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) -> app2_out_gga(.(X0, X), Y, .(X0, Z)) U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) -> U5_ga(X, X0, Y, perm_in_ga(Z, Y)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(X, X0, Y, perm_out_ga(Z, Y)) -> perm_out_ga(X, .(X0, Y)) 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) app1_in_aag(x1, x2, x3) = app1_in_aag(x3) .(x1, x2) = .(x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) app1_out_aag(x1, x2, x3) = app1_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app2_in_gga(x1, x2, x3) = app2_in_gga(x1, x2) U2_gga(x1, x2, x3, x4, x5) = U2_gga(x5) [] = [] app2_out_gga(x1, x2, x3) = app2_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(X, .(X0, Y)) -> U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X)) app1_in_aag(.(X0, X), Y, .(X0, Z)) -> U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z)) app1_in_aag([], Y, Y) -> app1_out_aag([], Y, Y) U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) -> app1_out_aag(.(X0, X), Y, .(X0, Z)) U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) -> U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z)) app2_in_gga(.(X0, X), Y, .(X0, Z)) -> U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z)) app2_in_gga([], Y, Y) -> app2_out_gga([], Y, Y) U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) -> app2_out_gga(.(X0, X), Y, .(X0, Z)) U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) -> U5_ga(X, X0, Y, perm_in_ga(Z, Y)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(X, X0, Y, perm_out_ga(Z, Y)) -> perm_out_ga(X, .(X0, Y)) 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) app1_in_aag(x1, x2, x3) = app1_in_aag(x3) .(x1, x2) = .(x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) app1_out_aag(x1, x2, x3) = app1_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app2_in_gga(x1, x2, x3) = app2_in_gga(x1, x2) U2_gga(x1, x2, x3, x4, x5) = U2_gga(x5) [] = [] app2_out_gga(x1, x2, x3) = app2_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(X, .(X0, Y)) -> U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X)) PERM_IN_GA(X, .(X0, Y)) -> APP1_IN_AAG(X1, .(X0, X2), X) APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) -> U1_AAG(X0, X, Y, Z, app1_in_aag(X, Y, Z)) APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) -> APP1_IN_AAG(X, Y, Z) U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) -> U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z)) U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) -> APP2_IN_GGA(X1, X2, Z) APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) -> U2_GGA(X0, X, Y, Z, app2_in_gga(X, Y, Z)) APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) -> APP2_IN_GGA(X, Y, Z) U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) -> U5_GA(X, X0, Y, perm_in_ga(Z, Y)) U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) -> PERM_IN_GA(Z, Y) The TRS R consists of the following rules: perm_in_ga(X, .(X0, Y)) -> U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X)) app1_in_aag(.(X0, X), Y, .(X0, Z)) -> U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z)) app1_in_aag([], Y, Y) -> app1_out_aag([], Y, Y) U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) -> app1_out_aag(.(X0, X), Y, .(X0, Z)) U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) -> U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z)) app2_in_gga(.(X0, X), Y, .(X0, Z)) -> U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z)) app2_in_gga([], Y, Y) -> app2_out_gga([], Y, Y) U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) -> app2_out_gga(.(X0, X), Y, .(X0, Z)) U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) -> U5_ga(X, X0, Y, perm_in_ga(Z, Y)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(X, X0, Y, perm_out_ga(Z, Y)) -> perm_out_ga(X, .(X0, Y)) 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) app1_in_aag(x1, x2, x3) = app1_in_aag(x3) .(x1, x2) = .(x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) app1_out_aag(x1, x2, x3) = app1_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app2_in_gga(x1, x2, x3) = app2_in_gga(x1, x2) U2_gga(x1, x2, x3, x4, x5) = U2_gga(x5) [] = [] app2_out_gga(x1, x2, x3) = app2_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) APP1_IN_AAG(x1, x2, x3) = APP1_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x5) U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6) APP2_IN_GGA(x1, x2, x3) = APP2_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4, x5) = U2_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(X, .(X0, Y)) -> U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X)) PERM_IN_GA(X, .(X0, Y)) -> APP1_IN_AAG(X1, .(X0, X2), X) APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) -> U1_AAG(X0, X, Y, Z, app1_in_aag(X, Y, Z)) APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) -> APP1_IN_AAG(X, Y, Z) U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) -> U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z)) U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) -> APP2_IN_GGA(X1, X2, Z) APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) -> U2_GGA(X0, X, Y, Z, app2_in_gga(X, Y, Z)) APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) -> APP2_IN_GGA(X, Y, Z) U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) -> U5_GA(X, X0, Y, perm_in_ga(Z, Y)) U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) -> PERM_IN_GA(Z, Y) The TRS R consists of the following rules: perm_in_ga(X, .(X0, Y)) -> U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X)) app1_in_aag(.(X0, X), Y, .(X0, Z)) -> U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z)) app1_in_aag([], Y, Y) -> app1_out_aag([], Y, Y) U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) -> app1_out_aag(.(X0, X), Y, .(X0, Z)) U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) -> U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z)) app2_in_gga(.(X0, X), Y, .(X0, Z)) -> U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z)) app2_in_gga([], Y, Y) -> app2_out_gga([], Y, Y) U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) -> app2_out_gga(.(X0, X), Y, .(X0, Z)) U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) -> U5_ga(X, X0, Y, perm_in_ga(Z, Y)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(X, X0, Y, perm_out_ga(Z, Y)) -> perm_out_ga(X, .(X0, Y)) 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) app1_in_aag(x1, x2, x3) = app1_in_aag(x3) .(x1, x2) = .(x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) app1_out_aag(x1, x2, x3) = app1_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app2_in_gga(x1, x2, x3) = app2_in_gga(x1, x2) U2_gga(x1, x2, x3, x4, x5) = U2_gga(x5) [] = [] app2_out_gga(x1, x2, x3) = app2_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) APP1_IN_AAG(x1, x2, x3) = APP1_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x5) U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6) APP2_IN_GGA(x1, x2, x3) = APP2_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4, x5) = U2_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: APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) -> APP2_IN_GGA(X, Y, Z) The TRS R consists of the following rules: perm_in_ga(X, .(X0, Y)) -> U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X)) app1_in_aag(.(X0, X), Y, .(X0, Z)) -> U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z)) app1_in_aag([], Y, Y) -> app1_out_aag([], Y, Y) U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) -> app1_out_aag(.(X0, X), Y, .(X0, Z)) U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) -> U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z)) app2_in_gga(.(X0, X), Y, .(X0, Z)) -> U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z)) app2_in_gga([], Y, Y) -> app2_out_gga([], Y, Y) U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) -> app2_out_gga(.(X0, X), Y, .(X0, Z)) U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) -> U5_ga(X, X0, Y, perm_in_ga(Z, Y)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(X, X0, Y, perm_out_ga(Z, Y)) -> perm_out_ga(X, .(X0, Y)) 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) app1_in_aag(x1, x2, x3) = app1_in_aag(x3) .(x1, x2) = .(x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) app1_out_aag(x1, x2, x3) = app1_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app2_in_gga(x1, x2, x3) = app2_in_gga(x1, x2) U2_gga(x1, x2, x3, x4, x5) = U2_gga(x5) [] = [] app2_out_gga(x1, x2, x3) = app2_out_gga(x3) U5_ga(x1, x2, x3, x4) = U5_ga(x4) perm_out_ga(x1, x2) = perm_out_ga(x2) APP2_IN_GGA(x1, x2, x3) = APP2_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: APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) -> APP2_IN_GGA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APP2_IN_GGA(x1, x2, x3) = APP2_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: APP2_IN_GGA(.(X), Y) -> APP2_IN_GGA(X, Y) 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: *APP2_IN_GGA(.(X), Y) -> APP2_IN_GGA(X, Y) 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: APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) -> APP1_IN_AAG(X, Y, Z) The TRS R consists of the following rules: perm_in_ga(X, .(X0, Y)) -> U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X)) app1_in_aag(.(X0, X), Y, .(X0, Z)) -> U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z)) app1_in_aag([], Y, Y) -> app1_out_aag([], Y, Y) U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) -> app1_out_aag(.(X0, X), Y, .(X0, Z)) U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) -> U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z)) app2_in_gga(.(X0, X), Y, .(X0, Z)) -> U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z)) app2_in_gga([], Y, Y) -> app2_out_gga([], Y, Y) U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) -> app2_out_gga(.(X0, X), Y, .(X0, Z)) U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) -> U5_ga(X, X0, Y, perm_in_ga(Z, Y)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(X, X0, Y, perm_out_ga(Z, Y)) -> perm_out_ga(X, .(X0, Y)) 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) app1_in_aag(x1, x2, x3) = app1_in_aag(x3) .(x1, x2) = .