/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern count(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND,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, 15 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) UsableRulesReductionPairsProof [EQUIVALENT, 20 ms] (13) QDP (14) PisEmptyProof [EQUIVALENT, 0 ms] (15) YES (16) PiDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) PiDP (19) PiDPToQDPProof [SOUND, 0 ms] (20) QDP (21) UsableRulesReductionPairsProof [EQUIVALENT, 32 ms] (22) QDP (23) DependencyGraphProof [EQUIVALENT, 0 ms] (24) TRUE ---------------------------------------- (0) Obligation: Clauses: flatten(atom(X), .(X, [])). flatten(cons(atom(X), U), .(X, Y)) :- flatten(U, Y). flatten(cons(cons(U, V), W), X) :- flatten(cons(U, cons(V, W)), X). count(atom(X), s(0)). count(cons(atom(X), Y), s(Z)) :- count(Y, Z). count(cons(cons(U, V), W), Z) :- ','(flatten(cons(cons(U, V), W), X), count(X, Z)). Query: count(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: count_in_2: (b,f) flatten_in_2: (b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: count_in_ga(atom(X), s(0)) -> count_out_ga(atom(X), s(0)) count_in_ga(cons(atom(X), Y), s(Z)) -> U3_ga(X, Y, Z, count_in_ga(Y, Z)) count_in_ga(cons(cons(U, V), W), Z) -> U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X)) flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, [])) flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y)) flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X)) U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X) U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y)) U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_ga(U, V, W, Z, count_in_ga(X, Z)) U5_ga(U, V, W, Z, count_out_ga(X, Z)) -> count_out_ga(cons(cons(U, V), W), Z) U3_ga(X, Y, Z, count_out_ga(Y, Z)) -> count_out_ga(cons(atom(X), Y), s(Z)) The argument filtering Pi contains the following mapping: count_in_ga(x1, x2) = count_in_ga(x1) atom(x1) = atom(x1) count_out_ga(x1, x2) = count_out_ga(x2) cons(x1, x2) = cons(x1, x2) U3_ga(x1, x2, x3, x4) = U3_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5) flatten_in_ga(x1, x2) = flatten_in_ga(x1) flatten_out_ga(x1, x2) = flatten_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) U5_ga(x1, x2, x3, x4, x5) = U5_ga(x5) .(x1, x2) = .(x1, x2) s(x1) = s(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: count_in_ga(atom(X), s(0)) -> count_out_ga(atom(X), s(0)) count_in_ga(cons(atom(X), Y), s(Z)) -> U3_ga(X, Y, Z, count_in_ga(Y, Z)) count_in_ga(cons(cons(U, V), W), Z) -> U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X)) flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, [])) flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y)) flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X)) U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X) U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y)) U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_ga(U, V, W, Z, count_in_ga(X, Z)) U5_ga(U, V, W, Z, count_out_ga(X, Z)) -> count_out_ga(cons(cons(U, V), W), Z) U3_ga(X, Y, Z, count_out_ga(Y, Z)) -> count_out_ga(cons(atom(X), Y), s(Z)) The argument filtering Pi contains the following mapping: count_in_ga(x1, x2) = count_in_ga(x1) atom(x1) = atom(x1) count_out_ga(x1, x2) = count_out_ga(x2) cons(x1, x2) = cons(x1, x2) U3_ga(x1, x2, x3, x4) = U3_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5) flatten_in_ga(x1, x2) = flatten_in_ga(x1) flatten_out_ga(x1, x2) = flatten_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) U5_ga(x1, x2, x3, x4, x5) = U5_ga(x5) .(x1, x2) = .(x1, x2) s(x1) = s(x1) ---------------------------------------- (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: COUNT_IN_GA(cons(atom(X), Y), s(Z)) -> U3_GA(X, Y, Z, count_in_ga(Y, Z)) COUNT_IN_GA(cons(atom(X), Y), s(Z)) -> COUNT_IN_GA(Y, Z) COUNT_IN_GA(cons(cons(U, V), W), Z) -> U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X)) COUNT_IN_GA(cons(cons(U, V), W), Z) -> FLATTEN_IN_GA(cons(cons(U, V), W), X) FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> U1_GA(X, U, Y, flatten_in_ga(U, Y)) FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> FLATTEN_IN_GA(U, Y) FLATTEN_IN_GA(cons(cons(U, V), W), X) -> U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X)) FLATTEN_IN_GA(cons(cons(U, V), W), X) -> FLATTEN_IN_GA(cons(U, cons(V, W)), X) U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_GA(U, V, W, Z, count_in_ga(X, Z)) U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> COUNT_IN_GA(X, Z) The TRS R consists of the following rules: count_in_ga(atom(X), s(0)) -> count_out_ga(atom(X), s(0)) count_in_ga(cons(atom(X), Y), s(Z)) -> U3_ga(X, Y, Z, count_in_ga(Y, Z)) count_in_ga(cons(cons(U, V), W), Z) -> U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X)) flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, [])) flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y)) flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X)) U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X) U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y)) U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_ga(U, V, W, Z, count_in_ga(X, Z)) U5_ga(U, V, W, Z, count_out_ga(X, Z)) -> count_out_ga(cons(cons(U, V), W), Z) U3_ga(X, Y, Z, count_out_ga(Y, Z)) -> count_out_ga(cons(atom(X), Y), s(Z)) The argument filtering Pi contains the following mapping: count_in_ga(x1, x2) = count_in_ga(x1) atom(x1) = atom(x1) count_out_ga(x1, x2) = count_out_ga(x2) cons(x1, x2) = cons(x1, x2) U3_ga(x1, x2, x3, x4) = U3_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5) flatten_in_ga(x1, x2) = flatten_in_ga(x1) flatten_out_ga(x1, x2) = flatten_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) U5_ga(x1, x2, x3, x4, x5) = U5_ga(x5) .(x1, x2) = .(x1, x2) s(x1) = s(x1) COUNT_IN_GA(x1, x2) = COUNT_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x4) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5) FLATTEN_IN_GA(x1, x2) = FLATTEN_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5) U5_GA(x1, x2, x3, x4, x5) = U5_GA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: COUNT_IN_GA(cons(atom(X), Y), s(Z)) -> U3_GA(X, Y, Z, count_in_ga(Y, Z)) COUNT_IN_GA(cons(atom(X), Y), s(Z)) -> COUNT_IN_GA(Y, Z) COUNT_IN_GA(cons(cons(U, V), W), Z) -> U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X)) COUNT_IN_GA(cons(cons(U, V), W), Z) -> FLATTEN_IN_GA(cons(cons(U, V), W), X) FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> U1_GA(X, U, Y, flatten_in_ga(U, Y)) FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> FLATTEN_IN_GA(U, Y) FLATTEN_IN_GA(cons(cons(U, V), W), X) -> U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X)) FLATTEN_IN_GA(cons(cons(U, V), W), X) -> FLATTEN_IN_GA(cons(U, cons(V, W)), X) U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_GA(U, V, W, Z, count_in_ga(X, Z)) U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> COUNT_IN_GA(X, Z) The TRS R consists of the following rules: count_in_ga(atom(X), s(0)) -> count_out_ga(atom(X), s(0)) count_in_ga(cons(atom(X), Y), s(Z)) -> U3_ga(X, Y, Z, count_in_ga(Y, Z)) count_in_ga(cons(cons(U, V), W), Z) -> U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X)) flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, [])) flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y)) flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X)) U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X) U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y)) U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_ga(U, V, W, Z, count_in_ga(X, Z)) U5_ga(U, V, W, Z, count_out_ga(X, Z)) -> count_out_ga(cons(cons(U, V), W), Z) U3_ga(X, Y, Z, count_out_ga(Y, Z)) -> count_out_ga(cons(atom(X), Y), s(Z)) The argument filtering Pi contains the following mapping: count_in_ga(x1, x2) = count_in_ga(x1) atom(x1) = atom(x1) count_out_ga(x1, x2) = count_out_ga(x2) cons(x1, x2) = cons(x1, x2) U3_ga(x1, x2, x3, x4) = U3_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5) flatten_in_ga(x1, x2) = flatten_in_ga(x1) flatten_out_ga(x1, x2) = flatten_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) U5_ga(x1, x2, x3, x4, x5) = U5_ga(x5) .