/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern in(a,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 10 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, 1 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [EQUIVALENT, 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 (28) PiDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) PiDP (31) PiDPToQDPProof [SOUND, 0 ms] (32) QDP (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] (34) YES (35) PiDP (36) UsableRulesProof [EQUIVALENT, 0 ms] (37) PiDP (38) PiDPToQDPProof [SOUND, 0 ms] (39) QDP (40) QDPSizeChangeProof [EQUIVALENT, 0 ms] (41) YES ---------------------------------------- (0) Obligation: Clauses: in(X, tree(X, X1, X2)). in(X, tree(Y, Left, X3)) :- ','(less(X, Y), in(X, Left)). in(X, tree(Y, X4, Right)) :- ','(less(Y, X), in(X, Right)). less(0, s(X5)). less(s(X), s(Y)) :- less(X, Y). Query: in(a,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: in_in_2: (f,b) (b,b) less_in_2: (f,b) (b,b) (b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 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_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) The argument filtering Pi contains the following mapping: in_in_ag(x1, x2) = in_in_ag(x2) tree(x1, x2, x3) = tree(x1, x2, x3) in_out_ag(x1, x2) = in_out_ag(x1) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) less_in_ag(x1, x2) = less_in_ag(x2) s(x1) = s(x1) less_out_ag(x1, x2) = less_out_ag(x1) U5_ag(x1, x2, x3) = U5_ag(x3) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) in_in_gg(x1, x2) = in_in_gg(x1, x2) in_out_gg(x1, x2) = in_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 less_out_gg(x1, x2) = less_out_gg U5_gg(x1, x2, x3) = U5_gg(x3) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) less_in_ga(x1, x2) = less_in_ga(x1) less_out_ga(x1, x2) = less_out_ga U5_ga(x1, x2, x3) = U5_ga(x3) U4_ag(x1, x2, x3, x4, x5) = U4_ag(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: in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 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_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) The argument filtering Pi contains the following mapping: in_in_ag(x1, x2) = in_in_ag(x2) tree(x1, x2, x3) = tree(x1, x2, x3) in_out_ag(x1, x2) = in_out_ag(x1) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) less_in_ag(x1, x2) = less_in_ag(x2) s(x1) = s(x1) less_out_ag(x1, x2) = less_out_ag(x1) U5_ag(x1, x2, x3) = U5_ag(x3) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) in_in_gg(x1, x2) = in_in_gg(x1, x2) in_out_gg(x1, x2) = in_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 less_out_gg(x1, x2) = less_out_gg U5_gg(x1, x2, x3) = U5_gg(x3) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) less_in_ga(x1, x2) = less_in_ga(x1) less_out_ga(x1, x2) = less_out_ga U5_ga(x1, x2, x3) = U5_ga(x3) U4_ag(x1, x2, x3, x4, x5) = U4_ag(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: IN_IN_AG(X, tree(Y, Left, X3)) -> U1_AG(X, Y, Left, X3, less_in_ag(X, Y)) IN_IN_AG(X, tree(Y, Left, X3)) -> LESS_IN_AG(X, Y) LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_AG(X, Y, Left, X3, in_in_gg(X, Left)) U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) -> IN_IN_GG(X, Left) IN_IN_GG(X, tree(Y, Left, X3)) -> U1_GG(X, Y, Left, X3, less_in_gg(X, Y)) IN_IN_GG(X, tree(Y, Left, X3)) -> 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_GG(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_GG(X, Y, Left, X3, in_in_gg(X, Left)) U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) -> IN_IN_GG(X, Left) IN_IN_GG(X, tree(Y, X4, Right)) -> U3_GG(X, Y, X4, Right, less_in_gg(Y, X)) IN_IN_GG(X, tree(Y, X4, Right)) -> LESS_IN_GG(Y, X) U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_GG(X, Y, X4, Right, in_in_gg(X, Right)) U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) -> IN_IN_GG(X, Right) IN_IN_AG(X, tree(Y, X4, Right)) -> U3_AG(X, Y, X4, Right, less_in_ga(Y, X)) IN_IN_AG(X, tree(Y, X4, Right)) -> LESS_IN_GA(Y, X) LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_AG(X, Y, X4, Right, in_in_ag(X, Right)) U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) -> IN_IN_AG(X, Right) The TRS R consists of the following rules: in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 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_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) The argument filtering Pi contains the following mapping: in_in_ag(x1, x2) = in_in_ag(x2) tree(x1, x2, x3) = tree(x1, x2, x3) in_out_ag(x1, x2) = in_out_ag(x1) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) less_in_ag(x1, x2) = less_in_ag(x2) s(x1) = s(x1) less_out_ag(x1, x2) = less_out_ag(x1) U5_ag(x1, x2, x3) = U5_ag(x3) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) in_in_gg(x1, x2) = in_in_gg(x1, x2) in_out_gg(x1, x2) = in_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 