/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern cnfequiv(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof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, 4 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 2 ms] (18) QDP (19) QDPOrderProof [EQUIVALENT, 213 ms] (20) QDP (21) PisEmptyProof [EQUIVALENT, 0 ms] (22) YES ---------------------------------------- (0) Obligation: Clauses: cnfequiv(X, Y) :- ','(transform(X, Z), cnfequiv(Z, Y)). cnfequiv(X, X). transform(n(n(X)), X). transform(n(a(X, Y)), o(n(X), n(Y))). transform(n(o(X, Y)), a(n(X), n(Y))). transform(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))). transform(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))). transform(o(X1, Y), o(X2, Y)) :- transform(X1, X2). transform(o(X, Y1), o(X, Y2)) :- transform(Y1, Y2). transform(a(X1, Y), a(X2, Y)) :- transform(X1, X2). transform(a(X, Y1), a(X, Y2)) :- transform(Y1, Y2). transform(n(X1), n(X2)) :- transform(X1, X2). Query: cnfequiv(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: cnfequiv_in_2: (b,f) transform_in_2: (b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: cnfequiv_in_ga(X, Y) -> U1_ga(X, Y, transform_in_ga(X, Z)) transform_in_ga(n(n(X)), X) -> transform_out_ga(n(n(X)), X) transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) -> transform_out_ga(n(a(X, Y)), o(n(X), n(Y))) transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) -> transform_out_ga(n(o(X, Y)), a(n(X), n(Y))) transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) -> transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) -> transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) transform_in_ga(o(X1, Y), o(X2, Y)) -> U3_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(o(X, Y1), o(X, Y2)) -> U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(a(X1, Y), a(X2, Y)) -> U5_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(a(X, Y1), a(X, Y2)) -> U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(n(X1), n(X2)) -> U7_ga(X1, X2, transform_in_ga(X1, X2)) U7_ga(X1, X2, transform_out_ga(X1, X2)) -> transform_out_ga(n(X1), n(X2)) U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(a(X, Y1), a(X, Y2)) U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(a(X1, Y), a(X2, Y)) U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(o(X, Y1), o(X, Y2)) U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(o(X1, Y), o(X2, Y)) U1_ga(X, Y, transform_out_ga(X, Z)) -> U2_ga(X, Y, cnfequiv_in_ga(Z, Y)) cnfequiv_in_ga(X, X) -> cnfequiv_out_ga(X, X) U2_ga(X, Y, cnfequiv_out_ga(Z, Y)) -> cnfequiv_out_ga(X, Y) The argument filtering Pi contains the following mapping: cnfequiv_in_ga(x1, x2) = cnfequiv_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) transform_in_ga(x1, x2) = transform_in_ga(x1) n(x1) = n(x1) transform_out_ga(x1, x2) = transform_out_ga(x2) a(x1, x2) = a(x1, x2) o(x1, x2) = o(x1, x2) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4) U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U7_ga(x1, x2, x3) = U7_ga(x3) U2_ga(x1, x2, x3) = U2_ga(x3) cnfequiv_out_ga(x1, x2) = cnfequiv_out_ga(x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: cnfequiv_in_ga(X, Y) -> U1_ga(X, Y, transform_in_ga(X, Z)) transform_in_ga(n(n(X)), X) -> transform_out_ga(n(n(X)), X) transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) -> transform_out_ga(n(a(X, Y)), o(n(X), n(Y))) transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) -> transform_out_ga(n(o(X, Y)), a(n(X), n(Y))) transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) -> transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) -> transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) transform_in_ga(o(X1, Y), o(X2, Y)) -> U3_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(o(X, Y1), o(X, Y2)) -> U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(a(X1, Y), a(X2, Y)) -> U5_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(a(X, Y1), a(X, Y2)) -> U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(n(X1), n(X2)) -> U7_ga(X1, X2, transform_in_ga(X1, X2)) U7_ga(X1, X2, transform_out_ga(X1, X2)) -> transform_out_ga(n(X1), n(X2)) U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(a(X, Y1), a(X, Y2)) U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(a(X1, Y), a(X2, Y)) U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(o(X, Y1), o(X, Y2)) U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(o(X1, Y), o(X2, Y)) U1_ga(X, Y, transform_out_ga(X, Z)) -> U2_ga(X, Y, cnfequiv_in_ga(Z, Y)) cnfequiv_in_ga(X, X) -> cnfequiv_out_ga(X, X) U2_ga(X, Y, cnfequiv_out_ga(Z, Y)) -> cnfequiv_out_ga(X, Y) The argument filtering Pi contains the following mapping: cnfequiv_in_ga(x1, x2) = cnfequiv_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) transform_in_ga(x1, x2) = transform_in_ga(x1) n(x1) = n(x1) transform_out_ga(x1, x2) = transform_out_ga(x2) a(x1, x2) = a(x1, x2) o(x1, x2) = o(x1, x2) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4) U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U7_ga(x1, x2, x3) = U7_ga(x3) U2_ga(x1, x2, x3) = U2_ga(x3) cnfequiv_out_ga(x1, x2) = cnfequiv_out_ga(x2) ---------------------------------------- (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: CNFEQUIV_IN_GA(X, Y) -> U1_GA(X, Y, transform_in_ga(X, Z)) CNFEQUIV_IN_GA(X, Y) -> TRANSFORM_IN_GA(X, Z) TRANSFORM_IN_GA(o(X1, Y), o(X2, Y)) -> U3_GA(X1, Y, X2, transform_in_ga(X1, X2)) TRANSFORM_IN_GA(o(X1, Y), o(X2, Y)) -> TRANSFORM_IN_GA(X1, X2) TRANSFORM_IN_GA(o(X, Y1), o(X, Y2)) -> U4_GA(X, Y1, Y2, transform_in_ga(Y1, Y2)) TRANSFORM_IN_GA(o(X, Y1), o(X, Y2)) -> TRANSFORM_IN_GA(Y1, Y2) TRANSFORM_IN_GA(a(X1, Y), a(X2, Y)) -> U5_GA(X1, Y, X2, transform_in_ga(X1, X2)) TRANSFORM_IN_GA(a(X1, Y), a(X2, Y)) -> TRANSFORM_IN_GA(X1, X2) TRANSFORM_IN_GA(a(X, Y1), a(X, Y2)) -> U6_GA(X, Y1, Y2, transform_in_ga(Y1, Y2)) TRANSFORM_IN_GA(a(X, Y1), a(X, Y2)) -> TRANSFORM_IN_GA(Y1, Y2) TRANSFORM_IN_GA(n(X1), n(X2)) -> U7_GA(X1, X2, transform_in_ga(X1, X2)) TRANSFORM_IN_GA(n(X1), n(X2)) -> TRANSFORM_IN_GA(X1, X2) U1_GA(X, Y, transform_out_ga(X, Z)) -> U2_GA(X, Y, cnfequiv_in_ga(Z, Y)) U1_GA(X, Y, transform_out_ga(X, Z)) -> CNFEQUIV_IN_GA(Z, Y) The TRS R consists of the following rules: cnfequiv_in_ga(X, Y) -> U1_ga(X, Y, transform_in_ga(X, Z)) transform_in_ga(n(n(X)), X) -> transform_out_ga(n(n(X)), X) transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) -> transform_out_ga(n(a(X, Y)), o(n(X), n(Y))) transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) -> transform_out_ga(n(o(X, Y)), a(n(X), n(Y))) transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) -> transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) -> transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) transform_in_ga(o(X1, Y), o(X2, Y)) -> U3_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(o(X, Y1), o(X, Y2)) -> U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(