/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern sameleaves(g,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 2 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) UsableRulesReductionPairsProof [EQUIVALENT, 31 ms] (27) QDP (28) DependencyGraphProof [EQUIVALENT, 0 ms] (29) TRUE ---------------------------------------- (0) Obligation: Clauses: sameleaves(leaf(L), leaf(L)). sameleaves(tree(T1, T2), tree(S1, S2)) :- ','(getleave(T1, T2, L, T), ','(getleave(S1, S2, L, S), sameleaves(T, S))). getleave(leaf(A), C, A, C). getleave(tree(A, B), C, L, O) :- getleave(A, tree(B, C), L, O). Query: sameleaves(g,g) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: sameleaves_in_2: (b,b) getleave_in_4: (b,b,f,f) (b,b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L)) sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T)) getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C) getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O)) U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O) U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S)) getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C) getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O)) U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O) U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S)) U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2)) The argument filtering Pi contains the following mapping: sameleaves_in_gg(x1, x2) = sameleaves_in_gg(x1, x2) leaf(x1) = leaf(x1) sameleaves_out_gg(x1, x2) = sameleaves_out_gg tree(x1, x2) = tree(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x3, x4, x5) getleave_in_ggaa(x1, x2, x3, x4) = getleave_in_ggaa(x1, x2) getleave_out_ggaa(x1, x2, x3, x4) = getleave_out_ggaa(x3, x4) U4_ggaa(x1, x2, x3, x4, x5, x6) = U4_ggaa(x6) U2_gg(x1, x2, x3, x4, x5, x6) = U2_gg(x5, x6) getleave_in_ggga(x1, x2, x3, x4) = getleave_in_ggga(x1, x2, x3) getleave_out_ggga(x1, x2, x3, x4) = getleave_out_ggga(x4) U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6) U3_gg(x1, x2, x3, x4, x5) = U3_gg(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: sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L)) sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T)) getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C) getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O)) U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O) U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S)) getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C) getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O)) U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O) U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S)) U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2)) The argument filtering Pi contains the following mapping: sameleaves_in_gg(x1, x2) = sameleaves_in_gg(x1, x2) leaf(x1) = leaf(x1) sameleaves_out_gg(x1, x2) = sameleaves_out_gg tree(x1, x2) = tree(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x3, x4, x5) getleave_in_ggaa(x1, x2, x3, x4) = getleave_in_ggaa(x1, x2) getleave_out_ggaa(x1, x2, x3, x4) = getleave_out_ggaa(x3, x4) U4_ggaa(x1, x2, x3, x4, x5, x6) = U4_ggaa(x6) U2_gg(x1, x2, x3, x4, x5, x6) = U2_gg(x5, x6) getleave_in_ggga(x1, x2, x3, x4) = getleave_in_ggga(x1, x2, x3) getleave_out_ggga(x1, x2, x3, x4) = getleave_out_ggga(x4) U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6) U3_gg(x1, x2, x3, x4, x5) = U3_gg(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: SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> U1_GG(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T)) SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> GETLEAVE_IN_GGAA(T1, T2, L, T) GETLEAVE_IN_GGAA(tree(A, B), C, L, O) -> U4_GGAA(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O)) GETLEAVE_IN_GGAA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGAA(A, tree(B, C), L, O) U1_GG(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_GG(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S)) U1_GG(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> GETLEAVE_IN_GGGA(S1, S2, L, S) GETLEAVE_IN_GGGA(tree(A, B), C, L, O) -> U4_GGGA(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O)) GETLEAVE_IN_GGGA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGGA(A, tree(B, C), L, O) U2_GG(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_GG(T1, T2, S1, S2, sameleaves_in_gg(T, S)) U2_GG(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> SAMELEAVES_IN_GG(T, S) The TRS R consists of the following rules: sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L)) sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T)) getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C) getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O)) U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O) U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S)) getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C) getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O)) U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O) U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S)) U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2)) The argument filtering Pi contains the following mapping: sameleaves_in_gg(x1, x2) = sameleaves_in_gg(x1, x2) leaf(x1) = leaf(x1) sameleaves_out_gg(x1, x2) = sameleaves_out_gg tree(x1, x2) = tree(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x3, x4, x5) getleave_in_ggaa(x1, x2, x3, x4) = getleave_in_ggaa(x1, x2) getleave_out_ggaa(x1, x2, x3, x4) = getleave_out_ggaa(x3, x4) U4_ggaa(x1, x2, x3, x4, x5, x6) = U4_ggaa(x6) U2_gg(x1, x2, x3, x4, x5, x6) = U2_gg(x5, x6) getleave_in_ggga(x1, x2, x3, x4) = getleave_in_ggga(x1, x2, x3) getleave_out_ggga(x1, x2, x3, x4) = getleave_out_ggga(x4) U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x5) SAMELEAVES_IN_GG(x1, x2) = SAMELEAVES_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x3, x4, x5) GETLEAVE_IN_GGAA(x1, x2, x3, x4) = GETLEAVE_IN_GGAA(x1, x2) U4_GGAA(x1, x2, x3, x4, x5, x6) = U4_GGAA(x6) U2_GG(x1, x2, x3, x4, x5, x6) = U2_GG(x5, x6) GETLEAVE_IN_GGGA(x1, x2, x3, x4) = GETLEAVE_IN_GGGA(x1, x2, x3) U4_GGGA(x1, x2, x3, x4, x5, x6) = U4_GGGA(x6) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> U1_GG(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T)) SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> GETLEAVE_IN_GGAA(T1, T2, L, T) GETLEAVE_IN_GGAA(tree(A, B), C, L, O) -> U4_GGAA(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O)) GETLEAVE_IN_GGAA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGAA(A, tree(B, C), L, O) U1_GG(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_GG(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S)) U1_GG(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> GETLEAVE_IN_GGGA(S1, S2, L, S) GETLEAVE_IN_GGGA(tree(A, B), C, L, O) -> U4_GGGA(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O)) GETLEAVE_IN_GGGA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGGA(A, tree(B, C), L, O) U2_GG(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_GG(T1, T2, S1, S2, sameleaves_in_gg(T, S)) U2_GG(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> SAMELEAVES_IN_GG(T, S) The TRS R consists of the following rules: sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L)) sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T)) getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C) getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O)) U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O) U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S)) getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C) getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O)) U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O) U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S)) U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2)) The argument filtering Pi contains the following mapping: sameleaves_in_gg(x1, x2) = sameleaves_in_gg(x1, x2) leaf(x1) = leaf(x1) sameleaves_out_gg(x1, x2) = sameleaves_out_gg tree(x1, x2) = tree(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x3, x4, x5) getleave_in_ggaa(x1, x2, x3, x4) = getleave_in_ggaa(x1, x2) getleave_out_ggaa(x1, x2, x3, x4) = getleave_out_ggaa(x3, x4) U4_ggaa(x1, x2, x3, x4, x5, x6) = U4_ggaa(x6) U2_gg(x1, x2, x3, x4, x5, x6) = U2_gg(x5, x6) getleave_in_ggga(x1, x2, x3, x4) = getleave_in_ggga(x1, x2, x3) getleave_out_ggga(x1, x2, x3, x4) = getleave_out_ggga(x4) U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x5) SAMELEAVES_IN_GG(x1, x2) = SAMELEAVES_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x3, x4, x5) GETLEAVE_IN_GGAA(x1, x2, x3, x4) = GETLEAVE_IN_GGAA(x1, x2) U4_GGAA(x1, x2, x3, x4, x5, x6) = U4_GGAA(x6) U2_GG(x1, x2, x3, x4, x5, x6) = U2_GG(x5, x6) GETLEAVE_IN_GGGA(x1, x2, x3, x4) = GETLEAVE_IN_GGGA(x1, x2, x3) U4_GGGA(x1, x2, x3, x4, x5, x6) = U4_GGGA(x6) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x5) 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: GETLEAVE_IN_GGGA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGGA(A, tree(B, C), L, O) The TRS R consists of the following rules: sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L)) sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T)) getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C) getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O)) U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O) U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S)) getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C) getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O)) U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O) U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S)) U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2)) The argument filtering Pi contains the following mapping: sameleaves_in_gg(x1, x2) = sameleaves_in_gg(x1, x2) leaf(x1) = leaf(x1) sameleaves_out_gg(x1, x2) = sameleaves_out_gg tree(x1, x2) = tree(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x3, x4, x5) getleave_in_ggaa(x1, x2, x3, x4) = getleave_in_ggaa(x1, x2) getleave_out_ggaa(x1, x2, x3, x4) = getleave_out_ggaa(x3, x4) U4_ggaa(x1, x2, x3, x4, x5, x6) = U4_ggaa(x6) U2_gg(x1, x2, x3, x4, x5, x6) = U2_gg(x5, x6) getleave_in_ggga(x1, x2, x3, x4) = getleave_in_ggga(x1, x2, x3) getleave_out_ggga(x1, x2, x3, x4) = getleave_out_ggga(x4) U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x5) GETLEAVE_IN_GGGA(x1, x2, x3, x4) = GETLEAVE_IN_GGGA(x1, x2, x3) 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: GETLEAVE_IN_GGGA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGGA(A, tree(B, C), L, O) R is empty. The argument filtering Pi contains the following mapping: tree(x1, x2) = tree(x1, x2) GETLEAVE_IN_GGGA(x1, x2, x3, x4) = GETLEAVE_IN_GGGA(x1, x2, x3) 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: GETLEAVE_IN_GGGA(tree(A, B), C, L) -> GETLEAVE_IN_GGGA(A, tree(B, C), L) 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: *GETLEAVE_IN_GGGA(tree(A, B), C, L) -> GETLEAVE_IN_GGGA(A, tree(B, C), L) The graph contains the following edges 1 > 1, 3 >= 3 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: GETLEAVE_IN_GGAA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGAA(A, tree(B, C), L, O) The TRS R consists of the following rules: sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L)) sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T)) getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C) getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O)) U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O) U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S)) getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C) getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O)) U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O) U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S)) U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2)) The argument filtering Pi contains the following mapping: sameleaves_in_gg(x1, x2) = sameleaves_in_gg(x1, x2) leaf(x1) = leaf(x1) sameleaves_out_gg(x1, x2) = sameleaves_out_gg tree(x1, x2) = tree(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x3, x4, x5) getleave_in_ggaa(x1, x2, x3, x4) = getleave_in_ggaa(x1, x2) getleave_out_ggaa(x1, x2, x3, x4) = getleave_out_ggaa(x3, x4) U4_ggaa(x1, x2, x3, x4, x5, x6) = U4_ggaa(x6) U2_gg(x1, x2, x3, x4, x5, x6) = U2_gg(x5, x6) getleave_in_ggga(x1, x2, x3, x4) = getleave_in_ggga(x1, x2, x3) getleave_out_ggga(x1, x2, x3, x4) = getleave_out_ggga(x4) U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x5) GETLEAVE_IN_GGAA(x1, x2, x3, x4) = GETLEAVE_IN_GGAA(x1, x2) 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: GETLEAVE_IN_GGAA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGAA(A, tree(B, C), L, O) R is empty. The argument filtering Pi contains the following mapping: tree(x1, x2) = tree(x1, x2) GETLEAVE_IN_GGAA(x1, x2, x3, x4) = GETLEAVE_IN_GGAA(x1, x2) 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: GETLEAVE_IN_GGAA(tree(A, B), C) -> GETLEAVE_IN_GGAA(A, tree(B, C)) 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: *GETLEAVE_IN_GGAA(tree(A, B), C) -> GETLEAVE_IN_GGAA(A, tree(B, C)) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GG(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_GG(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S)) U2_GG(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> SAMELEAVES_IN_GG(T, S) SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> U1_GG(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T)) The TRS R consists of the following rules: sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L)) sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T)) getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C) getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O)) U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O) U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S)) getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C) getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O)) U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O) U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S)) U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2)) The