/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern queens(a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 48 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 [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 (42) PiDP (43) UsableRulesProof [EQUIVALENT, 0 ms] (44) PiDP (45) PiDPToQDPProof [SOUND, 0 ms] (46) QDP (47) MRRProof [EQUIVALENT, 57 ms] (48) QDP (49) DependencyGraphProof [EQUIVALENT, 0 ms] (50) TRUE ---------------------------------------- (0) Obligation: Clauses: queens(Y) :- ','(perm(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y), safe(Y)). perm([], []). perm(.(X, Y), .(V, Res)) :- ','(delete(V, .(X, Y), Rest), perm(Rest, Res)). delete(X, .(X, Y), Y). delete(X, .(F, T), .(F, R)) :- delete(X, T, R). safe([]). safe(.(X, Y)) :- ','(noattack(X, Y, s(0)), safe(Y)). noattack(X, [], N). noattack(X, .(F, T), N) :- ','(notEq(X, F), ','(add(F, N, FplusN), ','(notEq(X, FplusN), ','(add(X, N, XplusN), ','(notEq(F, XplusN), noattack(X, T, s(N))))))). add(0, X, X). add(s(X), Y, s(Z)) :- add(X, Y, Z). notEq(0, s(X)). notEq(s(X), 0). notEq(s(X), s(Y)) :- notEq(X, Y). Query: queens(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: queens_in_1: (f) perm_in_2: (b,f) delete_in_3: (f,b,f) safe_in_1: (b) noattack_in_3: (b,b,b) notEq_in_2: (b,b) add_in_3: (b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: queens_in_a(Y) -> U1_a(Y, perm_in_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(.(X, Y), .(V, Res)) -> U3_ga(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(X, .(F, T), .(F, R)) -> U5_aga(X, F, T, R, delete_in_aga(X, T, R)) U5_aga(X, F, T, R, delete_out_aga(X, T, R)) -> delete_out_aga(X, .(F, T), .(F, R)) U3_ga(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> U4_ga(X, Y, V, Res, perm_in_ga(Rest, Res)) U4_ga(X, Y, V, Res, perm_out_ga(Rest, Res)) -> perm_out_ga(.(X, Y), .(V, Res)) U1_a(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> U2_a(Y, safe_in_g(Y)) safe_in_g([]) -> safe_out_g([]) safe_in_g(.(X, Y)) -> U6_g(X, Y, noattack_in_ggg(X, Y, s(0))) noattack_in_ggg(X, [], N) -> noattack_out_ggg(X, [], N) noattack_in_ggg(X, .(F, T), N) -> U8_ggg(X, F, T, N, notEq_in_gg(X, F)) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(s(X), s(Y)) -> U15_gg(X, Y, notEq_in_gg(X, Y)) U15_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U8_ggg(X, F, T, N, notEq_out_gg(X, F)) -> U9_ggg(X, F, T, N, add_in_gga(F, N, FplusN)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U14_gga(X, Y, Z, add_in_gga(X, Y, Z)) U14_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U9_ggg(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_ggg(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U10_ggg(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_ggg(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U11_ggg(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_ggg(X, F, T, N, notEq_in_gg(F, XplusN)) U12_ggg(X, F, T, N, notEq_out_gg(F, XplusN)) -> U13_ggg(X, F, T, N, noattack_in_ggg(X, T, s(N))) U13_ggg(X, F, T, N, noattack_out_ggg(X, T, s(N))) -> noattack_out_ggg(X, .(F, T), N) U6_g(X, Y, noattack_out_ggg(X, Y, s(0))) -> U7_g(X, Y, safe_in_g(Y)) U7_g(X, Y, safe_out_g(Y)) -> safe_out_g(.(X, Y)) U2_a(Y, safe_out_g(Y)) -> queens_out_a(Y) The argument filtering Pi contains the following mapping: queens_in_a(x1) = queens_in_a U1_a(x1, x2) = U1_a(x2) perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) .(x1, x2) = .(x1, x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5) s(x1) = s(x1) 0 = 0 U2_a(x1, x2) = U2_a(x1, x2) safe_in_g(x1) = safe_in_g(x1) safe_out_g(x1) = safe_out_g U6_g(x1, x2, x3) = U6_g(x2, x3) noattack_in_ggg(x1, x2, x3) = noattack_in_ggg(x1, x2, x3) noattack_out_ggg(x1, x2, x3) = noattack_out_ggg U8_ggg(x1, x2, x3, x4, x5) = U8_ggg(x1, x2, x3, x4, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) notEq_out_gg(x1, x2) = notEq_out_gg U15_gg(x1, x2, x3) = U15_gg(x3) U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x1, x2, x3, x4, x5) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U14_gga(x1, x2, x3, x4) = U14_gga(x4) U10_ggg(x1, x2, x3, x4, x5, x6) = U10_ggg(x1, x2, x3, x4, x6) U11_ggg(x1, x2, x3, x4, x5, x6) = U11_ggg(x1, x2, x3, x4, x6) U12_ggg(x1, x2, x3, x4, x5) = U12_ggg(x1, x3, x4, x5) U13_ggg(x1, x2, x3, x4, x5) = U13_ggg(x5) U7_g(x1, x2, x3) = U7_g(x3) queens_out_a(x1) = queens_out_a(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: queens_in_a(Y) -> U1_a(Y, perm_in_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(.(X, Y), .(V, Res)) -> U3_ga(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(X, .(F, T), .(F, R)) -> U5_aga(X, F, T, R, delete_in_aga(X, T, R)) U5_aga(X, F, T, R, delete_out_aga(X, T, R)) -> delete_out_aga(X, .(F, T), .(F, R)) U3_ga(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> U4_ga(X, Y, V, Res, perm_in_ga(Rest, Res)) U4_ga(X, Y, V, Res, perm_out_ga(Rest, Res)) -> perm_out_ga(.(X, Y), .(V, Res)) U1_a(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> U2_a(Y, safe_in_g(Y)) safe_in_g([]) -> safe_out_g([]) safe_in_g(.(X, Y)) -> U6_g(X, Y, noattack_in_ggg(X, Y, s(0))) noattack_in_ggg(X, [], N) -> noattack_out_ggg(X, [], N) noattack_in_ggg(X, .(F, T), N) -> U8_ggg(X, F, T, N, notEq_in_gg(X, F)) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(s(X), s(Y)) -> U15_gg(X, Y, notEq_in_gg(X, Y)) U15_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U8_ggg(X, F, T, N, notEq_out_gg(X, F)) -> U9_ggg(X, F, T, N, add_in_gga(F, N, FplusN)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U14_gga(X, Y, Z, add_in_gga(X, Y, Z)) U14_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U9_ggg(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_ggg(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U10_ggg(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_ggg(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U11_ggg(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_ggg(X, F, T, N, notEq_in_gg(F, XplusN)) U12_ggg(X, F, T, N, notEq_out_gg(F, XplusN)) -> U13_ggg(X, F, T, N, noattack_in_ggg(X, T, s(N))) U13_ggg(X, F, T, N, noattack_out_ggg(X, T, s(N))) -> noattack_out_ggg(X, .(F, T), N) U6_g(X, Y, noattack_out_ggg(X, Y, s(0))) -> U7_g(X, Y, safe_in_g(Y)) U7_g(X, Y, safe_out_g(Y)) -> safe_out_g(.(X, Y)) U2_a(Y, safe_out_g(Y)) -> queens_out_a(Y) The argument filtering Pi contains the following mapping: queens_in_a(x1) = queens_in_a U1_a(x1, x2) = U1_a(x2) perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) .