a(X1, Y), a(X2, Y)) -> U5_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(a(X, Y1), a(X, Y2)) -> U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(n(X1), n(X2)) -> U7_ga(X1, X2, transform_in_ga(X1, X2)) U7_ga(X1, X2, transform_out_ga(X1, X2)) -> transform_out_ga(n(X1), n(X2)) U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(a(X, Y1), a(X, Y2)) U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(a(X1, Y), a(X2, Y)) U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(o(X, Y1), o(X, Y2)) U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(o(X1, Y), o(X2, Y)) U1_ga(X, Y, transform_out_ga(X, Z)) -> U2_ga(X, Y, cnfequiv_in_ga(Z, Y)) cnfequiv_in_ga(X, X) -> cnfequiv_out_ga(X, X) U2_ga(X, Y, cnfequiv_out_ga(Z, Y)) -> cnfequiv_out_ga(X, Y) The argument filtering Pi contains the following mapping: cnfequiv_in_ga(x1, x2) = cnfequiv_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) transform_in_ga(x1, x2) = transform_in_ga(x1) n(x1) = n(x1) transform_out_ga(x1, x2) = transform_out_ga(x2) a(x1, x2) = a(x1, x2) o(x1, x2) = o(x1, x2) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4) U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U7_ga(x1, x2, x3) = U7_ga(x3) U2_ga(x1, x2, x3) = U2_ga(x3) cnfequiv_out_ga(x1, x2) = cnfequiv_out_ga(x2) CNFEQUIV_IN_GA(x1, x2) = CNFEQUIV_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) TRANSFORM_IN_GA(x1, x2) = TRANSFORM_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x2, x4) U4_GA(x1, x2, x3, x4) = U4_GA(x1, x4) U5_GA(x1, x2, x3, x4) = U5_GA(x2, x4) U6_GA(x1, x2, x3, x4) = U6_GA(x1, x4) U7_GA(x1, x2, x3) = U7_GA(x3) U2_GA(x1, x2, x3) = U2_GA(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: CNFEQUIV_IN_GA(X, Y) -> U1_GA(X, Y, transform_in_ga(X, Z)) CNFEQUIV_IN_GA(X, Y) -> TRANSFORM_IN_GA(X, Z) TRANSFORM_IN_GA(o(X1, Y), o(X2, Y)) -> U3_GA(X1, Y, X2, transform_in_ga(X1, X2)) TRANSFORM_IN_GA(o(X1, Y), o(X2, Y)) -> TRANSFORM_IN_GA(X1, X2) TRANSFORM_IN_GA(o(X, Y1), o(X, Y2)) -> U4_GA(X, Y1, Y2, transform_in_ga(Y1, Y2)) TRANSFORM_IN_GA(o(X, Y1), o(X, Y2)) -> TRANSFORM_IN_GA(Y1, Y2) TRANSFORM_IN_GA(a(X1, Y), a(X2, Y)) -> U5_GA(X1, Y, X2, transform_in_ga(X1, X2)) TRANSFORM_IN_GA(a(X1, Y), a(X2, Y)) -> TRANSFORM_IN_GA(X1, X2) TRANSFORM_IN_GA(a(X, Y1), a(X, Y2)) -> U6_GA(X, Y1, Y2, transform_in_ga(Y1, Y2)) TRANSFORM_IN_GA(a(X, Y1), a(X, Y2)) -> TRANSFORM_IN_GA(Y1, Y2) TRANSFORM_IN_GA(n(X1), n(X2)) -> U7_GA(X1, X2, transform_in_ga(X1, X2)) TRANSFORM_IN_GA(n(X1), n(X2)) -> TRANSFORM_IN_GA(X1, X2) U1_GA(X, Y, transform_out_ga(X, Z)) -> U2_GA(X, Y, cnfequiv_in_ga(Z, Y)) U1_GA(X, Y, transform_out_ga(X, Z)) -> CNFEQUIV_IN_GA(Z, Y) The TRS R consists of the following rules: cnfequiv_in_ga(X, Y) -> U1_ga(X, Y, transform_in_ga(X, Z)) transform_in_ga(n(n(X)), X) -> transform_out_ga(n(n(X)), X) transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) -> transform_out_ga(n(a(X, Y)), o(n(X), n(Y))) transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) -> transform_out_ga(n(o(X, Y)), a(n(X), n(Y))) transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) -> transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) -> transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) transform_in_ga(o(X1, Y), o(X2, Y)) -> U3_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(o(X, Y1), o(X, Y2)) -> U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(a(X1, Y), a(X2, Y)) -> U5_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(a(X, Y1), a(X, Y2)) -> U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(n(X1), n(X2)) -> U7_ga(X1, X2, transform_in_ga(X1, X2)) U7_ga(X1, X2, transform_out_ga(X1, X2)) -> transform_out_ga(n(X1), n(X2)) U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(a(X, Y1), a(X, Y2)) U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(a(X1, Y), a(X2, Y)) U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(o(X, Y1), o(X, Y2)) U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(o(X1, Y), o(X2, Y)) U1_ga(X, Y, transform_out_ga(X, Z)) -> U2_ga(X, Y, cnfequiv_in_ga(Z, Y)) cnfequiv_in_ga(X, X) -> cnfequiv_out_ga(X, X) U2_ga(X, Y, cnfequiv_out_ga(Z, Y)) -> cnfequiv_out_ga(X, Y) The argument filtering Pi contains the following mapping: cnfequiv_in_ga(x1, x2) = cnfequiv_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) transform_in_ga(x1, x2) = transform_in_ga(x1) n(x1) = n(x1) transform_out_ga(x1, x2) = transform_out_ga(x2) a(x1, x2) = a(x1, x2) o(x1, x2) = o(x1, x2) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4) U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U7_ga(x1, x2, x3) = U7_ga(x3) U2_ga(x1, x2, x3) = U2_ga(x3) cnfequiv_out_ga(x1, x2) = cnfequiv_out_ga(x2) CNFEQUIV_IN_GA(x1, x2) = CNFEQUIV_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) TRANSFORM_IN_GA(x1, x2) = TRANSFORM_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x2, x4) U4_GA(x1, x2, x3, x4) = U4_GA(x1, x4) U5_GA(x1, x2, x3, x4) = U5_GA(x2, x4) U6_GA(x1, x2, x3, x4) = U6_GA(x1, x4) U7_GA(x1, x2, x3) = U7_GA(x3) U2_GA(x1, x2, x3) = U2_GA(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: TRANSFORM_IN_GA(o(X, Y1), o(X, Y2)) -> TRANSFORM_IN_GA(Y1, Y2) TRANSFORM_IN_GA(o(X1, Y), o(X2, Y)) -> TRANSFORM_IN_GA(X1, X2) TRANSFORM_IN_GA(a(X1, Y), a(X2, Y)) -> TRANSFORM_IN_GA(X1, X2) TRANSFORM_IN_GA(a(X, Y1), a(X, Y2)) -> TRANSFORM_IN_GA(Y1, Y2) TRANSFORM_IN_GA(n(X1), n(X2)) -> TRANSFORM_IN_GA(X1, X2) The TRS R consists of the following rules: cnfequiv_in_ga(X, Y) -> U1_ga(X, Y, transform_in_ga(X, Z)) transform_in_ga(n(n(X)), X) -> transform_out_ga(n(n(X)), X) transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) -> transform_out_ga(n(a(X, Y)), o(n(X), n(Y))) transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) -> transform_out_ga(n(o(X, Y)), a(n(X), n(Y))) transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) -> transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) -> transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) transform_in_ga(o(X1, Y), o(X2, Y)) -> U3_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(o(X, Y1), o(X, Y2)) -> U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(a(X1, Y), a(X2, Y)) -> U5_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(a(X, Y1), a(X, Y2)) -> U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(n(X1), n(X2)) -> U7_ga(X1, X2, transform_in_ga(X1, X2)) U7_ga(X1, X2, transform_out_ga(X1, X2)) -> transform_out_ga(n(X1), n(X2)) U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(a(X, Y1), a(X, Y2)) U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(a(X1, Y), a(X2, Y)) U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(o(X, Y1), o(X, Y2)) U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(o(X1, Y), o(X2, Y)) U1_ga(X, Y, transform_out_ga(X, Z)) -> U2_ga(X, Y, cnfequiv_in_ga(Z, Y)) cnfequiv_in_ga(X, X) -> cnfequiv_out_ga(X, X) U2_ga(X, Y, cnfequiv_out_ga(Z, Y)) -> cnfequiv_out_ga(X, Y) The argument filtering Pi contains the