less_out_gg(x1, x2) = less_out_gg U5_gg(x1, x2, x3) = U5_gg(x3) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) less_in_ga(x1, x2) = less_in_ga(x1) less_out_ga(x1, x2) = less_out_ga U5_ga(x1, x2, x3) = U5_ga(x3) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) IN_IN_AG(x1, x2) = IN_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x3, x5) LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) U5_AG(x1, x2, x3) = U5_AG(x3) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x1, x5) IN_IN_GG(x1, x2) = IN_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x3, x5) LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) U5_GG(x1, x2, x3) = U5_GG(x3) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x5) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x4, x5) U4_GG(x1, x2, x3, x4, x5) = U4_GG(x5) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x4, x5) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U5_GA(x1, x2, x3) = U5_GA(x3) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: IN_IN_AG(X, tree(Y, Left, X3)) -> U1_AG(X, Y, Left, X3, less_in_ag(X, Y)) IN_IN_AG(X, tree(Y, Left, X3)) -> LESS_IN_AG(X, Y) LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_AG(X, Y, Left, X3, in_in_gg(X, Left)) U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) -> IN_IN_GG(X, Left) IN_IN_GG(X, tree(Y, Left, X3)) -> U1_GG(X, Y, Left, X3, less_in_gg(X, Y)) IN_IN_GG(X, tree(Y, Left, X3)) -> 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_GG(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_GG(X, Y, Left, X3, in_in_gg(X, Left)) U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) -> IN_IN_GG(X, Left) IN_IN_GG(X, tree(Y, X4, Right)) -> U3_GG(X, Y, X4, Right, less_in_gg(Y, X)) IN_IN_GG(X, tree(Y, X4, Right)) -> LESS_IN_GG(Y, X) U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_GG(X, Y, X4, Right, in_in_gg(X, Right)) U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) -> IN_IN_GG(X, Right) IN_IN_AG(X, tree(Y, X4, Right)) -> U3_AG(X, Y, X4, Right, less_in_ga(Y, X)) IN_IN_AG(X, tree(Y, X4, Right)) -> LESS_IN_GA(Y, X) LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_AG(X, Y, X4, Right, in_in_ag(X, Right)) U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) -> IN_IN_AG(X, Right) The TRS R consists of the following rules: in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 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_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) The argument filtering Pi contains the following mapping: in_in_ag(x1, x2) = in_in_ag(x2) tree(x1, x2, x3) = tree(x1, x2, x3) in_out_ag(x1, x2) = in_out_ag(x1) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) less_in_ag(x1, x2) = less_in_ag(x2) s(x1) = s(x1) less_out_ag(x1, x2) = less_out_ag(x1) U5_ag(x1, x2, x3) = U5_ag(x3) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) in_in_gg(x1, x2) = in_in_gg(x1, x2) in_out_gg(x1, x2) = in_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 less_out_gg(x1, x2) = less_out_gg U5_gg(x1, x2, x3) = U5_gg(x3) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) less_in_ga(x1, x2) = less_in_ga(x1) less_out_ga(x1, x2) = less_out_ga U5_ga(x1, x2, x3) = U5_ga(x3) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) IN_IN_AG(x1, x2) = IN_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x3, x5) LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) U5_AG(x1, x2, x3) = U5_AG(x3) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x1, x5) IN_IN_GG(x1, x2) = IN_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x3, x5) LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) U5_GG(x1, x2, x3) = U5_GG(x3) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x5) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x4, x5) U4_GG(x1, x2, x3, x4, x5) = U4_GG(x5) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x4, x5) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U5_GA(x1, x2, x3) = U5_GA(x3) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 13 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) The TRS R consists of the following rules: in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 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_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) The argument filtering Pi contains the following mapping: in_in_ag(x1, x2) = in_in_ag(x2) tree(x1, x2, x3) = tree(x1, x2, x3) in_out_ag(x1, x2) = in_out_ag(x1) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) less_in_ag(x1, x2) = less_in_ag(x2) s(x1) = s(x1) less_out_ag(x1, x2) = less_out_ag(x1) U5_ag(x1, x2, x3) = U5_ag(x3) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) in_in_gg(x1, x2) = in_in_gg(x1, x2) in_out_gg(x1, x2) = in_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 