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

0 Prolog
↳1 UndefinedPredicateHandlerProof (⇒, 0 ms)
↳2 Prolog
↳3 PrologToPiTRSProof (⇒, 63 ms)
↳4 PiTRS
↳5 DependencyPairsProof (⇔, 287 ms)
↳6 PiDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 AND
    ↳9 PiDP
        ↳10 UsableRulesProof (⇔, 0 ms)
        ↳11 PiDP
        ↳12 PiDPToQDPProof (⇒, 0 ms)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔, 0 ms)
        ↳15 YES
    ↳16 PiDP
        ↳17 UsableRulesProof (⇔, 0 ms)
        ↳18 PiDP
        ↳19 PiDPToQDPProof (⇒, 15 ms)
        ↳20 QDP
        ↳21 QDPSizeChangeProof (⇔, 0 ms)
        ↳22 YES
    ↳23 PiDP
        ↳24 UsableRulesProof (⇔, 0 ms)
        ↳25 PiDP
        ↳26 PiDPToQDPProof (⇒, 0 ms)
        ↳27 QDP
        ↳28 QDPSizeChangeProof (⇔, 0 ms)
        ↳29 YES
    ↳30 PiDP
        ↳31 UsableRulesProof (⇔, 0 ms)
        ↳32 PiDP
        ↳33 PiDPToQDPProof (⇒, 0 ms)
        ↳34 QDP
        ↳35 MRRProof (⇔, 23 ms)
        ↳36 QDP
        ↳37 PisEmptyProof (⇔, 0 ms)
        ↳38 YES

(0) Obligation:

Clauses:

queens(X, Y) :- ','(perm(X, 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, 1), safe(Y)).
noattack(X, [], N).
noattack(X, .(F, T), N) :- ','(=\=(X, F), ','(=\=(X, +(F, N)), ','(=\=(F, +(X, N)), ','(is(N1, +(N, 1)), noattack(X, T, N1))))).

Query: queens(g,a)

(1) UndefinedPredicateHandlerProof (SOUND transformation)

Added facts for all undefined predicates [PROLOG].

(2) Obligation:

Clauses:

queens(X, Y) :- ','(perm(X, 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, 1), safe(Y)).
noattack(X, [], N).
noattack(X, .(F, T), N) :- ','(=\=(X, F), ','(=\=(X, +(F, N)), ','(=\=(F, +(X, N)), ','(is(N1, +(N, 1)), noattack(X, T, N1))))).
=\=(X0, X1).
is(X0, X1).

Query: queens(g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
queens_in: (b,f)
perm_in: (b,f)
delete_in: (f,b,f)
safe_in: (b)
noattack_in: (b,b,b) (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