following mapping: cnfequiv_in_ga(x1, x2) = cnfequiv_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) transform_in_ga(x1, x2) = transform_in_ga(x1) n(x1) = n(x1) transform_out_ga(x1, x2) = transform_out_ga(x2) a(x1, x2) = a(x1, x2) o(x1, x2) = o(x1, x2) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4) U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U7_ga(x1, x2, x3) = U7_ga(x3) U2_ga(x1, x2, x3) = U2_ga(x3) cnfequiv_out_ga(x1, x2) = cnfequiv_out_ga(x2) TRANSFORM_IN_GA(x1, x2) = TRANSFORM_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: TRANSFORM_IN_GA(o(X, Y1), o(X, Y2)) -> TRANSFORM_IN_GA(Y1, Y2) TRANSFORM_IN_GA(o(X1, Y), o(X2, Y)) -> TRANSFORM_IN_GA(X1, X2) TRANSFORM_IN_GA(a(X1, Y), a(X2, Y)) -> TRANSFORM_IN_GA(X1, X2) TRANSFORM_IN_GA(a(X, Y1), a(X, Y2)) -> TRANSFORM_IN_GA(Y1, Y2) TRANSFORM_IN_GA(n(X1), n(X2)) -> TRANSFORM_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: n(x1) = n(x1) a(x1, x2) = a(x1, x2) o(x1, x2) = o(x1, x2) TRANSFORM_IN_GA(x1, x2) = TRANSFORM_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: TRANSFORM_IN_GA(o(X, Y1)) -> TRANSFORM_IN_GA(Y1) TRANSFORM_IN_GA(o(X1, Y)) -> TRANSFORM_IN_GA(X1) TRANSFORM_IN_GA(a(X1, Y)) -> TRANSFORM_IN_GA(X1) TRANSFORM_IN_GA(a(X, Y1)) -> TRANSFORM_IN_GA(Y1) TRANSFORM_IN_GA(n(X1)) -> TRANSFORM_IN_GA(X1) 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: *TRANSFORM_IN_GA(o(X, Y1)) -> TRANSFORM_IN_GA(Y1) The graph contains the following edges 1 > 1 *TRANSFORM_IN_GA(o(X1, Y)) -> TRANSFORM_IN_GA(X1) The graph contains the following edges 1 > 1 *TRANSFORM_IN_GA(a(X1, Y)) -> TRANSFORM_IN_GA(X1) The graph contains the following edges 1 > 1 *TRANSFORM_IN_GA(a(X, Y1)) -> TRANSFORM_IN_GA(Y1) The graph contains the following edges 1 > 1 *TRANSFORM_IN_GA(n(X1)) -> TRANSFORM_IN_GA(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(X, Y, transform_out_ga(X, Z)) -> CNFEQUIV_IN_GA(Z, Y) CNFEQUIV_IN_GA(X, Y) -> U1_GA(X, Y, transform_in_ga(X, Z)) The TRS R consists of the following rules: cnfequiv_in_ga(X, Y) -> U1_ga(X, Y, transform_in_ga(X, Z)) transform_in_ga(n(n(X)), X) -> transform_out_ga(n(n(X)), X) transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) -> transform_out_ga(n(a(X, Y)), o(n(X), n(Y))) transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) -> transform_out_ga(n(o(X, Y)), a(n(X), n(Y))) transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) -> transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) -> transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) transform_in_ga(o(X1, Y), o(X2, Y)) -> U3_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(o(X, Y1), o(X, Y2)) -> U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(a(X1, Y), a(X2, Y)) -> U5_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(a(X, Y1), a(X, Y2)) -> U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(n(X1), n(X2)) -> U7_ga(X1, X2, transform_in_ga(X1, X2)) U7_ga(X1, X2, transform_out_ga(X1, X2)) -> transform_out_ga(n(X1), n(X2)) U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(a(X, Y1), a(X, Y2)) U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(a(X1, Y), a(X2, Y)) U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(o(X, Y1), o(X, Y2)) U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(o(X1, Y), o(X2, Y)) U1_ga(X, Y, transform_out_ga(X, Z)) -> U2_ga(X, Y, cnfequiv_in_ga(Z, Y)) cnfequiv_in_ga(X, X) -> cnfequiv_out_ga(X, X) U2_ga(X, Y, cnfequiv_out_ga(Z, Y)) -> cnfequiv_out_ga(X, Y) The argument filtering Pi contains the following mapping: cnfequiv_in_ga(x1, x2) = cnfequiv_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) transform_in_ga(x1, x2) = transform_in_ga(x1) n(x1) = n(x1) transform_out_ga(x1, x2) = transform_out_ga(x2) a(x1, x2) = a(x1, x2) o(x1, x2) = o(x1, x2) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4) U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U7_ga(x1, x2, x3) = U7_ga(x3) U2_ga(x1, x2, x3) = U2_ga(x3) cnfequiv_out_ga(x1, x2) = cnfequiv_out_ga(x2) CNFEQUIV_IN_GA(x1, x2) = CNFEQUIV_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) 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_GA(X, Y, transform_out_ga(X, Z)) -> CNFEQUIV_IN_GA(Z, Y) CNFEQUIV_IN_GA(X, Y) -> U1_GA(X, Y, transform_in_ga(X, Z)) The TRS R consists of the following rules: transform_in_ga(n(n(X)), X) -> transform_out_ga(n(n(X)), X) transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) -> transform_out_ga(n(a(X, Y)), o(n(X), n(Y))) transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) -> transform_out_ga(n(o(X, Y)), a(n(X), n(Y))) transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) -> transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) -> transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) transform_in_ga(o(X1, Y), o(X2, Y)) -> U3_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(o(X, Y1), o(X, Y2)) -> U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(a(X1, Y), a(X2, Y)) -> U5_ga(X1, Y, X2, transform_in_ga(X1, X2)) transform_in_ga(a(X, Y1), a(X, Y2)) -> U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2)) transform_in_ga(n(X1), n(X2)) -> U7_ga(X1, X2, transform_in_ga(X1, X2)) U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(o(X1, Y), o(X2, Y)) U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(o(X, Y1), o(X, Y2)) U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) -> transform_out_ga(a(X1, Y), a(X2, Y)) U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) -> transform_out_ga(a(X, Y1), a(X, Y2)) U7_ga(X1, X2, transform_out_ga(X1, X2)) -> transform_out_ga(n(X1), n(X2)) The argument filtering Pi contains the following mapping: transform_in_ga(x1, x2) = transform_in_ga(x1) n(x1) = n(x1) transform_out_ga(x1, x2) = transform_out_ga(x2) a(x1, x2) = a(x1, x2) o(x1, x2) = o(x1, x2) U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4) U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4) U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U7_ga(x1, x2, x3) = U7_ga(x3) CNFEQUIV_IN_GA(x1, x2) = CNFEQUIV_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) 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_GA(transform_out_ga(Z)) -> CNFEQUIV_IN_GA(Z) CNFEQUIV_IN_GA(X) -> U1_GA(transform_in_ga(X)) The TRS R consists of the following rules: transform_in_ga(n(n(X))) -> transform_out_ga(X) transform_in_ga(n(a(X, Y))) -> transform_out_ga(o(n(X), n(Y))) transform_in_ga(n(o(X, Y))) -> transform_out_ga(a(n(X), n(Y))) transform_in_ga(o(X, a(Y, Z))) -> transform_out_ga(a(o(X, Y), o(X, Z))) transform_in_ga(o(a(X, Y), Z)) -> transform_out_ga(a(o(X, Z), o(Y, Z))) transform_in_ga(o(X1, Y)) -> U3_ga(Y, transform_in_ga(X1)) transform_in_ga(o(X, Y1)) -> U4_ga(X, transform_in_ga(Y1)) transform_in_ga(a(X1, Y)) -> U5_ga(Y, transform_in_ga(X1)) transform_in_ga(a(X, Y1)) -> U6_ga(X, transform_in_ga(Y1)) transform_in_ga(n(X1)) -> U7_ga(transform_in_ga(X1)) U3_ga(Y, transform_out_ga(X2)) -> transform_out_ga(o(X2, Y)) U4_ga(X, transform_out_ga(Y2)) -> transform_out_ga(o(X, Y2)) U5_ga(Y, transform_out_ga(X2)) -> transform_out_ga(a(X2, Y)) U6_ga(X, transform_out_ga(Y2)) -> transform_out_ga(a(X, Y2)) U7_ga(transform_out_ga(X2)) -> transform_out_ga(n(X2)) The set Q consists of the following terms: transform_in_ga(x0) U3_ga(x0, x1) U4_ga(x0, x1) U5_ga(x0, x1) U6_ga(x0, x1) U7_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U1_GA(transform_out_ga(Z)) -> CNFEQUIV_IN_GA(Z) CNFEQUIV_IN_GA(X) -> U1_GA(transform_in_ga(X)) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. U1_GA(x1) = U1_GA(x1) transform_out_ga(x1) = transform_out_ga(x1) CNFEQUIV_IN_GA(x1) = CNFEQUIV_IN_GA(x1) transform_in_ga(x1) = x1 n(x1) = n(x1) a(x1, x2) = a(x1, x2) o(x1, x2) = o(x1, x2) U3_ga(x1, x2) = U3_ga(x1, x2) U4_ga(x1, x2) = U4_ga(x1, x2) U5_ga(x1, x2) = U5_ga(x1, x2) U6_ga(x1, x2) = U6_ga(x1, x2) U7_ga(x1) = U7_ga(x1) Recursive path order with status [RPO]. Quasi-Precedence: [n_1, U7_ga_1] > [o_2, U3_ga_2, U4_ga_2] > [a_2, U5_ga_2, U6_ga_2] > transform_out_ga_1 > CNFEQUIV_IN_GA_1 > U1_GA_1 Status: U1_GA_1: multiset status transform_out_ga_1: multiset status CNFEQUIV_IN_GA_1: multiset status n_1: [1] a_2: [2,1] o_2: multiset status U3_ga_2: multiset status U4_ga_2: multiset status U5_ga_2: [1,2] U6_ga_2: [2,1] U7_ga_1: [1] The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: transform_in_ga(n(n(X))) -> transform_out_ga(X) transform_in_ga(n(a(X, Y))) -> transform_out_ga(o(n(X), n(Y))) transform_in_ga(n(o(X, Y))) -> transform_out_ga(a(n(X), n(Y))) transform_in_ga(o(X, a(Y, Z))) -> transform_out_ga(a(o(X, Y), o(X, Z))) transform_in_ga(o(a(X, Y), Z)) -> transform_out_ga(a(o(X, Z), o(Y, Z))) transform_in_ga(o(X1, Y)) -> U3_ga(Y, transform_in_ga(X1)) transform_in_ga(o(X, Y1)) -> U4_ga(X, transform_in_ga(Y1)) transform_in_ga(a(X1, Y)) -> U5_ga(Y, transform_in_ga(X1)) transform_in_ga(a(X, Y1)) -> U6_ga(X, transform_in_ga(Y1)) transform_in_ga(n(X1)) -> U7_ga(transform_in_ga(X1)) U4_ga(X, transform_out_ga(Y2)) -> transform_out_ga(o(X, Y2)) U3_ga(Y, transform_out_ga(X2)) -> transform_out_ga(o(X2, Y)) U5_ga(Y, transform_out_ga(X2)) -> transform_out_ga(a(X2, Y)) U6_ga(X, transform_out_ga(Y2)) -> transform_out_ga(a(X, Y2)) U7_ga(transform_out_ga(X2)) -> transform_out_ga(n(X2)) ---------------------------------------- (20) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: transform_in_ga(n(n(X))) -> transform_out_ga(X) transform_in_ga(n(a(X, Y))) -> transform_out_ga(o(n(X), n(Y))) transform_in_ga(n(o(X, Y))) -> transform_out_ga(a(n(X), n(Y))) transform_in_ga(o(X, a(Y, Z))) -> transform_out_ga(a(o(X, Y), o(X, Z))) transform_in_ga(o(a(X, Y), Z)) -> transform_out_ga(a(o(X, Z), o(Y, Z))) transform_in_ga(o(X1, Y)) -> U3_ga(Y, transform_in_ga(X1)) transform_in_ga(o(X, Y1)) -> U4_ga(X, transform_in_ga(Y1)) transform_in_ga(a(X1, Y)) -> U5_ga(Y, transform_in_ga(X1)) transform_in_ga(a(X, Y1)) -> U6_ga(X, transform_in_ga(Y1)) transform_in_ga(n(X1)) -> U7_ga(transform_in_ga(X1)) U3_ga(Y, transform_out_ga(X2)) -> transform_out_ga(o(X2, Y)) U4_ga(X, transform_out_ga(Y2)) -> transform_out_ga(o(X, Y2)) U5_ga(Y, transform_out_ga(X2)) -> transform_out_ga(a(X2, Y)) U6_ga(X, transform_out_ga(Y2)) -> transform_out_ga(a(X, Y2)) U7_ga(transform_out_ga(X2)) -> transform_out_ga(n(X2)) The set Q consists of the following terms: transform_in_ga(x0) U3_ga(x0, x1) U4_ga(x0, x1) U5_ga(x0, x1) U6_ga(x0, x1) U7_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (22) YES