(x1, x2) = .(x1, x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5) s(x1) = s(x1) 0 = 0 U2_a(x1, x2) = U2_a(x1, x2) safe_in_g(x1) = safe_in_g(x1) safe_out_g(x1) = safe_out_g U6_g(x1, x2, x3) = U6_g(x2, x3) noattack_in_ggg(x1, x2, x3) = noattack_in_ggg(x1, x2, x3) noattack_out_ggg(x1, x2, x3) = noattack_out_ggg U8_ggg(x1, x2, x3, x4, x5) = U8_ggg(x1, x2, x3, x4, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) notEq_out_gg(x1, x2) = notEq_out_gg U15_gg(x1, x2, x3) = U15_gg(x3) U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x1, x2, x3, x4, x5) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U14_gga(x1, x2, x3, x4) = U14_gga(x4) U10_ggg(x1, x2, x3, x4, x5, x6) = U10_ggg(x1, x2, x3, x4, x6) U11_ggg(x1, x2, x3, x4, x5, x6) = U11_ggg(x1, x2, x3, x4, x6) U12_ggg(x1, x2, x3, x4, x5) = U12_ggg(x1, x3, x4, x5) U13_ggg(x1, x2, x3, x4, x5) = U13_ggg(x5) U7_g(x1, x2, x3) = U7_g(x3) queens_out_a(x1) = queens_out_a(x1) ---------------------------------------- (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: QUEENS_IN_A(Y) -> U1_A(Y, perm_in_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) QUEENS_IN_A(Y) -> PERM_IN_GA(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y) PERM_IN_GA(.(X, Y), .(V, Res)) -> U3_GA(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) PERM_IN_GA(.(X, Y), .(V, Res)) -> DELETE_IN_AGA(V, .(X, Y), Rest) DELETE_IN_AGA(X, .(F, T), .(F, R)) -> U5_AGA(X, F, T, R, delete_in_aga(X, T, R)) DELETE_IN_AGA(X, .(F, T), .(F, R)) -> DELETE_IN_AGA(X, T, R) U3_GA(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> U4_GA(X, Y, V, Res, perm_in_ga(Rest, Res)) U3_GA(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> PERM_IN_GA(Rest, Res) U1_A(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> U2_A(Y, safe_in_g(Y)) U1_A(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> SAFE_IN_G(Y) SAFE_IN_G(.(X, Y)) -> U6_G(X, Y, noattack_in_ggg(X, Y, s(0))) SAFE_IN_G(.(X, Y)) -> NOATTACK_IN_GGG(X, Y, s(0)) NOATTACK_IN_GGG(X, .(F, T), N) -> U8_GGG(X, F, T, N, notEq_in_gg(X, F)) NOATTACK_IN_GGG(X, .(F, T), N) -> NOTEQ_IN_GG(X, F) NOTEQ_IN_GG(s(X), s(Y)) -> U15_GG(X, Y, notEq_in_gg(X, Y)) NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) U8_GGG(X, F, T, N, notEq_out_gg(X, F)) -> U9_GGG(X, F, T, N, add_in_gga(F, N, FplusN)) U8_GGG(X, F, T, N, notEq_out_gg(X, F)) -> ADD_IN_GGA(F, N, FplusN) ADD_IN_GGA(s(X), Y, s(Z)) -> U14_GGA(X, Y, Z, add_in_gga(X, Y, Z)) ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) U9_GGG(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_GGG(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U9_GGG(X, F, T, N, add_out_gga(F, N, FplusN)) -> NOTEQ_IN_GG(X, FplusN) U10_GGG(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_GGG(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U10_GGG(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> ADD_IN_GGA(X, N, XplusN) U11_GGG(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_GGG(X, F, T, N, notEq_in_gg(F, XplusN)) U11_GGG(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> NOTEQ_IN_GG(F, XplusN) U12_GGG(X, F, T, N, notEq_out_gg(F, XplusN)) -> U13_GGG(X, F, T, N, noattack_in_ggg(X, T, s(N))) U12_GGG(X, F, T, N, notEq_out_gg(F, XplusN)) -> NOATTACK_IN_GGG(X, T, s(N)) U6_G(X, Y, noattack_out_ggg(X, Y, s(0))) -> U7_G(X, Y, safe_in_g(Y)) U6_G(X, Y, noattack_out_ggg(X, Y, s(0))) -> SAFE_IN_G(Y) The TRS R consists of the following rules: queens_in_a(Y) -> U1_a(Y, perm_in_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(.(X, Y), .(V, Res)) -> U3_ga(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(X, .(F, T), .(F, R)) -> U5_aga(X, F, T, R, delete_in_aga(X, T, R)) U5_aga(X, F, T, R, delete_out_aga(X, T, R)) -> delete_out_aga(X, .(F, T), .(F, R)) U3_ga(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> U4_ga(X, Y, V, Res, perm_in_ga(Rest, Res)) U4_ga(X, Y, V, Res, perm_out_ga(Rest, Res)) -> perm_out_ga(.(X, Y), .(V, Res)) U1_a(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> U2_a(Y, safe_in_g(Y)) safe_in_g([]) -> safe_out_g([]) safe_in_g(.(X, Y)) -> U6_g(X, Y, noattack_in_ggg(X, Y, s(0))) noattack_in_ggg(X, [], N) -> noattack_out_ggg(X, [], N) noattack_in_ggg(X, .(F, T), N) -> U8_ggg(X, F, T, N, notEq_in_gg(X, F)) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(s(X), s(Y)) -> U15_gg(X, Y, notEq_in_gg(X, Y)) U15_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U8_ggg(X, F, T, N, notEq_out_gg(X, F)) -> U9_ggg(X, F, T, N, add_in_gga(F, N, FplusN)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U14_gga(X, Y, Z, add_in_gga(X, Y, Z)) U14_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U9_ggg(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_ggg(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U10_ggg(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_ggg(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U11_ggg(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_ggg(X, F, T, N, notEq_in_gg(F, XplusN)) U12_ggg(X, F, T, N, notEq_out_gg(F, XplusN)) -> U13_ggg(X, F, T, N, noattack_in_ggg(X, T, s(N))) U13_ggg(X, F, T, N, noattack_out_ggg(X, T, s(N))) -> noattack_out_ggg(X, .(F, T), N) U6_g(X, Y, noattack_out_ggg(X, Y, s(0))) -> U7_g(X, Y, safe_in_g(Y)) U7_g(X, Y, safe_out_g(Y)) -> safe_out_g(.(X, Y)) U2_a(Y, safe_out_g(Y)) -> queens_out_a(Y) The argument filtering Pi contains the following mapping: queens_in_a(x1) = queens_in_a U1_a(x1, x2) = U1_a(x2) perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) .(x1, x2) = .