queens_in_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, 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_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(X, 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, 1))
noattack_in_ggg(X, [], N) → noattack_out_ggg(X, [], N)
noattack_in_ggg(X, .(F, T), N) → U8_ggg(X, F, T, N, =\=_in_gg(X, F))
=\=_in_gg(X0, X1) → =\=_out_gg(X0, X1)
U8_ggg(X, F, T, N, =\=_out_gg(X, F)) → U9_ggg(X, F, T, N, =\=_in_gg(X, +(F, N)))
U9_ggg(X, F, T, N, =\=_out_gg(X, +(F, N))) → U10_ggg(X, F, T, N, =\=_in_gg(F, +(X, N)))
U10_ggg(X, F, T, N, =\=_out_gg(F, +(X, N))) → U11_ggg(X, F, T, N, is_in_ag(N1, +(N, 1)))
is_in_ag(X0, X1) → is_out_ag(X0, X1)
U11_ggg(X, F, T, N, is_out_ag(N1, +(N, 1))) → U12_ggg(X, F, T, N, noattack_in_gga(X, T, N1))
noattack_in_gga(X, [], N) → noattack_out_gga(X, [], N)
noattack_in_gga(X, .(F, T), N) → U8_gga(X, F, T, N, =\=_in_gg(X, F))
U8_gga(X, F, T, N, =\=_out_gg(X, F)) → U9_gga(X, F, T, N, =\=_in_ga(X, +(F, N)))
=\=_in_ga(X0, X1) → =\=_out_ga(X0, X1)
U9_gga(X, F, T, N, =\=_out_ga(X, +(F, N))) → U10_gga(X, F, T, N, =\=_in_ga(F, +(X, N)))
U10_gga(X, F, T, N, =\=_out_ga(F, +(X, N))) → U11_gga(X, F, T, N, is_in_aa(N1, +(N, 1)))
is_in_aa(X0, X1) → is_out_aa(X0, X1)
U11_gga(X, F, T, N, is_out_aa(N1, +(N, 1))) → U12_gga(X, F, T, N, noattack_in_gga(X, T, N1))
U12_gga(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_gga(X, .(F, T), N)
U12_ggg(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_ggg(X, .(F, T), N)
U6_g(X, Y, noattack_out_ggg(X, Y, 1)) → U7_g(X, Y, safe_in_g(Y))
U7_g(X, Y, safe_out_g(Y)) → safe_out_g(.(X, Y))
U2_ga(X, Y, safe_out_g(Y)) → queens_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
queens_in_ga(x1, x2)  =  queens_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
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)
U2_ga(x1, x2, x3)  =  U2_ga(x2, x3)
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)
=\=_in_gg(x1, x2)  =  =\=_in_gg(x1, x2)
=\=_out_gg(x1, x2)  =  =\=_out_gg
U9_ggg(x1, x2, x3, x4, x5)  =  U9_ggg(x1, x2, x3, x4, x5)
+(x1, x2)  =  +(x1, x2)
U10_ggg(x1, x2, x3, x4, x5)  =  U10_ggg(x1, x3, x4, x5)
U11_ggg(x1, x2, x3, x4, x5)  =  U11_ggg(x1, x3, x5)
is_in_ag(x1, x2)  =  is_in_ag(x2)
is_out_ag(x1, x2)  =  is_out_ag
1  =  1
U12_ggg(x1, x2, x3, x4, x5)  =  U12_ggg(x5)
noattack_in_gga(x1, x2, x3)  =  noattack_in_gga(x1, x2)
noattack_out_gga(x1, x2, x3)  =  noattack_out_gga
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
=\=_in_ga(x1, x2)  =  =\=_in_ga(x1)
=\=_out_ga(x1, x2)  =  =\=_out_ga
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x3, x5)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x3, x5)
is_in_aa(x1, x2)  =  is_in_aa
is_out_aa(x1, x2)  =  is_out_aa
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U7_g(x1, x2, x3)  =  U7_g(x3)
queens_out_ga(x1, x2)  =  queens_out_ga(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

queens_in_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, 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_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(X, 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, 1))
noattack_in_ggg(X, [], N) → noattack_out_ggg(X, [], N)
noattack_in_ggg(X, .(F, T), N) → U8_ggg(X, F, T, N, =\=_in_gg(X, F))
=\=_in_gg(X0, X1) → =\=_out_gg(X0, X1)
U8_ggg(X, F, T, N, =\=_out_gg(X, F)) → U9_ggg(X, F, T, N, =\=_in_gg(X, +(F, N)))
U9_ggg(X, F, T, N, =\=_out_gg(X, +(F, N))) → U10_ggg(X, F, T, N, =\=_in_gg(F, +(X, N)))
U10_ggg(X, F, T, N, =\=_out_gg(F, +(X, N))) → U11_ggg(X, F, T, N, is_in_ag(N1, +(N, 1)))
is_in_ag(X0, X1) → is_out_ag(X0, X1)
U11_ggg(X, F, T, N, is_out_ag(N1, +(N, 1))) → U12_ggg(X, F, T, N, noattack_in_gga(X, T, N1))
noattack_in_gga(X, [], N) → noattack_out_gga(X, [], N)
noattack_in_gga(X, .(F, T), N) → U8_gga(X, F, T, N, =\=_in_gg(X, F))
U8_gga(X, F, T, N, =\=_out_gg(X, F)) → U9_gga(X, F, T, N, =\=_in_ga(X, +(F, N)))
=\=_in_ga(X0, X1) → =\=_out_ga(X0, X1)
U9_gga(X, F, T, N, =\=_out_ga(X, +(F, N))) → U10_gga(X, F, T, N, =\=_in_ga(F, +(X, N)))
U10_gga(X, F, T, N, =\=_out_ga(F, +(X, N))) → U11_gga(X, F, T, N, is_in_aa(N1, +(N, 1)))
is_in_aa(X0, X1) → is_out_aa(X0, X1)
U11_gga(X, F, T, N, is_out_aa(N1, +(N, 1))) → U12_gga(X, F, T, N, noattack_in_gga(X, T, N1))
U12_gga(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_gga(X, .(F, T), N)
U12_ggg(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_ggg(X, .(F, T), N)
U6_g(X, Y, noattack_out_ggg(X, Y, 1)) → U7_g(X, Y, safe_in_g(Y))
U7_g(X, Y, safe_out_g(Y)) → safe_out_g(.(X, Y))
U2_ga(X, Y, safe_out_g(Y)) → queens_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
queens_in_ga(x1, x2)  =  queens_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
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)
U2_ga(x1, x2, x3)  =  U2_ga(x2, x3)
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)
=\=_in_gg(x1, x2)  =  =\=_in_gg(x1, x2)
=\=_out_gg(x1, x2)  =  =\=_out_gg
U9_ggg(x1, x2, x3, x4, x5)  =  U9_ggg(x1, x2, x3, x4, x5)
+(x1, x2)  =  +(x1, x2)
U10_ggg(x1, x2, x3, x4, x5)  =  U10_ggg(x1, x3, x4, x5)
U11_ggg(x1, x2, x3, x4, x5)  =  U11_ggg(x1, x3, x5)
is_in_ag(x1, x2)  =  is_in_ag(x2)
is_out_ag(x1, x2)  =  is_out_ag
1  =  1
U12_ggg(x1, x2, x3, x4, x5)  =  U12_ggg(x5)
noattack_in_gga(x1, x2, x3)  =  noattack_in_gga(x1, x2)
noattack_out_gga(x1, x2, x3)  =  noattack_out_gga
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
=\=_in_ga(x1, x2)  =  =\=_in_ga(x1)
=\=_out_ga(x1, x2)  =  =\=_out_ga
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x3, x5)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x3, x5)
is_in_aa(x1, x2)  =  is_in_aa
is_out_aa(x1, x2)  =  is_out_aa
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U7_g(x1, x2, x3)  =  U7_g(x3)
queens_out_ga(x1, x2)  =  queens_out_ga(x2)

(5) DependencyPairsProof (EQUIVALENT transformation)

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_GA(X, Y) → U1_GA(X, Y, perm_in_ga(X, Y))
QUEENS_IN_GA(X, Y) → PERM_IN_GA(X, 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_GA(X, Y, perm_out_ga(X, Y)) → U2_GA(X, Y, safe_in_g(Y))
U1_GA(X, Y, perm_out_ga(X, Y)) → SAFE_IN_G(Y)
SAFE_IN_G(.(X, Y)) → U6_G(X, Y, noattack_in_ggg(X, Y, 1))
SAFE_IN_G(.(X, Y)) → NOATTACK_IN_GGG(X, Y, 1)
NOATTACK_IN_GGG(X, .(F, T), N) → U8_GGG(X, F, T, N, =\=_in_gg(X, F))
NOATTACK_IN_GGG(X, .(F, T), N) → =\=_IN_GG(X, F)
U8_GGG(X, F, T, N, =\=_out_gg(X, F)) → U9_GGG(X, F, T, N, =\=_in_gg(X, +(F, N)))
U8_GGG(X, F, T, N, =\=_out_gg(X, F)) → =\=_IN_GG(X, +(F, N))
U9_GGG(X, F, T, N, =\=_out_gg(X, +(F, N))) → U10_GGG(X, F, T, N, =\=_in_gg(F, +(X, N)))
U9_GGG(X, F, T, N, =\=_out_gg(X, +(F, N))) → =\=_IN_GG(F, +(X, N))
U10_GGG(X, F, T, N, =\=_out_gg(F, +(X, N))) → U11_GGG(X, F, T, N, is_in_ag(N1, +(N, 1)))
U10_GGG(X, F, T, N, =\=_out_gg(F, +(X, N))) → IS_IN_AG(N1, +(N, 1))
U11_GGG(X, F, T, N, is_out_ag(N1, +(N, 1))) → U12_GGG(X, F, T, N, noattack_in_gga(X, T, N1))
U11_GGG(X, F, T, N, is_out_ag(N1, +(N, 1))) → NOATTACK_IN_GGA(X, T, N1)
NOATTACK_IN_GGA(X, .(F, T), N) → U8_GGA(X, F, T, N, =\=_in_gg(X, F))
NOATTACK_IN_GGA(X, .(F, T), N) → =\=_IN_GG(X, F)
U8_GGA(X, F, T, N, =\=_out_gg(X, F)) → U9_GGA(X, F, T, N, =\=_in_ga(X, +(F, N)))
U8_GGA(X, F, T, N, =\=_out_gg(X, F)) → =\=_IN_GA(X, +(F, N))
U9_GGA(X, F, T, N, =\=_out_ga(X, +(F, N))) → U10_GGA(X, F, T, N, =\=_in_ga(F, +(X, N)))
U9_GGA(X, F, T, N, =\=_out_ga(X, +(F, N))) → =\=_IN_GA(F, +(X, N))
U10_GGA(X, F, T, N, =\=_out_ga(F, +(X, N))) → U11_GGA(X, F, T, N, is_in_aa(N1, +(N, 1)))
U10_GGA(X, F, T, N, =\=_out_ga(F, +(X, N))) → IS_IN_AA(N1, +(N, 1))
U11_GGA(X, F, T, N, is_out_aa(N1, +(N, 1))) → U12_GGA(X, F, T, N, noattack_in_gga(X, T, N1))
U11_GGA(X, F, T, N, is_out_aa(N1, +(N, 1))) → NOATTACK_IN_GGA(X, T, N1)
U6_G(X, Y, noattack_out_ggg(X, Y, 1)) → U7_G(X, Y, safe_in_g(Y))
U6_G(X, Y, noattack_out_ggg(X, Y, 1)) → SAFE_IN_G(Y)

