/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern prime(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 13 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 36 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 (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, 2 ms] (39) QDP (40) QDPQMonotonicMRRProof [EQUIVALENT, 62 ms] (41) QDP (42) DependencyGraphProof [EQUIVALENT, 0 ms] (43) TRUE ---------------------------------------- (0) Obligation: Clauses: div(X, Y, Z) :- quot(X, Y, Y, Z). quot(0, s(Y), s(Z), 0). quot(s(X), s(Y), Z, U) :- quot(X, Y, Z, U). quot(X, 0, s(Z), s(U)) :- quot(X, s(Z), s(Z), U). prime(s(s(X))) :- pr(s(s(X)), s(X)). pr(X, s(0)). pr(X, s(s(Y))) :- ','(not_divides(s(s(Y)), X), pr(X, s(Y))). not_divides(Y, X) :- ','(div(X, Y, U), ','(times(U, Y, Z), neq(X, Z))). neq(s(X), 0). neq(0, s(X)). neq(s(X), s(Y)) :- neq(X, Y). times(0, Y, 0). times(s(X), Y, Z) :- ','(times(X, Y, U), add(U, Y, Z)). add(X, 0, X). add(0, X, X). add(s(X), Y, s(Z)) :- add(X, Y, Z). Query: prime(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: prime_in_1: (b) pr_in_2: (b,b) not_divides_in_2: (b,b) div_in_3: (b,b,f) quot_in_4: (b,b,b,f) times_in_3: (b,b,f) add_in_3: (b,b,f) neq_in_2: (b,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: prime_in_g(s(s(X))) -> U4_g(X, pr_in_gg(s(s(X)), s(X))) pr_in_gg(X, s(0)) -> pr_out_gg(X, s(0)) pr_in_gg(X, s(s(Y))) -> U5_gg(X, Y, not_divides_in_gg(s(s(Y)), X)) not_divides_in_gg(Y, X) -> U7_gg(Y, X, div_in_gga(X, Y, U)) div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z)) quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0) quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U)) quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U)) U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U)) U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U) U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z) U7_gg(Y, X, div_out_gga(X, Y, U)) -> U8_gg(Y, X, times_in_gga(U, Y, Z)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U11_gga(X, Y, Z, times_in_gga(X, Y, U)) U11_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U12_gga(X, Y, Z, add_in_gga(U, Y, Z)) add_in_gga(X, 0, X) -> add_out_gga(X, 0, X) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z)) U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U12_gga(X, Y, Z, add_out_gga(U, Y, Z)) -> times_out_gga(s(X), Y, Z) U8_gg(Y, X, times_out_gga(U, Y, Z)) -> U9_gg(Y, X, neq_in_gg(X, Z)) neq_in_gg(s(X), 0) -> neq_out_gg(s(X), 0) neq_in_gg(0, s(X)) -> neq_out_gg(0, s(X)) neq_in_gg(s(X), s(Y)) -> U10_gg(X, Y, neq_in_gg(X, Y)) U10_gg(X, Y, neq_out_gg(X, Y)) -> neq_out_gg(s(X), s(Y)) U9_gg(Y, X, neq_out_gg(X, Z)) -> not_divides_out_gg(Y, X) U5_gg(X, Y, not_divides_out_gg(s(s(Y)), X)) -> U6_gg(X, Y, pr_in_gg(X, s(Y))) U6_gg(X, Y, pr_out_gg(X, s(Y))) -> pr_out_gg(X, s(s(Y))) U4_g(X, pr_out_gg(s(s(X)), s(X))) -> prime_out_g(s(s(X))) The argument filtering Pi contains the following mapping: prime_in_g(x1) = prime_in_g(x1) s(x1) = s(x1) U4_g(x1, x2) = U4_g(x2) pr_in_gg(x1, x2) = pr_in_gg(x1, x2) 0 = 0 pr_out_gg(x1, x2) = pr_out_gg U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) not_divides_in_gg(x1, x2) = not_divides_in_gg(x1, x2) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) quot_in_ggga(x1, x2, x3, x4) = quot_in_ggga(x1, x2, x3) quot_out_ggga(x1, x2, x3, x4) = quot_out_ggga(x4) U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) U8_gg(x1, x2, x3) = U8_gg(x2, x3) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x3) U11_gga(x1, x2, x3, x4) = U11_gga(x2, x4) U12_gga(x1, x2, x3, x4) = U12_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U13_gga(x1, x2, x3, x4) = U13_gga(x4) U9_gg(x1, x2, x3) = U9_gg(x3) neq_in_gg(x1, x2) = neq_in_gg(x1, x2) neq_out_gg(x1, x2) = neq_out_gg U10_gg(x1, x2, x3) = U10_gg(x3) not_divides_out_gg(x1, x2) = not_divides_out_gg U6_gg(x1, x2, x3) = U6_gg(x3) prime_out_g(x1) = prime_out_g Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: prime_in_g(s(s(X))) -> U4_g(X, pr_in_gg(s(s(X)), s(X))) pr_in_gg(X, s(0)) -> pr_out_gg(X, s(0)) pr_in_gg(X, s(s(Y))) -> U5_gg(X, Y, not_divides_in_gg(s(s(Y)), X)) not_divides_in_gg(Y, X) -> U7_gg(Y, X, div_in_gga(X, Y, U)) div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z)) quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0) quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U)) quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U)) U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U)) U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U) U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z) U7_gg(Y, X, div_out_gga(X, Y, U)) -> U8_gg(Y, X, times_in_gga(U, Y, Z)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U11_gga(X, Y, Z, times_in_gga(X, Y, U)) U11_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U12_gga(X, Y, Z, add_in_gga(U, Y, Z)) add_in_gga(X, 0, X) -> add_out_gga(X, 0, X) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z)) U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U12_gga(X, Y, Z, add_out_gga(U, Y, Z)) -> times_out_gga(s(X), Y, Z) U8_gg(Y, X, times_out_gga(U, Y, Z)) -> U9_gg(Y, X, neq_in_gg(X, Z)) neq_in_gg(s(X), 0) -> neq_out_gg(s(X), 0) neq_in_gg(0, s(X)) -> neq_out_gg(0, s(X)) neq_in_gg(s(X), s(Y)) -> U10_gg(X, Y, neq_in_gg(X, Y)) U10_gg(X, Y, neq_out_gg(X, Y)) -> neq_out_gg(s(X), s(Y)) U9_gg(Y, X, neq_out_gg(X, Z)) -> not_divides_out_gg(Y, X) U5_gg(X, Y, not_divides_out_gg(s(s(Y)), X)) -> U6_gg(X, Y, pr_in_gg(X, s(Y))) U6_gg(X, Y, pr_out_gg(X, s(Y))) -> pr_out_gg(X, s(s(Y))) U4_g(X, pr_out_gg(s(s(X)), s(X))) -> prime_out_g(s(s(X))) The argument filtering Pi contains the following mapping: prime_in_g(x1) = prime_in_g(x1) s(x1) = s(x1) U4_g(x1, x2) = U4_g(x2) pr_in_gg(x1, x2) = pr_in_gg(x1, x2) 0 = 0 pr_out_gg(x1, x2) = pr_out_gg U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) not_divides_in_gg(x1, x2) = not_divides_in_gg(x1, x2) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) quot_in_ggga(x1, x2, x3, x4) = quot_in_ggga(x1, x2, x3) quot_out_ggga(x1, x2, x3, x4) = quot_out_ggga(x4) U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) U8_gg(x1, x2, x3) = U8_gg(x2, x3) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x3) U11_gga(x1, x2, x3, x4) = U11_gga(x2, x4) U12_gga(x1, x2, x3, x4) = U12_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U13_gga(x1, x2, x3, x4) = U13_gga(x4) U9_gg(x1, x2, x3) = U9_gg(x3) neq_in_gg(x1, x2) = neq_in_gg(x1, x2) neq_out_gg(x1, x2) = neq_out_gg U10_gg(x1, x2, x3) = U10_gg(x3) not_divides_out_gg(x1, x2) = not_divides_out_gg U6_gg(x1, x2, x3) = U6_gg(x3) prime_out_g(x1) = prime_out_g ---------------------------------------- (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: PRIME_IN_G(s(s(X))) -> U4_G(X, pr_in_gg(s(s(X)), s(X))) PRIME_IN_G(s(s(X))) -> PR_IN_GG(s(s(X)), s(X)) PR_IN_GG(X, s(s(Y))) -> U5_GG(X, Y, not_divides_in_gg(s(s(Y)), X)) PR_IN_GG(X, s(s(Y))) -> NOT_DIVIDES_IN_GG(s(s(Y)), X) NOT_DIVIDES_IN_GG(Y, X) -> U7_GG(Y, X, div_in_gga(X, Y, U)) NOT_DIVIDES_IN_GG(Y, X) -> DIV_IN_GGA(X, Y, U) DIV_IN_GGA(X, Y, Z) -> U1_GGA(X, Y, Z, quot_in_ggga(X, Y, Y, Z)) DIV_IN_GGA(X, Y, Z) -> QUOT_IN_GGGA(X, Y, Y, Z) QUOT_IN_GGGA(s(X), s(Y), Z, U) -> U2_GGGA(X, Y, Z, U, quot_in_ggga(X, Y, Z, U)) QUOT_IN_GGGA(s(X), s(Y), Z, U) -> QUOT_IN_GGGA(X, Y, Z, U) QUOT_IN_GGGA(X, 0, s(Z), s(U)) -> U3_GGGA(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U)) QUOT_IN_GGGA(X, 0, s(Z), s(U)) -> QUOT_IN_GGGA(X, s(Z), s(Z), U) U7_GG(Y, X, div_out_gga(X, Y, U)) -> U8_GG(Y, X, times_in_gga(U, Y, Z)) U7_GG(Y, X, div_out_gga(X, Y, U)) -> TIMES_IN_GGA(U, Y, Z) TIMES_IN_GGA(s(X), Y, Z) -> U11_GGA(X, Y, Z, times_in_gga(X, Y, U)) TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) U11_GGA(X, Y, Z, times_out_gga(X, Y, U)) -> U12_GGA(X, Y, Z, add_in_gga(U, Y, Z)) U11_GGA(X, Y, Z, times_out_gga(X, Y, U)) -> ADD_IN_GGA(U, Y, Z) ADD_IN_GGA(s(X), Y, s(Z)) -> U13_GGA(X, Y, Z, add_in_gga(X, Y, Z)) ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) U8_GG(Y, X, times_out_gga(U, Y, Z)) -> U9_GG(Y, X, neq_in_gg(X, Z)) U8_GG(Y, X, times_out_gga(U, Y, Z)) -> NEQ_IN_GG(X, Z) NEQ_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, neq_in_gg(X, Y)) NEQ_IN_GG(s(X), s(Y)) -> NEQ_IN_GG(X, Y) U5_GG(X, Y, not_divides_out_gg(s(s(Y)), X)) -> U6_GG(X, Y, pr_in_gg(X, s(Y))) U5_GG(X, Y, not_divides_out_gg(s(s(Y)), X)) -> PR_IN_GG(X, s(Y)) The TRS R consists of the following rules: prime_in_g(s(s(X))) -> U4_g(X, pr_in_gg(s(s(X)), s(X))) pr_in_gg(X, s(0)) -> pr_out_gg(X, s(0)) pr_in_gg(X, s(s(Y))) -> U5_gg(X, Y, not_divides_in_gg(s(s(Y)), X)) not_divides_in_gg(Y, X) -> U7_gg(Y, X, div_in_gga(X, Y, U)) div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z)) quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0) quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U)) quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U)) U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U)) U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U) U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z) U7_gg(Y, X, div_out_gga(X, Y, U)) -> U8_gg(Y, X, times_in_gga(U, Y, Z)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U11_gga(X, Y, Z, times_in_gga(X, Y, U)) U11_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U12_gga(X, Y, Z, add_in_gga(U, Y, Z)) add_in_gga(X, 0, X) -> add_out_gga(X, 0, X) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z)) U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U12_gga(X, Y, Z, add_out_gga(U, Y, Z)) -> times_out_gga(s(X), Y, Z) U8_gg(Y, X, times_out_gga(U, Y, Z)) -> U9_gg(Y, X, neq_in_gg(X, Z)) neq_in_gg(s(X), 0) -> neq_out_gg(s(X), 0) neq_in_gg(0, s(X)) -> neq_out_gg(0, s(X)) neq_in_gg(s(X), s(Y)) -> U10_gg(X, Y, neq_in_gg(X, Y)) U10_gg(X, Y, neq_out_gg(X, Y)) -> neq_out_gg(s(X), s(Y)) U9_gg(Y, X, neq_out_gg(X, Z)) -> not_divides_out_gg(Y, X) U5_gg(X, Y, not_divides_out_gg(s(s(Y)), X)) -> U6_gg(X, Y, pr_in_gg(X, s(Y))) U6_gg(X, Y, pr_out_gg(X, s(Y))) -> pr_out_gg(X, s(s(Y))) U4_g(X, pr_out_gg(s(s(X)), s(X))) -> prime_out_g(s(s(X))) The argument filtering Pi contains the following mapping: prime_in_g(x1) = prime_in_g(x1) s(x1) = s(x1) U4_g(x1, x2) = U4_g(x2) pr_in_gg(x1, x2) = pr_in_gg(x1, x2) 0 = 0 pr_out_gg(x1, x2) = pr_out_gg U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) not_divides_in_gg(x1, x2) = not_divides_in_gg(x1, x2) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) quot_in_ggga(x1, x2, x3, x4) = quot_in_ggga(x1, x2, x3) quot_out_ggga(x1, x2, x3, x4) = quot_out_ggga(x4) U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) U8_gg(x1, x2, x3) = U8_gg(x2, x3) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x3) U11_gga(x1, x2, x3, x4) = U11_gga(x2, x4) U12_gga(x1, x2, x3, x4) = U12_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U13_gga(x1, x2, x3, x4) = U13_gga(x4) U9_gg(x1, x2, x3) = U9_gg(x3) neq_in_gg(x1, x2) = neq_in_gg(x1, x2) neq_out_gg(x1, x2) = neq_out_gg U10_gg(x1, x2, x3) = U10_gg(x3) not_divides_out_gg(x1, x2) = not_divides_out_gg U6_gg(x1, x2, x3) = U6_gg(x3) prime_out_g(x1) = prime_out_g PRIME_IN_G(x1) = PRIME_IN_G(x1) U4_G(x1, x2) = U4_G(x2) PR_IN_GG(x1, x2) = PR_IN_GG(x1, x2) U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3) NOT_DIVIDES_IN_GG(x1, x2) = NOT_DIVIDES_IN_GG(x1, x2) U7_GG(x1, x2, x3) = U7_GG(x1, x2, x3) DIV_IN_GGA(x1, x2, x3) = DIV_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x4) QUOT_IN_GGGA(x1, x2, x3, x4) = QUOT_IN_GGGA(x1, x2, x3) U2_GGGA(x1, x2, x3, x4, x5) = U2_GGGA(x5) U3_GGGA(x1, x2, x3, x4) = U3_GGGA(x4) U8_GG(x1, x2, x3) = U8_GG(x2, x3) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U11_GGA(x1, x2, x3, x4) = U11_GGA(x2, x4) U12_GGA(x1, x2, x3, x4) = U12_GGA(x4) ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) U13_GGA(x1, x2, x3, x4) = U13_GGA(x4) U9_GG(x1, x2, x3) = U9_GG(x3) NEQ_IN_GG(x1, x2) = NEQ_IN_GG(x1, x2) U10_GG(x1, x2, x3) = U10_GG(x3) U6_GG(x1, x2, x3) = U6_GG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: PRIME_IN_G(s(s(X))) -> U4_G(X, pr_in_gg(s(s(X)), s(X))) PRIME_IN_G(s(s(X))) -> PR_IN_GG(s(s(X)), s(X)) PR_IN_GG(X, s(s(Y))) -> U5_GG(X, Y, not_divides_in_gg(s(s(Y)), X)) PR_IN_GG(X, s(s(Y))) -> NOT_DIVIDES_IN_GG(s(s(Y)), X) NOT_DIVIDES_IN_GG(Y, X) -> U7_GG(Y, X, div_in_gga(X, Y, U)) NOT_DIVIDES_IN_GG(Y, X) -> DIV_IN_GGA(X, Y, U) DIV_IN_GGA(X, Y, Z) -> U1_GGA(X, Y, Z, quot_in_ggga(X, Y, Y, Z)) DIV_IN_GGA(X, Y, Z) -> QUOT_IN_GGGA(X, Y, Y, Z) QUOT_IN_GGGA(s(X), s(Y), Z, U) -> U2_GGGA(X, Y, Z, U, quot_in_ggga(X, Y, Z, U)) QUOT_IN_GGGA(s(X), s(Y), Z, U) -> QUOT_IN_GGGA(X, Y, Z, U) QUOT_IN_GGGA(X, 0, s(Z), s(U)) -> U3_GGGA(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U)) QUOT_IN_GGGA(X, 0, s(Z), s(U)) -> QUOT_IN_GGGA(X, s(Z), s(Z), U) U7_GG(Y, X, div_out_gga(X, Y, U)) -> U8_GG(Y, X, times_in_gga(U, Y, Z)) U7_GG(Y, X, div_out_gga(X, Y, U)) -> TIMES_IN_GGA(U, Y, Z) TIMES_IN_GGA(s(X), Y, Z) -> U11_GGA(X, Y, Z, times_in_gga(X, Y, U)) TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) U11_GGA(X, Y, Z, times_out_gga(X, Y, U)) -> U12_GGA(X, Y, Z, add_in_gga(U, Y, Z)) U11_GGA(X, Y, Z, times_out_gga(X, Y, U)) -> ADD_IN_GGA(U, Y, Z) ADD_IN_GGA(s(X), Y, s(Z)) -> U13_GGA(X, Y, Z, add_in_gga(X, Y, Z)) ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) U8_GG(Y, X, times_out_gga(U, Y, Z)) -> U9_GG(Y, X, neq_in_gg(X, Z)) U8_GG(Y, X, times_out_gga(U, Y, Z)) -> NEQ_IN_GG(X, Z) NEQ_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, neq_in_gg(X, Y)) NEQ_IN_GG(s(X), s(Y)) -> NEQ_IN_GG(X, Y) U5_GG(X, Y, not_divides_out_gg(s(s(Y)), X)) -> U6_GG(X, Y, pr_in_gg(X, s(Y))) U5_GG(X, Y, not_divides_out_gg(s(s(Y)), X)) -> PR_IN_GG(X, s(Y)) The TRS R consists of the following rules: prime_in_g(s(s(X))) -> U4_g(X, pr_in_gg(s(s(X)), s(X))) pr_in_gg(X, s(0)) -> pr_out_gg(X, s(0)) pr_in_gg(X, s(s(Y))) -> U5_gg(X, Y, not_divides_in_gg(s(s(Y)), X)) not_divides_in_gg(Y, X) -> U7_gg(Y, X, div_in_gga(X, Y, U)) div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z)) quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0) quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U)) quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U)) U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U)) U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U) U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z) U7_gg(Y, X, div_out_gga(X, Y, U)) -> U8_gg(Y, X, times_in_gga(U, Y, Z)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U11_gga(X, Y, Z, times_in_gga(X, Y, U)) U11_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U12_gga(X, Y, Z, add_in_gga(U, Y, Z)) add_in_gga(X, 0, X) -> add_out_gga(X, 0, X) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z)) U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U12_gga(X, Y, Z, add_out_gga(U, Y, Z)) -> times_out_gga(s(X), Y, Z) U8_gg(Y, X, times_out_gga(U, Y, Z)) -> U9_gg(Y, X, neq_in_gg(X, Z)) neq_in_gg(s(X), 0) -> neq_out_gg(s(X), 0) neq_in_gg(0, s(X)) -> neq_out_gg(0, s(X)) neq_in_gg(s(X), s(Y)) -> U10_gg(X, Y, neq_in_gg(X, Y)) U10_gg(X, Y, neq_out_gg(X, Y)) -> neq_out_gg(s(X), s(Y)) U9_gg(Y, X, neq_out_gg(X, Z)) -> not_divides_out_gg(Y, X) U5_gg(X, Y, not_divides_out_gg(s(s(Y)), X)) -> U6_gg(X, Y, pr_in_gg(X, s(Y))) U6_gg(X, Y, pr_out_gg(X, s(Y))) -> pr_out_gg(X, s(s(Y))) U4_g(X, pr_out_gg(s(s(X)), s(X))) -> prime_out_g(s(s(X))) The argument filtering Pi contains the following mapping: prime_in_g(x1) = prime_in_g(x1) s(x1) = s(x1) U4_g(x1, x2) = U4_g(x2) pr_in_gg(x1, x2) = pr_in_gg(x1, x2) 0 = 0 pr_out_gg(x1, x2) = pr_out_gg U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) not_divides_in_gg(x1, x2) = not_divides_in_gg(x1, x2) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) quot_in_ggga(x1, x2, x3, x4) = quot_in_ggga(x1, x2, x3) quot_out_ggga(x1, x2, x3, x4) = quot_out_ggga(x4) U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) U8_gg(x1, x2, x3) = U8_gg(x2, x3) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x3) U11_gga(x1, x2, x3, x4) = U11_gga(x2, x4) U12_gga(x1, x2, x3, x4) = U12_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U13_gga(x1, x2, x3, x4) = U13_gga(x4) U9_gg(x1, x2, x3) = U9_gg(x3) neq_in_gg(x1, x2) = neq_in_gg(x1, x2) neq_out_gg(x1, x2) = neq_out_gg U10_gg(x1, x2, x3) = U10_gg(x3) not_divides_out_gg(x1, x2) = not_divides_out_gg U6_gg(x1, x2, x3) = U6_gg(x3) prime_out_g(x1) = prime_out_g PRIME_IN_G(x1) = PRIME_IN_G(x1) U4_G(x1, x2) = U4_G(x2) PR_IN_GG(x1, x2) = PR_IN_GG(x1, x2) U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3) NOT_DIVIDES_IN_GG(x1, x2) = NOT_DIVIDES_IN_GG(x1, x2) U7_GG(x1, x2, x3) = U7_GG(x1, x2, x3) DIV_IN_GGA(x1, x2, x3) = DIV_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x4) QUOT_IN_GGGA(x1, x2, x3, x4) = QUOT_IN_GGGA(x1, x2, x3) U2_GGGA(x1, x2, x3, x4, x5) = U2_GGGA(x5) U3_GGGA(x1, x2, x3, x4) = U3_GGGA(x4) U8_GG(x1, x2, x3) = U8_GG(x2, x3) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U11_GGA(x1, x2, x3, x4) = U11_GGA(x2, x4) U12_GGA(x1, x2, x3, x4) = U12_GGA(x4) ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) U13_GGA(x1, x2, x3, x4) = U13_GGA(x4) U9_GG(x1, x2, x3) = U9_GG(x3) NEQ_IN_GG(x1, x2) = NEQ_IN_GG(x1, x2) U10_GG(x1, x2, x3) = U10_GG(x3) U6_GG(x1, x2, x3) = U6_GG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 19 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: NEQ_IN_GG(s(X), s(Y)) -> NEQ_IN_GG(X, Y) The TRS R consists of the following rules: prime_in_g(s(s(X))) -> U4_g(X, pr_in_gg(s(s(X)), s(X))) pr_in_gg(X, s(0)) -> pr_out_gg(X, s(0)) pr_in_gg(X, s(s(Y))) -> U5_gg(X, Y, not_divides_in_gg(s(s(Y)), X)) not_divides_in_gg(Y, X) -> U7_gg(Y, X, div_in_gga(X, Y, U)) div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z)) quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0) quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U)) quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U)) U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U)) U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U) U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z) U7_gg(Y, X, div_out_gga(X, Y, U)) -> U8_gg(Y, X, times_in_gga(U, Y, Z)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U11_gga(X, Y, Z, times_in_gga(X, Y, U)) U11_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U12_gga(X, Y, Z, add_in_gga(U, Y, Z)) add_in_gga(X, 0, X) -> add_out_gga(X, 0, X) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z)) U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U12_gga(X, Y, Z, add_out_gga(U, Y, Z)) -> times_out_gga(s(X), Y, Z) U8_gg(Y, X, times_out_gga(U, Y, Z)) -> U9_gg(Y, X, neq_in_gg(X, Z)) neq_in_gg(s(X), 0) -> neq_out_gg(s(X), 0) neq_in_gg(0, s(X)) -> neq_out_gg(0, s(X)) neq_in_gg(s(X), s(Y)) -> U10_gg(X, Y, neq_in_gg(X, Y)) U10_gg(X, Y, neq_out_gg(X, Y)) -> neq_out_gg(s(X), s(Y)) U9_gg(Y, X, neq_out_gg(X, Z)) -> not_divides_out_gg(Y, X) U5_gg(X, Y, not_divides_out_gg(s(s(Y)), X)) -> U6_gg(X, Y, pr_in_gg(X, s(Y))) U6_gg(X, Y, pr_out_gg(X, s(Y))) -> pr_out_gg(X, s(s(Y))) U4_g(X, pr_out_gg(s(s(X)), s(X))) -> prime_out_g(s(s(X))) The argument filtering Pi contains the following mapping: prime_in_g(x1) = prime_in_g(x1) s(x1) = s(x1) U4_g(x1, x2) = U4_g(x2) pr_in_gg(x1, x2) = pr_in_gg(x1, x2) 0 = 0 pr_out_gg(x1, x2) = pr_out_gg U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) not_divides_in_gg(x1, x2) = not_divides_in_gg(x1, x2) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) quot_in_ggga(x1, x2, x3, x4) = quot_in_ggga(x1, x2, x3) quot_out_ggga(x1, x2, x3, x4) = quot_out_ggga(x4) U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) U8_gg(x1, x2, x3) = U8_gg(x2, x3) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x3) U11_gga(x1, x2, x3, x4) = U11_gga(x2, x4) U12_gga(x1, x2, x3, x4) = U12_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U13_gga(x1, x2, x3, x4) = U13_gga(x4) U9_gg(x1, x2, x3) = U9_gg(x3) neq_in_gg(x1, x2) = neq_in_gg(x1, x2) neq_out_gg(x1, x2) = neq_out_gg U10_gg(x1, x2, x3) = U10_gg(x3) not_divides_out_gg(x1, x2) = not_divides_out_gg U6_gg(x1, x2, x3) = U6_gg(x3) prime_out_g(x1) = prime_out_g NEQ_IN_GG(x1, x2) = NEQ_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: NEQ_IN_GG(s(X), s(Y)) -> NEQ_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: NEQ_IN_GG(s(X), s(Y)) -> NEQ_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: *NEQ_IN_GG(s(X), s(Y)) -> NEQ_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: ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) The TRS R consists of the following rules: prime_in_g(s(s(X))) -> U4_g(X, pr_in_gg(s(s(X)), s(X))) pr_in_gg(X, s(0)) -> pr_out_gg(X, s(0)) pr_in_gg(X, s(s(Y))) -> U5_gg(X, Y, not_divides_in_gg(s(s(Y)), X)) not_divides_in_gg(Y, X) -> U7_gg(Y, X, div_in_gga(X, Y, U)) div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z)) quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0) quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U)) quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U)) U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U)) U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U) U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z) U7_gg(Y, X, div_out_gga(X, Y, U)) -> U8_gg(Y, X, times_in_gga(U, Y, Z)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U11_gga(X, Y, Z, times_in_gga(X, Y, U)) U11_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U12_gga(X, Y, Z, add_in_gga(U, Y, Z)) add_in_gga(X, 0, X) -> add_out_gga(X, 0, X) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z)) U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U12_gga(X, Y, Z, add_out_gga(U, Y, Z)) -> times_out_gga(s(X), Y, Z) U8_gg(Y, X, times_out_gga(U, Y, Z)) -> U9_gg(Y, X, neq_in_gg(X, Z)) neq_in_gg(s(X), 0) -> neq_out_gg(s(X), 0) neq_in_gg(0, s(X)) -> neq_out_gg(0, s(X)) neq_in_gg(s(X), s(Y)) -> U10_gg(X, Y, neq_in_gg(X, Y)) U10_gg(X, Y, neq_out_gg(X, Y)) -> neq_out_gg(s(X), s(Y)) U9_gg(Y, X, neq_out_gg(X, Z)) -> not_divides_out_gg(Y, X) U5_gg(X, Y, not_divides_out_gg(s(s(Y)), X)) -> U6_gg(X, Y, pr_in_gg(X, s(Y))) U6_gg(X, Y, pr_out_gg(X, s(Y))) -> pr_out_gg(X, s(s(Y))) U4_g(X, pr_out_gg(s(s(X)), s(X))) -> prime_out_g(s(s(X))) The argument filtering Pi contains the following mapping: prime_in_g(x1) = prime_in_g(x1) s(x1) = s(x1) U4_g(x1, x2) = U4_g(x2) pr_in_gg(x1, x2) = pr_in_gg(x1, x2) 0 = 0 pr_out_gg(x1, x2) = pr_out_gg U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) not_divides_in_gg(x1, x2) = not_divides_in_gg(x1, x2) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) quot_in_ggga(x1, x2, x3, x4) = quot_in_ggga(x1, x2, x3) quot_out_ggga(x1, x2, x3, x4) = quot_out_ggga(x4) U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) U8_gg(x1, x2, x3) = U8_gg(x2, x3) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x3) U11_gga(x1, x2, x3, x4) = U11_gga(x2, x4) U12_gga(x1, x2, x3, x4) = U12_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U13_gga(x1, x2, x3, x4) = U13_gga(x4) U9_gg(x1, x2, x3) = U9_gg(x3) neq_in_gg(x1, x2) = neq_in_gg(x1, x2) neq_out_gg(x1, x2) = neq_out_gg U10_gg(x1, x2, x3) = U10_gg(x3) not_divides_out_gg(x1, x2) = not_divides_out_gg U6_gg(x1, x2, x3) = U6_gg(x3) prime_out_g(x1) = prime_out_g ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(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: ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) 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: ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(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: *ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(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: TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) The TRS R consists of the following rules: prime_in_g(s(s(X))) -> U4_g(X, pr_in_gg(s(s(X)), s(X))) pr_in_gg(X, s(0)) -> pr_out_gg(X, s(0)) pr_in_gg(X, s(s(Y))) -> U5_gg(X, Y, not_divides_in_gg(s(s(Y)), X)) not_divides_in_gg(Y, X) -> U7_gg(Y, X, div_in_gga(X, Y, U)) div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z)) quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0) quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U)) quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U)) U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U)) U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U) U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z) U7_gg(Y, X, div_out_gga(X, Y, U)) -> U8_gg(Y, X, times_in_gga(U, Y, Z)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U11_gga(X, Y, Z, times_in_gga(X, Y, U)) U11_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U12_gga(X, Y, Z, add_in_gga(U, Y, Z)) add_in_gga(X, 0, X) -> add_out_gga(X, 0, X) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z)) U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U12_gga(X, Y, Z, add_out_gga(U, Y, Z)) -> times_out_gga(s(X), Y, Z) U8_gg(Y, X, times_out_gga(U, Y, Z)) -> U9_gg(Y, X, neq_in_gg(X, Z)) neq_in_gg(s(X), 0) -> neq_out_gg(s(X), 0) neq_in_gg(0, s(X)) -> neq_out_gg(0, s(X)) neq_in_gg(s(X), s(Y)) -> U10_gg(X, Y, neq_in_gg(X, Y)) U10_gg(X, Y, neq_out_gg(X, Y)) -> neq_out_gg(s(X), s(Y)) U9_gg(Y, X, neq_out_gg(X, Z)) -> not_divides_out_gg(Y, X) U5_gg(X, Y, not_divides_out_gg(s(s(Y)), X)) -> U6_gg(X, Y, pr_in_gg(X, s(Y))) U6_gg(X, Y, pr_out_gg(X, s(Y))) -> pr_out_gg(X, s(s(Y))) U4_g(X, pr_out_gg(s(s(X)), s(X))) -> prime_out_g(s(s(X))) The argument filtering Pi contains the following mapping: prime_in_g(x1) = prime_in_g(x1) s(x1) = s(x1) U4_g(x1, x2) = U4_g(x2) pr_in_gg(x1, x2) = pr_in_gg(x1, x2) 0 = 0 pr_out_gg(x1, x2) = pr_out_gg U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) not_divides_in_gg(x1, x2) = not_divides_in_gg(x1, x2) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) quot_in_ggga(x1, x2, x3, x4) = quot_in_ggga(x1, x2, x3) quot_out_ggga(x1, x2, x3, x4) = quot_out_ggga(x4) U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) U8_gg(x1, x2, x3) = U8_gg(x2, x3) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x3) U11_gga(x1, x2, x3, x4) = U11_gga(x2, x4) U12_gga(x1, x2, x3, x4) = U12_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U13_gga(x1, x2, x3, x4) = U13_gga(x4) U9_gg(x1, x2, x3) = U9_gg(x3) neq_in_gg(x1, x2) = neq_in_gg(x1, x2) neq_out_gg(x1, x2) = neq_out_gg U10_gg(x1, x2, x3) = U10_gg(x3) not_divides_out_gg(x1, x2) = not_divides_out_gg U6_gg(x1, x2, x3) = U6_gg(x3) prime_out_g(x1) = prime_out_g TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) 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: TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) 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: TIMES_IN_GGA(s(X), Y) -> TIMES_IN_GGA(X, Y) R is empty. Q is empty. 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: *TIMES_IN_GGA(s(X), Y) -> TIMES_IN_GGA(X, Y) 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: QUOT_IN_GGGA(X, 0, s(Z), s(U)) -> QUOT_IN_GGGA(X, s(Z), s(Z), U) QUOT_IN_GGGA(s(X), s(Y), Z, U) -> QUOT_IN_GGGA(X, Y, Z, U) The TRS R consists of the following rules: prime_in_g(s(s(X))) -> U4_g(X, pr_in_gg(s(s(X)), s(X))) pr_in_gg(X, s(0)) -> pr_out_gg(X, s(0)) pr_in_gg(X, s(s(Y))) -> U5_gg(X, Y, not_divides_in_gg(s(s(Y)), X)) not_divides_in_gg(Y, X) -> U7_gg(Y, X, div_in_gga(X, Y, U)) div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z)) quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0) quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U)) quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U)) U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U)) U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U) U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z) U7_gg(Y, X, div_out_gga(X, Y, U)) -> U8_gg(Y, X, times_in_gga(U, Y, Z)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U11_gga(X, Y, Z, times_in_gga(X, Y, U)) U11_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U12_gga(X, Y, Z, add_in_gga(U, Y, Z)) add_in_gga(X, 0, X) -> add_out_gga(X, 0, X) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z)) U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U12_gga(X, Y, Z, add_out_gga(U, Y, Z)) -> times_out_gga(s(X), Y, Z) U8_gg(Y, X, times_out_gga(U, Y, Z)) -> U9_gg(Y, X, neq_in_gg(X, Z)) neq_in_gg(s(X), 0) -> neq_out_gg(s(X), 0) neq_in_gg(0, s(X)) -> neq_out_gg(0, s(X)) neq_in_gg(s(X), s(Y)) -> U10_gg(X, Y, neq_in_gg(X, Y)) U10_gg(X, Y, neq_out_gg(X, Y)) -> neq_out_gg(s(X), s(Y)) U9_gg(Y, X, neq_out_gg(X, Z)) -> not_divides_out_gg(Y, X) U5_gg(X, Y, not_divides_out_gg(s(s(Y)), X)) -> U6_gg(X, Y, pr_in_gg(X, s(Y))) U6_gg(X, Y, pr_out_gg(X, s(Y))) -> pr_out_gg(X, s(s(Y))) U4_g(X, pr_out_gg(s(s(X)), s(X))) -> prime_out_g(s(s(X))) The argument filtering Pi contains the following mapping: prime_in_g(x1) = prime_in_g(x1) s(x1) = s(x1) U4_g(x1, x2) = U4_g(x2) pr_in_gg(x1, x2) = pr_in_gg(x1, x2) 0 = 0 pr_out_gg(x1, x2) = pr_out_gg U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) not_divides_in_gg(x1, x2) = not_divides_in_gg(x1, x2) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) quot_in_ggga(x1, x2, x3, x4) = quot_in_ggga(x1, x2, x3) quot_out_ggga(x1, x2, x3, x4) = quot_out_ggga(x4) U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) U8_gg(x1, x2, x3) = U8_gg(x2, x3) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x3) U11_gga(x1, x2, x3, x4) = U11_gga(x2, x4) U12_gga(x1, x2, x3, x4) = U12_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U13_gga(x1, x2, x3, x4) = U13_gga(x4) U9_gg(x1, x2, x3) = U9_gg(x3) neq_in_gg(x1, x2) = neq_in_gg(x1, x2) neq_out_gg(x1, x2) = neq_out_gg U10_gg(x1, x2, x3) = U10_gg(x3) not_divides_out_gg(x1, x2) = not_divides_out_gg U6_gg(x1, x2, x3) = U6_gg(x3) prime_out_g(x1) = prime_out_g QUOT_IN_GGGA(x1, x2, x3, x4) = QUOT_IN_GGGA(x1, x2, x3) 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: QUOT_IN_GGGA(X, 0, s(Z), s(U)) -> QUOT_IN_GGGA(X, s(Z), s(Z), U) QUOT_IN_GGGA(s(X), s(Y), Z, U) -> QUOT_IN_GGGA(X, Y, Z, U) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) 0 = 0 QUOT_IN_GGGA(x1, x2, x3, x4) = QUOT_IN_GGGA(x1, x2, x3) 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: QUOT_IN_GGGA(X, 0, s(Z)) -> QUOT_IN_GGGA(X, s(Z), s(Z)) QUOT_IN_GGGA(s(X), s(Y), Z) -> QUOT_IN_GGGA(X, Y, Z) 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: *QUOT_IN_GGGA(s(X), s(Y), Z) -> QUOT_IN_GGGA(X, Y, Z) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 *QUOT_IN_GGGA(X, 0, s(Z)) -> QUOT_IN_GGGA(X, s(Z), s(Z)) The graph contains the following edges 1 >= 1, 3 >= 2, 3 >= 3 ---------------------------------------- (34) YES ---------------------------------------- (35) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GG(X, Y, not_divides_out_gg(s(s(Y)), X)) -> PR_IN_GG(X, s(Y)) PR_IN_GG(X, s(s(Y))) -> U5_GG(X, Y, not_divides_in_gg(s(s(Y)), X)) The TRS R consists of the following rules: prime_in_g(s(s(X))) -> U4_g(X, pr_in_gg(s(s(X)), s(X))) pr_in_gg(X, s(0)) -> pr_out_gg(X, s(0)) pr_in_gg(X, s(s(Y))) -> U5_gg(X, Y, not_divides_in_gg(s(s(Y)), X)) not_divides_in_gg(Y, X) -> U7_gg(Y, X, div_in_gga(X, Y, U)) div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z)) quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0) quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U)) quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U)) U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U)) U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U) U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z) U7_gg(Y, X, div_out_gga(X, Y, U)) -> U8_gg(Y, X, times_in_gga(U, Y, Z)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U11_gga(X, Y, Z, times_in_gga(X, Y, U)) U11_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U12_gga(X, Y, Z, add_in_gga(U, Y, Z)) add_in_gga(X, 0, X) -> add_out_gga(X, 0, X) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z)) U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U12_gga(X, Y, Z, add_out_gga(U, Y, Z)) -> times_out_gga(s(X), Y, Z) U8_gg(Y, X, times_out_gga(U, Y, Z)) -> U9_gg(Y, X, neq_in_gg(X, Z)) neq_in_gg(s(X), 0) -> neq_out_gg(s(X), 0) neq_in_gg(0, s(X)) -> neq_out_gg(0, s(X)) neq_in_gg(s(X), s(Y)) -> U10_gg(X, Y, neq_in_gg(X, Y)) U10_gg(X, Y, neq_out_gg(X, Y)) -> neq_out_gg(s(X), s(Y)) U9_gg(Y, X, neq_out_gg(X, Z)) -> not_divides_out_gg(Y, X) U5_gg(X, Y, not_divides_out_gg(s(s(Y)), X)) -> U6_gg(X, Y, pr_in_gg(X, s(Y))) U6_gg(X, Y, pr_out_gg(X, s(Y))) -> pr_out_gg(X, s(s(Y))) U4_g(X, pr_out_gg(s(s(X)), s(X))) -> prime_out_g(s(s(X))) The argument filtering Pi contains the following mapping: prime_in_g(x1) = prime_in_g(x1) s(x1) = s(x1) U4_g(x1, x2) = U4_g(x2) pr_in_gg(x1, x2) = pr_in_gg(x1, x2) 0 = 0 pr_out_gg(x1, x2) = pr_out_gg U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) not_divides_in_gg(x1, x2) = not_divides_in_gg(x1, x2) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) quot_in_ggga(x1, x2, x3, x4) = quot_in_ggga(x1, x2, x3) quot_out_ggga(x1, x2, x3, x4) = quot_out_ggga(x4) U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) U8_gg(x1, x2, x3) = U8_gg(x2, x3) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x3) U11_gga(x1, x2, x3, x4) = U11_gga(x2, x4) U12_gga(x1, x2, x3, x4) = U12_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U13_gga(x1, x2, x3, x4) = U13_gga(x4) U9_gg(x1, x2, x3) = U9_gg(x3) neq_in_gg(x1, x2) = neq_in_gg(x1, x2) neq_out_gg(x1, x2) = neq_out_gg U10_gg(x1, x2, x3) = U10_gg(x3) not_divides_out_gg(x1, x2) = not_divides_out_gg U6_gg(x1, x2, x3) = U6_gg(x3) prime_out_g(x1) = prime_out_g PR_IN_GG(x1, x2) = PR_IN_GG(x1, x2) U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3) 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: U5_GG(X, Y, not_divides_out_gg(s(s(Y)), X)) -> PR_IN_GG(X, s(Y)) PR_IN_GG(X, s(s(Y))) -> U5_GG(X, Y, not_divides_in_gg(s(s(Y)), X)) The TRS R consists of the following rules: not_divides_in_gg(Y, X) -> U7_gg(Y, X, div_in_gga(X, Y, U)) U7_gg(Y, X, div_out_gga(X, Y, U)) -> U8_gg(Y, X, times_in_gga(U, Y, Z)) div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z)) U8_gg(Y, X, times_out_gga(U, Y, Z)) -> U9_gg(Y, X, neq_in_gg(X, Z)) U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U11_gga(X, Y, Z, times_in_gga(X, Y, U)) U9_gg(Y, X, neq_out_gg(X, Z)) -> not_divides_out_gg(Y, X) quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0) quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U)) quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U)) U11_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U12_gga(X, Y, Z, add_in_gga(U, Y, Z)) neq_in_gg(s(X), 0) -> neq_out_gg(s(X), 0) neq_in_gg(0, s(X)) -> neq_out_gg(0, s(X)) neq_in_gg(s(X), s(Y)) -> U10_gg(X, Y, neq_in_gg(X, Y)) U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U) U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U)) U12_gga(X, Y, Z, add_out_gga(U, Y, Z)) -> times_out_gga(s(X), Y, Z) U10_gg(X, Y, neq_out_gg(X, Y)) -> neq_out_gg(s(X), s(Y)) add_in_gga(X, 0, X) -> add_out_gga(X, 0, X) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z)) U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) 0 = 0 not_divides_in_gg(x1, x2) = not_divides_in_gg(x1, x2) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) div_in_gga(x1, x2, x3) = div_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) quot_in_ggga(x1, x2, x3, x4) = quot_in_ggga(x1, x2, x3) quot_out_ggga(x1, x2, x3, x4) = quot_out_ggga(x4) U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5) U3_ggga(x1, x2, x3, x4) = U3_ggga(x4) div_out_gga(x1, x2, x3) = div_out_gga(x3) U8_gg(x1, x2, x3) = U8_gg(x2, x3) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x3) U11_gga(x1, x2, x3, x4) = U11_gga(x2, x4) U12_gga(x1, x2, x3, x4) = U12_gga(x4) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U13_gga(x1, x2, x3, x4) = U13_gga(x4) U9_gg(x1, x2, x3) = U9_gg(x3) neq_in_gg(x1, x2) = neq_in_gg(x1, x2) neq_out_gg(x1, x2) = neq_out_gg U10_gg(x1, x2, x3) = U10_gg(x3) not_divides_out_gg(x1, x2) = not_divides_out_gg PR_IN_GG(x1, x2) = PR_IN_GG(x1, x2) U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3) 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: U5_GG(X, Y, not_divides_out_gg) -> PR_IN_GG(X, s(Y)) PR_IN_GG(X, s(s(Y))) -> U5_GG(X, Y, not_divides_in_gg(s(s(Y)), X)) The TRS R consists of the following rules: not_divides_in_gg(Y, X) -> U7_gg(Y, X, div_in_gga(X, Y)) U7_gg(Y, X, div_out_gga(U)) -> U8_gg(X, times_in_gga(U, Y)) div_in_gga(X, Y) -> U1_gga(quot_in_ggga(X, Y, Y)) U8_gg(X, times_out_gga(Z)) -> U9_gg(neq_in_gg(X, Z)) U1_gga(quot_out_ggga(Z)) -> div_out_gga(Z) times_in_gga(0, Y) -> times_out_gga(0) times_in_gga(s(X), Y) -> U11_gga(Y, times_in_gga(X, Y)) U9_gg(neq_out_gg) -> not_divides_out_gg quot_in_ggga(0, s(Y), s(Z)) -> quot_out_ggga(0) quot_in_ggga(s(X), s(Y), Z) -> U2_ggga(quot_in_ggga(X, Y, Z)) quot_in_ggga(X, 0, s(Z)) -> U3_ggga(quot_in_ggga(X, s(Z), s(Z))) U11_gga(Y, times_out_gga(U)) -> U12_gga(add_in_gga(U, Y)) neq_in_gg(s(X), 0) -> neq_out_gg neq_in_gg(0, s(X)) -> neq_out_gg neq_in_gg(s(X), s(Y)) -> U10_gg(neq_in_gg(X, Y)) U2_ggga(quot_out_ggga(U)) -> quot_out_ggga(U) U3_ggga(quot_out_ggga(U)) -> quot_out_ggga(s(U)) U12_gga(add_out_gga(Z)) -> times_out_gga(Z) U10_gg(neq_out_gg) -> neq_out_gg add_in_gga(X, 0) -> add_out_gga(X) add_in_gga(0, X) -> add_out_gga(X) add_in_gga(s(X), Y) -> U13_gga(add_in_gga(X, Y)) U13_gga(add_out_gga(Z)) -> add_out_gga(s(Z)) The set Q consists of the following terms: not_divides_in_gg(x0, x1) U7_gg(x0, x1, x2) div_in_gga(x0, x1) U8_gg(x0, x1) U1_gga(x0) times_in_gga(x0, x1) U9_gg(x0) quot_in_ggga(x0, x1, x2) U11_gga(x0, x1) neq_in_gg(x0, x1) U2_ggga(x0) U3_ggga(x0) U12_gga(x0) U10_gg(x0) add_in_gga(x0, x1) U13_gga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (40) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: PR_IN_GG(X, s(s(Y))) -> U5_GG(X, Y, not_divides_in_gg(s(s(Y)), X)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(PR_IN_GG(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(U10_gg(x_1)) = 2 POL(U11_gga(x_1, x_2)) = 2 + x_1 POL(U12_gga(x_1)) = 2 POL(U13_gga(x_1)) = 2 POL(U1_gga(x_1)) = 0 POL(U2_ggga(x_1)) = 0 POL(U3_ggga(x_1)) = 2 POL(U5_GG(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 POL(U7_gg(x_1, x_2, x_3)) = 2 POL(U8_gg(x_1, x_2)) = 1 POL(U9_gg(x_1)) = 0 POL(add_in_gga(x_1, x_2)) = 2 + x_1 POL(add_out_gga(x_1)) = 0 POL(div_in_gga(x_1, x_2)) = 0 POL(div_out_gga(x_1)) = 0 POL(neq_in_gg(x_1, x_2)) = 2*x_1 + 2*x_2 POL(neq_out_gg) = 0 POL(not_divides_in_gg(x_1, x_2)) = 2 POL(not_divides_out_gg) = 0 POL(quot_in_ggga(x_1, x_2, x_3)) = 2 + 2*x_2 + 2*x_3 POL(quot_out_ggga(x_1)) = 0 POL(s(x_1)) = 1 + 2*x_1 POL(times_in_gga(x_1, x_2)) = 2*x_1 + x_2 POL(times_out_gga(x_1)) = 0 ---------------------------------------- (41) Obligation: Q DP problem: The TRS P consists of the following rules: U5_GG(X, Y, not_divides_out_gg) -> PR_IN_GG(X, s(Y)) The TRS R consists of the following rules: not_divides_in_gg(Y, X) -> U7_gg(Y, X, div_in_gga(X, Y)) U7_gg(Y, X, div_out_gga(U)) -> U8_gg(X, times_in_gga(U, Y)) div_in_gga(X, Y) -> U1_gga(quot_in_ggga(X, Y, Y)) U8_gg(X, times_out_gga(Z)) -> U9_gg(neq_in_gg(X, Z)) U1_gga(quot_out_ggga(Z)) -> div_out_gga(Z) times_in_gga(0, Y) -> times_out_gga(0) times_in_gga(s(X), Y) -> U11_gga(Y, times_in_gga(X, Y)) U9_gg(neq_out_gg) -> not_divides_out_gg quot_in_ggga(0, s(Y), s(Z)) -> quot_out_ggga(0) quot_in_ggga(s(X), s(Y), Z) -> U2_ggga(quot_in_ggga(X, Y, Z)) quot_in_ggga(X, 0, s(Z)) -> U3_ggga(quot_in_ggga(X, s(Z), s(Z))) U11_gga(Y, times_out_gga(U)) -> U12_gga(add_in_gga(U, Y)) neq_in_gg(s(X), 0) -> neq_out_gg neq_in_gg(0, s(X)) -> neq_out_gg neq_in_gg(s(X), s(Y)) -> U10_gg(neq_in_gg(X, Y)) U2_ggga(quot_out_ggga(U)) -> quot_out_ggga(U) U3_ggga(quot_out_ggga(U)) -> quot_out_ggga(s(U)) U12_gga(add_out_gga(Z)) -> times_out_gga(Z) U10_gg(neq_out_gg) -> neq_out_gg add_in_gga(X, 0) -> add_out_gga(X) add_in_gga(0, X) -> add_out_gga(X) add_in_gga(s(X), Y) -> U13_gga(add_in_gga(X, Y)) U13_gga(add_out_gga(Z)) -> add_out_gga(s(Z)) The set Q consists of the following terms: not_divides_in_gg(x0, x1) U7_gg(x0, x1, x2) div_in_gga(x0, x1) U8_gg(x0, x1) U1_gga(x0) times_in_gga(x0, x1) U9_gg(x0) quot_in_ggga(x0, x1, x2) U11_gga(x0, x1) neq_in_gg(x0, x1) U2_ggga(x0) U3_ggga(x0) U12_gga(x0) U10_gg(x0) add_in_gga(x0, x1) U13_gga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (42) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (43) TRUE