argument filtering Pi contains the following mapping: sameleaves_in_gg(x1, x2) = sameleaves_in_gg(x1, x2) leaf(x1) = leaf(x1) sameleaves_out_gg(x1, x2) = sameleaves_out_gg tree(x1, x2) = tree(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x3, x4, x5) getleave_in_ggaa(x1, x2, x3, x4) = getleave_in_ggaa(x1, x2) getleave_out_ggaa(x1, x2, x3, x4) = getleave_out_ggaa(x3, x4) U4_ggaa(x1, x2, x3, x4, x5, x6) = U4_ggaa(x6) U2_gg(x1, x2, x3, x4, x5, x6) = U2_gg(x5, x6) getleave_in_ggga(x1, x2, x3, x4) = getleave_in_ggga(x1, x2, x3) getleave_out_ggga(x1, x2, x3, x4) = getleave_out_ggga(x4) U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x5) SAMELEAVES_IN_GG(x1, x2) = SAMELEAVES_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x3, x4, x5) U2_GG(x1, x2, x3, x4, x5, x6) = U2_GG(x5, 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: U1_GG(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_GG(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S)) U2_GG(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> SAMELEAVES_IN_GG(T, S) SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> U1_GG(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T)) The TRS R consists of the following rules: getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C) getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O)) getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C) getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O)) U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O) U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O) The argument filtering Pi contains the following mapping: leaf(x1) = leaf(x1) tree(x1, x2) = tree(x1, x2) getleave_in_ggaa(x1, x2, x3, x4) = getleave_in_ggaa(x1, x2) getleave_out_ggaa(x1, x2, x3, x4) = getleave_out_ggaa(x3, x4) U4_ggaa(x1, x2, x3, x4, x5, x6) = U4_ggaa(x6) getleave_in_ggga(x1, x2, x3, x4) = getleave_in_ggga(x1, x2, x3) getleave_out_ggga(x1, x2, x3, x4) = getleave_out_ggga(x4) U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x6) SAMELEAVES_IN_GG(x1, x2) = SAMELEAVES_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x3, x4, x5) U2_GG(x1, x2, x3, x4, x5, x6) = U2_GG(x5, 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: U1_GG(S1, S2, getleave_out_ggaa(L, T)) -> U2_GG(T, getleave_in_ggga(S1, S2, L)) U2_GG(T, getleave_out_ggga(S)) -> SAMELEAVES_IN_GG(T, S) SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> U1_GG(S1, S2, getleave_in_ggaa(T1, T2)) The TRS R consists of the following rules: getleave_in_ggga(leaf(A), C, A) -> getleave_out_ggga(C) getleave_in_ggga(tree(A, B), C, L) -> U4_ggga(getleave_in_ggga(A, tree(B, C), L)) getleave_in_ggaa(leaf(A), C) -> getleave_out_ggaa(A, C) getleave_in_ggaa(tree(A, B), C) -> U4_ggaa(getleave_in_ggaa(A, tree(B, C))) U4_ggga(getleave_out_ggga(O)) -> getleave_out_ggga(O) U4_ggaa(getleave_out_ggaa(L, O)) -> getleave_out_ggaa(L, O) The set Q consists of the following terms: getleave_in_ggga(x0, x1, x2) getleave_in_ggaa(x0, x1) U4_ggga(x0) U4_ggaa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) 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: SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> U1_GG(S1, S2, getleave_in_ggaa(T1, T2)) The following rules are removed from R: getleave_in_ggga(leaf(A), C, A) -> getleave_out_ggga(C) getleave_in_ggaa(leaf(A), C) -> getleave_out_ggaa(A, C) Used ordering: POLO with Polynomial interpretation [POLO]: POL(SAMELEAVES_IN_GG(x_1, x_2)) = 2*x_1 + x_2 POL(U1_GG(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3 POL(U2_GG(x_1, x_2)) = 2*x_1 + x_2 POL(U4_ggaa(x_1)) = x_1 POL(U4_ggga(x_1)) = x_1 POL(getleave_in_ggaa(x_1, x_2)) = x_1 + x_2 POL(getleave_in_ggga(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3 POL(getleave_out_ggaa(x_1, x_2)) = 2*x_1 + x_2 POL(getleave_out_ggga(x_1)) = x_1 POL(leaf(x_1)) = 2*x_1 POL(tree(x_1, x_2)) = 1 + x_1 + x_2 ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GG(S1, S2, getleave_out_ggaa(L, T)) -> U2_GG(T, getleave_in_ggga(S1, S2, L)) U2_GG(T, getleave_out_ggga(S)) -> SAMELEAVES_IN_GG(T, S) The TRS R consists of the following rules: getleave_in_ggaa(tree(A, B), C) -> U4_ggaa(getleave_in_ggaa(A, tree(B, C))) U4_ggaa(getleave_out_ggaa(L, O)) -> getleave_out_ggaa(L, O) getleave_in_ggga(tree(A, B), C, L) -> U4_ggga(getleave_in_ggga(A, tree(B, C), L)) U4_ggga(getleave_out_ggga(O)) -> getleave_out_ggga(O) The set Q consists of the following terms: getleave_in_ggga(x0, x1, x2) getleave_in_ggaa(x0, x1) U4_ggga(x0) U4_ggaa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (28) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (29) TRUE