(x1, x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5) s(x1) = s(x1) 0 = 0 U2_a(x1, x2) = U2_a(x1, x2) safe_in_g(x1) = safe_in_g(x1) safe_out_g(x1) = safe_out_g U6_g(x1, x2, x3) = U6_g(x2, x3) noattack_in_ggg(x1, x2, x3) = noattack_in_ggg(x1, x2, x3) noattack_out_ggg(x1, x2, x3) = noattack_out_ggg U8_ggg(x1, x2, x3, x4, x5) = U8_ggg(x1, x2, x3, x4, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) notEq_out_gg(x1, x2) = notEq_out_gg U15_gg(x1, x2, x3) = U15_gg(x3) U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x1, x2, x3, x4, x5) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U14_gga(x1, x2, x3, x4) = U14_gga(x4) U10_ggg(x1, x2, x3, x4, x5, x6) = U10_ggg(x1, x2, x3, x4, x6) U11_ggg(x1, x2, x3, x4, x5, x6) = U11_ggg(x1, x2, x3, x4, x6) U12_ggg(x1, x2, x3, x4, x5) = U12_ggg(x1, x3, x4, x5) U13_ggg(x1, x2, x3, x4, x5) = U13_ggg(x5) U7_g(x1, x2, x3) = U7_g(x3) queens_out_a(x1) = queens_out_a(x1) QUEENS_IN_A(x1) = QUEENS_IN_A U1_A(x1, x2) = U1_A(x2) PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5) DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x2, x5) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x3, x5) U2_A(x1, x2) = U2_A(x1, x2) SAFE_IN_G(x1) = SAFE_IN_G(x1) U6_G(x1, x2, x3) = U6_G(x2, x3) NOATTACK_IN_GGG(x1, x2, x3) = NOATTACK_IN_GGG(x1, x2, x3) U8_GGG(x1, x2, x3, x4, x5) = U8_GGG(x1, x2, x3, x4, x5) NOTEQ_IN_GG(x1, x2) = NOTEQ_IN_GG(x1, x2) U15_GG(x1, x2, x3) = U15_GG(x3) U9_GGG(x1, x2, x3, x4, x5) = U9_GGG(x1, x2, x3, x4, x5) ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) U14_GGA(x1, x2, x3, x4) = U14_GGA(x4) U10_GGG(x1, x2, x3, x4, x5, x6) = U10_GGG(x1, x2, x3, x4, x6) U11_GGG(x1, x2, x3, x4, x5, x6) = U11_GGG(x1, x2, x3, x4, x6) U12_GGG(x1, x2, x3, x4, x5) = U12_GGG(x1, x3, x4, x5) U13_GGG(x1, x2, x3, x4, x5) = U13_GGG(x5) U7_G(x1, x2, x3) = U7_G(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: QUEENS_IN_A(Y) -> U1_A(Y, perm_in_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) QUEENS_IN_A(Y) -> PERM_IN_GA(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y) PERM_IN_GA(.(X, Y), .(V, Res)) -> U3_GA(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) PERM_IN_GA(.(X, Y), .(V, Res)) -> DELETE_IN_AGA(V, .(X, Y), Rest) DELETE_IN_AGA(X, .(F, T), .(F, R)) -> U5_AGA(X, F, T, R, delete_in_aga(X, T, R)) DELETE_IN_AGA(X, .(F, T), .(F, R)) -> DELETE_IN_AGA(X, T, R) U3_GA(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> U4_GA(X, Y, V, Res, perm_in_ga(Rest, Res)) U3_GA(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> PERM_IN_GA(Rest, Res) U1_A(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> U2_A(Y, safe_in_g(Y)) U1_A(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> SAFE_IN_G(Y) SAFE_IN_G(.(X, Y)) -> U6_G(X, Y, noattack_in_ggg(X, Y, s(0))) SAFE_IN_G(.(X, Y)) -> NOATTACK_IN_GGG(X, Y, s(0)) NOATTACK_IN_GGG(X, .(F, T), N) -> U8_GGG(X, F, T, N, notEq_in_gg(X, F)) NOATTACK_IN_GGG(X, .(F, T), N) -> NOTEQ_IN_GG(X, F) NOTEQ_IN_GG(s(X), s(Y)) -> U15_GG(X, Y, notEq_in_gg(X, Y)) NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) U8_GGG(X, F, T, N, notEq_out_gg(X, F)) -> U9_GGG(X, F, T, N, add_in_gga(F, N, FplusN)) U8_GGG(X, F, T, N, notEq_out_gg(X, F)) -> ADD_IN_GGA(F, N, FplusN) ADD_IN_GGA(s(X), Y, s(Z)) -> U14_GGA(X, Y, Z, add_in_gga(X, Y, Z)) ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z) U9_GGG(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_GGG(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U9_GGG(X, F, T, N, add_out_gga(F, N, FplusN)) -> NOTEQ_IN_GG(X, FplusN) U10_GGG(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_GGG(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U10_GGG(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> ADD_IN_GGA(X, N, XplusN) U11_GGG(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_GGG(X, F, T, N, notEq_in_gg(F, XplusN)) U11_GGG(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> NOTEQ_IN_GG(F, XplusN) U12_GGG(X, F, T, N, notEq_out_gg(F, XplusN)) -> U13_GGG(X, F, T, N, noattack_in_ggg(X, T, s(N))) U12_GGG(X, F, T, N, notEq_out_gg(F, XplusN)) -> NOATTACK_IN_GGG(X, T, s(N)) U6_G(X, Y, noattack_out_ggg(X, Y, s(0))) -> U7_G(X, Y, safe_in_g(Y)) U6_G(X, Y, noattack_out_ggg(X, Y, s(0))) -> SAFE_IN_G(Y) The TRS R consists of the following rules: queens_in_a(Y) -> U1_a(Y, perm_in_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(.(X, Y), .(V, Res)) -> U3_ga(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(X, .(F, T), .(F, R)) -> U5_aga(X, F, T, R, delete_in_aga(X, T, R)) U5_aga(X, F, T, R, delete_out_aga(X, T, R)) -> delete_out_aga(X, .(F, T), .(F, R)) U3_ga(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> U4_ga(X, Y, V, Res, perm_in_ga(Rest, Res)) U4_ga(X, Y, V, Res, perm_out_ga(Rest, Res)) -> perm_out_ga(.(X, Y), .(V, Res)) U1_a(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> U2_a(Y, safe_in_g(Y)) safe_in_g([]) -> safe_out_g([]) safe_in_g(.(X, Y)) -> U6_g(X, Y, noattack_in_ggg(X, Y, s(0))) noattack_in_ggg(X, [], N) -> noattack_out_ggg(X, [], N) noattack_in_ggg(X, .(F, T), N) -> U8_ggg(X, F, T, N, notEq_in_gg(X, F)) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(s(X), s(Y)) -> U15_gg(X, Y, notEq_in_gg(X, Y)) U15_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U8_ggg(X, F, T, N, notEq_out_gg(X, F)) -> U9_ggg(X, F, T, N, add_in_gga(F, N, FplusN)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U14_gga(X, Y, Z, add_in_gga(X, Y, Z)) U14_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U9_ggg(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_ggg(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U10_ggg(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_ggg(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U11_ggg(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_ggg(X, F, T, N, notEq_in_gg(F, XplusN)) U12_ggg(X, F, T, N, notEq_out_gg(F, XplusN)) -> U13_ggg(X, F, T, N, noattack_in_ggg(X, T, s(N))) U13_ggg(X, F, T, N, noattack_out_ggg(X, T, s(N))) -> noattack_out_ggg(X, .(F, T), N) U6_g(X, Y, noattack_out_ggg(X, Y, s(0))) -> U7_g(X, Y, safe_in_g(Y)) U7_g(X, Y, safe_out_g(Y)) -> safe_out_g(.(X, Y)) U2_a(Y, safe_out_g(Y)) -> queens_out_a(Y) The argument filtering Pi contains the following mapping: queens_in_a(x1) = queens_in_a U1_a(x1, x2) = U1_a(x2) perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) .(x1, x2) = .(x1, x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5) s(x1) = s(x1) 0 = 0 U2_a(x1, x2) = U2_a(x1, x2) safe_in_g(x1) = safe_in_g(x1) safe_out_g(x1) = safe_out_g U6_g(x1, x2, x3) = U6_g(x2, x3) noattack_in_ggg(x1, x2, x3) = noattack_in_ggg(x1, x2, x3) noattack_out_ggg(x1, x2, x3) = noattack_out_ggg U8_ggg(x1, x2, x3, x4, x5) = U8_ggg(x1, x2, x3, x4, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) notEq_out_gg(x1, x2) = notEq_out_gg U15_gg(x1, x2, x3) = U15_gg(x3) U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x1, x2, x3, x4, x5) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U14_gga(x1, x2, x3, x4) = U14_gga(x4) U10_ggg(x1, x2, x3, x4, x5, x6) = U10_ggg(x1, x2, x3, x4, x6) U11_ggg(x1, x2, x3, x4, x5, x6) = U11_ggg(x1, x2, x3, x4, x6) U12_ggg(x1, x2, x3, x4, x5) = U12_ggg(x1, x3, x4, x5) U13_ggg(x1, x2, x3, x4, x5) = U13_ggg(x5) U7_g(x1, x2, x3) = U7_g(x3) queens_out_a(x1) = queens_out_a(x1) QUEENS_IN_A(x1) = QUEENS_IN_A U1_A(x1, x2) = U1_A(x2) PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5) DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) U5_AGA(x1, x2, x3, x4, x5) = U5_AGA(x2, x5) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x3, x5) U2_A(x1, x2) = U2_A(x1, x2) SAFE_IN_G(x1) = SAFE_IN_G(x1) U6_G(x1, x2, x3) = U6_G(x2, x3) NOATTACK_IN_GGG(x1, x2, x3) = NOATTACK_IN_GGG(x1, x2, x3) U8_GGG(x1, x2, x3, x4, x5) = U8_GGG(x1, x2, x3, x4, x5) NOTEQ_IN_GG(x1, x2) = NOTEQ_IN_GG(x1, x2) U15_GG(x1, x2, x3) = U15_GG(x3) U9_GGG(x1, x2, x3, x4, x5) = U9_GGG(x1, x2, x3, x4, x5) ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) U14_GGA(x1, x2, x3, x4) = U14_GGA(x4) U10_GGG(x1, x2, x3, x4, x5, x6) = U10_GGG(x1, x2, x3, x4, x6) U11_GGG(x1, x2, x3, x4, x5, x6) = U11_GGG(x1, x2, x3, x4, x6) U12_GGG(x1, x2, x3, x4, x5) = U12_GGG(x1, x3, x4, x5) U13_GGG(x1, x2, x3, x4, x5) = U13_GGG(x5) U7_G(x1, x2, x3) = U7_G(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 17 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) 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: queens_in_a(Y) -> U1_a(Y, perm_in_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(.(X, Y), .(V, Res)) -> U3_ga(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(X, .(F, T), .(F, R)) -> U5_aga(X, F, T, R, delete_in_aga(X, T, R)) U5_aga(X, F, T, R, delete_out_aga(X, T, R)) -> delete_out_aga(X, .(F, T), .(F, R)) U3_ga(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> U4_ga(X, Y, V, Res, perm_in_ga(Rest, Res)) U4_ga(X, Y, V, Res, perm_out_ga(Rest, Res)) -> perm_out_ga(.(X, Y), .(V, Res)) U1_a(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> U2_a(Y, safe_in_g(Y)) safe_in_g([]) -> safe_out_g([]) safe_in_g(.(X, Y)) -> U6_g(X, Y, noattack_in_ggg(X, Y, s(0))) noattack_in_ggg(X, [], N) -> noattack_out_ggg(X, [], N) noattack_in_ggg(X, .(F, T), N) -> U8_ggg(X, F, T, N, notEq_in_gg(X, F)) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(s(X), s(Y)) -> U15_gg(X, Y, notEq_in_gg(X, Y)) U15_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U8_ggg(X, F, T, N, notEq_out_gg(X, F)) -> U9_ggg(X, F, T, N, add_in_gga(F, N, FplusN)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U14_gga(X, Y, Z, add_in_gga(X, Y, Z)) U14_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U9_ggg(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_ggg(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U10_ggg(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_ggg(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U11_ggg(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_ggg(X, F, T, N, notEq_in_gg(F, XplusN)) U12_ggg(X, F, T, N, notEq_out_gg(F, XplusN)) -> U13_ggg(X, F, T, N, noattack_in_ggg(X, T, s(N))) U13_ggg(X, F, T, N, noattack_out_ggg(X, T, s(N))) -> noattack_out_ggg(X, .(F, T), N) U6_g(X, Y, noattack_out_ggg(X, Y, s(0))) -> U7_g(X, Y, safe_in_g(Y)) U7_g(X, Y, safe_out_g(Y)) -> safe_out_g(.(X, Y)) U2_a(Y, safe_out_g(Y)) -> queens_out_a(Y) The argument filtering Pi contains the following mapping: queens_in_a(x1) = queens_in_a U1_a(x1, x2) = U1_a(x2) perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) .(x1, x2) = .(x1, x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5) s(x1) = s(x1) 0 = 0 U2_a(x1, x2) = U2_a(x1, x2) safe_in_g(x1) = safe_in_g(x1) safe_out_g(x1) = safe_out_g U6_g(x1, x2, x3) = U6_g(x2, x3) noattack_in_ggg(x1, x2, x3) = noattack_in_ggg(x1, x2, x3) noattack_out_ggg(x1, x2, x3) = noattack_out_ggg U8_ggg(x1, x2, x3, x4, x5) = U8_ggg(x1, x2, x3, x4, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) notEq_out_gg(x1, x2) = notEq_out_gg U15_gg(x1, x2, x3) = U15_gg(x3) U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x1, x2, x3, x4, x5) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U14_gga(x1, x2, x3, x4) = U14_gga(x4) U10_ggg(x1, x2, x3, x4, x5, x6) = U10_ggg(x1, x2, x3, x4, x6) U11_ggg(x1, x2, x3, x4, x5, x6) = U11_ggg(x1, x2, x3, x4, x6) U12_ggg(x1, x2, x3, x4, x5) = U12_ggg(x1, x3, x4, x5) U13_ggg(x1, x2, x3, x4, x5) = U13_ggg(x5) U7_g(x1, x2, x3) = U7_g(x3) queens_out_a(x1) = queens_out_a(x1) ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(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: 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 ---------------------------------------- (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: 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. ---------------------------------------- (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: *ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(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: NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_IN_GG(X, Y) The TRS R consists of the following rules: queens_in_a(Y) -> U1_a(Y, perm_in_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(.(X, Y), .(V, Res)) -> U3_ga(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(X, .(F, T), .(F, R)) -> U5_aga(X, F, T, R, delete_in_aga(X, T, R)) U5_aga(X, F, T, R, delete_out_aga(X, T, R)) -> delete_out_aga(X, .(F, T), .(F, R)) U3_ga(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> U4_ga(X, Y, V, Res, perm_in_ga(Rest, Res)) U4_ga(X, Y, V, Res, perm_out_ga(Rest, Res)) -> perm_out_ga(.(X, Y), .(V, Res)) U1_a(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> U2_a(Y, safe_in_g(Y)) safe_in_g([]) -> safe_out_g([]) safe_in_g(.(X, Y)) -> U6_g(X, Y, noattack_in_ggg(X, Y, s(0))) noattack_in_ggg(X, [], N) -> noattack_out_ggg(X, [], N) noattack_in_ggg(X, .(F, T), N) -> U8_ggg(X, F, T, N, notEq_in_gg(X, F)) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(s(X), s(Y)) -> U15_gg(X, Y, notEq_in_gg(X, Y)) U15_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U8_ggg(X, F, T, N, notEq_out_gg(X, F)) -> U9_ggg(X, F, T, N, add_in_gga(F, N, FplusN)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U14_gga(X, Y, Z, add_in_gga(X, Y, Z)) U14_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U9_ggg(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_ggg(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U10_ggg(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_ggg(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U11_ggg(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_ggg(X, F, T, N, notEq_in_gg(F, XplusN)) U12_ggg(X, F, T, N, notEq_out_gg(F, XplusN)) -> U13_ggg(X, F, T, N, noattack_in_ggg(X, T, s(N))) U13_ggg(X, F, T, N, noattack_out_ggg(X, T, s(N))) -> noattack_out_ggg(X, .(F, T), N) U6_g(X, Y, noattack_out_ggg(X, Y, s(0))) -> U7_g(X, Y, safe_in_g(Y)) U7_g(X, Y, safe_out_g(Y)) -> safe_out_g(.(X, Y)) U2_a(Y, safe_out_g(Y)) -> queens_out_a(Y) The argument filtering Pi contains the following mapping: queens_in_a(x1) = queens_in_a U1_a(x1, x2) = U1_a(x2) perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) .(x1, x2) = .(x1, x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5) s(x1) = s(x1) 0 = 0 U2_a(x1, x2) = U2_a(x1, x2) safe_in_g(x1) = safe_in_g(x1) safe_out_g(x1) = safe_out_g U6_g(x1, x2, x3) = U6_g(x2, x3) noattack_in_ggg(x1, x2, x3) = noattack_in_ggg(x1, x2, x3) noattack_out_ggg(x1, x2, x3) = noattack_out_ggg U8_ggg(x1, x2, x3, x4, x5) = U8_ggg(x1, x2, x3, x4, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) notEq_out_gg(x1, x2) = notEq_out_gg U15_gg(x1, x2, x3) = U15_gg(x3) U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x1, x2, x3, x4, x5) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U14_gga(x1, x2, x3, x4) = U14_gga(x4) U10_ggg(x1, x2, x3, x4, x5, x6) = U10_ggg(x1, x2, x3, x4, x6) U11_ggg(x1, x2, x3, x4, x5, x6) = U11_ggg(x1, x2, x3, x4, x6) U12_ggg(x1, x2, x3, x4, x5) = U12_ggg(x1, x3, x4, x5) U13_ggg(x1, x2, x3, x4, x5) = U13_ggg(x5) U7_g(x1, x2, x3) = U7_g(x3) queens_out_a(x1) = queens_out_a(x1) NOTEQ_IN_GG(x1, x2) = NOTEQ_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: NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_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: NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_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: *NOTEQ_IN_GG(s(X), s(Y)) -> NOTEQ_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: U8_GGG(X, F, T, N, notEq_out_gg(X, F)) -> U9_GGG(X, F, T, N, add_in_gga(F, N, FplusN)) U9_GGG(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_GGG(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U10_GGG(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_GGG(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U11_GGG(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_GGG(X, F, T, N, notEq_in_gg(F, XplusN)) U12_GGG(X, F, T, N, notEq_out_gg(F, XplusN)) -> NOATTACK_IN_GGG(X, T, s(N)) NOATTACK_IN_GGG(X, .(F, T), N) -> U8_GGG(X, F, T, N, notEq_in_gg(X, F)) The TRS R consists of the following rules: queens_in_a(Y) -> U1_a(Y, perm_in_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(.(X, Y), .(V, Res)) -> U3_ga(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(X, .(F, T), .(F, R)) -> U5_aga(X, F, T, R, delete_in_aga(X, T, R)) U5_aga(X, F, T, R, delete_out_aga(X, T, R)) -> delete_out_aga(X, .(F, T), .(F, R)) U3_ga(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> U4_ga(X, Y, V, Res, perm_in_ga(Rest, Res)) U4_ga(X, Y, V, Res, perm_out_ga(Rest, Res)) -> perm_out_ga(.(X, Y), .(V, Res)) U1_a(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> U2_a(Y, safe_in_g(Y)) safe_in_g([]) -> safe_out_g([]) safe_in_g(.(X, Y)) -> U6_g(X, Y, noattack_in_ggg(X, Y, s(0))) noattack_in_ggg(X, [], N) -> noattack_out_ggg(X, [], N) noattack_in_ggg(X, .(F, T), N) -> U8_ggg(X, F, T, N, notEq_in_gg(X, F)) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(s(X), s(Y)) -> U15_gg(X, Y, notEq_in_gg(X, Y)) U15_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U8_ggg(X, F, T, N, notEq_out_gg(X, F)) -> U9_ggg(X, F, T, N, add_in_gga(F, N, FplusN)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U14_gga(X, Y, Z, add_in_gga(X, Y, Z)) U14_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U9_ggg(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_ggg(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U10_ggg(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_ggg(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U11_ggg(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_ggg(X, F, T, N, notEq_in_gg(F, XplusN)) U12_ggg(X, F, T, N, notEq_out_gg(F, XplusN)) -> U13_ggg(X, F, T, N, noattack_in_ggg(X, T, s(N))) U13_ggg(X, F, T, N, noattack_out_ggg(X, T, s(N))) -> noattack_out_ggg(X, .(F, T), N) U6_g(X, Y, noattack_out_ggg(X, Y, s(0))) -> U7_g(X, Y, safe_in_g(Y)) U7_g(X, Y, safe_out_g(Y)) -> safe_out_g(.(X, Y)) U2_a(Y, safe_out_g(Y)) -> queens_out_a(Y) The argument filtering Pi contains the following mapping: queens_in_a(x1) = queens_in_a U1_a(x1, x2) = U1_a(x2) perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) .(x1, x2) = .(x1, x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5) s(x1) = s(x1) 0 = 0 U2_a(x1, x2) = U2_a(x1, x2) safe_in_g(x1) = safe_in_g(x1) safe_out_g(x1) = safe_out_g U6_g(x1, x2, x3) = U6_g(x2, x3) noattack_in_ggg(x1, x2, x3) = noattack_in_ggg(x1, x2, x3) noattack_out_ggg(x1, x2, x3) = noattack_out_ggg U8_ggg(x1, x2, x3, x4, x5) = U8_ggg(x1, x2, x3, x4, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) notEq_out_gg(x1, x2) = notEq_out_gg U15_gg(x1, x2, x3) = U15_gg(x3) U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x1, x2, x3, x4, x5) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U14_gga(x1, x2, x3, x4) = U14_gga(x4) U10_ggg(x1, x2, x3, x4, x5, x6) = U10_ggg(x1, x2, x3, x4, x6) U11_ggg(x1, x2, x3, x4, x5, x6) = U11_ggg(x1, x2, x3, x4, x6) U12_ggg(x1, x2, x3, x4, x5) = U12_ggg(x1, x3, x4, x5) U13_ggg(x1, x2, x3, x4, x5) = U13_ggg(x5) U7_g(x1, x2, x3) = U7_g(x3) queens_out_a(x1) = queens_out_a(x1) NOATTACK_IN_GGG(x1, x2, x3) = NOATTACK_IN_GGG(x1, x2, x3) U8_GGG(x1, x2, x3, x4, x5) = U8_GGG(x1, x2, x3, x4, x5) U9_GGG(x1, x2, x3, x4, x5) = U9_GGG(x1, x2, x3, x4, x5) U10_GGG(x1, x2, x3, x4, x5, x6) = U10_GGG(x1, x2, x3, x4, x6) U11_GGG(x1, x2, x3, x4, x5, x6) = U11_GGG(x1, x2, x3, x4, x6) U12_GGG(x1, x2, x3, x4, x5) = U12_GGG(x1, x3, 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: U8_GGG(X, F, T, N, notEq_out_gg(X, F)) -> U9_GGG(X, F, T, N, add_in_gga(F, N, FplusN)) U9_GGG(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_GGG(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U10_GGG(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_GGG(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U11_GGG(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_GGG(X, F, T, N, notEq_in_gg(F, XplusN)) U12_GGG(X, F, T, N, notEq_out_gg(F, XplusN)) -> NOATTACK_IN_GGG(X, T, s(N)) NOATTACK_IN_GGG(X, .(F, T), N) -> U8_GGG(X, F, T, N, notEq_in_gg(X, F)) The TRS R consists of the following rules: add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U14_gga(X, Y, Z, add_in_gga(X, Y, Z)) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(s(X), s(Y)) -> U15_gg(X, Y, notEq_in_gg(X, Y)) U14_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U15_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) s(x1) = s(x1) 0 = 0 notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) notEq_out_gg(x1, x2) = notEq_out_gg U15_gg(x1, x2, x3) = U15_gg(x3) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U14_gga(x1, x2, x3, x4) = U14_gga(x4) NOATTACK_IN_GGG(x1, x2, x3) = NOATTACK_IN_GGG(x1, x2, x3) U8_GGG(x1, x2, x3, x4, x5) = U8_GGG(x1, x2, x3, x4, x5) U9_GGG(x1, x2, x3, x4, x5) = U9_GGG(x1, x2, x3, x4, x5) U10_GGG(x1, x2, x3, x4, x5, x6) = U10_GGG(x1, x2, x3, x4, x6) U11_GGG(x1, x2, x3, x4, x5, x6) = U11_GGG(x1, x2, x3, x4, x6) U12_GGG(x1, x2, x3, x4, x5) = U12_GGG(x1, x3, 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: U8_GGG(X, F, T, N, notEq_out_gg) -> U9_GGG(X, F, T, N, add_in_gga(F, N)) U9_GGG(X, F, T, N, add_out_gga(FplusN)) -> U10_GGG(X, F, T, N, notEq_in_gg(X, FplusN)) U10_GGG(X, F, T, N, notEq_out_gg) -> U11_GGG(X, F, T, N, add_in_gga(X, N)) U11_GGG(X, F, T, N, add_out_gga(XplusN)) -> U12_GGG(X, T, N, notEq_in_gg(F, XplusN)) U12_GGG(X, T, N, notEq_out_gg) -> NOATTACK_IN_GGG(X, T, s(N)) NOATTACK_IN_GGG(X, .(F, T), N) -> U8_GGG(X, F, T, N, notEq_in_gg(X, F)) The TRS R consists of the following rules: add_in_gga(0, X) -> add_out_gga(X) add_in_gga(s(X), Y) -> U14_gga(add_in_gga(X, Y)) notEq_in_gg(0, s(X)) -> notEq_out_gg notEq_in_gg(s(X), 0) -> notEq_out_gg notEq_in_gg(s(X), s(Y)) -> U15_gg(notEq_in_gg(X, Y)) U14_gga(add_out_gga(Z)) -> add_out_gga(s(Z)) U15_gg(notEq_out_gg) -> notEq_out_gg The set Q consists of the following terms: add_in_gga(x0, x1) notEq_in_gg(x0, x1) U14_gga(x0) U15_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: *U9_GGG(X, F, T, N, add_out_gga(FplusN)) -> U10_GGG(X, F, T, N, notEq_in_gg(X, FplusN)) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4 *NOATTACK_IN_GGG(X, .(F, T), N) -> U8_GGG(X, F, T, N, notEq_in_gg(X, F)) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4 *U10_GGG(X, F, T, N, notEq_out_gg) -> U11_GGG(X, F, T, N, add_in_gga(X, N)) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4 *U8_GGG(X, F, T, N, notEq_out_gg) -> U9_GGG(X, F, T, N, add_in_gga(F, N)) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4 *U11_GGG(X, F, T, N, add_out_gga(XplusN)) -> U12_GGG(X, T, N, notEq_in_gg(F, XplusN)) The graph contains the following edges 1 >= 1, 3 >= 2, 4 >= 3 *U12_GGG(X, T, N, notEq_out_gg) -> NOATTACK_IN_GGG(X, T, s(N)) 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: U6_G(X, Y, noattack_out_ggg(X, Y, s(0))) -> SAFE_IN_G(Y) SAFE_IN_G(.(X, Y)) -> U6_G(X, Y, noattack_in_ggg(X, Y, s(0))) The TRS R consists of the following rules: queens_in_a(Y) -> U1_a(Y, perm_in_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(.(X, Y), .(V, Res)) -> U3_ga(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(X, .(F, T), .(F, R)) -> U5_aga(X, F, T, R, delete_in_aga(X, T, R)) U5_aga(X, F, T, R, delete_out_aga(X, T, R)) -> delete_out_aga(X, .(F, T), .(F, R)) U3_ga(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> U4_ga(X, Y, V, Res, perm_in_ga(Rest, Res)) U4_ga(X, Y, V, Res, perm_out_ga(Rest, Res)) -> perm_out_ga(.(X, Y), .(V, Res)) U1_a(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> U2_a(Y, safe_in_g(Y)) safe_in_g([]) -> safe_out_g([]) safe_in_g(.(X, Y)) -> U6_g(X, Y, noattack_in_ggg(X, Y, s(0))) noattack_in_ggg(X, [], N) -> noattack_out_ggg(X, [], N) noattack_in_ggg(X, .(F, T), N) -> U8_ggg(X, F, T, N, notEq_in_gg(X, F)) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(s(X), s(Y)) -> U15_gg(X, Y, notEq_in_gg(X, Y)) U15_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U8_ggg(X, F, T, N, notEq_out_gg(X, F)) -> U9_ggg(X, F, T, N, add_in_gga(F, N, FplusN)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U14_gga(X, Y, Z, add_in_gga(X, Y, Z)) U14_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U9_ggg(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_ggg(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U10_ggg(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_ggg(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U11_ggg(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_ggg(X, F, T, N, notEq_in_gg(F, XplusN)) U12_ggg(X, F, T, N, notEq_out_gg(F, XplusN)) -> U13_ggg(X, F, T, N, noattack_in_ggg(X, T, s(N))) U13_ggg(X, F, T, N, noattack_out_ggg(X, T, s(N))) -> noattack_out_ggg(X, .(F, T), N) U6_g(X, Y, noattack_out_ggg(X, Y, s(0))) -> U7_g(X, Y, safe_in_g(Y)) U7_g(X, Y, safe_out_g(Y)) -> safe_out_g(.(X, Y)) U2_a(Y, safe_out_g(Y)) -> queens_out_a(Y) The argument filtering Pi contains the following mapping: queens_in_a(x1) = queens_in_a U1_a(x1, x2) = U1_a(x2) perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) .(x1, x2) = .(x1, x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5) s(x1) = s(x1) 0 = 0 U2_a(x1, x2) = U2_a(x1, x2) safe_in_g(x1) = safe_in_g(x1) safe_out_g(x1) = safe_out_g U6_g(x1, x2, x3) = U6_g(x2, x3) noattack_in_ggg(x1, x2, x3) = noattack_in_ggg(x1, x2, x3) noattack_out_ggg(x1, x2, x3) = noattack_out_ggg U8_ggg(x1, x2, x3, x4, x5) = U8_ggg(x1, x2, x3, x4, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) notEq_out_gg(x1, x2) = notEq_out_gg U15_gg(x1, x2, x3) = U15_gg(x3) U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x1, x2, x3, x4, x5) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U14_gga(x1, x2, x3, x4) = U14_gga(x4) U10_ggg(x1, x2, x3, x4, x5, x6) = U10_ggg(x1, x2, x3, x4, x6) U11_ggg(x1, x2, x3, x4, x5, x6) = U11_ggg(x1, x2, x3, x4, x6) U12_ggg(x1, x2, x3, x4, x5) = U12_ggg(x1, x3, x4, x5) U13_ggg(x1, x2, x3, x4, x5) = U13_ggg(x5) U7_g(x1, x2, x3) = U7_g(x3) queens_out_a(x1) = queens_out_a(x1) SAFE_IN_G(x1) = SAFE_IN_G(x1) U6_G(x1, x2, x3) = U6_G(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: U6_G(X, Y, noattack_out_ggg(X, Y, s(0))) -> SAFE_IN_G(Y) SAFE_IN_G(.(X, Y)) -> U6_G(X, Y, noattack_in_ggg(X, Y, s(0))) The TRS R consists of the following rules: noattack_in_ggg(X, [], N) -> noattack_out_ggg(X, [], N) noattack_in_ggg(X, .