(x1, x2) = .(x1, x2) s(x1) = s(x1) COUNT_IN_GA(x1, x2) = COUNT_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x4) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5) FLATTEN_IN_GA(x1, x2) = FLATTEN_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5) U5_GA(x1, x2, x3, x4, x5) = U5_GA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: FLATTEN_IN_GA(cons(cons(U, V), W), X) -> FLATTEN_IN_GA(cons(U, cons(V, W)), X) FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> FLATTEN_IN_GA(U, Y) The TRS R consists of the following rules: count_in_ga(atom(X), s(0)) -> count_out_ga(atom(X), s(0)) count_in_ga(cons(atom(X), Y), s(Z)) -> U3_ga(X, Y, Z, count_in_ga(Y, Z)) count_in_ga(cons(cons(U, V), W), Z) -> U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X)) flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, [])) flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y)) flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X)) U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X) U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y)) U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_ga(U, V, W, Z, count_in_ga(X, Z)) U5_ga(U, V, W, Z, count_out_ga(X, Z)) -> count_out_ga(cons(cons(U, V), W), Z) U3_ga(X, Y, Z, count_out_ga(Y, Z)) -> count_out_ga(cons(atom(X), Y), s(Z)) The argument filtering Pi contains the following mapping: count_in_ga(x1, x2) = count_in_ga(x1) atom(x1) = atom(x1) count_out_ga(x1, x2) = count_out_ga(x2) cons(x1, x2) = cons(x1, x2) U3_ga(x1, x2, x3, x4) = U3_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5) flatten_in_ga(x1, x2) = flatten_in_ga(x1) flatten_out_ga(x1, x2) = flatten_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) U5_ga(x1, x2, x3, x4, x5) = U5_ga(x5) .(x1, x2) = .(x1, x2) s(x1) = s(x1) FLATTEN_IN_GA(x1, x2) = FLATTEN_IN_GA(x1) 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: FLATTEN_IN_GA(cons(cons(U, V), W), X) -> FLATTEN_IN_GA(cons(U, cons(V, W)), X) FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) -> FLATTEN_IN_GA(U, Y) R is empty. The argument filtering Pi contains the following mapping: atom(x1) = atom(x1) cons(x1, x2) = cons(x1, x2) .(x1, x2) = .(x1, x2) FLATTEN_IN_GA(x1, x2) = FLATTEN_IN_GA(x1) 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: FLATTEN_IN_GA(cons(cons(U, V), W)) -> FLATTEN_IN_GA(cons(U, cons(V, W))) FLATTEN_IN_GA(cons(atom(X), U)) -> FLATTEN_IN_GA(U) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. The following dependency pairs can be deleted: FLATTEN_IN_GA(cons(cons(U, V), W)) -> FLATTEN_IN_GA(cons(U, cons(V, W))) FLATTEN_IN_GA(cons(atom(X), U)) -> FLATTEN_IN_GA(U) No rules are removed from R. Used ordering: POLO with Polynomial interpretation [POLO]: POL(FLATTEN_IN_GA(x_1)) = 2*x_1 POL(atom(x_1)) = x_1 POL(cons(x_1, x_2)) = 1 + 2*x_1 + x_2 ---------------------------------------- (13) Obligation: Q DP problem: P is empty. R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (14) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: COUNT_IN_GA(cons(cons(U, V), W), Z) -> U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X)) U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> COUNT_IN_GA(X, Z) COUNT_IN_GA(cons(atom(X), Y), s(Z)) -> COUNT_IN_GA(Y, Z) The TRS R consists of the following rules: count_in_ga(atom(X), s(0)) -> count_out_ga(atom(X), s(0)) count_in_ga(cons(atom(X), Y), s(Z)) -> U3_ga(X, Y, Z, count_in_ga(Y, Z)) count_in_ga(cons(cons(U, V), W), Z) -> U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X)) flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, [])) flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y)) flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X)) U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X) U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y)) U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> U5_ga(U, V, W, Z, count_in_ga(X, Z)) U5_ga(U, V, W, Z, count_out_ga(X, Z)) -> count_out_ga(cons(cons(U, V), W), Z) U3_ga(X, Y, Z, count_out_ga(Y, Z)) -> count_out_ga(cons(atom(X), Y), s(Z)) The argument filtering Pi contains the following mapping: count_in_ga(x1, x2) = count_in_ga(x1) atom(x1) = atom(x1) count_out_ga(x1, x2) = count_out_ga(x2) cons(x1, x2) = cons(x1, x2) U3_ga(x1, x2, x3, x4) = U3_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5) flatten_in_ga(x1, x2) = flatten_in_ga(x1) flatten_out_ga(x1, x2) = flatten_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) U5_ga(x1, x2, x3, x4, x5) = U5_ga(x5) .(x1, x2) = .(x1, x2) s(x1) = s(x1) COUNT_IN_GA(x1, x2) = COUNT_IN_GA(x1) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: COUNT_IN_GA(cons(cons(U, V), W), Z) -> U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X)) U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) -> COUNT_IN_GA(X, Z) COUNT_IN_GA(cons(atom(X), Y), s(Z)) -> COUNT_IN_GA(Y, Z) The TRS R consists of the following rules: flatten_in_ga(cons(cons(U, V), W), X) -> U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X)) U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) -> flatten_out_ga(cons(cons(U, V), W), X) flatten_in_ga(cons(atom(X), U), .(X, Y)) -> U1_ga(X, U, Y, flatten_in_ga(U, Y)) U1_ga(X, U, Y, flatten_out_ga(U, Y)) -> flatten_out_ga(cons(atom(X), U), .(X, Y)) flatten_in_ga(atom(X), .(X, [])) -> flatten_out_ga(atom(X), .(X, [])) The argument filtering Pi contains the following mapping: atom(x1) = atom(x1) cons(x1, x2) = cons(x1, x2) flatten_in_ga(x1, x2) = flatten_in_ga(x1) flatten_out_ga(x1, x2) = flatten_out_ga(x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) .(x1, x2) = .(x1, x2) s(x1) = s(x1) COUNT_IN_GA(x1, x2) = COUNT_IN_GA(x1) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: COUNT_IN_GA(cons(cons(U, V), W)) -> U4_GA(flatten_in_ga(cons(cons(U, V), W))) U4_GA(flatten_out_ga(X)) -> COUNT_IN_GA(X) COUNT_IN_GA(cons(atom(X), Y)) -> COUNT_IN_GA(Y) The TRS R consists of the following rules: flatten_in_ga(cons(cons(U, V), W)) -> U2_ga(flatten_in_ga(cons(U, cons(V, W)))) U2_ga(flatten_out_ga(X)) -> flatten_out_ga(X) flatten_in_ga(cons(atom(X), U)) -> U1_ga(X, flatten_in_ga(U)) U1_ga(X, flatten_out_ga(Y)) -> flatten_out_ga(.(X, Y)) flatten_in_ga(atom(X)) -> flatten_out_ga(.(X, [])) The set Q consists of the following terms: flatten_in_ga(x0) U2_ga(x0) U1_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. The following dependency pairs can be deleted: COUNT_IN_GA(cons(cons(U, V), W)) -> U4_GA(flatten_in_ga(cons(cons(U, V), W))) COUNT_IN_GA(cons(atom(X), Y)) -> COUNT_IN_GA(Y) The following rules are removed from R: flatten_in_ga(cons(atom(X), U)) -> U1_ga(X, flatten_in_ga(U)) flatten_in_ga(atom(X)) -> flatten_out_ga(.(X, [])) Used ordering: POLO with Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 1 + x_1 + x_2 POL(COUNT_IN_GA(x_1)) = 1 + x_1 POL(U1_ga(x_1, x_2)) = 1 + x_1 + x_2 POL(U2_ga(x_1)) = x_1 POL(U4_GA(x_1)) = x_1 POL([]) = 0 POL(atom(x_1)) = 2 + x_1 POL(cons(x_1, x_2)) = x_1 + x_2 POL(flatten_in_ga(x_1)) = x_1 POL(flatten_out_ga(x_1)) = 1 + x_1 ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GA(flatten_out_ga(X)) -> COUNT_IN_GA(X) The TRS R consists of the following rules: flatten_in_ga(cons(cons(U, V), W)) -> U2_ga(flatten_in_ga(cons(U, cons(V, W)))) U2_ga(flatten_out_ga(X)) -> flatten_out_ga(X) U1_ga(X, flatten_out_ga(Y)) -> flatten_out_ga(.(X, Y)) The set Q consists of the following terms: flatten_in_ga(x0) U2_ga(x0) U1_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (24) TRUE