less_out_gg(x1, x2) = less_out_gg U5_gg(x1, x2, x3) = U5_gg(x3) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) less_in_ga(x1, x2) = less_in_ga(x1) less_out_ga(x1, x2) = less_out_ga U5_ga(x1, x2, x3) = U5_ga(x3) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 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_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 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: LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 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_GA(s(X)) -> LESS_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) 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: in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 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_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) The argument filtering Pi contains the following mapping: in_in_ag(x1, x2) = in_in_ag(x2) tree(x1, x2, x3) = tree(x1, x2, x3) in_out_ag(x1, x2) = in_out_ag(x1) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) less_in_ag(x1, x2) = less_in_ag(x2) s(x1) = s(x1) less_out_ag(x1, x2) = less_out_ag(x1) U5_ag(x1, x2, x3) = U5_ag(x3) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) in_in_gg(x1, x2) = in_in_gg(x1, x2) in_out_gg(x1, x2) = in_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 less_out_gg(x1, x2) = less_out_gg U5_gg(x1, x2, x3) = U5_gg(x3) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) less_in_ga(x1, x2) = less_in_ga(x1) less_out_ga(x1, x2) = less_out_ga U5_ga(x1, x2, x3) = U5_ga(x3) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) LESS_IN_GG(x1, x2) = LESS_IN_GG(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: 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 ---------------------------------------- (17) PiDPToQDPProof (EQUIVALENT) 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: 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. ---------------------------------------- (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: *LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) -> IN_IN_GG(X, Left) IN_IN_GG(X, tree(Y, Left, X3)) -> U1_GG(X, Y, Left, X3, less_in_gg(X, Y)) IN_IN_GG(X, tree(Y, X4, Right)) -> U3_GG(X, Y, X4, Right, less_in_gg(Y, X)) U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) -> IN_IN_GG(X, Right) The TRS R consists of the following rules: in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 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_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) The argument filtering Pi contains the following mapping: in_in_ag(x1, x2) = in_in_ag(x2) tree(x1, x2, x3) = tree(x1, x2, x3) in_out_ag(x1, x2) = in_out_ag(x1) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) less_in_ag(x1, x2) = less_in_ag(x2) s(x1) = s(x1) less_out_ag(x1, x2) = less_out_ag(x1) U5_ag(x1, x2, x3) = U5_ag(x3) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) in_in_gg(x1, x2) = in_in_gg(x1, x2) in_out_gg(x1, x2) = in_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 less_out_gg(x1, x2) = less_out_gg U5_gg(x1, x2, x3) = U5_gg(x3) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) less_in_ga(x1, x2) = less_in_ga(x1) less_out_ga(x1, x2) = less_out_ga U5_ga(x1, x2, x3) = U5_ga(x3) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) IN_IN_GG(x1, x2) = IN_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x3, x5) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x4, x5) 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(X, Y, Left, X3, less_out_gg(X, Y)) -> IN_IN_GG(X, Left) IN_IN_GG(X, tree(Y, Left, X3)) -> U1_GG(X, Y, Left, X3, less_in_gg(X, Y)) IN_IN_GG(X, tree(Y, X4, Right)) -> U3_GG(X, Y, X4, Right, less_in_gg(Y, X)) U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) -> IN_IN_GG(X, Right) The TRS R consists of the following rules: less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 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) s(x1) = s(x1) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 less_out_gg(x1, x2) = less_out_gg U5_gg(x1, x2, x3) = U5_gg(x3) IN_IN_GG(x1, x2) = IN_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x3, x5) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x4, x5) 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(X, Left, less_out_gg) -> IN_IN_GG(X, Left) IN_IN_GG(X, tree(Y, Left, X3)) -> U1_GG(X, Left, less_in_gg(X, Y)) IN_IN_GG(X, tree(Y, X4, Right)) -> U3_GG(X, Right, less_in_gg(Y, X)) U3_GG(X, Right, less_out_gg) -> IN_IN_GG(X, Right) The