/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- Graph construction failed Graph construction failed Graph construction failed YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern encode_direct(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) UndefinedPredicateHandlerProof [SOUND, 0 ms] (2) Prolog (3) PrologToPiTRSProof [SOUND, 2 ms] (4) PiTRS (5) DependencyPairsProof [EQUIVALENT, 16 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) AND (9) PiDP (10) UsableRulesProof [EQUIVALENT, 0 ms] (11) PiDP (12) PiDPToQDPProof [SOUND, 19 ms] (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] (15) YES (16) PiDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) PiDP (19) PiDPToQDPProof [SOUND, 0 ms] (20) QDP (21) MRRProof [EQUIVALENT, 153 ms] (22) QDP (23) PisEmptyProof [EQUIVALENT, 0 ms] (24) YES ---------------------------------------- (0) Obligation: Clauses: encode_direct([], []). encode_direct(.(X, Xs), .(Z, Zs)) :- ','(count(X, Xs, Ys, 1, Z), encode_direct(Ys, Zs)). count(X, [], [], 1, X). count(X, [], [], N, .(N, .(X, []))) :- >(N, 1). count(X, .(Y, Ys), .(Y, Ys), 1, X) :- \=(X, Y). count(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) :- ','(>(N, 1), \=(X, Y)). count(X, .(X, Xs), Ys, K, T) :- ','(is(K1, +(K, 1)), count(X, Xs, Ys, K1, T)). Query: encode_direct(g,a) ---------------------------------------- (1) UndefinedPredicateHandlerProof (SOUND) Added facts for all undefined predicates [PROLOG]. ---------------------------------------- (2) Obligation: Clauses: encode_direct([], []). encode_direct(.(X, Xs), .(Z, Zs)) :- ','(count(X, Xs, Ys, 1, Z), encode_direct(Ys, Zs)). count(X, [], [], 1, X). count(X, [], [], N, .(N, .(X, []))) :- >(N, 1). count(X, .(Y, Ys), .(Y, Ys), 1, X) :- \=(X, Y). count(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) :- ','(>(N, 1), \=(X, Y)). count(X, .(X, Xs), Ys, K, T) :- ','(is(K1, +(K, 1)), count(X, Xs, Ys, K1, T)). >(X0, X1). \=(X0, X1). is(X0, X1). Query: encode_direct(g,a) ---------------------------------------- (3) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: encode_direct_in_2: (b,f) count_in_5: (b,b,f,b,f) (b,b,f,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: encode_direct_in_ga([], []) -> encode_direct_out_ga([], []) encode_direct_in_ga(.(X, Xs), .(Z, Zs)) -> U1_ga(X, Xs, Z, Zs, count_in_ggaga(X, Xs, Ys, 1, Z)) count_in_ggaga(X, [], [], 1, X) -> count_out_ggaga(X, [], [], 1, X) count_in_ggaga(X, [], [], N, .(N, .(X, []))) -> U3_ggaga(X, N, >_in_gg(N, 1)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U3_ggaga(X, N, >_out_gg(N, 1)) -> count_out_ggaga(X, [], [], N, .(N, .(X, []))) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaga(X, Y, Ys, \=_in_gg(X, Y)) \=_in_gg(X0, X1) -> \=_out_gg(X0, X1) U4_ggaga(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaga(X, Y, Ys, N, >_in_gg(N, 1)) U5_ggaga(X, Y, Ys, N, >_out_gg(N, 1)) -> U6_ggaga(X, Y, Ys, N, \=_in_gg(X, Y)) U6_ggaga(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) count_in_ggaga(X, .(X, Xs), Ys, K, T) -> U7_ggaga(X, Xs, Ys, K, T, is_in_ag(K1, +(K, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U7_ggaga(X, Xs, Ys, K, T, is_out_ag(K1, +(K, 1))) -> U8_ggaga(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) count_in_ggaaa(X, [], [], 1, X) -> count_out_ggaaa(X, [], [], 1, X) count_in_ggaaa(X, [], [], N, .(N, .(X, []))) -> U3_ggaaa(X, N, >_in_ag(N, 1)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U3_ggaaa(X, N, >_out_ag(N, 1)) -> count_out_ggaaa(X, [], [], N, .(N, .(X, []))) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaaa(X, Y, Ys, \=_in_gg(X, Y)) U4_ggaaa(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaaa(X, Y, Ys, N, >_in_ag(N, 1)) U5_ggaaa(X, Y, Ys, N, >_out_ag(N, 1)) -> U6_ggaaa(X, Y, Ys, N, \=_in_gg(X, Y)) U6_ggaaa(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) count_in_ggaaa(X, .(X, Xs), Ys, K, T) -> U7_ggaaa(X, Xs, Ys, K, T, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U7_ggaaa(X, Xs, Ys, K, T, is_out_aa(K1, +(K, 1))) -> U8_ggaaa(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) U8_ggaaa(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaaa(X, .(X, Xs), Ys, K, T) U8_ggaga(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaga(X, .(X, Xs), Ys, K, T) U1_ga(X, Xs, Z, Zs, count_out_ggaga(X, Xs, Ys, 1, Z)) -> U2_ga(X, Xs, Z, Zs, encode_direct_in_ga(Ys, Zs)) U2_ga(X, Xs, Z, Zs, encode_direct_out_ga(Ys, Zs)) -> encode_direct_out_ga(.(X, Xs), .(Z, Zs)) The argument filtering Pi contains the following mapping: encode_direct_in_ga(x1, x2) = encode_direct_in_ga(x1) [] = [] encode_direct_out_ga(x1, x2) = encode_direct_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) count_in_ggaga(x1, x2, x3, x4, x5) = count_in_ggaga(x1, x2, x4) 1 = 1 count_out_ggaga(x1, x2, x3, x4, x5) = count_out_ggaga(x3) U3_ggaga(x1, x2, x3) = U3_ggaga(x3) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg U4_ggaga(x1, x2, x3, x4) = U4_ggaga(x2, x3, x4) \=_in_gg(x1, x2) = \=_in_gg(x1, x2) \=_out_gg(x1, x2) = \=_out_gg U5_ggaga(x1, x2, x3, x4, x5) = U5_ggaga(x1, x2, x3, x5) U6_ggaga(x1, x2, x3, x4, x5) = U6_ggaga(x2, x3, x5) U7_ggaga(x1, x2, x3, x4, x5, x6) = U7_ggaga(x1, x2, x6) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) U8_ggaga(x1, x2, x3, x4, x5, x6) = U8_ggaga(x6) count_in_ggaaa(x1, x2, x3, x4, x5) = count_in_ggaaa(x1, x2) count_out_ggaaa(x1, x2, x3, x4, x5) = count_out_ggaaa(x3) U3_ggaaa(x1, x2, x3) = U3_ggaaa(x3) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag U4_ggaaa(x1, x2, x3, x4) = U4_ggaaa(x2, x3, x4) U5_ggaaa(x1, x2, x3, x4, x5) = U5_ggaaa(x1, x2, x3, x5) U6_ggaaa(x1, x2, x3, x4, x5) = U6_ggaaa(x2, x3, x5) U7_ggaaa(x1, x2, x3, x4, x5, x6) = U7_ggaaa(x1, x2, x6) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U8_ggaaa(x1, x2, x3, x4, x5, x6) = U8_ggaaa(x6) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (4) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: encode_direct_in_ga([], []) -> encode_direct_out_ga([], []) encode_direct_in_ga(.