(F, T), N) -> U8_ggg(X, F, T, N, notEq_in_gg(X, F)) U8_ggg(X, F, T, N, notEq_out_gg(X, F)) -> U9_ggg(X, F, T, N, add_in_gga(F, N, FplusN)) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(s(X), s(Y)) -> U15_gg(X, Y, notEq_in_gg(X, Y)) U9_ggg(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_ggg(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U15_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U14_gga(X, Y, Z, add_in_gga(X, Y, Z)) U10_ggg(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_ggg(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U14_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U11_ggg(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_ggg(X, F, T, N, notEq_in_gg(F, XplusN)) U12_ggg(X, F, T, N, notEq_out_gg(F, XplusN)) -> U13_ggg(X, F, T, N, noattack_in_ggg(X, T, s(N))) U13_ggg(X, F, T, N, noattack_out_ggg(X, T, s(N))) -> noattack_out_ggg(X, .(F, T), N) The argument filtering Pi contains the following mapping: [] = [] .(x1, x2) = .(x1, x2) s(x1) = s(x1) 0 = 0 noattack_in_ggg(x1, x2, x3) = noattack_in_ggg(x1, x2, x3) noattack_out_ggg(x1, x2, x3) = noattack_out_ggg U8_ggg(x1, x2, x3, x4, x5) = U8_ggg(x1, x2, x3, x4, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) notEq_out_gg(x1, x2) = notEq_out_gg U15_gg(x1, x2, x3) = U15_gg(x3) U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x1, x2, x3, x4, x5) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U14_gga(x1, x2, x3, x4) = U14_gga(x4) U10_ggg(x1, x2, x3, x4, x5, x6) = U10_ggg(x1, x2, x3, x4, x6) U11_ggg(x1, x2, x3, x4, x5, x6) = U11_ggg(x1, x2, x3, x4, x6) U12_ggg(x1, x2, x3, x4, x5) = U12_ggg(x1, x3, x4, x5) U13_ggg(x1, x2, x3, x4, x5) = U13_ggg(x5) SAFE_IN_G(x1) = SAFE_IN_G(x1) U6_G(x1, x2, x3) = U6_G(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: U6_G(Y, noattack_out_ggg) -> SAFE_IN_G(Y) SAFE_IN_G(.(X, Y)) -> U6_G(Y, noattack_in_ggg(X, Y, s(0))) The TRS R consists of the following rules: noattack_in_ggg(X, [], N) -> noattack_out_ggg noattack_in_ggg(X, .(F, T), N) -> U8_ggg(X, F, T, N, notEq_in_gg(X, F)) U8_ggg(X, F, T, N, notEq_out_gg) -> U9_ggg(X, F, T, N, add_in_gga(F, N)) notEq_in_gg(0, s(X)) -> notEq_out_gg notEq_in_gg(s(X), 0) -> notEq_out_gg notEq_in_gg(s(X), s(Y)) -> U15_gg(notEq_in_gg(X, Y)) U9_ggg(X, F, T, N, add_out_gga(FplusN)) -> U10_ggg(X, F, T, N, notEq_in_gg(X, FplusN)) U15_gg(notEq_out_gg) -> notEq_out_gg add_in_gga(0, X) -> add_out_gga(X) add_in_gga(s(X), Y) -> U14_gga(add_in_gga(X, Y)) U10_ggg(X, F, T, N, notEq_out_gg) -> U11_ggg(X, F, T, N, add_in_gga(X, N)) U14_gga(add_out_gga(Z)) -> add_out_gga(s(Z)) U11_ggg(X, F, T, N, add_out_gga(XplusN)) -> U12_ggg(X, T, N, notEq_in_gg(F, XplusN)) U12_ggg(X, T, N, notEq_out_gg) -> U13_ggg(noattack_in_ggg(X, T, s(N))) U13_ggg(noattack_out_ggg) -> noattack_out_ggg The set Q consists of the following terms: noattack_in_ggg(x0, x1, x2) U8_ggg(x0, x1, x2, x3, x4) notEq_in_gg(x0, x1) U9_ggg(x0, x1, x2, x3, x4) U15_gg(x0) add_in_gga(x0, x1) U10_ggg(x0, x1, x2, x3, x4) U14_gga(x0) U11_ggg(x0, x1, x2, x3, x4) U12_ggg(x0, x1, x2, x3) U13_ggg(x0) 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: *SAFE_IN_G(.(X, Y)) -> U6_G(Y, noattack_in_ggg(X, Y, s(0))) The graph contains the following edges 1 > 1 *U6_G(Y, noattack_out_ggg) -> SAFE_IN_G(Y) The graph contains the following edges 1 >= 1 ---------------------------------------- (34) YES ---------------------------------------- (35) Obligation: Pi DP problem: The TRS P consists of the following rules: DELETE_IN_AGA(X, .(F, T), .(F, R)) -> DELETE_IN_AGA(X, T, R) The TRS R consists of the following rules: queens_in_a(Y) -> U1_a(Y, perm_in_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(.(X, Y), .(V, Res)) -> U3_ga(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(X, .(F, T), .(F, R)) -> U5_aga(X, F, T, R, delete_in_aga(X, T, R)) U5_aga(X, F, T, R, delete_out_aga(X, T, R)) -> delete_out_aga(X, .(F, T), .(F, R)) U3_ga(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> U4_ga(X, Y, V, Res, perm_in_ga(Rest, Res)) U4_ga(X, Y, V, Res, perm_out_ga(Rest, Res)) -> perm_out_ga(.(X, Y), .(V, Res)) U1_a(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> U2_a(Y, safe_in_g(Y)) safe_in_g([]) -> safe_out_g([]) safe_in_g(.(X, Y)) -> U6_g(X, Y, noattack_in_ggg(X, Y, s(0))) noattack_in_ggg(X, [], N) -> noattack_out_ggg(X, [], N) noattack_in_ggg(X, .(F, T), N) -> U8_ggg(X, F, T, N, notEq_in_gg(X, F)) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(s(X), s(Y)) -> U15_gg(X, Y, notEq_in_gg(X, Y)) U15_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U8_ggg(X, F, T, N, notEq_out_gg(X, F)) -> U9_ggg(X, F, T, N, add_in_gga(F, N, FplusN)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U14_gga(X, Y, Z, add_in_gga(X, Y, Z)) U14_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U9_ggg(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_ggg(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U10_ggg(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_ggg(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U11_ggg(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_ggg(X, F, T, N, notEq_in_gg(F, XplusN)) U12_ggg(X, F, T, N, notEq_out_gg(F, XplusN)) -> U13_ggg(X, F, T, N, noattack_in_ggg(X, T, s(N))) U13_ggg(X, F, T, N, noattack_out_ggg(X, T, s(N))) -> noattack_out_ggg(X, .(F, T), N) U6_g(X, Y, noattack_out_ggg(X, Y, s(0))) -> U7_g(X, Y, safe_in_g(Y)) U7_g(X, Y, safe_out_g(Y)) -> safe_out_g(.(X, Y)) U2_a(Y, safe_out_g(Y)) -> queens_out_a(Y) The argument filtering Pi contains the following mapping: queens_in_a(x1) = queens_in_a U1_a(x1, x2) = U1_a(x2) perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) .(x1, x2) = .(x1, x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5) s(x1) = s(x1) 0 = 0 U2_a(x1, x2) = U2_a(x1, x2) safe_in_g(x1) = safe_in_g(x1) safe_out_g(x1) = safe_out_g U6_g(x1, x2, x3) = U6_g(x2, x3) noattack_in_ggg(x1, x2, x3) = noattack_in_ggg(x1, x2, x3) noattack_out_ggg(x1, x2, x3) = noattack_out_ggg U8_ggg(x1, x2, x3, x4, x5) = U8_ggg(x1, x2, x3, x4, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) notEq_out_gg(x1, x2) = notEq_out_gg U15_gg(x1, x2, x3) = U15_gg(x3) U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x1, x2, x3, x4, x5) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U14_gga(x1, x2, x3, x4) = U14_gga(x4) U10_ggg(x1, x2, x3, x4, x5, x6) = U10_ggg(x1, x2, x3, x4, x6) U11_ggg(x1, x2, x3, x4, x5, x6) = U11_ggg(x1, x2, x3, x4, x6) U12_ggg(x1, x2, x3, x4, x5) = U12_ggg(x1, x3, x4, x5) U13_ggg(x1, x2, x3, x4, x5) = U13_ggg(x5) U7_g(x1, x2, x3) = U7_g(x3) queens_out_a(x1) = queens_out_a(x1) DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) 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: DELETE_IN_AGA(X, .(F, T), .(F, R)) -> DELETE_IN_AGA(X, T, R) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) DELETE_IN_AGA(x1, x2, x3) = DELETE_IN_AGA(x2) 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: DELETE_IN_AGA(.(F, T)) -> DELETE_IN_AGA(T) R is empty. Q is empty. 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: *DELETE_IN_AGA(.(F, T)) -> DELETE_IN_AGA(T) The graph contains the following edges 1 > 1 ---------------------------------------- (41) YES ---------------------------------------- (42) Obligation: Pi DP problem: The TRS P consists of the following rules: U3_GA(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> PERM_IN_GA(Rest, Res) PERM_IN_GA(.(X, Y), .(V, Res)) -> U3_GA(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) The TRS R consists of the following rules: queens_in_a(Y) -> U1_a(Y, perm_in_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) perm_in_ga([], []) -> perm_out_ga([], []) perm_in_ga(.(X, Y), .(V, Res)) -> U3_ga(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(X, .(F, T), .(F, R)) -> U5_aga(X, F, T, R, delete_in_aga(X, T, R)) U5_aga(X, F, T, R, delete_out_aga(X, T, R)) -> delete_out_aga(X, .(F, T), .(F, R)) U3_ga(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> U4_ga(X, Y, V, Res, perm_in_ga(Rest, Res)) U4_ga(X, Y, V, Res, perm_out_ga(Rest, Res)) -> perm_out_ga(.(X, Y), .(V, Res)) U1_a(Y, perm_out_ga(.(s(0), .(s(s(0)), .(s(s(s(0))), .(s(s(s(s(0)))), .(s(s(s(s(s(0))))), .(s(s(s(s(s(s(0)))))), .(s(s(s(s(s(s(s(0))))))), .(s(s(s(s(s(s(s(s(0)))))))), [])))))))), Y)) -> U2_a(Y, safe_in_g(Y)) safe_in_g([]) -> safe_out_g([]) safe_in_g(.(X, Y)) -> U6_g(X, Y, noattack_in_ggg(X, Y, s(0))) noattack_in_ggg(X, [], N) -> noattack_out_ggg(X, [], N) noattack_in_ggg(X, .(F, T), N) -> U8_ggg(X, F, T, N, notEq_in_gg(X, F)) notEq_in_gg(0, s(X)) -> notEq_out_gg(0, s(X)) notEq_in_gg(s(X), 0) -> notEq_out_gg(s(X), 0) notEq_in_gg(s(X), s(Y)) -> U15_gg(X, Y, notEq_in_gg(X, Y)) U15_gg(X, Y, notEq_out_gg(X, Y)) -> notEq_out_gg(s(X), s(Y)) U8_ggg(X, F, T, N, notEq_out_gg(X, F)) -> U9_ggg(X, F, T, N, add_in_gga(F, N, FplusN)) add_in_gga(0, X, X) -> add_out_gga(0, X, X) add_in_gga(s(X), Y, s(Z)) -> U14_gga(X, Y, Z, add_in_gga(X, Y, Z)) U14_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z)) U9_ggg(X, F, T, N, add_out_gga(F, N, FplusN)) -> U10_ggg(X, F, T, N, FplusN, notEq_in_gg(X, FplusN)) U10_ggg(X, F, T, N, FplusN, notEq_out_gg(X, FplusN)) -> U11_ggg(X, F, T, N, FplusN, add_in_gga(X, N, XplusN)) U11_ggg(X, F, T, N, FplusN, add_out_gga(X, N, XplusN)) -> U12_ggg(X, F, T, N, notEq_in_gg(F, XplusN)) U12_ggg(X, F, T, N, notEq_out_gg(F, XplusN)) -> U13_ggg(X, F, T, N, noattack_in_ggg(X, T, s(N))) U13_ggg(X, F, T, N, noattack_out_ggg(X, T, s(N))) -> noattack_out_ggg(X, .(F, T), N) U6_g(X, Y, noattack_out_ggg(X, Y, s(0))) -> U7_g(X, Y, safe_in_g(Y)) U7_g(X, Y, safe_out_g(Y)) -> safe_out_g(.(X, Y)) U2_a(Y, safe_out_g(Y)) -> queens_out_a(Y) The argument filtering Pi contains the following mapping: queens_in_a(x1) = queens_in_a U1_a(x1, x2) = U1_a(x2) perm_in_ga(x1, x2) = perm_in_ga(x1) [] = [] perm_out_ga(x1, x2) = perm_out_ga(x2) .(x1, x2) = .(x1, x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5) s(x1) = s(x1) 0 = 0 U2_a(x1, x2) = U2_a(x1, x2) safe_in_g(x1) = safe_in_g(x1) safe_out_g(x1) = safe_out_g U6_g(x1, x2, x3) = U6_g(x2, x3) noattack_in_ggg(x1, x2, x3) = noattack_in_ggg(x1, x2, x3) noattack_out_ggg(x1, x2, x3) = noattack_out_ggg U8_ggg(x1, x2, x3, x4, x5) = U8_ggg(x1, x2, x3, x4, x5) notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2) notEq_out_gg(x1, x2) = notEq_out_gg U15_gg(x1, x2, x3) = U15_gg(x3) U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x1, x2, x3, x4, x5) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U14_gga(x1, x2, x3, x4) = U14_gga(x4) U10_ggg(x1, x2, x3, x4, x5, x6) = U10_ggg(x1, x2, x3, x4, x6) U11_ggg(x1, x2, x3, x4, x5, x6) = U11_ggg(x1, x2, x3, x4, x6) U12_ggg(x1, x2, x3, x4, x5) = U12_ggg(x1, x3, x4, x5) U13_ggg(x1, x2, x3, x4, x5) = U13_ggg(x5) U7_g(x1, x2, x3) = U7_g(x3) queens_out_a(x1) = queens_out_a(x1) PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (43) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (44) Obligation: Pi DP problem: The TRS P consists of the following rules: U3_GA(X, Y, V, Res, delete_out_aga(V, .(X, Y), Rest)) -> PERM_IN_GA(Rest, Res) PERM_IN_GA(.(X, Y), .(V, Res)) -> U3_GA(X, Y, V, Res, delete_in_aga(V, .(X, Y), Rest)) The TRS R consists of the following rules: delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y) delete_in_aga(X, .(F, T), .(F, R)) -> U5_aga(X, F, T, R, delete_in_aga(X, T, R)) U5_aga(X, F, T, R, delete_out_aga(X, T, R)) -> delete_out_aga(X, .(F, T), .(F, R)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) delete_in_aga(x1, x2, x3) = delete_in_aga(x2) delete_out_aga(x1, x2, x3) = delete_out_aga(x1, x3) U5_aga(x1, x2, x3, x4, x5) = U5_aga(x2, x5) PERM_IN_GA(x1, x2) = PERM_IN_GA(x1) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (45) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: U3_GA(delete_out_aga(V, Rest)) -> PERM_IN_GA(Rest) PERM_IN_GA(.(X, Y)) -> U3_GA(delete_in_aga(.(X, Y))) The TRS R consists of the following rules: delete_in_aga(.(X, Y)) -> delete_out_aga(X, Y) delete_in_aga(.(F, T)) -> U5_aga(F, delete_in_aga(T)) U5_aga(F, delete_out_aga(X, R)) -> delete_out_aga(X, .(F, R)) The set Q consists of the following terms: delete_in_aga(x0) U5_aga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (47) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: U3_GA(delete_out_aga(V, Rest)) -> PERM_IN_GA(Rest) Strictly oriented rules of the TRS R: delete_in_aga(.(F, T)) -> U5_aga(F, delete_in_aga(T)) U5_aga(F, delete_out_aga(X, R)) -> delete_out_aga(X, .(F, R)) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(PERM_IN_GA(x_1)) = 2*x_1 POL(U3_GA(x_1)) = x_1 POL(U5_aga(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(delete_in_aga(x_1)) = 2*x_1 POL(delete_out_aga(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: PERM_IN_GA(.(X, Y)) -> U3_GA(delete_in_aga(.(X, Y))) The TRS R consists of the following rules: delete_in_aga(.(X, Y)) -> delete_out_aga(X, Y) The set Q consists of the following terms: delete_in_aga(x0) U5_aga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (49) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (50) TRUE