/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern search_tree(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [EQUIVALENT, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 1 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES ---------------------------------------- (0) Obligation: Clauses: search_tree(void). search_tree(T) :- search_tree(T, X1, X2). search_tree(tree(X, void, void), X, X). search_tree(tree(X, void, Right), X, Max) :- ','(search_tree(Right, Min, Max), less(X, Min)). search_tree(tree(X, Left, void), Min, X) :- ','(search_tree(Left, Min, Max), less(Max, X)). search_tree(tree(X, Left, Right), Min1, Max2) :- ','(search_tree(Left, Min1, Max1), ','(less(Max1, X), ','(search_tree(Right, Min2, Max2), less(X, Min2)))). less(0, s(X3)). less(s(X), s(Y)) :- less(X, Y). Query: search_tree(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: search_tree_in_1: (b) search_tree_in_3: (b,f,f) less_in_2: (b,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: search_tree_in_g(void) -> search_tree_out_g(void) search_tree_in_g(T) -> U1_g(T, search_tree_in_gaa(T, X1, X2)) search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X) search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max)) search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max)) search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1)) U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y)) U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2)) U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2)) U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2) U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X)) U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X) U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min)) U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max) U1_g(T, search_tree_out_gaa(T, X1, X2)) -> search_tree_out_g(T) The argument filtering Pi contains the following mapping: search_tree_in_g(x1) = search_tree_in_g(x1) void = void search_tree_out_g(x1) = search_tree_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) search_tree_in_gaa(x1, x2, x3) = search_tree_in_gaa(x1) tree(x1, x2, x3) = tree(x1, x2, x3) search_tree_out_gaa(x1, x2, x3) = search_tree_out_gaa(x1, x2, x3) U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x2, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x2, x4) U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x1, x2, x3, x6) U7_gaa(x1, x2, x3, x4, x5, x6, x7) = U7_gaa(x1, x2, x3, x4, x7) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 s(x1) = s(x1) less_out_gg(x1, x2) = less_out_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x1, x2, x3) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x4, x6) U9_gaa(x1, x2, x3, x4, x5, x6) = U9_gaa(x1, x2, x3, x4, x5, x6) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x2, x3, x4) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x2, x3, x4) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: search_tree_in_g(void) -> search_tree_out_g(void) search_tree_in_g(T) -> U1_g(T, search_tree_in_gaa(T, X1, X2)) search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X) search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max)) search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max)) search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1)) U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y)) U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2)) U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2)) U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2) U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X)) U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X) U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min)) U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max) U1_g(T, search_tree_out_gaa(T, X1, X2)) -> search_tree_out_g(T) The argument filtering Pi contains the following mapping: search_tree_in_g(x1) = search_tree_in_g(x1) void = void search_tree_out_g(x1) = search_tree_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) search_tree_in_gaa(x1, x2, x3) = search_tree_in_gaa(x1) tree(x1, x2, x3) = tree(x1, x2, x3) search_tree_out_gaa(x1, x2, x3) = search_tree_out_gaa(x1, x2, x3) U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x2, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x2, x4) U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x1, x2, x3, x6) U7_gaa(x1, x2, x3, x4, x5, x6, x7) = U7_gaa(x1, x2, x3, x4, x7) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 s(x1) = s(x1) less_out_gg(x1, x2) = less_out_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x1, x2, x3) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x4, x6) U9_gaa(x1, x2, x3, x4, x5, x6) = U9_gaa(x1, x2, x3, x4, x5, x6) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x2, x3, x4) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x2, x3, x4) ---------------------------------------- (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: SEARCH_TREE_IN_G(T) -> U1_G(T, search_tree_in_gaa(T, X1, X2)) SEARCH_TREE_IN_G(T) -> SEARCH_TREE_IN_GAA(T, X1, X2) SEARCH_TREE_IN_GAA(tree(X, void, Right), X, Max) -> U2_GAA(X, Right, Max, search_tree_in_gaa(Right, Min, Max)) SEARCH_TREE_IN_GAA(tree(X, void, Right), X, Max) -> SEARCH_TREE_IN_GAA(Right, Min, Max) SEARCH_TREE_IN_GAA(tree(X, Left, void), Min, X) -> U4_GAA(X, Left, Min, search_tree_in_gaa(Left, Min, Max)) SEARCH_TREE_IN_GAA(tree(X, Left, void), Min, X) -> SEARCH_TREE_IN_GAA(Left, Min, Max) SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> U6_GAA(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1)) SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> SEARCH_TREE_IN_GAA(Left, Min1, Max1) U6_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_GAA(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X)) U6_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> LESS_IN_GG(Max1, X) LESS_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, less_in_gg(X, Y)) LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) U7_GAA(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_GAA(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2)) U7_GAA(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> SEARCH_TREE_IN_GAA(Right, Min2, Max2) U8_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_GAA(X, Left, Right, Min1, Max2, less_in_gg(X, Min2)) U8_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> LESS_IN_GG(X, Min2) U4_GAA(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_GAA(X, Left, Min, less_in_gg(Max, X)) U4_GAA(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> LESS_IN_GG(Max, X) U2_GAA(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_GAA(X, Right, Max, less_in_gg(X, Min)) U2_GAA(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> LESS_IN_GG(X, Min) The TRS R consists of the following rules: search_tree_in_g(void) -> search_tree_out_g(void) search_tree_in_g(T) -> U1_g(T, search_tree_in_gaa(T, X1, X2)) search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X) search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max)) search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max)) search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1)) U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y)) U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2)) U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2)) U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2) U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X)) U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X) U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min)) U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max) U1_g(T, search_tree_out_gaa(T, X1, X2)) -> search_tree_out_g(T) The argument filtering Pi contains the following mapping: search_tree_in_g(x1) = search_tree_in_g(x1) void = void search_tree_out_g(x1) = search_tree_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) search_tree_in_gaa(x1, x2, x3) = search_tree_in_gaa(x1) tree(x1, x2, x3) = tree(x1, x2, x3) search_tree_out_gaa(x1, x2, x3) = search_tree_out_gaa(x1, x2, x3) U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x2, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x2, x4) U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x1, x2, x3, x6) U7_gaa(x1, x2, x3, x4, x5, x6, x7) = U7_gaa(x1, x2, x3, x4, x7) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 s(x1) = s(x1) less_out_gg(x1, x2) = less_out_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x1, x2, x3) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x4, x6) U9_gaa(x1, x2, x3, x4, x5, x6) = U9_gaa(x1, x2, x3, x4, x5, x6) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x2, x3, x4) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x2, x3, x4) SEARCH_TREE_IN_G(x1) = SEARCH_TREE_IN_G(x1) U1_G(x1, x2) = U1_G(x1, x2) SEARCH_TREE_IN_GAA(x1, x2, x3) = SEARCH_TREE_IN_GAA(x1) U2_GAA(x1, x2, x3, x4) = U2_GAA(x1, x2, x4) U4_GAA(x1, x2, x3, x4) = U4_GAA(x1, x2, x4) U6_GAA(x1, x2, x3, x4, x5, x6) = U6_GAA(x1, x2, x3, x6) U7_GAA(x1, x2, x3, x4, x5, x6, x7) = U7_GAA(x1, x2, x3, x4, x7) LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) U10_GG(x1, x2, x3) = U10_GG(x1, x2, x3) U8_GAA(x1, x2, x3, x4, x5, x6) = U8_GAA(x1, x2, x3, x4, x6) U9_GAA(x1, x2, x3, x4, x5, x6) = U9_GAA(x1, x2, x3, x4, x5, x6) U5_GAA(x1, x2, x3, x4) = U5_GAA(x1, x2, x3, x4) U3_GAA(x1, x2, x3, x4) = U3_GAA(x1, x2, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: SEARCH_TREE_IN_G(T) -> U1_G(T, search_tree_in_gaa(T, X1, X2)) SEARCH_TREE_IN_G(T) -> SEARCH_TREE_IN_GAA(T, X1, X2) SEARCH_TREE_IN_GAA(tree(X, void, Right), X, Max) -> U2_GAA(X, Right, Max, search_tree_in_gaa(Right, Min, Max)) SEARCH_TREE_IN_GAA(tree(X, void, Right), X, Max) -> SEARCH_TREE_IN_GAA(Right, Min, Max) SEARCH_TREE_IN_GAA(tree(X, Left, void), Min, X) -> U4_GAA(X, Left, Min, search_tree_in_gaa(Left, Min, Max)) SEARCH_TREE_IN_GAA(tree(X, Left, void), Min, X) -> SEARCH_TREE_IN_GAA(Left, Min, Max) SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> U6_GAA(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1)) SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> SEARCH_TREE_IN_GAA(Left, Min1, Max1) U6_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_GAA(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X)) U6_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> LESS_IN_GG(Max1, X) LESS_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, less_in_gg(X, Y)) LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) U7_GAA(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_GAA(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2)) U7_GAA(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> SEARCH_TREE_IN_GAA(Right, Min2, Max2) U8_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_GAA(X, Left, Right, Min1, Max2, less_in_gg(X, Min2)) U8_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> LESS_IN_GG(X, Min2) U4_GAA(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_GAA(X, Left, Min, less_in_gg(Max, X)) U4_GAA(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> LESS_IN_GG(Max, X) U2_GAA(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_GAA(X, Right, Max, less_in_gg(X, Min)) U2_GAA(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> LESS_IN_GG(X, Min) The TRS R consists of the following rules: search_tree_in_g(void) -> search_tree_out_g(void) search_tree_in_g(T) -> U1_g(T, search_tree_in_gaa(T, X1, X2)) search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X) search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max)) search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max)) search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1)) U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y)) U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2)) U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2)) U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2) U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X)) U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X) U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min)) U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max) U1_g(T, search_tree_out_gaa(T, X1, X2)) -> search_tree_out_g(T) The argument filtering Pi contains the following mapping: search_tree_in_g(x1) = search_tree_in_g(x1) void = void search_tree_out_g(x1) = search_tree_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) search_tree_in_gaa(x1, x2, x3) = search_tree_in_gaa(x1) tree(x1, x2, x3) = tree(x1, x2, x3) search_tree_out_gaa(x1, x2, x3) = search_tree_out_gaa(x1, x2, x3) U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x2, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x2, x4) U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x1, x2, x3, x6) U7_gaa(x1, x2, x3, x4, x5, x6, x7) = U7_gaa(x1, x2, x3, x4, x7) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 s(x1) = s(x1) less_out_gg(x1, x2) = less_out_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x1, x2, x3) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x4, x6) U9_gaa(x1, x2, x3, x4, x5, x6) = U9_gaa(x1, x2, x3, x4, x5, x6) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x2, x3, x4) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x2, x3, x4) SEARCH_TREE_IN_G(x1) = SEARCH_TREE_IN_G(x1) U1_G(x1, x2) = U1_G(x1, x2) SEARCH_TREE_IN_GAA(x1, x2, x3) = SEARCH_TREE_IN_GAA(x1) U2_GAA(x1, x2, x3, x4) = U2_GAA(x1, x2, x4) U4_GAA(x1, x2, x3, x4) = U4_GAA(x1, x2, x4) U6_GAA(x1, x2, x3, x4, x5, x6) = U6_GAA(x1, x2, x3, x6) U7_GAA(x1, x2, x3, x4, x5, x6, x7) = U7_GAA(x1, x2, x3, x4, x7) LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) U10_GG(x1, x2, x3) = U10_GG(x1, x2, x3) U8_GAA(x1, x2, x3, x4, x5, x6) = U8_GAA(x1, x2, x3, x4, x6) U9_GAA(x1, x2, x3, x4, x5, x6) = U9_GAA(x1, x2, x3, x4, x5, x6) U5_GAA(x1, x2, x3, x4) = U5_GAA(x1, x2, x3, x4) U3_GAA(x1, x2, x3, x4) = U3_GAA(x1, x2, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 13 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) The TRS R consists of the following rules: search_tree_in_g(void) -> search_tree_out_g(void) search_tree_in_g(T) -> U1_g(T, search_tree_in_gaa(T, X1, X2)) search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X) search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max)) search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max)) search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1)) U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y)) U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2)) U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2)) U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2) U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X)) U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X) U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min)) U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max) U1_g(T, search_tree_out_gaa(T, X1, X2)) -> search_tree_out_g(T) The argument filtering Pi contains the following mapping: search_tree_in_g(x1) = search_tree_in_g(x1) void = void search_tree_out_g(x1) = search_tree_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) search_tree_in_gaa(x1, x2, x3) = search_tree_in_gaa(x1) tree(x1, x2, x3) = tree(x1, x2, x3) search_tree_out_gaa(x1, x2, x3) = search_tree_out_gaa(x1, x2, x3) U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x2, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x2, x4) U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x1, x2, x3, x6) U7_gaa(x1, x2, x3, x4, x5, x6, x7) = U7_gaa(x1, x2, x3, x4, x7) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 s(x1) = s(x1) less_out_gg(x1, x2) = less_out_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x1, x2, x3) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x4, x6) U9_gaa(x1, x2, x3, x4, x5, x6) = U9_gaa(x1, x2, x3, x4, x5, x6) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x2, x3, x4) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x2, x3, x4) LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (EQUIVALENT) 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: LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: SEARCH_TREE_IN_GAA(tree(X, Left, void), Min, X) -> SEARCH_TREE_IN_GAA(Left, Min, Max) SEARCH_TREE_IN_GAA(tree(X, void, Right), X, Max) -> SEARCH_TREE_IN_GAA(Right, Min, Max) SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> U6_GAA(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1)) U6_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_GAA(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X)) U7_GAA(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> SEARCH_TREE_IN_GAA(Right, Min2, Max2) SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> SEARCH_TREE_IN_GAA(Left, Min1, Max1) The TRS R consists of the following rules: search_tree_in_g(void) -> search_tree_out_g(void) search_tree_in_g(T) -> U1_g(T, search_tree_in_gaa(T, X1, X2)) search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X) search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max)) search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max)) search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1)) U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y)) U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2)) U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2)) U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2) U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X)) U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X) U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min)) U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max) U1_g(T, search_tree_out_gaa(T, X1, X2)) -> search_tree_out_g(T) The argument filtering Pi contains the following mapping: search_tree_in_g(x1) = search_tree_in_g(x1) void = void search_tree_out_g(x1) = search_tree_out_g(x1) U1_g(x1, x2) = U1_g(x1, x2) search_tree_in_gaa(x1, x2, x3) = search_tree_in_gaa(x1) tree(x1, x2, x3) = tree(x1, x2, x3) search_tree_out_gaa(x1, x2, x3) = search_tree_out_gaa(x1, x2, x3) U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x2, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x2, x4) U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x1, x2, x3, x6) U7_gaa(x1, x2, x3, x4, x5, x6, x7) = U7_gaa(x1, x2, x3, x4, x7) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 s(x1) = s(x1) less_out_gg(x1, x2) = less_out_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x1, x2, x3) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x4, x6) U9_gaa(x1, x2, x3, x4, x5, x6) = U9_gaa(x1, x2, x3, x4, x5, x6) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x2, x3, x4) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x2, x3, x4) SEARCH_TREE_IN_GAA(x1, x2, x3) = SEARCH_TREE_IN_GAA(x1) U6_GAA(x1, x2, x3, x4, x5, x6) = U6_GAA(x1, x2, x3, x6) U7_GAA(x1, x2, x3, x4, x5, x6, x7) = U7_GAA(x1, x2, x3, x4, x7) 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: SEARCH_TREE_IN_GAA(tree(X, Left, void), Min, X) -> SEARCH_TREE_IN_GAA(Left, Min, Max) SEARCH_TREE_IN_GAA(tree(X, void, Right), X, Max) -> SEARCH_TREE_IN_GAA(Right, Min, Max) SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> U6_GAA(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1)) U6_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_GAA(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X)) U7_GAA(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> SEARCH_TREE_IN_GAA(Right, Min2, Max2) SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> SEARCH_TREE_IN_GAA(Left, Min1, Max1) The