(X, Xs), .(Z, Zs)) -> U1_ga(X, Xs, Z, Zs, count_in_ggaga(X, Xs, Ys, 1, Z)) count_in_ggaga(X, [], [], 1, X) -> count_out_ggaga(X, [], [], 1, X) count_in_ggaga(X, [], [], N, .(N, .(X, []))) -> U3_ggaga(X, N, >_in_gg(N, 1)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U3_ggaga(X, N, >_out_gg(N, 1)) -> count_out_ggaga(X, [], [], N, .(N, .(X, []))) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaga(X, Y, Ys, \=_in_gg(X, Y)) \=_in_gg(X0, X1) -> \=_out_gg(X0, X1) U4_ggaga(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaga(X, Y, Ys, N, >_in_gg(N, 1)) U5_ggaga(X, Y, Ys, N, >_out_gg(N, 1)) -> U6_ggaga(X, Y, Ys, N, \=_in_gg(X, Y)) U6_ggaga(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) count_in_ggaga(X, .(X, Xs), Ys, K, T) -> U7_ggaga(X, Xs, Ys, K, T, is_in_ag(K1, +(K, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U7_ggaga(X, Xs, Ys, K, T, is_out_ag(K1, +(K, 1))) -> U8_ggaga(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) count_in_ggaaa(X, [], [], 1, X) -> count_out_ggaaa(X, [], [], 1, X) count_in_ggaaa(X, [], [], N, .(N, .(X, []))) -> U3_ggaaa(X, N, >_in_ag(N, 1)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U3_ggaaa(X, N, >_out_ag(N, 1)) -> count_out_ggaaa(X, [], [], N, .(N, .(X, []))) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaaa(X, Y, Ys, \=_in_gg(X, Y)) U4_ggaaa(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaaa(X, Y, Ys, N, >_in_ag(N, 1)) U5_ggaaa(X, Y, Ys, N, >_out_ag(N, 1)) -> U6_ggaaa(X, Y, Ys, N, \=_in_gg(X, Y)) U6_ggaaa(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) count_in_ggaaa(X, .(X, Xs), Ys, K, T) -> U7_ggaaa(X, Xs, Ys, K, T, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U7_ggaaa(X, Xs, Ys, K, T, is_out_aa(K1, +(K, 1))) -> U8_ggaaa(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) U8_ggaaa(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaaa(X, .(X, Xs), Ys, K, T) U8_ggaga(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaga(X, .(X, Xs), Ys, K, T) U1_ga(X, Xs, Z, Zs, count_out_ggaga(X, Xs, Ys, 1, Z)) -> U2_ga(X, Xs, Z, Zs, encode_direct_in_ga(Ys, Zs)) U2_ga(X, Xs, Z, Zs, encode_direct_out_ga(Ys, Zs)) -> encode_direct_out_ga(.(X, Xs), .(Z, Zs)) The argument filtering Pi contains the following mapping: encode_direct_in_ga(x1, x2) = encode_direct_in_ga(x1) [] = [] encode_direct_out_ga(x1, x2) = encode_direct_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) count_in_ggaga(x1, x2, x3, x4, x5) = count_in_ggaga(x1, x2, x4) 1 = 1 count_out_ggaga(x1, x2, x3, x4, x5) = count_out_ggaga(x3) U3_ggaga(x1, x2, x3) = U3_ggaga(x3) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg U4_ggaga(x1, x2, x3, x4) = U4_ggaga(x2, x3, x4) \=_in_gg(x1, x2) = \=_in_gg(x1, x2) \=_out_gg(x1, x2) = \=_out_gg U5_ggaga(x1, x2, x3, x4, x5) = U5_ggaga(x1, x2, x3, x5) U6_ggaga(x1, x2, x3, x4, x5) = U6_ggaga(x2, x3, x5) U7_ggaga(x1, x2, x3, x4, x5, x6) = U7_ggaga(x1, x2, x6) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) U8_ggaga(x1, x2, x3, x4, x5, x6) = U8_ggaga(x6) count_in_ggaaa(x1, x2, x3, x4, x5) = count_in_ggaaa(x1, x2) count_out_ggaaa(x1, x2, x3, x4, x5) = count_out_ggaaa(x3) U3_ggaaa(x1, x2, x3) = U3_ggaaa(x3) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag U4_ggaaa(x1, x2, x3, x4) = U4_ggaaa(x2, x3, x4) U5_ggaaa(x1, x2, x3, x4, x5) = U5_ggaaa(x1, x2, x3, x5) U6_ggaaa(x1, x2, x3, x4, x5) = U6_ggaaa(x2, x3, x5) U7_ggaaa(x1, x2, x3, x4, x5, x6) = U7_ggaaa(x1, x2, x6) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U8_ggaaa(x1, x2, x3, x4, x5, x6) = U8_ggaaa(x6) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) ---------------------------------------- (5) 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: ENCODE_DIRECT_IN_GA(.(X, Xs), .(Z, Zs)) -> U1_GA(X, Xs, Z, Zs, count_in_ggaga(X, Xs, Ys, 1, Z)) ENCODE_DIRECT_IN_GA(.(X, Xs), .(Z, Zs)) -> COUNT_IN_GGAGA(X, Xs, Ys, 1, Z) COUNT_IN_GGAGA(X, [], [], N, .(N, .(X, []))) -> U3_GGAGA(X, N, >_in_gg(N, 1)) COUNT_IN_GGAGA(X, [], [], N, .(N, .(X, []))) -> >_IN_GG(N, 1) COUNT_IN_GGAGA(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_GGAGA(X, Y, Ys, \=_in_gg(X, Y)) COUNT_IN_GGAGA(X, .(Y, Ys), .(Y, Ys), 1, X) -> \=_IN_GG(X, Y) COUNT_IN_GGAGA(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_GGAGA(X, Y, Ys, N, >_in_gg(N, 1)) COUNT_IN_GGAGA(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> >_IN_GG(N, 1) U5_GGAGA(X, Y, Ys, N, >_out_gg(N, 1)) -> U6_GGAGA(X, Y, Ys, N, \=_in_gg(X, Y)) U5_GGAGA(X, Y, Ys, N, >_out_gg(N, 1)) -> \=_IN_GG(X, Y) COUNT_IN_GGAGA(X, .(X, Xs), Ys, K, T) -> U7_GGAGA(X, Xs, Ys, K, T, is_in_ag(K1, +(K, 1))) COUNT_IN_GGAGA(X, .