Left Termination proof of /tmp/tmp3pWuw0/queens.pl

Left Termination of the query pattern queens(f) w.r.t. the given Prolog program could successfully be proven:

↳ PROLOG

  ↳ PrologToPiTRSProof

queens₁(Y) :- perm₂(.₂(s₁(0₀), .₂(s₁²(0₀), .₂(s₁³(0₀), .₂(s₁⁴(0₀), .₂(s₁⁵(0₀), .₂(s₁⁶(0₀), .₂(s₁⁷(0₀), .₂(s₁⁸(0₀), {}₀)))))))), Y), safe₁(Y).
perm₂({}₀, {}₀).
perm₂(.₂(X, Y), .₂(V, Res)) :- delete₃(V, .₂(X, Y), Rest), perm₂(Rest, Res).
delete₃(X, .₂(X, Y), Y).
delete₃(X, .₂(F, T), .₂(F, R)) :- delete₃(X, T, R).
safe₁({}₀).
safe₁(.₂(X, Y)) :- noattack₃(X, Y, s₁(0₀)), safe₁(Y).
noattack₃(X, {}₀, N).
noattack₃(X, .₂(F, T), N) :- notEq₂(X, F), add₃(F, N, FplusN), notEq₂(X, FplusN), add₃(X, N, XplusN), notEq₂(F, XplusN), noattack₃(X, T, s₁(N)).
add₃(0₀, X, X).
add₃(s₁(X), Y, s₁(Z)) :- add₃(X, Y, Z).
notEq₂(0₀, s₁(X)).
notEq₂(s₁(X), 0₀).
notEq₂(s₁(X), s₁(Y)) :- notEq₂(X, Y).

With regard to the inferred argument filtering the predicates were used in the following modes:
queens₁: (f)
perm₂: (b,f)
delete₃: (f,b,f)
safe₁: (b)
noattack₃: (b,b,b)
notEq₂: (b,b)
add₃: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

queens_1_in_a₁(Y) -> if_queens_1_in_1_a₂(Y, perm_2_in_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y))
perm_2_in_ga₂([]_0, []_0) -> perm_2_out_ga₂([]_0, []_0)
perm_2_in_ga₂(._2₂(X, Y), ._2₂(V, Res)) -> if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))
delete_3_in_aga₃(X, ._2₂(X, Y), Y) -> delete_3_out_aga₃(X, ._2₂(X, Y), Y)
delete_3_in_aga₃(X, ._2₂(F, T), ._2₂(F, R)) -> if_delete_3_in_1_aga₅(X, F, T, R, delete_3_in_aga₃(X, T, R))
if_delete_3_in_1_aga₅(X, F, T, R, delete_3_out_aga₃(X, T, R)) -> delete_3_out_aga₃(X, ._2₂(F, T), ._2₂(F, R))
if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_in_ga₂(Rest, Res))
if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_out_ga₂(Rest, Res)) -> perm_2_out_ga₂(._2₂(X, Y), ._2₂(V, Res))
if_queens_1_in_1_a₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> if_queens_1_in_2_a₂(Y, safe_1_in_g₁(Y))
safe_1_in_g₁([]_0) -> safe_1_out_g₁([]_0)
safe_1_in_g₁(._2₂(X, Y)) -> if_safe_1_in_1_g₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
noattack_3_in_ggg₃(X, []_0, N) -> noattack_3_out_ggg₃(X, []_0, N)
noattack_3_in_ggg₃(X, ._2₂(F, T), N) -> if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_in_gg₂(X, F))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))
if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_in_ggg₃(X, T, s_1₁(N)))
if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_out_ggg₃(X, T, s_1₁(N))) -> noattack_3_out_ggg₃(X, ._2₂(F, T), N)
if_safe_1_in_1_g₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> if_safe_1_in_2_g₃(X, Y, safe_1_in_g₁(Y))
if_safe_1_in_2_g₃(X, Y, safe_1_out_g₁(Y)) -> safe_1_out_g₁(._2₂(X, Y))
if_queens_1_in_2_a₂(Y, safe_1_out_g₁(Y)) -> queens_1_out_a₁(Y)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

queens_1_in_a₁(Y) -> if_queens_1_in_1_a₂(Y, perm_2_in_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y))
perm_2_in_ga₂([]_0, []_0) -> perm_2_out_ga₂([]_0, []_0)
perm_2_in_ga₂(._2₂(X, Y), ._2₂(V, Res)) -> if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))
delete_3_in_aga₃(X, ._2₂(X, Y), Y) -> delete_3_out_aga₃(X, ._2₂(X, Y), Y)
delete_3_in_aga₃(X, ._2₂(F, T), ._2₂(F, R)) -> if_delete_3_in_1_aga₅(X, F, T, R, delete_3_in_aga₃(X, T, R))
if_delete_3_in_1_aga₅(X, F, T, R, delete_3_out_aga₃(X, T, R)) -> delete_3_out_aga₃(X, ._2₂(F, T), ._2₂(F, R))
if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_in_ga₂(Rest, Res))
if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_out_ga₂(Rest, Res)) -> perm_2_out_ga₂(._2₂(X, Y), ._2₂(V, Res))
if_queens_1_in_1_a₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> if_queens_1_in_2_a₂(Y, safe_1_in_g₁(Y))
safe_1_in_g₁([]_0) -> safe_1_out_g₁([]_0)
safe_1_in_g₁(._2₂(X, Y)) -> if_safe_1_in_1_g₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
noattack_3_in_ggg₃(X, []_0, N) -> noattack_3_out_ggg₃(X, []_0, N)
noattack_3_in_ggg₃(X, ._2₂(F, T), N) -> if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_in_gg₂(X, F))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))
if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_in_ggg₃(X, T, s_1₁(N)))
if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_out_ggg₃(X, T, s_1₁(N))) -> noattack_3_out_ggg₃(X, ._2₂(F, T), N)
if_safe_1_in_1_g₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> if_safe_1_in_2_g₃(X, Y, safe_1_in_g₁(Y))
if_safe_1_in_2_g₃(X, Y, safe_1_out_g₁(Y)) -> safe_1_out_g₁(._2₂(X, Y))
if_queens_1_in_2_a₂(Y, safe_1_out_g₁(Y)) -> queens_1_out_a₁(Y)

QUEENS_1_IN_A₁(Y) -> IF_QUEENS_1_IN_1_A₂(Y, perm_2_in_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y))
QUEENS_1_IN_A₁(Y) -> PERM_2_IN_GA₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)
PERM_2_IN_GA₂(._2₂(X, Y), ._2₂(V, Res)) -> IF_PERM_2_IN_1_GA₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))
PERM_2_IN_GA₂(._2₂(X, Y), ._2₂(V, Res)) -> DELETE_3_IN_AGA₃(V, ._2₂(X, Y), Rest)
DELETE_3_IN_AGA₃(X, ._2₂(F, T), ._2₂(F, R)) -> IF_DELETE_3_IN_1_AGA₅(X, F, T, R, delete_3_in_aga₃(X, T, R))
DELETE_3_IN_AGA₃(X, ._2₂(F, T), ._2₂(F, R)) -> DELETE_3_IN_AGA₃(X, T, R)
IF_PERM_2_IN_1_GA₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> IF_PERM_2_IN_2_GA₆(X, Y, V, Res, Rest, perm_2_in_ga₂(Rest, Res))
IF_PERM_2_IN_1_GA₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> PERM_2_IN_GA₂(Rest, Res)
IF_QUEENS_1_IN_1_A₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> IF_QUEENS_1_IN_2_A₂(Y, safe_1_in_g₁(Y))
IF_QUEENS_1_IN_1_A₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> SAFE_1_IN_G₁(Y)
SAFE_1_IN_G₁(._2₂(X, Y)) -> IF_SAFE_1_IN_1_G₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
SAFE_1_IN_G₁(._2₂(X, Y)) -> NOATTACK_3_IN_GGG₃(X, Y, s_1₁(0_0))
NOATTACK_3_IN_GGG₃(X, ._2₂(F, T), N) -> IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_in_gg₂(X, F))
NOATTACK_3_IN_GGG₃(X, ._2₂(F, T), N) -> NOTEQ_2_IN_GG₂(X, F)
NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_NOTEQ_2_IN_1_GG₃(X, Y, notEq_2_in_gg₂(X, Y))
NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NOTEQ_2_IN_GG₂(X, Y)
IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> ADD_3_IN_GGA₃(F, N, FplusN)
ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> IF_ADD_3_IN_1_GGA₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GGA₃(X, Y, Z)
IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> IF_NOATTACK_3_IN_3_GGG₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))
IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> NOTEQ_2_IN_GG₂(X, FplusN)
IF_NOATTACK_3_IN_3_GGG₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
IF_NOATTACK_3_IN_3_GGG₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> ADD_3_IN_GGA₃(X, N, XplusN)
IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> IF_NOATTACK_3_IN_5_GGG₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> NOTEQ_2_IN_GG₂(F, XplusN)
IF_NOATTACK_3_IN_5_GGG₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> IF_NOATTACK_3_IN_6_GGG₅(X, F, T, N, noattack_3_in_ggg₃(X, T, s_1₁(N)))
IF_NOATTACK_3_IN_5_GGG₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> NOATTACK_3_IN_GGG₃(X, T, s_1₁(N))
IF_SAFE_1_IN_1_G₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> IF_SAFE_1_IN_2_G₃(X, Y, safe_1_in_g₁(Y))
IF_SAFE_1_IN_1_G₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> SAFE_1_IN_G₁(Y)

The TRS R consists of the following rules:

queens_1_in_a₁(Y) -> if_queens_1_in_1_a₂(Y, perm_2_in_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y))
perm_2_in_ga₂([]_0, []_0) -> perm_2_out_ga₂([]_0, []_0)
perm_2_in_ga₂(._2₂(X, Y), ._2₂(V, Res)) -> if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))
delete_3_in_aga₃(X, ._2₂(X, Y), Y) -> delete_3_out_aga₃(X, ._2₂(X, Y), Y)
delete_3_in_aga₃(X, ._2₂(F, T), ._2₂(F, R)) -> if_delete_3_in_1_aga₅(X, F, T, R, delete_3_in_aga₃(X, T, R))
if_delete_3_in_1_aga₅(X, F, T, R, delete_3_out_aga₃(X, T, R)) -> delete_3_out_aga₃(X, ._2₂(F, T), ._2₂(F, R))
if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_in_ga₂(Rest, Res))
if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_out_ga₂(Rest, Res)) -> perm_2_out_ga₂(._2₂(X, Y), ._2₂(V, Res))
if_queens_1_in_1_a₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> if_queens_1_in_2_a₂(Y, safe_1_in_g₁(Y))
safe_1_in_g₁([]_0) -> safe_1_out_g₁([]_0)
safe_1_in_g₁(._2₂(X, Y)) -> if_safe_1_in_1_g₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
noattack_3_in_ggg₃(X, []_0, N) -> noattack_3_out_ggg₃(X, []_0, N)
noattack_3_in_ggg₃(X, ._2₂(F, T), N) -> if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_in_gg₂(X, F))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))
if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_in_ggg₃(X, T, s_1₁(N)))
if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_out_ggg₃(X, T, s_1₁(N))) -> noattack_3_out_ggg₃(X, ._2₂(F, T), N)
if_safe_1_in_1_g₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> if_safe_1_in_2_g₃(X, Y, safe_1_in_g₁(Y))
if_safe_1_in_2_g₃(X, Y, safe_1_out_g₁(Y)) -> safe_1_out_g₁(._2₂(X, Y))
if_queens_1_in_2_a₂(Y, safe_1_out_g₁(Y)) -> queens_1_out_a₁(Y)

The argument filtering Pi contains the following mapping:
queens_1_in_a₁(x1) = queens_1_in_a
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
[]_0 = []_0
if_queens_1_in_1_a₂(x1, x2) = if_queens_1_in_1_a₁(x2)
perm_2_in_ga₂(x1, x2) = perm_2_in_ga₁(x1)
perm_2_out_ga₂(x1, x2) = perm_2_out_ga₁(x2)
if_perm_2_in_1_ga₅(x1, x2, x3, x4, x5) = if_perm_2_in_1_ga₁(x5)
delete_3_in_aga₃(x1, x2, x3) = delete_3_in_aga₁(x2)
delete_3_out_aga₃(x1, x2, x3) = delete_3_out_aga₂(x1, x3)
if_delete_3_in_1_aga₅(x1, x2, x3, x4, x5) = if_delete_3_in_1_aga₂(x2, x5)
if_perm_2_in_2_ga₆(x1, x2, x3, x4, x5, x6) = if_perm_2_in_2_ga₂(x3, x6)
if_queens_1_in_2_a₂(x1, x2) = if_queens_1_in_2_a₂(x1, x2)
safe_1_in_g₁(x1) = safe_1_in_g₁(x1)
safe_1_out_g₁(x1) = safe_1_out_g
if_safe_1_in_1_g₃(x1, x2, x3) = if_safe_1_in_1_g₂(x2, x3)
noattack_3_in_ggg₃(x1, x2, x3) = noattack_3_in_ggg₃(x1, x2, x3)
noattack_3_out_ggg₃(x1, x2, x3) = noattack_3_out_ggg
if_noattack_3_in_1_ggg₅(x1, x2, x3, x4, x5) = if_noattack_3_in_1_ggg₅(x1, x2, x3, x4, x5)
notEq_2_in_gg₂(x1, x2) = notEq_2_in_gg₂(x1, x2)
notEq_2_out_gg₂(x1, x2) = notEq_2_out_gg
if_notEq_2_in_1_gg₃(x1, x2, x3) = if_notEq_2_in_1_gg₁(x3)
if_noattack_3_in_2_ggg₅(x1, x2, x3, x4, x5) = if_noattack_3_in_2_ggg₅(x1, x2, x3, x4, x5)
add_3_in_gga₃(x1, x2, x3) = add_3_in_gga₂(x1, x2)
add_3_out_gga₃(x1, x2, x3) = add_3_out_gga₁(x3)
if_add_3_in_1_gga₄(x1, x2, x3, x4) = if_add_3_in_1_gga₁(x4)
if_noattack_3_in_3_ggg₆(x1, x2, x3, x4, x5, x6) = if_noattack_3_in_3_ggg₅(x1, x2, x3, x4, x6)
if_noattack_3_in_4_ggg₅(x1, x2, x3, x4, x5) = if_noattack_3_in_4_ggg₅(x1, x2, x3, x4, x5)
if_noattack_3_in_5_ggg₆(x1, x2, x3, x4, x5, x6) = if_noattack_3_in_5_ggg₄(x1, x3, x4, x6)
if_noattack_3_in_6_ggg₅(x1, x2, x3, x4, x5) = if_noattack_3_in_6_ggg₁(x5)
if_safe_1_in_2_g₃(x1, x2, x3) = if_safe_1_in_2_g₁(x3)
queens_1_out_a₁(x1) = queens_1_out_a₁(x1)
IF_NOATTACK_3_IN_5_GGG₆(x1, x2, x3, x4, x5, x6) = IF_NOATTACK_3_IN_5_GGG₄(x1, x3, x4, x6)
ADD_3_IN_GGA₃(x1, x2, x3) = ADD_3_IN_GGA₂(x1, x2)
IF_DELETE_3_IN_1_AGA₅(x1, x2, x3, x4, x5) = IF_DELETE_3_IN_1_AGA₂(x2, x5)
IF_NOATTACK_3_IN_4_GGG₅(x1, x2, x3, x4, x5) = IF_NOATTACK_3_IN_4_GGG₅(x1, x2, x3, x4, x5)
IF_NOTEQ_2_IN_1_GG₃(x1, x2, x3) = IF_NOTEQ_2_IN_1_GG₁(x3)
PERM_2_IN_GA₂(x1, x2) = PERM_2_IN_GA₁(x1)
IF_PERM_2_IN_1_GA₅(x1, x2, x3, x4, x5) = IF_PERM_2_IN_1_GA₁(x5)
IF_SAFE_1_IN_2_G₃(x1, x2, x3) = IF_SAFE_1_IN_2_G₁(x3)
QUEENS_1_IN_A₁(x1) = QUEENS_1_IN_A
IF_QUEENS_1_IN_2_A₂(x1, x2) = IF_QUEENS_1_IN_2_A₂(x1, x2)
NOATTACK_3_IN_GGG₃(x1, x2, x3) = NOATTACK_3_IN_GGG₃(x1, x2, x3)
NOTEQ_2_IN_GG₂(x1, x2) = NOTEQ_2_IN_GG₂(x1, x2)
IF_NOATTACK_3_IN_1_GGG₅(x1, x2, x3, x4, x5) = IF_NOATTACK_3_IN_1_GGG₅(x1, x2, x3, x4, x5)
IF_NOATTACK_3_IN_6_GGG₅(x1, x2, x3, x4, x5) = IF_NOATTACK_3_IN_6_GGG₁(x5)
IF_ADD_3_IN_1_GGA₄(x1, x2, x3, x4) = IF_ADD_3_IN_1_GGA₁(x4)
DELETE_3_IN_AGA₃(x1, x2, x3) = DELETE_3_IN_AGA₁(x2)
IF_SAFE_1_IN_1_G₃(x1, x2, x3) = IF_SAFE_1_IN_1_G₂(x2, x3)
IF_NOATTACK_3_IN_3_GGG₆(x1, x2, x3, x4, x5, x6) = IF_NOATTACK_3_IN_3_GGG₅(x1, x2, x3, x4, x6)
SAFE_1_IN_G₁(x1) = SAFE_1_IN_G₁(x1)
IF_QUEENS_1_IN_1_A₂(x1, x2) = IF_QUEENS_1_IN_1_A₁(x2)
IF_NOATTACK_3_IN_2_GGG₅(x1, x2, x3, x4, x5) = IF_NOATTACK_3_IN_2_GGG₅(x1, x2, x3, x4, x5)
IF_PERM_2_IN_2_GA₆(x1, x2, x3, x4, x5, x6) = IF_PERM_2_IN_2_GA₂(x3, x6)

We have to consider all (P,R,Pi)-chains

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

Pi DP problem:
The TRS P consists of the following rules:

QUEENS_1_IN_A₁(Y) -> IF_QUEENS_1_IN_1_A₂(Y, perm_2_in_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y))
QUEENS_1_IN_A₁(Y) -> PERM_2_IN_GA₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)
PERM_2_IN_GA₂(._2₂(X, Y), ._2₂(V, Res)) -> IF_PERM_2_IN_1_GA₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))
PERM_2_IN_GA₂(._2₂(X, Y), ._2₂(V, Res)) -> DELETE_3_IN_AGA₃(V, ._2₂(X, Y), Rest)
DELETE_3_IN_AGA₃(X, ._2₂(F, T), ._2₂(F, R)) -> IF_DELETE_3_IN_1_AGA₅(X, F, T, R, delete_3_in_aga₃(X, T, R))
DELETE_3_IN_AGA₃(X, ._2₂(F, T), ._2₂(F, R)) -> DELETE_3_IN_AGA₃(X, T, R)
IF_PERM_2_IN_1_GA₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> IF_PERM_2_IN_2_GA₆(X, Y, V, Res, Rest, perm_2_in_ga₂(Rest, Res))
IF_PERM_2_IN_1_GA₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> PERM_2_IN_GA₂(Rest, Res)
IF_QUEENS_1_IN_1_A₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> IF_QUEENS_1_IN_2_A₂(Y, safe_1_in_g₁(Y))
IF_QUEENS_1_IN_1_A₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> SAFE_1_IN_G₁(Y)
SAFE_1_IN_G₁(._2₂(X, Y)) -> IF_SAFE_1_IN_1_G₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
SAFE_1_IN_G₁(._2₂(X, Y)) -> NOATTACK_3_IN_GGG₃(X, Y, s_1₁(0_0))
NOATTACK_3_IN_GGG₃(X, ._2₂(F, T), N) -> IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_in_gg₂(X, F))
NOATTACK_3_IN_GGG₃(X, ._2₂(F, T), N) -> NOTEQ_2_IN_GG₂(X, F)
NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_NOTEQ_2_IN_1_GG₃(X, Y, notEq_2_in_gg₂(X, Y))
NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NOTEQ_2_IN_GG₂(X, Y)
IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> ADD_3_IN_GGA₃(F, N, FplusN)
ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> IF_ADD_3_IN_1_GGA₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GGA₃(X, Y, Z)
IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> IF_NOATTACK_3_IN_3_GGG₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))
IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> NOTEQ_2_IN_GG₂(X, FplusN)
IF_NOATTACK_3_IN_3_GGG₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
IF_NOATTACK_3_IN_3_GGG₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> ADD_3_IN_GGA₃(X, N, XplusN)
IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> IF_NOATTACK_3_IN_5_GGG₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> NOTEQ_2_IN_GG₂(F, XplusN)
IF_NOATTACK_3_IN_5_GGG₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> IF_NOATTACK_3_IN_6_GGG₅(X, F, T, N, noattack_3_in_ggg₃(X, T, s_1₁(N)))
IF_NOATTACK_3_IN_5_GGG₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> NOATTACK_3_IN_GGG₃(X, T, s_1₁(N))
IF_SAFE_1_IN_1_G₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> IF_SAFE_1_IN_2_G₃(X, Y, safe_1_in_g₁(Y))
IF_SAFE_1_IN_1_G₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> SAFE_1_IN_G₁(Y)

The TRS R consists of the following rules:

queens_1_in_a₁(Y) -> if_queens_1_in_1_a₂(Y, perm_2_in_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y))
perm_2_in_ga₂([]_0, []_0) -> perm_2_out_ga₂([]_0, []_0)
perm_2_in_ga₂(._2₂(X, Y), ._2₂(V, Res)) -> if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))
delete_3_in_aga₃(X, ._2₂(X, Y), Y) -> delete_3_out_aga₃(X, ._2₂(X, Y), Y)
delete_3_in_aga₃(X, ._2₂(F, T), ._2₂(F, R)) -> if_delete_3_in_1_aga₅(X, F, T, R, delete_3_in_aga₃(X, T, R))
if_delete_3_in_1_aga₅(X, F, T, R, delete_3_out_aga₃(X, T, R)) -> delete_3_out_aga₃(X, ._2₂(F, T), ._2₂(F, R))
if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_in_ga₂(Rest, Res))
if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_out_ga₂(Rest, Res)) -> perm_2_out_ga₂(._2₂(X, Y), ._2₂(V, Res))
if_queens_1_in_1_a₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> if_queens_1_in_2_a₂(Y, safe_1_in_g₁(Y))
safe_1_in_g₁([]_0) -> safe_1_out_g₁([]_0)
safe_1_in_g₁(._2₂(X, Y)) -> if_safe_1_in_1_g₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
noattack_3_in_ggg₃(X, []_0, N) -> noattack_3_out_ggg₃(X, []_0, N)
noattack_3_in_ggg₃(X, ._2₂(F, T), N) -> if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_in_gg₂(X, F))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))
if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_in_ggg₃(X, T, s_1₁(N)))
if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_out_ggg₃(X, T, s_1₁(N))) -> noattack_3_out_ggg₃(X, ._2₂(F, T), N)
if_safe_1_in_1_g₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> if_safe_1_in_2_g₃(X, Y, safe_1_in_g₁(Y))
if_safe_1_in_2_g₃(X, Y, safe_1_out_g₁(Y)) -> safe_1_out_g₁(._2₂(X, Y))
if_queens_1_in_2_a₂(Y, safe_1_out_g₁(Y)) -> queens_1_out_a₁(Y)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GGA₃(X, Y, Z)

The TRS R consists of the following rules:

queens_1_in_a₁(Y) -> if_queens_1_in_1_a₂(Y, perm_2_in_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y))
perm_2_in_ga₂([]_0, []_0) -> perm_2_out_ga₂([]_0, []_0)
perm_2_in_ga₂(._2₂(X, Y), ._2₂(V, Res)) -> if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))
delete_3_in_aga₃(X, ._2₂(X, Y), Y) -> delete_3_out_aga₃(X, ._2₂(X, Y), Y)
delete_3_in_aga₃(X, ._2₂(F, T), ._2₂(F, R)) -> if_delete_3_in_1_aga₅(X, F, T, R, delete_3_in_aga₃(X, T, R))
if_delete_3_in_1_aga₅(X, F, T, R, delete_3_out_aga₃(X, T, R)) -> delete_3_out_aga₃(X, ._2₂(F, T), ._2₂(F, R))
if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_in_ga₂(Rest, Res))
if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_out_ga₂(Rest, Res)) -> perm_2_out_ga₂(._2₂(X, Y), ._2₂(V, Res))
if_queens_1_in_1_a₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> if_queens_1_in_2_a₂(Y, safe_1_in_g₁(Y))
safe_1_in_g₁([]_0) -> safe_1_out_g₁([]_0)
safe_1_in_g₁(._2₂(X, Y)) -> if_safe_1_in_1_g₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
noattack_3_in_ggg₃(X, []_0, N) -> noattack_3_out_ggg₃(X, []_0, N)
noattack_3_in_ggg₃(X, ._2₂(F, T), N) -> if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_in_gg₂(X, F))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))
if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_in_ggg₃(X, T, s_1₁(N)))
if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_out_ggg₃(X, T, s_1₁(N))) -> noattack_3_out_ggg₃(X, ._2₂(F, T), N)
if_safe_1_in_1_g₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> if_safe_1_in_2_g₃(X, Y, safe_1_in_g₁(Y))
if_safe_1_in_2_g₃(X, Y, safe_1_out_g₁(Y)) -> safe_1_out_g₁(._2₂(X, Y))
if_queens_1_in_2_a₂(Y, safe_1_out_g₁(Y)) -> queens_1_out_a₁(Y)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

ADD_3_IN_GGA₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GGA₃(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s_1₁(x1) = s_1₁(x1)
ADD_3_IN_GGA₃(x1, x2, x3) = ADD_3_IN_GGA₂(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Q DP problem:
The TRS P consists of the following rules:

ADD_3_IN_GGA₂(s_1₁(X), Y) -> ADD_3_IN_GGA₂(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {ADD_3_IN_GGA₂}.
By using the subterm criterion together with the size-change analysis 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_3_IN_GGA₂(s_1₁(X), Y) -> ADD_3_IN_GGA₂(X, Y)
The graph contains the following edges 1 > 1, 2 >= 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NOTEQ_2_IN_GG₂(X, Y)

The TRS R consists of the following rules:

queens_1_in_a₁(Y) -> if_queens_1_in_1_a₂(Y, perm_2_in_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y))
perm_2_in_ga₂([]_0, []_0) -> perm_2_out_ga₂([]_0, []_0)
perm_2_in_ga₂(._2₂(X, Y), ._2₂(V, Res)) -> if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))
delete_3_in_aga₃(X, ._2₂(X, Y), Y) -> delete_3_out_aga₃(X, ._2₂(X, Y), Y)
delete_3_in_aga₃(X, ._2₂(F, T), ._2₂(F, R)) -> if_delete_3_in_1_aga₅(X, F, T, R, delete_3_in_aga₃(X, T, R))
if_delete_3_in_1_aga₅(X, F, T, R, delete_3_out_aga₃(X, T, R)) -> delete_3_out_aga₃(X, ._2₂(F, T), ._2₂(F, R))
if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_in_ga₂(Rest, Res))
if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_out_ga₂(Rest, Res)) -> perm_2_out_ga₂(._2₂(X, Y), ._2₂(V, Res))
if_queens_1_in_1_a₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> if_queens_1_in_2_a₂(Y, safe_1_in_g₁(Y))
safe_1_in_g₁([]_0) -> safe_1_out_g₁([]_0)
safe_1_in_g₁(._2₂(X, Y)) -> if_safe_1_in_1_g₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
noattack_3_in_ggg₃(X, []_0, N) -> noattack_3_out_ggg₃(X, []_0, N)
noattack_3_in_ggg₃(X, ._2₂(F, T), N) -> if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_in_gg₂(X, F))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))
if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_in_ggg₃(X, T, s_1₁(N)))
if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_out_ggg₃(X, T, s_1₁(N))) -> noattack_3_out_ggg₃(X, ._2₂(F, T), N)
if_safe_1_in_1_g₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> if_safe_1_in_2_g₃(X, Y, safe_1_in_g₁(Y))
if_safe_1_in_2_g₃(X, Y, safe_1_out_g₁(Y)) -> safe_1_out_g₁(._2₂(X, Y))
if_queens_1_in_2_a₂(Y, safe_1_out_g₁(Y)) -> queens_1_out_a₁(Y)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NOTEQ_2_IN_GG₂(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Q DP problem:
The TRS P consists of the following rules:

NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NOTEQ_2_IN_GG₂(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {NOTEQ_2_IN_GG₂}.
By using the subterm criterion together with the size-change analysis 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_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NOTEQ_2_IN_GG₂(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

IF_NOATTACK_3_IN_3_GGG₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
IF_NOATTACK_3_IN_5_GGG₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> NOATTACK_3_IN_GGG₃(X, T, s_1₁(N))
NOATTACK_3_IN_GGG₃(X, ._2₂(F, T), N) -> IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_in_gg₂(X, F))
IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> IF_NOATTACK_3_IN_5_GGG₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> IF_NOATTACK_3_IN_3_GGG₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))

The TRS R consists of the following rules:

queens_1_in_a₁(Y) -> if_queens_1_in_1_a₂(Y, perm_2_in_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y))
perm_2_in_ga₂([]_0, []_0) -> perm_2_out_ga₂([]_0, []_0)
perm_2_in_ga₂(._2₂(X, Y), ._2₂(V, Res)) -> if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))
delete_3_in_aga₃(X, ._2₂(X, Y), Y) -> delete_3_out_aga₃(X, ._2₂(X, Y), Y)
delete_3_in_aga₃(X, ._2₂(F, T), ._2₂(F, R)) -> if_delete_3_in_1_aga₅(X, F, T, R, delete_3_in_aga₃(X, T, R))
if_delete_3_in_1_aga₅(X, F, T, R, delete_3_out_aga₃(X, T, R)) -> delete_3_out_aga₃(X, ._2₂(F, T), ._2₂(F, R))
if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_in_ga₂(Rest, Res))
if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_out_ga₂(Rest, Res)) -> perm_2_out_ga₂(._2₂(X, Y), ._2₂(V, Res))
if_queens_1_in_1_a₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> if_queens_1_in_2_a₂(Y, safe_1_in_g₁(Y))
safe_1_in_g₁([]_0) -> safe_1_out_g₁([]_0)
safe_1_in_g₁(._2₂(X, Y)) -> if_safe_1_in_1_g₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
noattack_3_in_ggg₃(X, []_0, N) -> noattack_3_out_ggg₃(X, []_0, N)
noattack_3_in_ggg₃(X, ._2₂(F, T), N) -> if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_in_gg₂(X, F))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))
if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_in_ggg₃(X, T, s_1₁(N)))
if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_out_ggg₃(X, T, s_1₁(N))) -> noattack_3_out_ggg₃(X, ._2₂(F, T), N)
if_safe_1_in_1_g₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> if_safe_1_in_2_g₃(X, Y, safe_1_in_g₁(Y))
if_safe_1_in_2_g₃(X, Y, safe_1_out_g₁(Y)) -> safe_1_out_g₁(._2₂(X, Y))
if_queens_1_in_2_a₂(Y, safe_1_out_g₁(Y)) -> queens_1_out_a₁(Y)

The argument filtering Pi contains the following mapping:
queens_1_in_a₁(x1) = queens_1_in_a
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
[]_0 = []_0
if_queens_1_in_1_a₂(x1, x2) = if_queens_1_in_1_a₁(x2)
perm_2_in_ga₂(x1, x2) = perm_2_in_ga₁(x1)
perm_2_out_ga₂(x1, x2) = perm_2_out_ga₁(x2)
if_perm_2_in_1_ga₅(x1, x2, x3, x4, x5) = if_perm_2_in_1_ga₁(x5)
delete_3_in_aga₃(x1, x2, x3) = delete_3_in_aga₁(x2)
delete_3_out_aga₃(x1, x2, x3) = delete_3_out_aga₂(x1, x3)
if_delete_3_in_1_aga₅(x1, x2, x3, x4, x5) = if_delete_3_in_1_aga₂(x2, x5)
if_perm_2_in_2_ga₆(x1, x2, x3, x4, x5, x6) = if_perm_2_in_2_ga₂(x3, x6)
if_queens_1_in_2_a₂(x1, x2) = if_queens_1_in_2_a₂(x1, x2)
safe_1_in_g₁(x1) = safe_1_in_g₁(x1)
safe_1_out_g₁(x1) = safe_1_out_g
if_safe_1_in_1_g₃(x1, x2, x3) = if_safe_1_in_1_g₂(x2, x3)
noattack_3_in_ggg₃(x1, x2, x3) = noattack_3_in_ggg₃(x1, x2, x3)
noattack_3_out_ggg₃(x1, x2, x3) = noattack_3_out_ggg
if_noattack_3_in_1_ggg₅(x1, x2, x3, x4, x5) = if_noattack_3_in_1_ggg₅(x1, x2, x3, x4, x5)
notEq_2_in_gg₂(x1, x2) = notEq_2_in_gg₂(x1, x2)
notEq_2_out_gg₂(x1, x2) = notEq_2_out_gg
if_notEq_2_in_1_gg₃(x1, x2, x3) = if_notEq_2_in_1_gg₁(x3)
if_noattack_3_in_2_ggg₅(x1, x2, x3, x4, x5) = if_noattack_3_in_2_ggg₅(x1, x2, x3, x4, x5)
add_3_in_gga₃(x1, x2, x3) = add_3_in_gga₂(x1, x2)
add_3_out_gga₃(x1, x2, x3) = add_3_out_gga₁(x3)
if_add_3_in_1_gga₄(x1, x2, x3, x4) = if_add_3_in_1_gga₁(x4)
if_noattack_3_in_3_ggg₆(x1, x2, x3, x4, x5, x6) = if_noattack_3_in_3_ggg₅(x1, x2, x3, x4, x6)
if_noattack_3_in_4_ggg₅(x1, x2, x3, x4, x5) = if_noattack_3_in_4_ggg₅(x1, x2, x3, x4, x5)
if_noattack_3_in_5_ggg₆(x1, x2, x3, x4, x5, x6) = if_noattack_3_in_5_ggg₄(x1, x3, x4, x6)
if_noattack_3_in_6_ggg₅(x1, x2, x3, x4, x5) = if_noattack_3_in_6_ggg₁(x5)
if_safe_1_in_2_g₃(x1, x2, x3) = if_safe_1_in_2_g₁(x3)
queens_1_out_a₁(x1) = queens_1_out_a₁(x1)
IF_NOATTACK_3_IN_5_GGG₆(x1, x2, x3, x4, x5, x6) = IF_NOATTACK_3_IN_5_GGG₄(x1, x3, x4, x6)
IF_NOATTACK_3_IN_4_GGG₅(x1, x2, x3, x4, x5) = IF_NOATTACK_3_IN_4_GGG₅(x1, x2, x3, x4, x5)
NOATTACK_3_IN_GGG₃(x1, x2, x3) = NOATTACK_3_IN_GGG₃(x1, x2, x3)
IF_NOATTACK_3_IN_1_GGG₅(x1, x2, x3, x4, x5) = IF_NOATTACK_3_IN_1_GGG₅(x1, x2, x3, x4, x5)
IF_NOATTACK_3_IN_3_GGG₆(x1, x2, x3, x4, x5, x6) = IF_NOATTACK_3_IN_3_GGG₅(x1, x2, x3, x4, x6)
IF_NOATTACK_3_IN_2_GGG₅(x1, x2, x3, x4, x5) = IF_NOATTACK_3_IN_2_GGG₅(x1, x2, x3, x4, x5)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting we can delete all non-usable rules from R.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

IF_NOATTACK_3_IN_3_GGG₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
IF_NOATTACK_3_IN_5_GGG₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> NOATTACK_3_IN_GGG₃(X, T, s_1₁(N))
NOATTACK_3_IN_GGG₃(X, ._2₂(F, T), N) -> IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_in_gg₂(X, F))
IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> IF_NOATTACK_3_IN_5_GGG₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> IF_NOATTACK_3_IN_3_GGG₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))

The TRS R consists of the following rules:

add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))

The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
notEq_2_in_gg₂(x1, x2) = notEq_2_in_gg₂(x1, x2)
notEq_2_out_gg₂(x1, x2) = notEq_2_out_gg
if_notEq_2_in_1_gg₃(x1, x2, x3) = if_notEq_2_in_1_gg₁(x3)
add_3_in_gga₃(x1, x2, x3) = add_3_in_gga₂(x1, x2)
add_3_out_gga₃(x1, x2, x3) = add_3_out_gga₁(x3)
if_add_3_in_1_gga₄(x1, x2, x3, x4) = if_add_3_in_1_gga₁(x4)
IF_NOATTACK_3_IN_5_GGG₆(x1, x2, x3, x4, x5, x6) = IF_NOATTACK_3_IN_5_GGG₄(x1, x3, x4, x6)
IF_NOATTACK_3_IN_4_GGG₅(x1, x2, x3, x4, x5) = IF_NOATTACK_3_IN_4_GGG₅(x1, x2, x3, x4, x5)
NOATTACK_3_IN_GGG₃(x1, x2, x3) = NOATTACK_3_IN_GGG₃(x1, x2, x3)
IF_NOATTACK_3_IN_1_GGG₅(x1, x2, x3, x4, x5) = IF_NOATTACK_3_IN_1_GGG₅(x1, x2, x3, x4, x5)
IF_NOATTACK_3_IN_3_GGG₆(x1, x2, x3, x4, x5, x6) = IF_NOATTACK_3_IN_3_GGG₅(x1, x2, x3, x4, x6)
IF_NOATTACK_3_IN_2_GGG₅(x1, x2, x3, x4, x5) = IF_NOATTACK_3_IN_2_GGG₅(x1, x2, x3, x4, x5)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Q DP problem:
The TRS P consists of the following rules:

IF_NOATTACK_3_IN_3_GGG₅(X, F, T, N, notEq_2_out_gg) -> IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_in_gga₂(X, N))
IF_NOATTACK_3_IN_5_GGG₄(X, T, N, notEq_2_out_gg) -> NOATTACK_3_IN_GGG₃(X, T, s_1₁(N))
NOATTACK_3_IN_GGG₃(X, ._2₂(F, T), N) -> IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_in_gg₂(X, F))
IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_out_gg) -> IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_in_gga₂(F, N))
IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_out_gga₁(XplusN)) -> IF_NOATTACK_3_IN_5_GGG₄(X, T, N, notEq_2_in_gg₂(F, XplusN))
IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_out_gga₁(FplusN)) -> IF_NOATTACK_3_IN_3_GGG₅(X, F, T, N, notEq_2_in_gg₂(X, FplusN))

The TRS R consists of the following rules:

add_3_in_gga₂(0_0, X) -> add_3_out_gga₁(X)
add_3_in_gga₂(s_1₁(X), Y) -> if_add_3_in_1_gga₁(add_3_in_gga₂(X, Y))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₁(notEq_2_in_gg₂(X, Y))
if_add_3_in_1_gga₁(add_3_out_gga₁(Z)) -> add_3_out_gga₁(s_1₁(Z))
if_notEq_2_in_1_gg₁(notEq_2_out_gg) -> notEq_2_out_gg

The set Q consists of the following terms:

add_3_in_gga₂(x0, x1)
notEq_2_in_gg₂(x0, x1)
if_add_3_in_1_gga₁(x0)
if_notEq_2_in_1_gg₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_NOATTACK_3_IN_4_GGG₅, IF_NOATTACK_3_IN_3_GGG₅, NOATTACK_3_IN_GGG₃, IF_NOATTACK_3_IN_5_GGG₄, IF_NOATTACK_3_IN_1_GGG₅, IF_NOATTACK_3_IN_2_GGG₅}.
By using the subterm criterion together with the size-change analysis 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:

IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_out_gga₁(XplusN)) -> IF_NOATTACK_3_IN_5_GGG₄(X, T, N, notEq_2_in_gg₂(F, XplusN))
The graph contains the following edges 1 >= 1, 3 >= 2, 4 >= 3
IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_out_gga₁(FplusN)) -> IF_NOATTACK_3_IN_3_GGG₅(X, F, T, N, notEq_2_in_gg₂(X, FplusN))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
NOATTACK_3_IN_GGG₃(X, ._2₂(F, T), N) -> IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_in_gg₂(X, F))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4
IF_NOATTACK_3_IN_1_GGG₅(X, F, T, N, notEq_2_out_gg) -> IF_NOATTACK_3_IN_2_GGG₅(X, F, T, N, add_3_in_gga₂(F, N))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
IF_NOATTACK_3_IN_5_GGG₄(X, T, N, notEq_2_out_gg) -> NOATTACK_3_IN_GGG₃(X, T, s_1₁(N))
The graph contains the following edges 1 >= 1, 2 >= 2
IF_NOATTACK_3_IN_3_GGG₅(X, F, T, N, notEq_2_out_gg) -> IF_NOATTACK_3_IN_4_GGG₅(X, F, T, N, add_3_in_gga₂(X, N))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

SAFE_1_IN_G₁(._2₂(X, Y)) -> IF_SAFE_1_IN_1_G₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
IF_SAFE_1_IN_1_G₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> SAFE_1_IN_G₁(Y)

The TRS R consists of the following rules:

queens_1_in_a₁(Y) -> if_queens_1_in_1_a₂(Y, perm_2_in_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y))
perm_2_in_ga₂([]_0, []_0) -> perm_2_out_ga₂([]_0, []_0)
perm_2_in_ga₂(._2₂(X, Y), ._2₂(V, Res)) -> if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))
delete_3_in_aga₃(X, ._2₂(X, Y), Y) -> delete_3_out_aga₃(X, ._2₂(X, Y), Y)
delete_3_in_aga₃(X, ._2₂(F, T), ._2₂(F, R)) -> if_delete_3_in_1_aga₅(X, F, T, R, delete_3_in_aga₃(X, T, R))
if_delete_3_in_1_aga₅(X, F, T, R, delete_3_out_aga₃(X, T, R)) -> delete_3_out_aga₃(X, ._2₂(F, T), ._2₂(F, R))
if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_in_ga₂(Rest, Res))
if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_out_ga₂(Rest, Res)) -> perm_2_out_ga₂(._2₂(X, Y), ._2₂(V, Res))
if_queens_1_in_1_a₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> if_queens_1_in_2_a₂(Y, safe_1_in_g₁(Y))
safe_1_in_g₁([]_0) -> safe_1_out_g₁([]_0)
safe_1_in_g₁(._2₂(X, Y)) -> if_safe_1_in_1_g₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
noattack_3_in_ggg₃(X, []_0, N) -> noattack_3_out_ggg₃(X, []_0, N)
noattack_3_in_ggg₃(X, ._2₂(F, T), N) -> if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_in_gg₂(X, F))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))
if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_in_ggg₃(X, T, s_1₁(N)))
if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_out_ggg₃(X, T, s_1₁(N))) -> noattack_3_out_ggg₃(X, ._2₂(F, T), N)
if_safe_1_in_1_g₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> if_safe_1_in_2_g₃(X, Y, safe_1_in_g₁(Y))
if_safe_1_in_2_g₃(X, Y, safe_1_out_g₁(Y)) -> safe_1_out_g₁(._2₂(X, Y))
if_queens_1_in_2_a₂(Y, safe_1_out_g₁(Y)) -> queens_1_out_a₁(Y)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

SAFE_1_IN_G₁(._2₂(X, Y)) -> IF_SAFE_1_IN_1_G₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
IF_SAFE_1_IN_1_G₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> SAFE_1_IN_G₁(Y)

The TRS R consists of the following rules:

noattack_3_in_ggg₃(X, []_0, N) -> noattack_3_out_ggg₃(X, []_0, N)
noattack_3_in_ggg₃(X, ._2₂(F, T), N) -> if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_in_gg₂(X, F))
if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_in_ggg₃(X, T, s_1₁(N)))
if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_out_ggg₃(X, T, s_1₁(N))) -> noattack_3_out_ggg₃(X, ._2₂(F, T), N)

The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
[]_0 = []_0
noattack_3_in_ggg₃(x1, x2, x3) = noattack_3_in_ggg₃(x1, x2, x3)
noattack_3_out_ggg₃(x1, x2, x3) = noattack_3_out_ggg
if_noattack_3_in_1_ggg₅(x1, x2, x3, x4, x5) = if_noattack_3_in_1_ggg₅(x1, x2, x3, x4, x5)
notEq_2_in_gg₂(x1, x2) = notEq_2_in_gg₂(x1, x2)
notEq_2_out_gg₂(x1, x2) = notEq_2_out_gg
if_notEq_2_in_1_gg₃(x1, x2, x3) = if_notEq_2_in_1_gg₁(x3)
if_noattack_3_in_2_ggg₅(x1, x2, x3, x4, x5) = if_noattack_3_in_2_ggg₅(x1, x2, x3, x4, x5)
add_3_in_gga₃(x1, x2, x3) = add_3_in_gga₂(x1, x2)
add_3_out_gga₃(x1, x2, x3) = add_3_out_gga₁(x3)
if_add_3_in_1_gga₄(x1, x2, x3, x4) = if_add_3_in_1_gga₁(x4)
if_noattack_3_in_3_ggg₆(x1, x2, x3, x4, x5, x6) = if_noattack_3_in_3_ggg₅(x1, x2, x3, x4, x6)
if_noattack_3_in_4_ggg₅(x1, x2, x3, x4, x5) = if_noattack_3_in_4_ggg₅(x1, x2, x3, x4, x5)
if_noattack_3_in_5_ggg₆(x1, x2, x3, x4, x5, x6) = if_noattack_3_in_5_ggg₄(x1, x3, x4, x6)
if_noattack_3_in_6_ggg₅(x1, x2, x3, x4, x5) = if_noattack_3_in_6_ggg₁(x5)
IF_SAFE_1_IN_1_G₃(x1, x2, x3) = IF_SAFE_1_IN_1_G₂(x2, x3)
SAFE_1_IN_G₁(x1) = SAFE_1_IN_G₁(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

Q DP problem:
The TRS P consists of the following rules:

SAFE_1_IN_G₁(._2₂(X, Y)) -> IF_SAFE_1_IN_1_G₂(Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
IF_SAFE_1_IN_1_G₂(Y, noattack_3_out_ggg) -> SAFE_1_IN_G₁(Y)

The TRS R consists of the following rules:

noattack_3_in_ggg₃(X, []_0, N) -> noattack_3_out_ggg
noattack_3_in_ggg₃(X, ._2₂(F, T), N) -> if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_in_gg₂(X, F))
if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_out_gg) -> if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_in_gga₂(F, N))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₁(notEq_2_in_gg₂(X, Y))
if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_out_gga₁(FplusN)) -> if_noattack_3_in_3_ggg₅(X, F, T, N, notEq_2_in_gg₂(X, FplusN))
if_notEq_2_in_1_gg₁(notEq_2_out_gg) -> notEq_2_out_gg
add_3_in_gga₂(0_0, X) -> add_3_out_gga₁(X)
add_3_in_gga₂(s_1₁(X), Y) -> if_add_3_in_1_gga₁(add_3_in_gga₂(X, Y))
if_noattack_3_in_3_ggg₅(X, F, T, N, notEq_2_out_gg) -> if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_in_gga₂(X, N))
if_add_3_in_1_gga₁(add_3_out_gga₁(Z)) -> add_3_out_gga₁(s_1₁(Z))
if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_out_gga₁(XplusN)) -> if_noattack_3_in_5_ggg₄(X, T, N, notEq_2_in_gg₂(F, XplusN))
if_noattack_3_in_5_ggg₄(X, T, N, notEq_2_out_gg) -> if_noattack_3_in_6_ggg₁(noattack_3_in_ggg₃(X, T, s_1₁(N)))
if_noattack_3_in_6_ggg₁(noattack_3_out_ggg) -> noattack_3_out_ggg

The set Q consists of the following terms:

noattack_3_in_ggg₃(x0, x1, x2)
if_noattack_3_in_1_ggg₅(x0, x1, x2, x3, x4)
notEq_2_in_gg₂(x0, x1)
if_noattack_3_in_2_ggg₅(x0, x1, x2, x3, x4)
if_notEq_2_in_1_gg₁(x0)
add_3_in_gga₂(x0, x1)
if_noattack_3_in_3_ggg₅(x0, x1, x2, x3, x4)
if_add_3_in_1_gga₁(x0)
if_noattack_3_in_4_ggg₅(x0, x1, x2, x3, x4)
if_noattack_3_in_5_ggg₄(x0, x1, x2, x3)
if_noattack_3_in_6_ggg₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_SAFE_1_IN_1_G₂, SAFE_1_IN_G₁}.
By using the subterm criterion together with the size-change analysis 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:

IF_SAFE_1_IN_1_G₂(Y, noattack_3_out_ggg) -> SAFE_1_IN_G₁(Y)
The graph contains the following edges 1 >= 1
SAFE_1_IN_G₁(._2₂(X, Y)) -> IF_SAFE_1_IN_1_G₂(Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
The graph contains the following edges 1 > 1

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

DELETE_3_IN_AGA₃(X, ._2₂(F, T), ._2₂(F, R)) -> DELETE_3_IN_AGA₃(X, T, R)

The TRS R consists of the following rules:

queens_1_in_a₁(Y) -> if_queens_1_in_1_a₂(Y, perm_2_in_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y))
perm_2_in_ga₂([]_0, []_0) -> perm_2_out_ga₂([]_0, []_0)
perm_2_in_ga₂(._2₂(X, Y), ._2₂(V, Res)) -> if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))
delete_3_in_aga₃(X, ._2₂(X, Y), Y) -> delete_3_out_aga₃(X, ._2₂(X, Y), Y)
delete_3_in_aga₃(X, ._2₂(F, T), ._2₂(F, R)) -> if_delete_3_in_1_aga₅(X, F, T, R, delete_3_in_aga₃(X, T, R))
if_delete_3_in_1_aga₅(X, F, T, R, delete_3_out_aga₃(X, T, R)) -> delete_3_out_aga₃(X, ._2₂(F, T), ._2₂(F, R))
if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_in_ga₂(Rest, Res))
if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_out_ga₂(Rest, Res)) -> perm_2_out_ga₂(._2₂(X, Y), ._2₂(V, Res))
if_queens_1_in_1_a₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> if_queens_1_in_2_a₂(Y, safe_1_in_g₁(Y))
safe_1_in_g₁([]_0) -> safe_1_out_g₁([]_0)
safe_1_in_g₁(._2₂(X, Y)) -> if_safe_1_in_1_g₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
noattack_3_in_ggg₃(X, []_0, N) -> noattack_3_out_ggg₃(X, []_0, N)
noattack_3_in_ggg₃(X, ._2₂(F, T), N) -> if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_in_gg₂(X, F))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))
if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_in_ggg₃(X, T, s_1₁(N)))
if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_out_ggg₃(X, T, s_1₁(N))) -> noattack_3_out_ggg₃(X, ._2₂(F, T), N)
if_safe_1_in_1_g₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> if_safe_1_in_2_g₃(X, Y, safe_1_in_g₁(Y))
if_safe_1_in_2_g₃(X, Y, safe_1_out_g₁(Y)) -> safe_1_out_g₁(._2₂(X, Y))
if_queens_1_in_2_a₂(Y, safe_1_out_g₁(Y)) -> queens_1_out_a₁(Y)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

DELETE_3_IN_AGA₃(X, ._2₂(F, T), ._2₂(F, R)) -> DELETE_3_IN_AGA₃(X, T, R)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
DELETE_3_IN_AGA₃(x1, x2, x3) = DELETE_3_IN_AGA₁(x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

Q DP problem:
The TRS P consists of the following rules:

DELETE_3_IN_AGA₁(._2₂(F, T)) -> DELETE_3_IN_AGA₁(T)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {DELETE_3_IN_AGA₁}.
By using the subterm criterion together with the size-change analysis 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_3_IN_AGA₁(._2₂(F, T)) -> DELETE_3_IN_AGA₁(T)
The graph contains the following edges 1 > 1

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

Pi DP problem:
The TRS P consists of the following rules:

IF_PERM_2_IN_1_GA₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> PERM_2_IN_GA₂(Rest, Res)
PERM_2_IN_GA₂(._2₂(X, Y), ._2₂(V, Res)) -> IF_PERM_2_IN_1_GA₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))

The TRS R consists of the following rules:

queens_1_in_a₁(Y) -> if_queens_1_in_1_a₂(Y, perm_2_in_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y))
perm_2_in_ga₂([]_0, []_0) -> perm_2_out_ga₂([]_0, []_0)
perm_2_in_ga₂(._2₂(X, Y), ._2₂(V, Res)) -> if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))
delete_3_in_aga₃(X, ._2₂(X, Y), Y) -> delete_3_out_aga₃(X, ._2₂(X, Y), Y)
delete_3_in_aga₃(X, ._2₂(F, T), ._2₂(F, R)) -> if_delete_3_in_1_aga₅(X, F, T, R, delete_3_in_aga₃(X, T, R))
if_delete_3_in_1_aga₅(X, F, T, R, delete_3_out_aga₃(X, T, R)) -> delete_3_out_aga₃(X, ._2₂(F, T), ._2₂(F, R))
if_perm_2_in_1_ga₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_in_ga₂(Rest, Res))
if_perm_2_in_2_ga₆(X, Y, V, Res, Rest, perm_2_out_ga₂(Rest, Res)) -> perm_2_out_ga₂(._2₂(X, Y), ._2₂(V, Res))
if_queens_1_in_1_a₂(Y, perm_2_out_ga₂(._2₂(s_1₁(0_0), ._2₂(s_1₁(s_1₁(0_0)), ._2₂(s_1₁(s_1₁(s_1₁(0_0))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))), ._2₂(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))))), []_0)))))))), Y)) -> if_queens_1_in_2_a₂(Y, safe_1_in_g₁(Y))
safe_1_in_g₁([]_0) -> safe_1_out_g₁([]_0)
safe_1_in_g₁(._2₂(X, Y)) -> if_safe_1_in_1_g₃(X, Y, noattack_3_in_ggg₃(X, Y, s_1₁(0_0)))
noattack_3_in_ggg₃(X, []_0, N) -> noattack_3_out_ggg₃(X, []_0, N)
noattack_3_in_ggg₃(X, ._2₂(F, T), N) -> if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_in_gg₂(X, F))
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_noattack_3_in_1_ggg₅(X, F, T, N, notEq_2_out_gg₂(X, F)) -> if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_in_gga₃(F, N, FplusN))
add_3_in_gga₃(0_0, X, X) -> add_3_out_gga₃(0_0, X, X)
add_3_in_gga₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gga₄(X, Y, Z, add_3_in_gga₃(X, Y, Z))
if_add_3_in_1_gga₄(X, Y, Z, add_3_out_gga₃(X, Y, Z)) -> add_3_out_gga₃(s_1₁(X), Y, s_1₁(Z))
if_noattack_3_in_2_ggg₅(X, F, T, N, add_3_out_gga₃(F, N, FplusN)) -> if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_in_gg₂(X, FplusN))
if_noattack_3_in_3_ggg₆(X, F, T, N, FplusN, notEq_2_out_gg₂(X, FplusN)) -> if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_in_gga₃(X, N, XplusN))
if_noattack_3_in_4_ggg₅(X, F, T, N, add_3_out_gga₃(X, N, XplusN)) -> if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_in_gg₂(F, XplusN))
if_noattack_3_in_5_ggg₆(X, F, T, N, XplusN, notEq_2_out_gg₂(F, XplusN)) -> if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_in_ggg₃(X, T, s_1₁(N)))
if_noattack_3_in_6_ggg₅(X, F, T, N, noattack_3_out_ggg₃(X, T, s_1₁(N))) -> noattack_3_out_ggg₃(X, ._2₂(F, T), N)
if_safe_1_in_1_g₃(X, Y, noattack_3_out_ggg₃(X, Y, s_1₁(0_0))) -> if_safe_1_in_2_g₃(X, Y, safe_1_in_g₁(Y))
if_safe_1_in_2_g₃(X, Y, safe_1_out_g₁(Y)) -> safe_1_out_g₁(._2₂(X, Y))
if_queens_1_in_2_a₂(Y, safe_1_out_g₁(Y)) -> queens_1_out_a₁(Y)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

Pi DP problem:
The TRS P consists of the following rules:

IF_PERM_2_IN_1_GA₅(X, Y, V, Res, delete_3_out_aga₃(V, ._2₂(X, Y), Rest)) -> PERM_2_IN_GA₂(Rest, Res)
PERM_2_IN_GA₂(._2₂(X, Y), ._2₂(V, Res)) -> IF_PERM_2_IN_1_GA₅(X, Y, V, Res, delete_3_in_aga₃(V, ._2₂(X, Y), Rest))

The TRS R consists of the following rules:

delete_3_in_aga₃(X, ._2₂(X, Y), Y) -> delete_3_out_aga₃(X, ._2₂(X, Y), Y)
delete_3_in_aga₃(X, ._2₂(F, T), ._2₂(F, R)) -> if_delete_3_in_1_aga₅(X, F, T, R, delete_3_in_aga₃(X, T, R))
if_delete_3_in_1_aga₅(X, F, T, R, delete_3_out_aga₃(X, T, R)) -> delete_3_out_aga₃(X, ._2₂(F, T), ._2₂(F, R))

The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
delete_3_in_aga₃(x1, x2, x3) = delete_3_in_aga₁(x2)
delete_3_out_aga₃(x1, x2, x3) = delete_3_out_aga₂(x1, x3)
if_delete_3_in_1_aga₅(x1, x2, x3, x4, x5) = if_delete_3_in_1_aga₂(x2, x5)
PERM_2_IN_GA₂(x1, x2) = PERM_2_IN_GA₁(x1)
IF_PERM_2_IN_1_GA₅(x1, x2, x3, x4, x5) = IF_PERM_2_IN_1_GA₁(x5)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

Q DP problem:
The TRS P consists of the following rules:

IF_PERM_2_IN_1_GA₁(delete_3_out_aga₂(V, Rest)) -> PERM_2_IN_GA₁(Rest)
PERM_2_IN_GA₁(._2₂(X, Y)) -> IF_PERM_2_IN_1_GA₁(delete_3_in_aga₁(._2₂(X, Y)))

The TRS R consists of the following rules:

delete_3_in_aga₁(._2₂(X, Y)) -> delete_3_out_aga₂(X, Y)
delete_3_in_aga₁(._2₂(F, T)) -> if_delete_3_in_1_aga₂(F, delete_3_in_aga₁(T))
if_delete_3_in_1_aga₂(F, delete_3_out_aga₂(X, R)) -> delete_3_out_aga₂(X, ._2₂(F, R))

The set Q consists of the following terms:

delete_3_in_aga₁(x0)
if_delete_3_in_1_aga₂(x0, x1)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {PERM_2_IN_GA₁, IF_PERM_2_IN_1_GA₁}.
By using a polynomial ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

IF_PERM_2_IN_1_GA₁(delete_3_out_aga₂(V, Rest)) -> PERM_2_IN_GA₁(Rest)

Strictly oriented rules of the TRS R:

delete_3_in_aga₁(._2₂(X, Y)) -> delete_3_out_aga₂(X, Y)

Used ordering: POLO with Polynomial interpretation:

POL(._2₂(x₁, x₂)) = 1 + x₁ + x₂
POL(if_delete_3_in_1_aga₂(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(IF_PERM_2_IN_1_GA₁(x₁)) = x₁
POL(delete_3_in_aga₁(x₁)) = 2·x₁
POL(delete_3_out_aga₂(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(PERM_2_IN_GA₁(x₁)) = 2·x₁

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ RuleRemovalProof

Q DP problem:
The TRS P consists of the following rules:

PERM_2_IN_GA₁(._2₂(X, Y)) -> IF_PERM_2_IN_1_GA₁(delete_3_in_aga₁(._2₂(X, Y)))

The TRS R consists of the following rules:

delete_3_in_aga₁(._2₂(F, T)) -> if_delete_3_in_1_aga₂(F, delete_3_in_aga₁(T))
if_delete_3_in_1_aga₂(F, delete_3_out_aga₂(X, R)) -> delete_3_out_aga₂(X, ._2₂(F, R))

The set Q consists of the following terms:

delete_3_in_aga₁(x0)
if_delete_3_in_1_aga₂(x0, x1)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_PERM_2_IN_1_GA₁, PERM_2_IN_GA₁}.
By using a polynomial ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

delete_3_in_aga₁(._2₂(F, T)) -> if_delete_3_in_1_aga₂(F, delete_3_in_aga₁(T))

Used ordering: POLO with Polynomial interpretation:

POL(._2₂(x₁, x₂)) = 1 + x₁ + x₂
POL(if_delete_3_in_1_aga₂(x₁, x₂)) = 1 + x₁ + x₂
POL(IF_PERM_2_IN_1_GA₁(x₁)) = x₁
POL(delete_3_in_aga₁(x₁)) = 1 + 2·x₁
POL(delete_3_out_aga₂(x₁, x₂)) = x₁ + x₂
POL(PERM_2_IN_GA₁(x₁)) = 1 + 2·x₁

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ RuleRemovalProof

                              ↳ QDP

                                ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

PERM_2_IN_GA₁(._2₂(X, Y)) -> IF_PERM_2_IN_1_GA₁(delete_3_in_aga₁(._2₂(X, Y)))

The TRS R consists of the following rules:

if_delete_3_in_1_aga₂(F, delete_3_out_aga₂(X, R)) -> delete_3_out_aga₂(X, ._2₂(F, R))

The set Q consists of the following terms:

delete_3_in_aga₁(x0)
if_delete_3_in_1_aga₂(x0, x1)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_PERM_2_IN_1_GA₁, PERM_2_IN_GA₁}.
The approximation of the Dependency Graph contains 0 SCCs with 1 less node.