/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern delete(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 3 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, 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) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) YES ---------------------------------------- (0) Obligation: Clauses: delete(X, tree(X, void, Right), Right). delete(X, tree(X, Left, void), Left). delete(X, tree(X, Left, Right), tree(Y, Left, Right1)) :- delmin(Right, Y, Right1). delete(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), delete(X, Left, Left1)). delete(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), delete(X, Right, Right1)). delmin(tree(Y, void, Right), Y, Right). delmin(tree(X, Left, X1), Y, tree(X, Left1, X2)) :- delmin(Left, Y, Left1). less(0, s(X3)). less(s(X), s(Y)) :- less(X, Y). Query: delete(g,g,a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: delete_in_3: (b,b,f) delmin_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: delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right) delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left) delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1)) delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right) delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1)) U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y)) U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1)) U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gga(x1, x2, x3) = delete_in_gga(x1, x2) tree(x1, x2, x3) = tree(x1, x2, x3) void = void delete_out_gga(x1, x2, x3) = delete_out_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x6) delmin_in_gaa(x1, x2, x3) = delmin_in_gaa(x1) delmin_out_gaa(x1, x2, x3) = delmin_out_gaa(x1, x2) U6_gaa(x1, x2, x3, x4, x5, x6, x7) = U6_gaa(x1, x2, x3, x7) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 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) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) U5_gga(x1, x2, x3, x4, x5, x6) = U5_gga(x1, x2, x3, x4, x6) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right) delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left) delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1)) delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right) delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1)) U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y)) U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1)) U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gga(x1, x2, x3) = delete_in_gga(x1, x2) tree(x1, x2, x3) = tree(x1, x2, x3) void = void delete_out_gga(x1, x2, x3) = delete_out_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x6) delmin_in_gaa(x1, x2, x3) = delmin_in_gaa(x1) delmin_out_gaa(x1, x2, x3) = delmin_out_gaa(x1, x2) U6_gaa(x1, x2, x3, x4, x5, x6, x7) = U6_gaa(x1, x2, x3, x7) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 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) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) U5_gga(x1, x2, x3, x4, x5, x6) = U5_gga(x1, x2, x3, x4, x6) ---------------------------------------- (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: DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1)) DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_GAA(Right, Y, Right1) DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1)) DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_GAA(Left, Y, Left1) DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y) LESS_IN_GG(s(X), s(Y)) -> U7_GG(X, Y, less_in_gg(X, Y)) LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1)) U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GGA(X, Left, Left1) DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X) U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1)) U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GGA(X, Right, Right1) The TRS R consists of the following rules: delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right) delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left) delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1)) delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right) delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1)) U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y)) U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1)) U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gga(x1, x2, x3) = delete_in_gga(x1, x2) tree(x1, x2, x3) = tree(x1, x2, x3) void = void delete_out_gga(x1, x2, x3) = delete_out_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x6) delmin_in_gaa(x1, x2, x3) = delmin_in_gaa(x1) delmin_out_gaa(x1, x2, x3) = delmin_out_gaa(x1, x2) U6_gaa(x1, x2, x3, x4, x5, x6, x7) = U6_gaa(x1, x2, x3, x7) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 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) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) U5_gga(x1, x2, x3, x4, x5, x6) = U5_gga(x1, x2, x3, x4, x6) DELETE_IN_GGA(x1, x2, x3) = DELETE_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x6) DELMIN_IN_GAA(x1, x2, x3) = DELMIN_IN_GAA(x1) U6_GAA(x1, x2, x3, x4, x5, x6, x7) = U6_GAA(x1, x2, x3, x7) U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x1, x2, x3, x4, x6) LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) U7_GG(x1, x2, x3) = U7_GG(x1, x2, x3) U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x1, x2, x3, x4, x6) U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x4, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1)) DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_GAA(Right, Y, Right1) DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1)) DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_GAA(Left, Y, Left1) DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y) LESS_IN_GG(s(X), s(Y)) -> U7_GG(X, Y, less_in_gg(X, Y)) LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1)) U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GGA(X, Left, Left1) DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X) U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1)) U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GGA(X, Right, Right1) The TRS R consists of the following rules: delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right) delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left) delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1)) delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right) delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1)) U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y)) U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1)) U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gga(x1, x2, x3) = delete_in_gga(x1, x2) tree(x1, x2, x3) = tree(x1, x2, x3) void = void delete_out_gga(x1, x2, x3) = delete_out_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x6) delmin_in_gaa(x1, x2, x3) = delmin_in_gaa(x1) delmin_out_gaa(x1, x2, x3) = delmin_out_gaa(x1, x2) U6_gaa(x1, x2, x3, x4, x5, x6, x7) = U6_gaa(x1, x2, x3, x7) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 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) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) U5_gga(x1, x2, x3, x4, x5, x6) = U5_gga(x1, x2, x3, x4, x6) DELETE_IN_GGA(x1, x2, x3) = DELETE_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x6) DELMIN_IN_GAA(x1, x2, x3) = DELMIN_IN_GAA(x1) U6_GAA(x1, x2, x3, x4, x5, x6, x7) = U6_GAA(x1, x2, x3, x7) U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x1, x2, x3, x4, x6) LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) U7_GG(x1, x2, x3) = U7_GG(x1, x2, x3) U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x1, x2, x3, x4, x6) U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x4, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 8 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: delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right) delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left) delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1)) delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right) delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1)) U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y)) U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1)) U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gga(x1, x2, x3) = delete_in_gga(x1, x2) tree(x1, x2, x3) = tree(x1, x2, x3) void = void delete_out_gga(x1, x2, x3) = delete_out_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x6) delmin_in_gaa(x1, x2, x3) = delmin_in_gaa(x1) delmin_out_gaa(x1, x2, x3) = delmin_out_gaa(x1, x2) U6_gaa(x1, x2, x3, x4, x5, x6, x7) = U6_gaa(x1, x2, x3, x7) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 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) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) U5_gga(x1, x2, x3, x4, x5, x6) = U5_gga(x1, x2, x3, x4, x6) 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: DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_GAA(Left, Y, Left1) The TRS R consists of the following rules: delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right) delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left) delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1)) delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right) delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1)) U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y)) U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1)) U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gga(x1, x2, x3) = delete_in_gga(x1, x2) tree(x1, x2, x3) = tree(x1, x2, x3) void = void delete_out_gga(x1, x2, x3) = delete_out_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x6) delmin_in_gaa(x1, x2, x3) = delmin_in_gaa(x1) delmin_out_gaa(x1, x2, x3) = delmin_out_gaa(x1, x2) U6_gaa(x1, x2, x3, x4, x5, x6, x7) = U6_gaa(x1, x2, x3, x7) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 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) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) U5_gga(x1, x2, x3, x4, x5, x6) = U5_gga(x1, x2, x3, x4, x6) DELMIN_IN_GAA(x1, x2, x3) = DELMIN_IN_GAA(x1) 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: DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_GAA(Left, Y, Left1) R is empty. The argument filtering Pi contains the following mapping: tree(x1, x2, x3) = tree(x1, x2, x3) DELMIN_IN_GAA(x1, x2, x3) = DELMIN_IN_GAA(x1) 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: DELMIN_IN_GAA(tree(X, Left, X1)) -> DELMIN_IN_GAA(Left) 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: *DELMIN_IN_GAA(tree(X, Left, X1)) -> DELMIN_IN_GAA(Left) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GGA(X, Left, Left1) DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GGA(X, Right, Right1) The TRS R consists of the following rules: delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right) delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left) delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1)) delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right) delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1)) U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y)) U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1)) delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1)) U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gga(x1, x2, x3) = delete_in_gga(x1, x2) tree(x1, x2, x3) = tree(x1, x2, x3) void = void delete_out_gga(x1, x2, x3) = delete_out_gga(x1, x2) U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x6) delmin_in_gaa(x1, x2, x3) = delmin_in_gaa(x1) delmin_out_gaa(x1, x2, x3) = delmin_out_gaa(x1, x2) U6_gaa(x1, x2, x3, x4, x5, x6, x7) = U6_gaa(x1, x2, x3, x7) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 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) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) U5_gga(x1, x2, x3, x4, x5, x6) = U5_gga(x1, x2, x3, x4, x6) DELETE_IN_GGA(x1, x2, x3) = DELETE_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x1, x2, x3, x4, x6) U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x1, x2, x3, x4, 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: DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GGA(X, Left, Left1) DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GGA(X, Right, Right1) The TRS R consists of the following rules: less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y)) U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) The argument filtering Pi contains the following mapping: tree(x1, x2, x3) = tree(x1, x2, x3) 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) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) DELETE_IN_GGA(x1, x2, x3) = DELETE_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x1, x2, x3, x4, x6) U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x1, x2, x3, x4, 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: DELETE_IN_GGA(X, tree(Y, Left, Right)) -> U2_GGA(X, Y, Left, Right, less_in_gg(X, Y)) U2_GGA(X, Y, Left, Right, less_out_gg(X, Y)) -> DELETE_IN_GGA(X, Left) DELETE_IN_GGA(X, tree(Y, Left, Right)) -> U4_GGA(X, Y, Left, Right, less_in_gg(Y, X)) U4_GGA(X, Y, Left, Right, less_out_gg(Y, X)) -> DELETE_IN_GGA(X, Right) The TRS R consists of the following rules: less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3)) less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y)) U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) The set Q consists of the following terms: less_in_gg(x0, x1) U7_gg(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) 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: *U2_GGA(X, Y, Left, Right, less_out_gg(X, Y)) -> DELETE_IN_GGA(X, Left) The graph contains the following edges 1 >= 1, 5 > 1, 3 >= 2 *U4_GGA(X, Y, Left, Right, less_out_gg(Y, X)) -> DELETE_IN_GGA(X, Right) The graph contains the following edges 1 >= 1, 5 > 1, 4 >= 2 *DELETE_IN_GGA(X, tree(Y, Left, Right)) -> U2_GGA(X, Y, Left, Right, less_in_gg(X, Y)) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4 *DELETE_IN_GGA(X, tree(Y, Left, Right)) -> U4_GGA(X, Y, Left, Right, less_in_gg(Y, X)) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4 ---------------------------------------- (27) YES