(x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) app1_out_aag(x1, x2, x3) = app1_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app2_in_gga(x1, x2, x3) = app2_in_gga(x1, x2) U2_gga(x1, x2, x3, x4, x5) = U2_gga(x5) [] = [] app2_out_gga(x1, x2, x3) = app2_out_gga(x3) U5_ga(x1, x2, x3, x4) = U5_ga(x4) perm_out_ga(x1, x2) = perm_out_ga(x2) APP1_IN_AAG(x1, x2, x3) = APP1_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: APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) -> APP1_IN_AAG(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APP1_IN_AAG(x1, x2, x3) = APP1_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: APP1_IN_AAG(.(Z)) -> APP1_IN_AAG(Z) 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: *APP1_IN_AAG(.(Z)) -> APP1_IN_AAG(Z) 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(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) -> U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z)) U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) -> PERM_IN_GA(Z, Y) PERM_IN_GA(X, .(X0, Y)) -> U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X)) The TRS R consists of the following rules: perm_in_ga(X, .(X0, Y)) -> U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X)) app1_in_aag(.(X0, X), Y, .(X0, Z)) -> U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z)) app1_in_aag([], Y, Y) -> app1_out_aag([], Y, Y) U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) -> app1_out_aag(.(X0, X), Y, .(X0, Z)) U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) -> U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z)) app2_in_gga(.(X0, X), Y, .(X0, Z)) -> U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z)) app2_in_gga([], Y, Y) -> app2_out_gga([], Y, Y) U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) -> app2_out_gga(.(X0, X), Y, .(X0, Z)) U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) -> U5_ga(X, X0, Y, perm_in_ga(Z, Y)) perm_in_ga([], []) -> perm_out_ga([], []) U5_ga(X, X0, Y, perm_out_ga(Z, Y)) -> perm_out_ga(X, .(X0, Y)) 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) app1_in_aag(x1, x2, x3) = app1_in_aag(x3) .(x1, x2) = .(x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) app1_out_aag(x1, x2, x3) = app1_out_aag(x1, x2) U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6) app2_in_gga(x1, x2, x3) = app2_in_gga(x1, x2) U2_gga(x1, x2, x3, x4, x5) = U2_gga(x5) [] = [] app2_out_gga(x1, x2, x3) = app2_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(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) -> U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z)) U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) -> PERM_IN_GA(Z, Y) PERM_IN_GA(X, .(X0, Y)) -> U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X)) The TRS R consists of the following rules: app2_in_gga(.(X0, X), Y, .(X0, Z)) -> U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z)) app2_in_gga([], Y, Y) -> app2_out_gga([], Y, Y) app1_in_aag(.(X0, X), Y, .(X0, Z)) -> U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z)) app1_in_aag([], Y, Y) -> app1_out_aag([], Y, Y) U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) -> app2_out_gga(.(X0, X), Y, .(X0, Z)) U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) -> app1_out_aag(.(X0, X), Y, .(X0, Z)) The argument filtering Pi contains the following mapping: app1_in_aag(x1, x2, x3) = app1_in_aag(x3) .(x1, x2) = .(x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) app1_out_aag(x1, x2, x3) = app1_out_aag(x1, x2) app2_in_gga(x1, x2, x3) = app2_in_gga(x1, x2) U2_gga(x1, x2, x3, x4, x5) = U2_gga(x5) [] = [] app2_out_gga(x1, x2, x3) = app2_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(app1_out_aag(X1, .(X2))) -> U4_GA(app2_in_gga(X1, X2)) U4_GA(app2_out_gga(Z)) -> PERM_IN_GA(Z) PERM_IN_GA(X) -> U3_GA(app1_in_aag(X)) The TRS R consists of the following rules: app2_in_gga(.(X), Y) -> U2_gga(app2_in_gga(X, Y)) app2_in_gga([], Y) -> app2_out_gga(Y) app1_in_aag(.(Z)) -> U1_aag(app1_in_aag(Z)) app1_in_aag(Y) -> app1_out_aag([], Y) U2_gga(app2_out_gga(Z)) -> app2_out_gga(.(Z)) U1_aag(app1_out_aag(X, Y)) -> app1_out_aag(.(X), Y) The set Q consists of the following terms: app2_in_gga(x0, x1) app1_in_aag(x0) U2_gga(x0) U1_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(app1_out_aag(X1, .(X2))) -> U4_GA(app2_in_gga(X1, X2)) U4_GA(app2_out_gga(Z)) -> PERM_IN_GA(Z) PERM_IN_GA(X) -> U3_GA(app1_in_aag(X)) Strictly oriented rules of the TRS R: app2_in_gga(.(X), Y) -> U2_gga(app2_in_gga(X, Y)) app2_in_gga([], Y) -> app2_out_gga(Y) app1_in_aag(.(Z)) -> U1_aag(app1_in_aag(Z)) app1_in_aag(Y) -> app1_out_aag([], Y) U2_gga(app2_out_gga(Z)) -> app2_out_gga(.(Z)) U1_aag(app1_out_aag(X, Y)) -> app1_out_aag(.(X), Y) Used ordering: Knuth-Bendix order [KBO] with precedence:U3_GA_1 > app1_in_aag_1 > ._1 > app2_in_gga_2 > PERM_IN_GA_1 > U2_gga_1 > U1_aag_1 > U4_GA_1 > app1_out_aag_2 > app2_out_gga_1 > [] and weight map: []=2 ._1=3 U2_gga_1=3 app2_out_gga_1=7 app1_in_aag_1=4 U1_aag_1=3 U3_GA_1=2 U4_GA_1=1 PERM_IN_GA_1=7 app2_in_gga_2=5 app1_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: app2_in_gga(x0, x1) app1_in_aag(x0) U2_gga(x0) U1_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