(X, Xs), Ys, K, T) -> IS_IN_AG(K1, +(K, 1)) U7_GGAGA(X, Xs, Ys, K, T, is_out_ag(K1, +(K, 1))) -> U8_GGAGA(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) U7_GGAGA(X, Xs, Ys, K, T, is_out_ag(K1, +(K, 1))) -> COUNT_IN_GGAAA(X, Xs, Ys, K1, T) COUNT_IN_GGAAA(X, [], [], N, .(N, .(X, []))) -> U3_GGAAA(X, N, >_in_ag(N, 1)) COUNT_IN_GGAAA(X, [], [], N, .(N, .(X, []))) -> >_IN_AG(N, 1) COUNT_IN_GGAAA(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_GGAAA(X, Y, Ys, \=_in_gg(X, Y)) COUNT_IN_GGAAA(X, .(Y, Ys), .(Y, Ys), 1, X) -> \=_IN_GG(X, Y) COUNT_IN_GGAAA(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_GGAAA(X, Y, Ys, N, >_in_ag(N, 1)) COUNT_IN_GGAAA(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> >_IN_AG(N, 1) U5_GGAAA(X, Y, Ys, N, >_out_ag(N, 1)) -> U6_GGAAA(X, Y, Ys, N, \=_in_gg(X, Y)) U5_GGAAA(X, Y, Ys, N, >_out_ag(N, 1)) -> \=_IN_GG(X, Y) COUNT_IN_GGAAA(X, .(X, Xs), Ys, K, T) -> U7_GGAAA(X, Xs, Ys, K, T, is_in_aa(K1, +(K, 1))) COUNT_IN_GGAAA(X, .(X, Xs), Ys, K, T) -> IS_IN_AA(K1, +(K, 1)) U7_GGAAA(X, Xs, Ys, K, T, is_out_aa(K1, +(K, 1))) -> U8_GGAAA(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) U7_GGAAA(X, Xs, Ys, K, T, is_out_aa(K1, +(K, 1))) -> COUNT_IN_GGAAA(X, Xs, Ys, K1, T) U1_GA(X, Xs, Z, Zs, count_out_ggaga(X, Xs, Ys, 1, Z)) -> U2_GA(X, Xs, Z, Zs, encode_direct_in_ga(Ys, Zs)) U1_GA(X, Xs, Z, Zs, count_out_ggaga(X, Xs, Ys, 1, Z)) -> ENCODE_DIRECT_IN_GA(Ys, Zs) The TRS R consists of the following rules: encode_direct_in_ga([], []) -> encode_direct_out_ga([], []) encode_direct_in_ga(.(X, Xs), .(Z, Zs)) -> U1_ga(X, Xs, Z, Zs, count_in_ggaga(X, Xs, Ys, 1, Z)) count_in_ggaga(X, [], [], 1, X) -> count_out_ggaga(X, [], [], 1, X) count_in_ggaga(X, [], [], N, .(N, .(X, []))) -> U3_ggaga(X, N, >_in_gg(N, 1)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U3_ggaga(X, N, >_out_gg(N, 1)) -> count_out_ggaga(X, [], [], N, .(N, .(X, []))) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaga(X, Y, Ys, \=_in_gg(X, Y)) \=_in_gg(X0, X1) -> \=_out_gg(X0, X1) U4_ggaga(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaga(X, Y, Ys, N, >_in_gg(N, 1)) U5_ggaga(X, Y, Ys, N, >_out_gg(N, 1)) -> U6_ggaga(X, Y, Ys, N, \=_in_gg(X, Y)) U6_ggaga(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) count_in_ggaga(X, .(X, Xs), Ys, K, T) -> U7_ggaga(X, Xs, Ys, K, T, is_in_ag(K1, +(K, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U7_ggaga(X, Xs, Ys, K, T, is_out_ag(K1, +(K, 1))) -> U8_ggaga(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) count_in_ggaaa(X, [], [], 1, X) -> count_out_ggaaa(X, [], [], 1, X) count_in_ggaaa(X, [], [], N, .(N, .(X, []))) -> U3_ggaaa(X, N, >_in_ag(N, 1)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U3_ggaaa(X, N, >_out_ag(N, 1)) -> count_out_ggaaa(X, [], [], N, .(N, .(X, []))) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaaa(X, Y, Ys, \=_in_gg(X, Y)) U4_ggaaa(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaaa(X, Y, Ys, N, >_in_ag(N, 1)) U5_ggaaa(X, Y, Ys, N, >_out_ag(N, 1)) -> U6_ggaaa(X, Y, Ys, N, \=_in_gg(X, Y)) U6_ggaaa(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) count_in_ggaaa(X, .(X, Xs), Ys, K, T) -> U7_ggaaa(X, Xs, Ys, K, T, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U7_ggaaa(X, Xs, Ys, K, T, is_out_aa(K1, +(K, 1))) -> U8_ggaaa(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) U8_ggaaa(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaaa(X, .(X, Xs), Ys, K, T) U8_ggaga(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaga(X, .(X, Xs), Ys, K, T) U1_ga(X, Xs, Z, Zs, count_out_ggaga(X, Xs, Ys, 1, Z)) -> U2_ga(X, Xs, Z, Zs, encode_direct_in_ga(Ys, Zs)) U2_ga(X, Xs, Z, Zs, encode_direct_out_ga(Ys, Zs)) -> encode_direct_out_ga(.(X, Xs), .(Z, Zs)) The argument filtering Pi contains the following mapping: encode_direct_in_ga(x1, x2) = encode_direct_in_ga(x1) [] = [] encode_direct_out_ga(x1, x2) = encode_direct_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) count_in_ggaga(x1, x2, x3, x4, x5) = count_in_ggaga(x1, x2, x4) 1 = 1 count_out_ggaga(x1, x2, x3, x4, x5) = count_out_ggaga(x3) U3_ggaga(x1, x2, x3) = U3_ggaga(x3) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg U4_ggaga(x1, x2, x3, x4) = U4_ggaga(x2, x3, x4) \=_in_gg(x1, x2) = \=_in_gg(x1, x2) \=_out_gg(x1, x2) = \=_out_gg U5_ggaga(x1, x2, x3, x4, x5) = U5_ggaga(x1, x2, x3, x5) U6_ggaga(x1, x2, x3, x4, x5) = U6_ggaga(x2, x3, x5) U7_ggaga(x1, x2, x3, x4, x5, x6) = U7_ggaga(x1, x2, x6) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) U8_ggaga(x1, x2, x3, x4, x5, x6) = U8_ggaga(x6) count_in_ggaaa(x1, x2, x3, x4, x5) = count_in_ggaaa(x1, x2) count_out_ggaaa(x1, x2, x3, x4, x5) = count_out_ggaaa(x3) U3_ggaaa(x1, x2, x3) = U3_ggaaa(x3) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag U4_ggaaa(x1, x2, x3, x4) = U4_ggaaa(x2, x3, x4) U5_ggaaa(x1, x2, x3, x4, x5) = U5_ggaaa(x1, x2, x3, x5) U6_ggaaa(x1, x2, x3, x4, x5) = U6_ggaaa(x2, x3, x5) U7_ggaaa(x1, x2, x3, x4, x5, x6) = U7_ggaaa(x1, x2, x6) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U8_ggaaa(x1, x2, x3, x4, x5, x6) = U8_ggaaa(x6) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) ENCODE_DIRECT_IN_GA(x1, x2) = ENCODE_DIRECT_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) COUNT_IN_GGAGA(x1, x2, x3, x4, x5) = COUNT_IN_GGAGA(x1, x2, x4) U3_GGAGA(x1, x2, x3) = U3_GGAGA(x3) >_IN_GG(x1, x2) = >_IN_GG(x1, x2) U4_GGAGA(x1, x2, x3, x4) = U4_GGAGA(x2, x3, x4) \=_IN_GG(x1, x2) = \=_IN_GG(x1, x2) U5_GGAGA(x1, x2, x3, x4, x5) = U5_GGAGA(x1, x2, x3, x5) U6_GGAGA(x1, x2, x3, x4, x5) = U6_GGAGA(x2, x3, x5) U7_GGAGA(x1, x2, x3, x4, x5, x6) = U7_GGAGA(x1, x2, x6) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U8_GGAGA(x1, x2, x3, x4, x5, x6) = U8_GGAGA(x6) COUNT_IN_GGAAA(x1, x2, x3, x4, x5) = COUNT_IN_GGAAA(x1, x2) U3_GGAAA(x1, x2, x3) = U3_GGAAA(x3) >_IN_AG(x1, x2) = >_IN_AG(x2) U4_GGAAA(x1, x2, x3, x4) = U4_GGAAA(x2, x3, x4) U5_GGAAA(x1, x2, x3, x4, x5) = U5_GGAAA(x1, x2, x3, x5) U6_GGAAA(x1, x2, x3, x4, x5) = U6_GGAAA(x2, x3, x5) U7_GGAAA(x1, x2, x3, x4, x5, x6) = U7_GGAAA(x1, x2, x6) IS_IN_AA(x1, x2) = IS_IN_AA U8_GGAAA(x1, x2, x3, x4, x5, x6) = U8_GGAAA(x6) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: ENCODE_DIRECT_IN_GA(.(X, Xs), .(Z, Zs)) -> U1_GA(X, Xs, Z, Zs, count_in_ggaga(X, Xs, Ys, 1, Z)) ENCODE_DIRECT_IN_GA(.(X, Xs), .(Z, Zs)) -> COUNT_IN_GGAGA(X, Xs, Ys, 1, Z) COUNT_IN_GGAGA(X, [], [], N, .(N, .(X, []))) -> U3_GGAGA(X, N, >_in_gg(N, 1)) COUNT_IN_GGAGA(X, [], [], N, .(N, .(X, []))) -> >_IN_GG(N, 1) COUNT_IN_GGAGA(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_GGAGA(X, Y, Ys, \=_in_gg(X, Y)) COUNT_IN_GGAGA(X, .(Y, Ys), .(Y, Ys), 1, X) -> \=_IN_GG(X, Y) COUNT_IN_GGAGA(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_GGAGA(X, Y, Ys, N, >_in_gg(N, 1)) COUNT_IN_GGAGA(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> >_IN_GG(N, 1) U5_GGAGA(X, Y, Ys, N, >_out_gg(N, 1)) -> U6_GGAGA(X, Y, Ys, N, \=_in_gg(X, Y)) U5_GGAGA(X, Y, Ys, N, >_out_gg(N, 1)) -> \=_IN_GG(X, Y) COUNT_IN_GGAGA(X, .(X, Xs), Ys, K, T) -> U7_GGAGA(X, Xs, Ys, K, T, is_in_ag(K1, +(K, 1))) COUNT_IN_GGAGA(X, .(X, Xs), Ys, K, T) -> IS_IN_AG(K1, +(K, 1)) U7_GGAGA(X, Xs, Ys, K, T, is_out_ag(K1, +(K, 1))) -> U8_GGAGA(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) U7_GGAGA(X, Xs, Ys, K, T, is_out_ag(K1, +(K, 1))) -> COUNT_IN_GGAAA(X, Xs, Ys, K1, T) COUNT_IN_GGAAA(X, [], [], N, .(N, .(X, []))) -> U3_GGAAA(X, N, >_in_ag(N, 1)) COUNT_IN_GGAAA(X, [], [], N, .(N, .(X, []))) -> >_IN_AG(N, 1) COUNT_IN_GGAAA(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_GGAAA(X, Y, Ys, \=_in_gg(X, Y)) COUNT_IN_GGAAA(X, .(Y, Ys), .(Y, Ys), 1, X) -> \=_IN_GG(X, Y) COUNT_IN_GGAAA(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_GGAAA(X, Y, Ys, N, >_in_ag(N, 1)) COUNT_IN_GGAAA(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> >_IN_AG(N, 1) U5_GGAAA(X, Y, Ys, N, >_out_ag(N, 1)) -> U6_GGAAA(X, Y, Ys, N, \=_in_gg(X, Y)) U5_GGAAA(X, Y, Ys, N, >_out_ag(N, 1)) -> \=_IN_GG(X, Y) COUNT_IN_GGAAA(X, .(X, Xs), Ys, K, T) -> U7_GGAAA(X, Xs, Ys, K, T, is_in_aa(K1, +(K, 1))) COUNT_IN_GGAAA(X, .(X, Xs), Ys, K, T) -> IS_IN_AA(K1, +(K, 1)) U7_GGAAA(X, Xs, Ys, K, T, is_out_aa(K1, +(K, 1))) -> U8_GGAAA(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) U7_GGAAA(X, Xs, Ys, K, T, is_out_aa(K1, +(K, 1))) -> COUNT_IN_GGAAA(X, Xs, Ys, K1, T) U1_GA(X, Xs, Z, Zs, count_out_ggaga(X, Xs, Ys, 1, Z)) -> U2_GA(X, Xs, Z, Zs, encode_direct_in_ga(Ys, Zs)) U1_GA(X, Xs, Z, Zs, count_out_ggaga(X, Xs, Ys, 1, Z)) -> ENCODE_DIRECT_IN_GA(Ys, Zs) The TRS R consists of the following rules: encode_direct_in_ga([], []) -> encode_direct_out_ga([], []) encode_direct_in_ga(.(X, Xs), .(Z, Zs)) -> U1_ga(X, Xs, Z, Zs, count_in_ggaga(X, Xs, Ys, 1, Z)) count_in_ggaga(X, [], [], 1, X) -> count_out_ggaga(X, [], [], 1, X) count_in_ggaga(X, [], [], N, .(N, .(X, []))) -> U3_ggaga(X, N, >_in_gg(N, 1)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U3_ggaga(X, N, >_out_gg(N, 1)) -> count_out_ggaga(X, [], [], N, .(N, .(X, []))) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaga(X, Y, Ys, \=_in_gg(X, Y)) \=_in_gg(X0, X1) -> \=_out_gg(X0, X1) U4_ggaga(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaga(X, Y, Ys, N, >_in_gg(N, 1)) U5_ggaga(X, Y, Ys, N, >_out_gg(N, 1)) -> U6_ggaga(X, Y, Ys, N, \=_in_gg(X, Y)) U6_ggaga(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) count_in_ggaga(X, .(X, Xs), Ys, K, T) -> U7_ggaga(X, Xs, Ys, K, T, is_in_ag(K1, +(K, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U7_ggaga(X, Xs, Ys, K, T, is_out_ag(K1, +(K, 1))) -> U8_ggaga(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) count_in_ggaaa(X, [], [], 1, X) -> count_out_ggaaa(X, [], [], 1, X) count_in_ggaaa(X, [], [], N, .(N, .(X, []))) -> U3_ggaaa(X, N, >_in_ag(N, 1)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U3_ggaaa(X, N, >_out_ag(N, 1)) -> count_out_ggaaa(X, [], [], N, .(N, .(X, []))) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaaa(X, Y, Ys, \=_in_gg(X, Y)) U4_ggaaa(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaaa(X, Y, Ys, N, >_in_ag(N, 1)) U5_ggaaa(X, Y, Ys, N, >_out_ag(N, 1)) -> U6_ggaaa(X, Y, Ys, N, \=_in_gg(X, Y)) U6_ggaaa(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) count_in_ggaaa(X, .(X, Xs), Ys, K, T) -> U7_ggaaa(X, Xs, Ys, K, T, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U7_ggaaa(X, Xs, Ys, K, T, is_out_aa(K1, +(K, 1))) -> U8_ggaaa(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) U8_ggaaa(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaaa(X, .(X, Xs), Ys, K, T) U8_ggaga(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaga(X, .