TRS R consists of the following rules: search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X) search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max)) search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max)) search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y)) U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min)) U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X)) U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X)) U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max) U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X) U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2)) U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2)) U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2) The argument filtering Pi contains the following mapping: void = void search_tree_in_gaa(x1, x2, x3) = search_tree_in_gaa(x1) tree(x1, x2, x3) = tree(x1, x2, x3) search_tree_out_gaa(x1, x2, x3) = search_tree_out_gaa(x1, x2, x3) U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x2, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x2, x4) U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x1, x2, x3, x6) U7_gaa(x1, x2, x3, x4, x5, x6, x7) = U7_gaa(x1, x2, x3, x4, x7) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 s(x1) = s(x1) less_out_gg(x1, x2) = less_out_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x1, x2, x3) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x4, x6) U9_gaa(x1, x2, x3, x4, x5, x6) = U9_gaa(x1, x2, x3, x4, x5, x6) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x2, x3, x4) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x2, x3, x4) SEARCH_TREE_IN_GAA(x1, x2, x3) = SEARCH_TREE_IN_GAA(x1) U6_GAA(x1, x2, x3, x4, x5, x6) = U6_GAA(x1, x2, x3, x6) U7_GAA(x1, x2, x3, x4, x5, x6, x7) = U7_GAA(x1, x2, x3, x4, x7) 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: SEARCH_TREE_IN_GAA(tree(X, Left, void)) -> SEARCH_TREE_IN_GAA(Left) SEARCH_TREE_IN_GAA(tree(X, void, Right)) -> SEARCH_TREE_IN_GAA(Right) SEARCH_TREE_IN_GAA(tree(X, Left, Right)) -> U6_GAA(X, Left, Right, search_tree_in_gaa(Left)) U6_GAA(X, Left, Right, search_tree_out_gaa(Left, Min1, Max1)) -> U7_GAA(X, Left, Right, Min1, less_in_gg(Max1, X)) U7_GAA(X, Left, Right, Min1, less_out_gg(Max1, X)) -> SEARCH_TREE_IN_GAA(Right) SEARCH_TREE_IN_GAA(tree(X, Left, Right)) -> SEARCH_TREE_IN_GAA(Left) The TRS R consists of the following rules: search_tree_in_gaa(tree(X, void, void)) -> search_tree_out_gaa(tree(X, void, void), X, X) search_tree_in_gaa(tree(X, void, Right)) -> U2_gaa(X, Right, search_tree_in_gaa(Right)) search_tree_in_gaa(tree(X, Left, void)) -> U4_gaa(X, Left, search_tree_in_gaa(Left)) search_tree_in_gaa(tree(X, Left, Right)) -> U6_gaa(X, Left, Right, search_tree_in_gaa(Left)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y)) U2_gaa(X, Right, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min)) U4_gaa(X, Left, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X)) U6_gaa(X, Left, Right, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, less_in_gg(Max1, X)) U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max) U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X) U7_gaa(X, Left, Right, Min1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, search_tree_in_gaa(Right)) U8_gaa(X, Left, Right, Min1, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2)) U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2) The set Q consists of the following terms: search_tree_in_gaa(x0) less_in_gg(x0, x1) U2_gaa(x0, x1, x2) U4_gaa(x0, x1, x2) U6_gaa(x0, x1, x2, x3) U10_gg(x0, x1, x2) U3_gaa(x0, x1, x2, x3) U5_gaa(x0, x1, x2, x3) U7_gaa(x0, x1, x2, x3, x4) U8_gaa(x0, x1, x2, x3, x4) U9_gaa(x0, x1, x2, x3, x4, x5) 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: *SEARCH_TREE_IN_GAA(tree(X, Left, Right)) -> U6_GAA(X, Left, Right, search_tree_in_gaa(Left)) The graph contains the following edges 1 > 1, 1 > 2, 1 > 3 *U7_GAA(X, Left, Right, Min1, less_out_gg(Max1, X)) -> SEARCH_TREE_IN_GAA(Right) The graph contains the following edges 3 >= 1 *U6_GAA(X, Left, Right, search_tree_out_gaa(Left, Min1, Max1)) -> U7_GAA(X, Left, Right, Min1, less_in_gg(Max1, X)) The graph contains the following edges 1 >= 1, 2 >= 2, 4 > 2, 3 >= 3, 4 > 4 *SEARCH_TREE_IN_GAA(tree(X, Left, void)) -> SEARCH_TREE_IN_GAA(Left) The graph contains the following edges 1 > 1 *SEARCH_TREE_IN_GAA(tree(X, void, Right)) -> SEARCH_TREE_IN_GAA(Right) The graph contains the following edges 1 > 1 *SEARCH_TREE_IN_GAA(tree(X, Left, Right)) -> SEARCH_TREE_IN_GAA(Left) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES