/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern insert(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof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, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [EQUIVALENT, 1 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 ---------------------------------------- (0) Obligation: Clauses: insert(X, void, tree(X, void, void)). insert(X, tree(X, Left, Right), tree(X, Left, Right)). insert(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), insert(X, Left, Left1)). insert(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), insert(X, Right, Right1)). less(0, s(X1)). less(s(X), s(Y)) :- less(X, Y). Query: insert(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: insert_in_3: (b,b,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: insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) void = void insert_out_gga(x1, x2, x3) = insert_out_gga(x3) tree(x1, x2, x3) = tree(x1, x2, x3) U1_gga(x1, x2, x3, x4, x5, x6) = U1_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 U5_gg(x1, x2, x3) = U5_gg(x3) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 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(x2, x3, 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: insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) void = void insert_out_gga(x1, x2, x3) = insert_out_gga(x3) tree(x1, x2, x3) = tree(x1, x2, x3) U1_gga(x1, x2, x3, x4, x5, x6) = U1_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 U5_gg(x1, x2, x3) = U5_gg(x3) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 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(x2, x3, 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: INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y) LESS_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, less_in_gg(X, Y)) LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_GGA(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X) U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_GGA(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> INSERT_IN_GGA(X, Right, Right1) The TRS R consists of the following rules: insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) void = void insert_out_gga(x1, x2, x3) = insert_out_gga(x3) tree(x1, x2, x3) = tree(x1, x2, x3) U1_gga(x1, x2, x3, x4, x5, x6) = U1_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 U5_gg(x1, x2, x3) = U5_gg(x3) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 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(x2, x3, x6) INSERT_IN_GGA(x1, x2, x3) = INSERT_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) U5_GG(x1, x2, x3) = U5_GG(x3) U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x2, x4, x6) 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(x2, x3, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y) LESS_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, less_in_gg(X, Y)) LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_GGA(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X) U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_GGA(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> INSERT_IN_GGA(X, Right, Right1) The TRS R consists of the following rules: insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) void = void insert_out_gga(x1, x2, x3) = insert_out_gga(x3) tree(x1, x2, x3) = tree(x1, x2, x3) U1_gga(x1, x2, x3, x4, x5, x6) = U1_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 U5_gg(x1, x2, x3) = U5_gg(x3) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 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(x2, x3, x6) INSERT_IN_GGA(x1, x2, x3) = INSERT_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) U5_GG(x1, x2, x3) = U5_GG(x3) U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x2, x4, x6) 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(x2, x3, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) The TRS R consists of the following rules: insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) void = void insert_out_gga(x1, x2, x3) = insert_out_gga(x3) tree(x1, x2, x3) = tree(x1, x2, x3) U1_gga(x1, x2, x3, x4, x5, x6) = U1_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 U5_gg(x1, x2, x3) = U5_gg(x3) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 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(x2, x3, 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: U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> INSERT_IN_GGA(X, Right, Right1) The TRS R consists of the following rules: insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) void = void insert_out_gga(x1, x2, x3) = insert_out_gga(x3) tree(x1, x2, x3) = tree(x1, x2, x3) U1_gga(x1, x2, x3, x4, x5, x6) = U1_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 U5_gg(x1, x2, x3) = U5_gg(x3) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 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(x2, x3, x6) INSERT_IN_GGA(x1, x2, x3) = INSERT_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 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: U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> INSERT_IN_GGA(X, Right, Right1) The TRS R consists of the following rules: less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) U5_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 U5_gg(x1, x2, x3) = U5_gg(x3) INSERT_IN_GGA(x1, x2, x3) = INSERT_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 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: U1_GGA(X, Y, Left, Right, less_out_gg) -> INSERT_IN_GGA(X, Left) INSERT_IN_GGA(X, tree(Y, Left, Right)) -> U1_GGA(X, Y, Left, Right, less_in_gg(X, Y)) INSERT_IN_GGA(X, tree(Y, Left, Right)) -> U3_GGA(X, Y, Left, Right, less_in_gg(Y, X)) U3_GGA(X, Y, Left, Right, less_out_gg) -> INSERT_IN_GGA(X, Right) The TRS R consists of the following rules: less_in_gg(0, s(X1)) -> less_out_gg less_in_gg(s(X), s(Y)) -> U5_gg(less_in_gg(X, Y)) U5_gg(less_out_gg) -> less_out_gg The set Q consists of the following terms: less_in_gg(x0, x1) U5_gg(x0) 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: *INSERT_IN_GGA(X, tree(Y, Left, Right)) -> U1_GGA(X, Y, Left, Right, less_in_gg(X, Y)) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4 *INSERT_IN_GGA(X, tree(Y, Left, Right)) -> U3_GGA(X, Y, Left, Right, less_in_gg(Y, X)) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4 *U1_GGA(X, Y, Left, Right, less_out_gg) -> INSERT_IN_GGA(X, Left) The graph contains the following edges 1 >= 1, 3 >= 2 *U3_GGA(X, Y, Left, Right, less_out_gg) -> INSERT_IN_GGA(X, Right) The graph contains the following edges 1 >= 1, 4 >= 2 ---------------------------------------- (20) YES