(X, Xs), Ys, K, T) U1_ga(X, Xs, Z, Zs, count_out_ggaga(X, Xs, Ys, 1, Z)) -> U2_ga(X, Xs, Z, Zs, encode_direct_in_ga(Ys, Zs)) U2_ga(X, Xs, Z, Zs, encode_direct_out_ga(Ys, Zs)) -> encode_direct_out_ga(.(X, Xs), .(Z, Zs)) The argument filtering Pi contains the following mapping: encode_direct_in_ga(x1, x2) = encode_direct_in_ga(x1) [] = [] encode_direct_out_ga(x1, x2) = encode_direct_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) count_in_ggaga(x1, x2, x3, x4, x5) = count_in_ggaga(x1, x2, x4) 1 = 1 count_out_ggaga(x1, x2, x3, x4, x5) = count_out_ggaga(x3) U3_ggaga(x1, x2, x3) = U3_ggaga(x3) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg U4_ggaga(x1, x2, x3, x4) = U4_ggaga(x2, x3, x4) \=_in_gg(x1, x2) = \=_in_gg(x1, x2) \=_out_gg(x1, x2) = \=_out_gg U5_ggaga(x1, x2, x3, x4, x5) = U5_ggaga(x1, x2, x3, x5) U6_ggaga(x1, x2, x3, x4, x5) = U6_ggaga(x2, x3, x5) U7_ggaga(x1, x2, x3, x4, x5, x6) = U7_ggaga(x1, x2, x6) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) U8_ggaga(x1, x2, x3, x4, x5, x6) = U8_ggaga(x6) count_in_ggaaa(x1, x2, x3, x4, x5) = count_in_ggaaa(x1, x2) count_out_ggaaa(x1, x2, x3, x4, x5) = count_out_ggaaa(x3) U3_ggaaa(x1, x2, x3) = U3_ggaaa(x3) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag U4_ggaaa(x1, x2, x3, x4) = U4_ggaaa(x2, x3, x4) U5_ggaaa(x1, x2, x3, x4, x5) = U5_ggaaa(x1, x2, x3, x5) U6_ggaaa(x1, x2, x3, x4, x5) = U6_ggaaa(x2, x3, x5) U7_ggaaa(x1, x2, x3, x4, x5, x6) = U7_ggaaa(x1, x2, x6) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U8_ggaaa(x1, x2, x3, x4, x5, x6) = U8_ggaaa(x6) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) ENCODE_DIRECT_IN_GA(x1, x2) = ENCODE_DIRECT_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) COUNT_IN_GGAGA(x1, x2, x3, x4, x5) = COUNT_IN_GGAGA(x1, x2, x4) U3_GGAGA(x1, x2, x3) = U3_GGAGA(x3) >_IN_GG(x1, x2) = >_IN_GG(x1, x2) U4_GGAGA(x1, x2, x3, x4) = U4_GGAGA(x2, x3, x4) \=_IN_GG(x1, x2) = \=_IN_GG(x1, x2) U5_GGAGA(x1, x2, x3, x4, x5) = U5_GGAGA(x1, x2, x3, x5) U6_GGAGA(x1, x2, x3, x4, x5) = U6_GGAGA(x2, x3, x5) U7_GGAGA(x1, x2, x3, x4, x5, x6) = U7_GGAGA(x1, x2, x6) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U8_GGAGA(x1, x2, x3, x4, x5, x6) = U8_GGAGA(x6) COUNT_IN_GGAAA(x1, x2, x3, x4, x5) = COUNT_IN_GGAAA(x1, x2) U3_GGAAA(x1, x2, x3) = U3_GGAAA(x3) >_IN_AG(x1, x2) = >_IN_AG(x2) U4_GGAAA(x1, x2, x3, x4) = U4_GGAAA(x2, x3, x4) U5_GGAAA(x1, x2, x3, x4, x5) = U5_GGAAA(x1, x2, x3, x5) U6_GGAAA(x1, x2, x3, x4, x5) = U6_GGAAA(x2, x3, x5) U7_GGAAA(x1, x2, x3, x4, x5, x6) = U7_GGAAA(x1, x2, x6) IS_IN_AA(x1, x2) = IS_IN_AA U8_GGAAA(x1, x2, x3, x4, x5, x6) = U8_GGAAA(x6) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 24 less nodes. ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: COUNT_IN_GGAAA(X, .(X, Xs), Ys, K, T) -> U7_GGAAA(X, Xs, Ys, K, T, is_in_aa(K1, +(K, 1))) U7_GGAAA(X, Xs, Ys, K, T, is_out_aa(K1, +(K, 1))) -> COUNT_IN_GGAAA(X, Xs, Ys, K1, T) The TRS R consists of the following rules: encode_direct_in_ga([], []) -> encode_direct_out_ga([], []) encode_direct_in_ga(.(X, Xs), .(Z, Zs)) -> U1_ga(X, Xs, Z, Zs, count_in_ggaga(X, Xs, Ys, 1, Z)) count_in_ggaga(X, [], [], 1, X) -> count_out_ggaga(X, [], [], 1, X) count_in_ggaga(X, [], [], N, .(N, .(X, []))) -> U3_ggaga(X, N, >_in_gg(N, 1)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U3_ggaga(X, N, >_out_gg(N, 1)) -> count_out_ggaga(X, [], [], N, .(N, .(X, []))) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaga(X, Y, Ys, \=_in_gg(X, Y)) \=_in_gg(X0, X1) -> \=_out_gg(X0, X1) U4_ggaga(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaga(X, Y, Ys, N, >_in_gg(N, 1)) U5_ggaga(X, Y, Ys, N, >_out_gg(N, 1)) -> U6_ggaga(X, Y, Ys, N, \=_in_gg(X, Y)) U6_ggaga(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) count_in_ggaga(X, .(X, Xs), Ys, K, T) -> U7_ggaga(X, Xs, Ys, K, T, is_in_ag(K1, +(K, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U7_ggaga(X, Xs, Ys, K, T, is_out_ag(K1, +(K, 1))) -> U8_ggaga(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) count_in_ggaaa(X, [], [], 1, X) -> count_out_ggaaa(X, [], [], 1, X) count_in_ggaaa(X, [], [], N, .(N, .(X, []))) -> U3_ggaaa(X, N, >_in_ag(N, 1)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U3_ggaaa(X, N, >_out_ag(N, 1)) -> count_out_ggaaa(X, [], [], N, .(N, .(X, []))) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaaa(X, Y, Ys, \=_in_gg(X, Y)) U4_ggaaa(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaaa(X, Y, Ys, N, >_in_ag(N, 1)) U5_ggaaa(X, Y, Ys, N, >_out_ag(N, 1)) -> U6_ggaaa(X, Y, Ys, N, \=_in_gg(X, Y)) U6_ggaaa(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) count_in_ggaaa(X, .(X, Xs), Ys, K, T) -> U7_ggaaa(X, Xs, Ys, K, T, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U7_ggaaa(X, Xs, Ys, K, T, is_out_aa(K1, +(K, 1))) -> U8_ggaaa(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) U8_ggaaa(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaaa(X, .(X, Xs), Ys, K, T) U8_ggaga(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaga(X, .(X, Xs), Ys, K, T) U1_ga(X, Xs, Z, Zs, count_out_ggaga(X, Xs, Ys, 1, Z)) -> U2_ga(X, Xs, Z, Zs, encode_direct_in_ga(Ys, Zs)) U2_ga(X, Xs, Z, Zs, encode_direct_out_ga(Ys, Zs)) -> encode_direct_out_ga(.(X, Xs), .