TRS R consists of the following rules: less_in_gg(0, s(X5)) -> 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. ---------------------------------------- (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: *IN_IN_GG(X, tree(Y, Left, X3)) -> U1_GG(X, Left, less_in_gg(X, Y)) The graph contains the following edges 1 >= 1, 2 > 2 *IN_IN_GG(X, tree(Y, X4, Right)) -> U3_GG(X, Right, less_in_gg(Y, X)) The graph contains the following edges 1 >= 1, 2 > 2 *U1_GG(X, Left, less_out_gg) -> IN_IN_GG(X, Left) The graph contains the following edges 1 >= 1, 2 >= 2 *U3_GG(X, Right, less_out_gg) -> IN_IN_GG(X, Right) The graph contains the following edges 1 >= 1, 2 >= 2 ---------------------------------------- (27) YES ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) The TRS R consists of the following rules: in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 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_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) The argument filtering Pi contains the following mapping: in_in_ag(x1, x2) = in_in_ag(x2) tree(x1, x2, x3) = tree(x1, x2, x3) in_out_ag(x1, x2) = in_out_ag(x1) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) less_in_ag(x1, x2) = less_in_ag(x2) s(x1) = s(x1) less_out_ag(x1, x2) = less_out_ag(x1) U5_ag(x1, x2, x3) = U5_ag(x3) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) in_in_gg(x1, x2) = in_in_gg(x1, x2) in_out_gg(x1, x2) = in_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 less_out_gg(x1, x2) = less_out_gg U5_gg(x1, x2, x3) = U5_gg(x3) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) less_in_ga(x1, x2) = less_in_ga(x1) less_out_ga(x1, x2) = less_out_ga U5_ga(x1, x2, x3) = U5_ga(x3) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) 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_AG(s(Y)) -> LESS_IN_AG(Y) The graph contains the following edges 1 > 1 ---------------------------------------- (34) YES ---------------------------------------- (35) Obligation: Pi DP problem: The TRS P consists of the following rules: IN_IN_AG(X, tree(Y, X4, Right)) -> U3_AG(X, Y, X4, Right, less_in_ga(Y, X)) U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) -> IN_IN_AG(X, Right) The TRS R consists of the following rules: in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 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_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) The argument filtering Pi contains the following mapping: in_in_ag(x1, x2) = in_in_ag(x2) tree(x1, x2, x3) = tree(x1, x2, x3) in_out_ag(x1, x2) = in_out_ag(x1) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) less_in_ag(x1, x2) = less_in_ag(x2) s(x1) = s(x1) less_out_ag(x1, x2) = less_out_ag(x1) U5_ag(x1, x2, x3) = U5_ag(x3) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) in_in_gg(x1, x2) = in_in_gg(x1, x2) in_out_gg(x1, x2) = in_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) less_in_gg(x1, x2) = less_in_gg(x1, x2) 0 = 0 less_out_gg(x1, x2) = less_out_gg U5_gg(x1, x2, x3) = U5_gg(x3) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) less_in_ga(x1, x2) = less_in_ga(x1) less_out_ga(x1, x2) = less_out_ga U5_ga(x1, x2, x3) = U5_ga(x3) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) IN_IN_AG(x1, x2) = IN_IN_AG(x2) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (36) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (37) Obligation: Pi DP problem: The TRS P consists of the following rules: IN_IN_AG(X, tree(Y, X4, Right)) -> U3_AG(X, Y, X4, Right, less_in_ga(Y, X)) U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) -> IN_IN_AG(X, Right) The TRS R consists of the following rules: less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) The argument filtering Pi contains the following mapping: tree(x1, x2, x3) = tree(x1, x2, x3) s(x1) = s(x1) 0 = 0 less_in_ga(x1, x2) = less_in_ga(x1) less_out_ga(x1, x2) = less_out_ga U5_ga(x1, x2, x3) = U5_ga(x3) IN_IN_AG(x1, x2) = IN_IN_AG(x2) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (38) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: IN_IN_AG(tree(Y, X4, Right)) -> U3_AG(Right, less_in_ga(Y)) U3_AG(Right, less_out_ga) -> IN_IN_AG(Right) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) U5_ga(less_out_ga) -> less_out_ga The set Q consists of the following terms: less_in_ga(x0) U5_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (40) 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: *U3_AG(Right, less_out_ga) -> IN_IN_AG(Right) The graph contains the following edges 1 >= 1 *IN_IN_AG(tree(Y, X4, Right)) -> U3_AG(Right, less_in_ga(Y)) The graph contains the following edges 1 > 1 ---------------------------------------- (41) YES