/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern permute(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof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) MRRProof [EQUIVALENT, 3 ms] (20) QDP (21) PisEmptyProof [EQUIVALENT, 0 ms] (22) YES ---------------------------------------- (0) Obligation: Clauses: permute([], []). permute(.(X, Y), .(U, V)) :- ','(delete(U, .(X, Y), W), permute(W, V)). delete(X, .(X, Y), Y). delete(U, .(X, Y), .(X, Z)) :- delete(U, Y, Z). Query: permute(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: permute_in_2: (b,f) delete_in_3: (f,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: permute_in_ga([], []) -> permute_out_ga([], []) permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) The argument filtering Pi contains the following mapping: permute_in_ga(x1, x2) = permute_in_ga(x1) [] = [] permute_out_ga(x1, x2) = permute_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: permute_in_ga([], []) -> permute_out_ga([], []) permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) The argument filtering Pi contains the following mapping: permute_in_ga(x1, x2) = permute_in_ga(x1) [] = [] permute_out_ga(x1, x2) = permute_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) ---------------------------------------- (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: PERMUTE_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) PERMUTE_IN_GA(.(X, Y), .(U, V)) -> DELETE_IN_AGA(U, .(X, Y), W) DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> U3_AGA(U, X, Y, Z, delete_in_aga(U, Y, Z)) DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z) U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_GA(X, Y, U, V, permute_in_ga(W, V)) U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERMUTE_IN_GA(W, V) The TRS R consists of the following rules: permute_in_ga([], []) -> permute_out_ga([], []) permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) The argument filtering Pi contains the following mapping: permute_in_ga(x1, x2) = permute_in_ga(x1) [] = [] permute_out_ga(x1, x2) = permute_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) PERMUTE_IN_GA(x1, x2) = PERMUTE_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x2, x5) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: PERMUTE_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) PERMUTE_IN_GA(.(X, Y), .(U, V)) -> DELETE_IN_AGA(U, .(X, Y), W) DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> U3_AGA(U, X, Y, Z, delete_in_aga(U, Y, Z)) DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z) U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_GA(X, Y, U, V, permute_in_ga(W, V)) U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERMUTE_IN_GA(W, V) The TRS R consists of the following rules: permute_in_ga([], []) -> permute_out_ga([], []) permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) The argument filtering Pi contains the following mapping: permute_in_ga(x1, x2) = permute_in_ga(x1) [] = [] permute_out_ga(x1, x2) = permute_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) PERMUTE_IN_GA(x1, x2) = PERMUTE_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x2, x5) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z) The TRS R consists of the following rules: permute_in_ga([], []) -> permute_out_ga([], []) permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) The argument filtering Pi contains the following mapping: permute_in_ga(x1, x2) = permute_in_ga(x1) [] = [] permute_out_ga(x1, x2) = permute_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(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: DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(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: DELETE_IN_AGA(.(X, Y)) -> DELETE_IN_AGA(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: *DELETE_IN_AGA(.(X, Y)) -> DELETE_IN_AGA(Y) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERMUTE_IN_GA(W, V) PERMUTE_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) The TRS R consists of the following rules: permute_in_ga([], []) -> permute_out_ga([], []) permute_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, permute_in_ga(W, V)) U2_ga(X, Y, U, V, permute_out_ga(W, V)) -> permute_out_ga(.(X, Y), .(U, V)) The argument filtering Pi contains the following mapping: permute_in_ga(x1, x2) = permute_in_ga(x1) [] = [] permute_out_ga(x1, x2) = permute_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) PERMUTE_IN_GA(x1, x2) = PERMUTE_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) 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: U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERMUTE_IN_GA(W, V) PERMUTE_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W)) The TRS R consists of the following rules: delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z)) U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x5) PERMUTE_IN_GA(x1, x2) = PERMUTE_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) 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: U1_GA(delete_out_aga(U, W)) -> PERMUTE_IN_GA(W) PERMUTE_IN_GA(.(X, Y)) -> U1_GA(delete_in_aga(.(X, Y))) The TRS R consists of the following rules: delete_in_aga(.(X, Y)) -> delete_out_aga(X, Y) delete_in_aga(.(X, Y)) -> U3_aga(X, delete_in_aga(Y)) U3_aga(X, delete_out_aga(U, Z)) -> delete_out_aga(U, .(X, Z)) The set Q consists of the following terms: delete_in_aga(x0) U3_aga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) 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(delete_out_aga(U, W)) -> PERMUTE_IN_GA(W) PERMUTE_IN_GA(.(X, Y)) -> U1_GA(delete_in_aga(.(X, Y))) Strictly oriented rules of the TRS R: delete_in_aga(.(X, Y)) -> delete_out_aga(X, Y) delete_in_aga(.(X, Y)) -> U3_aga(X, delete_in_aga(Y)) U3_aga(X, delete_out_aga(U, Z)) -> delete_out_aga(U, .(X, Z)) Used ordering: Knuth-Bendix order [KBO] with precedence:._2 > delete_in_aga_1 > U3_aga_2 > U1_GA_1 > PERMUTE_IN_GA_1 > delete_out_aga_2 and weight map: delete_in_aga_1=1 U1_GA_1=1 PERMUTE_IN_GA_1=3 ._2=0 delete_out_aga_2=1 U3_aga_2=0 The variable weight is 1 ---------------------------------------- (20) Obligation: Q DP problem: P is empty. R is empty. The set Q consists of the following terms: delete_in_aga(x0) U3_aga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (22) YES