(Z, Zs)) The argument filtering Pi contains the following mapping: encode_direct_in_ga(x1, x2) = encode_direct_in_ga(x1) [] = [] encode_direct_out_ga(x1, x2) = encode_direct_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) count_in_ggaga(x1, x2, x3, x4, x5) = count_in_ggaga(x1, x2, x4) 1 = 1 count_out_ggaga(x1, x2, x3, x4, x5) = count_out_ggaga(x3) U3_ggaga(x1, x2, x3) = U3_ggaga(x3) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg U4_ggaga(x1, x2, x3, x4) = U4_ggaga(x2, x3, x4) \=_in_gg(x1, x2) = \=_in_gg(x1, x2) \=_out_gg(x1, x2) = \=_out_gg U5_ggaga(x1, x2, x3, x4, x5) = U5_ggaga(x1, x2, x3, x5) U6_ggaga(x1, x2, x3, x4, x5) = U6_ggaga(x2, x3, x5) U7_ggaga(x1, x2, x3, x4, x5, x6) = U7_ggaga(x1, x2, x6) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) U8_ggaga(x1, x2, x3, x4, x5, x6) = U8_ggaga(x6) count_in_ggaaa(x1, x2, x3, x4, x5) = count_in_ggaaa(x1, x2) count_out_ggaaa(x1, x2, x3, x4, x5) = count_out_ggaaa(x3) U3_ggaaa(x1, x2, x3) = U3_ggaaa(x3) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag U4_ggaaa(x1, x2, x3, x4) = U4_ggaaa(x2, x3, x4) U5_ggaaa(x1, x2, x3, x4, x5) = U5_ggaaa(x1, x2, x3, x5) U6_ggaaa(x1, x2, x3, x4, x5) = U6_ggaaa(x2, x3, x5) U7_ggaaa(x1, x2, x3, x4, x5, x6) = U7_ggaaa(x1, x2, x6) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U8_ggaaa(x1, x2, x3, x4, x5, x6) = U8_ggaaa(x6) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) COUNT_IN_GGAAA(x1, x2, x3, x4, x5) = COUNT_IN_GGAAA(x1, x2) U7_GGAAA(x1, x2, x3, x4, x5, x6) = U7_GGAAA(x1, x2, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (11) Obligation: Pi DP problem: The TRS P consists of the following rules: COUNT_IN_GGAAA(X, .(X, Xs), Ys, K, T) -> U7_GGAAA(X, Xs, Ys, K, T, is_in_aa(K1, +(K, 1))) U7_GGAAA(X, Xs, Ys, K, T, is_out_aa(K1, +(K, 1))) -> COUNT_IN_GGAAA(X, Xs, Ys, K1, T) The TRS R consists of the following rules: is_in_aa(X0, X1) -> is_out_aa(X0, X1) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) 1 = 1 +(x1, x2) = +(x1, x2) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa COUNT_IN_GGAAA(x1, x2, x3, x4, x5) = COUNT_IN_GGAAA(x1, x2) U7_GGAAA(x1, x2, x3, x4, x5, x6) = U7_GGAAA(x1, x2, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (12) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: COUNT_IN_GGAAA(X, .(X, Xs)) -> U7_GGAAA(X, Xs, is_in_aa) U7_GGAAA(X, Xs, is_out_aa) -> COUNT_IN_GGAAA(X, Xs) The TRS R consists of the following rules: is_in_aa -> is_out_aa The set Q consists of the following terms: is_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (14) 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: *U7_GGAAA(X, Xs, is_out_aa) -> COUNT_IN_GGAAA(X, Xs) The graph contains the following edges 1 >= 1, 2 >= 2 *COUNT_IN_GGAAA(X, .(X, Xs)) -> U7_GGAAA(X, Xs, is_in_aa) The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2 ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(X, Xs, Z, Zs, count_out_ggaga(X, Xs, Ys, 1, Z)) -> ENCODE_DIRECT_IN_GA(Ys, Zs) ENCODE_DIRECT_IN_GA(.(X, Xs), .(Z, Zs)) -> U1_GA(X, Xs, Z, Zs, count_in_ggaga(X, Xs, Ys, 1, Z)) The TRS R consists of the following rules: encode_direct_in_ga([], []) -> encode_direct_out_ga([], []) encode_direct_in_ga(.(X, Xs), .(Z, Zs)) -> U1_ga(X, Xs, Z, Zs, count_in_ggaga(X, Xs, Ys, 1, Z)) count_in_ggaga(X, [], [], 1, X) -> count_out_ggaga(X, [], [], 1, X) count_in_ggaga(X, [], [], N, .(N, .(X, []))) -> U3_ggaga(X, N, >_in_gg(N, 1)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) U3_ggaga(X, N, >_out_gg(N, 1)) -> count_out_ggaga(X, [], [], N, .(N, .(X, []))) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaga(X, Y, Ys, \=_in_gg(X, Y)) \=_in_gg(X0, X1) -> \=_out_gg(X0, X1) U4_ggaga(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaga(X, Y, Ys, N, >_in_gg(N, 1)) U5_ggaga(X, Y, Ys, N, >_out_gg(N, 1)) -> U6_ggaga(X, Y, Ys, N, \=_in_gg(X, Y)) U6_ggaga(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) count_in_ggaga(X, .(X, Xs), Ys, K, T) -> U7_ggaga(X, Xs, Ys, K, T, is_in_ag(K1, +(K, 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U7_ggaga(X, Xs, Ys, K, T, is_out_ag(K1, +(K, 1))) -> U8_ggaga(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) count_in_ggaaa(X, [], [], 1, X) -> count_out_ggaaa(X, [], [], 1, X) count_in_ggaaa(X, [], [], N, .(N, .(X, []))) -> U3_ggaaa(X, N, >_in_ag(N, 1)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U3_ggaaa(X, N, >_out_ag(N, 1)) -> count_out_ggaaa(X, [], [], N, .(N, .(X, []))) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaaa(X, Y, Ys, \=_in_gg(X, Y)) U4_ggaaa(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaaa(X, Y, Ys, N, >_in_ag(N, 1)) U5_ggaaa(X, Y, Ys, N, >_out_ag(N, 1)) -> U6_ggaaa(X, Y, Ys, N, \=_in_gg(X, Y)) U6_ggaaa(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) count_in_ggaaa(X, .(X, Xs), Ys, K, T) -> U7_ggaaa(X, Xs, Ys, K, T, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U7_ggaaa(X, Xs, Ys, K, T, is_out_aa(K1, +(K, 1))) -> U8_ggaaa(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) U8_ggaaa(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaaa(X, .(X, Xs), Ys, K, T) U8_ggaga(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaga(X, .(X, Xs), Ys, K, T) U1_ga(X, Xs, Z, Zs, count_out_ggaga(X, Xs, Ys, 1, Z)) -> U2_ga(X, Xs, Z, Zs, encode_direct_in_ga(Ys, Zs)) U2_ga(X, Xs, Z, Zs, encode_direct_out_ga(Ys, Zs)) -> encode_direct_out_ga(.(X, Xs), .(Z, Zs)) The argument filtering Pi contains the following mapping: encode_direct_in_ga(x1, x2) = encode_direct_in_ga(x1) [] = [] encode_direct_out_ga(x1, x2) = encode_direct_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) count_in_ggaga(x1, x2, x3, x4, x5) = count_in_ggaga(x1, x2, x4) 1 = 1 count_out_ggaga(x1, x2, x3, x4, x5) = count_out_ggaga(x3) U3_ggaga(x1, x2, x3) = U3_ggaga(x3) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg U4_ggaga(x1, x2, x3, x4) = U4_ggaga(x2, x3, x4) \=_in_gg(x1, x2) = \=_in_gg(x1, x2) \=_out_gg(x1, x2) = \=_out_gg U5_ggaga(x1, x2, x3, x4, x5) = U5_ggaga(x1, x2, x3, x5) U6_ggaga(x1, x2, x3, x4, x5) = U6_ggaga(x2, x3, x5) U7_ggaga(x1, x2, x3, x4, x5, x6) = U7_ggaga(x1, x2, x6) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) U8_ggaga(x1, x2, x3, x4, x5, x6) = U8_ggaga(x6) count_in_ggaaa(x1, x2, x3, x4, x5) = count_in_ggaaa(x1, x2) count_out_ggaaa(x1, x2, x3, x4, x5) = count_out_ggaaa(x3) U3_ggaaa(x1, x2, x3) = U3_ggaaa(x3) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag U4_ggaaa(x1, x2, x3, x4) = U4_ggaaa(x2, x3, x4) U5_ggaaa(x1, x2, x3, x4, x5) = U5_ggaaa(x1, x2, x3, x5) U6_ggaaa(x1, x2, x3, x4, x5) = U6_ggaaa(x2, x3, x5) U7_ggaaa(x1, x2, x3, x4, x5, x6) = U7_ggaaa(x1, x2, x6) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U8_ggaaa(x1, x2, x3, x4, x5, x6) = U8_ggaaa(x6) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) ENCODE_DIRECT_IN_GA(x1, x2) = ENCODE_DIRECT_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_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: U1_GA(X, Xs, Z, Zs, count_out_ggaga(X, Xs, Ys, 1, Z)) -> ENCODE_DIRECT_IN_GA(Ys, Zs) ENCODE_DIRECT_IN_GA(.(X, Xs), .(Z, Zs)) -> U1_GA(X, Xs, Z, Zs, count_in_ggaga(X, Xs, Ys, 1, Z)) The TRS R consists of the following rules: count_in_ggaga(X, [], [], 1, X) -> count_out_ggaga(X, [], [], 1, X) count_in_ggaga(X, [], [], N, .(N, .(X, []))) -> U3_ggaga(X, N, >_in_gg(N, 1)) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaga(X, Y, Ys, \=_in_gg(X, Y)) count_in_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaga(X, Y, Ys, N, >_in_gg(N, 1)) count_in_ggaga(X, .(X, Xs), Ys, K, T) -> U7_ggaga(X, Xs, Ys, K, T, is_in_ag(K1, +(K, 1))) U3_ggaga(X, N, >_out_gg(N, 1)) -> count_out_ggaga(X, [], [], N, .(N, .(X, []))) U4_ggaga(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), 1, X) U5_ggaga(X, Y, Ys, N, >_out_gg(N, 1)) -> U6_ggaga(X, Y, Ys, N, \=_in_gg(X, Y)) U7_ggaga(X, Xs, Ys, K, T, is_out_ag(K1, +(K, 1))) -> U8_ggaga(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) >_in_gg(X0, X1) -> >_out_gg(X0, X1) \=_in_gg(X0, X1) -> \=_out_gg(X0, X1) U6_ggaga(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaga(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U8_ggaga(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaga(X, .(X, Xs), Ys, K, T) count_in_ggaaa(X, [], [], 1, X) -> count_out_ggaaa(X, [], [], 1, X) count_in_ggaaa(X, [], [], N, .(N, .(X, []))) -> U3_ggaaa(X, N, >_in_ag(N, 1)) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) -> U4_ggaaa(X, Y, Ys, \=_in_gg(X, Y)) count_in_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) -> U5_ggaaa(X, Y, Ys, N, >_in_ag(N, 1)) count_in_ggaaa(X, .(X, Xs), Ys, K, T) -> U7_ggaaa(X, Xs, Ys, K, T, is_in_aa(K1, +(K, 1))) U3_ggaaa(X, N, >_out_ag(N, 1)) -> count_out_ggaaa(X, [], [], N, .(N, .(X, []))) U4_ggaaa(X, Y, Ys, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), 1, X) U5_ggaaa(X, Y, Ys, N, >_out_ag(N, 1)) -> U6_ggaaa(X, Y, Ys, N, \=_in_gg(X, Y)) U7_ggaaa(X, Xs, Ys, K, T, is_out_aa(K1, +(K, 1))) -> U8_ggaaa(X, Xs, Ys, K, T, count_in_ggaaa(X, Xs, Ys, K1, T)) >_in_ag(X0, X1) -> >_out_ag(X0, X1) U6_ggaaa(X, Y, Ys, N, \=_out_gg(X, Y)) -> count_out_ggaaa(X, .(Y, Ys), .(Y, Ys), N, .(N, .(X, []))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U8_ggaaa(X, Xs, Ys, K, T, count_out_ggaaa(X, Xs, Ys, K1, T)) -> count_out_ggaaa(X, .(X, Xs), Ys, K, T) The argument filtering Pi contains the following mapping: [] = [] .(x1, x2) = .(x1, x2) count_in_ggaga(x1, x2, x3, x4, x5) = count_in_ggaga(x1, x2, x4) 1 = 1 count_out_ggaga(x1, x2, x3, x4, x5) = count_out_ggaga(x3) U3_ggaga(x1, x2, x3) = U3_ggaga(x3) >_in_gg(x1, x2) = >_in_gg(x1, x2) >_out_gg(x1, x2) = >_out_gg U4_ggaga(x1, x2, x3, x4) = U4_ggaga(x2, x3, x4) \=_in_gg(x1, x2) = \=_in_gg(x1, x2) \=_out_gg(x1, x2) = \=_out_gg U5_ggaga(x1, x2, x3, x4, x5) = U5_ggaga(x1, x2, x3, x5) U6_ggaga(x1, x2, x3, x4, x5) = U6_ggaga(x2, x3, x5) U7_ggaga(x1, x2, x3, x4, x5, x6) = U7_ggaga(x1, x2, x6) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) U8_ggaga(x1, x2, x3, x4, x5, x6) = U8_ggaga(x6) count_in_ggaaa(x1, x2, x3, x4, x5) = count_in_ggaaa(x1, x2) count_out_ggaaa(x1, x2, x3, x4, x5) = count_out_ggaaa(x3) U3_ggaaa(x1, x2, x3) = U3_ggaaa(x3) >_in_ag(x1, x2) = >_in_ag(x2) >_out_ag(x1, x2) = >_out_ag U4_ggaaa(x1, x2, x3, x4) = U4_ggaaa(x2, x3, x4) U5_ggaaa(x1, x2, x3, x4, x5) = U5_ggaaa(x1, x2, x3, x5) U6_ggaaa(x1, x2, x3, x4, x5) = U6_ggaaa(x2, x3, x5) U7_ggaaa(x1, x2, x3, x4, x5, x6) = U7_ggaaa(x1, x2, x6) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U8_ggaaa(x1, x2, x3, x4, x5, x6) = U8_ggaaa(x6) ENCODE_DIRECT_IN_GA(x1, x2) = ENCODE_DIRECT_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_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: U1_GA(count_out_ggaga(Ys)) -> ENCODE_DIRECT_IN_GA(Ys) ENCODE_DIRECT_IN_GA(.(X, Xs)) -> U1_GA(count_in_ggaga(X, Xs, 1)) The TRS R consists of the following rules: count_in_ggaga(X, [], 1) -> count_out_ggaga([]) count_in_ggaga(X, [], N) -> U3_ggaga(>_in_gg(N, 1)) count_in_ggaga(X, .(Y, Ys), 1) -> U4_ggaga(Y, Ys, \=_in_gg(X, Y)) count_in_ggaga(X, .(Y, Ys), N) -> U5_ggaga(X, Y, Ys, >_in_gg(N, 1)) count_in_ggaga(X, .(X, Xs), K) -> U7_ggaga(X, Xs, is_in_ag(+(K, 1))) U3_ggaga(>_out_gg) -> count_out_ggaga([]) U4_ggaga(Y, Ys, \=_out_gg) -> count_out_ggaga(.(Y, Ys)) U5_ggaga(X, Y, Ys, >_out_gg) -> U6_ggaga(Y, Ys, \=_in_gg(X, Y)) U7_ggaga(X, Xs, is_out_ag) -> U8_ggaga(count_in_ggaaa(X, Xs)) >_in_gg(X0, X1) -> >_out_gg \=_in_gg(X0, X1) -> \=_out_gg U6_ggaga(Y, Ys, \=_out_gg) -> count_out_ggaga(.(Y, Ys)) is_in_ag(X1) -> is_out_ag U8_ggaga(count_out_ggaaa(Ys)) -> count_out_ggaga(Ys) count_in_ggaaa(X, []) -> count_out_ggaaa([]) count_in_ggaaa(X, []) -> U3_ggaaa(>_in_ag(1)) count_in_ggaaa(X, .(Y, Ys)) -> U4_ggaaa(Y, Ys, \=_in_gg(X, Y)) count_in_ggaaa(X, .(Y, Ys)) -> U5_ggaaa(X, Y, Ys, >_in_ag(1)) count_in_ggaaa(X, .(X, Xs)) -> U7_ggaaa(X, Xs, is_in_aa) U3_ggaaa(>_out_ag) -> count_out_ggaaa([]) U4_ggaaa(Y, Ys, \=_out_gg) -> count_out_ggaaa(.(Y, Ys)) U5_ggaaa(X, Y, Ys, >_out_ag) -> U6_ggaaa(Y, Ys, \=_in_gg(X, Y)) U7_ggaaa(X, Xs, is_out_aa) -> U8_ggaaa(count_in_ggaaa(X, Xs)) >_in_ag(X1) -> >_out_ag U6_ggaaa(Y, Ys, \=_out_gg) -> count_out_ggaaa(.(Y, Ys)) is_in_aa -> is_out_aa U8_ggaaa(count_out_ggaaa(Ys)) -> count_out_ggaaa(Ys) The set Q consists of the following terms: count_in_ggaga(x0, x1, x2) U3_ggaga(x0) U4_ggaga(x0, x1, x2) U5_ggaga(x0, x1, x2, x3) U7_ggaga(x0, x1, x2) >_in_gg(x0, x1) \=_in_gg(x0, x1) U6_ggaga(x0, x1, x2) is_in_ag(x0) U8_ggaga(x0) count_in_ggaaa(x0, x1) U3_ggaaa(x0) U4_ggaaa(x0, x1, x2) U5_ggaaa(x0, x1, x2, x3) U7_ggaaa(x0, x1, x2) >_in_ag(x0) U6_ggaaa(x0, x1, x2) is_in_aa U8_ggaaa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) 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(count_out_ggaga(Ys)) -> ENCODE_DIRECT_IN_GA(Ys) ENCODE_DIRECT_IN_GA(.(X, Xs)) -> U1_GA(count_in_ggaga(X, Xs, 1)) Strictly oriented rules of the TRS R: count_in_ggaga(X, .(Y, Ys), N) -> U5_ggaga(X, Y, Ys, >_in_gg(N, 1)) count_in_ggaga(X, .(X, Xs), K) -> U7_ggaga(X, Xs, is_in_ag(+(K, 1))) U4_ggaga(Y, Ys, \=_out_gg) -> count_out_ggaga(.(Y, Ys)) count_in_ggaaa(X, .(Y, Ys)) -> U4_ggaaa(Y, Ys, \=_in_gg(X, Y)) count_in_ggaaa(X, .(Y, Ys)) -> U5_ggaaa(X, Y, Ys, >_in_ag(1)) count_in_ggaaa(X, .(X, Xs)) -> U7_ggaaa(X, Xs, is_in_aa) Used ordering: Polynomial interpretation [POLO]: POL(+(x_1, x_2)) = x_1 + x_2 POL(.(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(1) = 0 POL(>_in_ag(x_1)) = x_1 POL(>_in_gg(x_1, x_2)) = x_1 + x_2 POL(>_out_ag) = 0 POL(>_out_gg) = 0 POL(ENCODE_DIRECT_IN_GA(x_1)) = 2*x_1 POL(U1_GA(x_1)) = 1 + 2*x_1 POL(U3_ggaaa(x_1)) = 2*x_1 POL(U3_ggaga(x_1)) = x_1 POL(U4_ggaaa(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 POL(U4_ggaga(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3 POL(U5_ggaaa(x_1, x_2, x_3, x_4)) = 1 + x_1 + 2*x_2 + 2*x_3 + 2*x_4 POL(U5_ggaga(x_1, x_2, x_3, x_4)) = 1 + x_1 + 2*x_2 + 2*x_3 + x_4 POL(U6_ggaaa(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 POL(U6_ggaga(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 POL(U7_ggaaa(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U7_ggaga(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U8_ggaaa(x_1)) = x_1 POL(U8_ggaga(x_1)) = x_1 POL([]) = 0 POL(\=_in_gg(x_1, x_2)) = x_1 + x_2 POL(\=_out_gg) = 0 POL(count_in_ggaaa(x_1, x_2)) = x_1 + 2*x_2 POL(count_in_ggaga(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3 POL(count_out_ggaaa(x_1)) = x_1 POL(count_out_ggaga(x_1)) = x_1 POL(is_in_aa) = 0 POL(is_in_ag(x_1)) = x_1 POL(is_out_aa) = 0 POL(is_out_ag) = 0 ---------------------------------------- (22) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: count_in_ggaga(X, [], 1) -> count_out_ggaga([]) count_in_ggaga(X, [], N) -> U3_ggaga(>_in_gg(N, 1)) count_in_ggaga(X, .(Y, Ys), 1) -> U4_ggaga(Y, Ys, \=_in_gg(X, Y)) U3_ggaga(>_out_gg) -> count_out_ggaga([]) U5_ggaga(X, Y, Ys, >_out_gg) -> U6_ggaga(Y, Ys, \=_in_gg(X, Y)) U7_ggaga(X, Xs, is_out_ag) -> U8_ggaga(count_in_ggaaa(X, Xs)) >_in_gg(X0, X1) -> >_out_gg \=_in_gg(X0, X1) -> \=_out_gg U6_ggaga(Y, Ys, \=_out_gg) -> count_out_ggaga(.(Y, Ys)) is_in_ag(X1) -> is_out_ag U8_ggaga(count_out_ggaaa(Ys)) -> count_out_ggaga(Ys) count_in_ggaaa(X, []) -> count_out_ggaaa([]) count_in_ggaaa(X, []) -> U3_ggaaa(>_in_ag(1)) U3_ggaaa(>_out_ag) -> count_out_ggaaa([]) U4_ggaaa(Y, Ys, \=_out_gg) -> count_out_ggaaa(.(Y, Ys)) U5_ggaaa(X, Y, Ys, >_out_ag) -> U6_ggaaa(Y, Ys, \=_in_gg(X, Y)) U7_ggaaa(X, Xs, is_out_aa) -> U8_ggaaa(count_in_ggaaa(X, Xs)) >_in_ag(X1) -> >_out_ag U6_ggaaa(Y, Ys, \=_out_gg) -> count_out_ggaaa(.(Y, Ys)) is_in_aa -> is_out_aa U8_ggaaa(count_out_ggaaa(Ys)) -> count_out_ggaaa(Ys) The set Q consists of the following terms: count_in_ggaga(x0, x1, x2) U3_ggaga(x0) U4_ggaga(x0, x1, x2) U5_ggaga(x0, x1, x2, x3) U7_ggaga(x0, x1, x2) >_in_gg(x0, x1) \=_in_gg(x0, x1) U6_ggaga(x0, x1, x2) is_in_ag(x0) U8_ggaga(x0) count_in_ggaaa(x0, x1) U3_ggaaa(x0) U4_ggaaa(x0, x1, x2) U5_ggaaa(x0, x1, x2, x3) U7_ggaaa(x0, x1, x2) >_in_ag(x0) U6_ggaaa(x0, x1, x2) is_in_aa U8_ggaaa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (24) YES