The TRS R consists of the following rules:

queens_in_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, 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_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(X, 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, 1))
noattack_in_ggg(X, [], N) → noattack_out_ggg(X, [], N)
noattack_in_ggg(X, .(F, T), N) → U8_ggg(X, F, T, N, =\=_in_gg(X, F))
=\=_in_gg(X0, X1) → =\=_out_gg(X0, X1)
U8_ggg(X, F, T, N, =\=_out_gg(X, F)) → U9_ggg(X, F, T, N, =\=_in_gg(X, +(F, N)))
U9_ggg(X, F, T, N, =\=_out_gg(X, +(F, N))) → U10_ggg(X, F, T, N, =\=_in_gg(F, +(X, N)))
U10_ggg(X, F, T, N, =\=_out_gg(F, +(X, N))) → U11_ggg(X, F, T, N, is_in_ag(N1, +(N, 1)))
is_in_ag(X0, X1) → is_out_ag(X0, X1)
U11_ggg(X, F, T, N, is_out_ag(N1, +(N, 1))) → U12_ggg(X, F, T, N, noattack_in_gga(X, T, N1))
noattack_in_gga(X, [], N) → noattack_out_gga(X, [], N)
noattack_in_gga(X, .(F, T), N) → U8_gga(X, F, T, N, =\=_in_gg(X, F))
U8_gga(X, F, T, N, =\=_out_gg(X, F)) → U9_gga(X, F, T, N, =\=_in_ga(X, +(F, N)))
=\=_in_ga(X0, X1) → =\=_out_ga(X0, X1)
U9_gga(X, F, T, N, =\=_out_ga(X, +(F, N))) → U10_gga(X, F, T, N, =\=_in_ga(F, +(X, N)))
U10_gga(X, F, T, N, =\=_out_ga(F, +(X, N))) → U11_gga(X, F, T, N, is_in_aa(N1, +(N, 1)))
is_in_aa(X0, X1) → is_out_aa(X0, X1)
U11_gga(X, F, T, N, is_out_aa(N1, +(N, 1))) → U12_gga(X, F, T, N, noattack_in_gga(X, T, N1))
U12_gga(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_gga(X, .(F, T), N)
U12_ggg(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_ggg(X, .(F, T), N)
U6_g(X, Y, noattack_out_ggg(X, Y, 1)) → U7_g(X, Y, safe_in_g(Y))
U7_g(X, Y, safe_out_g(Y)) → safe_out_g(.(X, Y))
U2_ga(X, Y, safe_out_g(Y)) → queens_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
queens_in_ga(x1, x2)  =  queens_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
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)
U2_ga(x1, x2, x3)  =  U2_ga(x2, x3)
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)
=\=_in_gg(x1, x2)  =  =\=_in_gg(x1, x2)
=\=_out_gg(x1, x2)  =  =\=_out_gg
U9_ggg(x1, x2, x3, x4, x5)  =  U9_ggg(x1, x2, x3, x4, x5)
+(x1, x2)  =  +(x1, x2)
U10_ggg(x1, x2, x3, x4, x5)  =  U10_ggg(x1, x3, x4, x5)
U11_ggg(x1, x2, x3, x4, x5)  =  U11_ggg(x1, x3, x5)
is_in_ag(x1, x2)  =  is_in_ag(x2)
is_out_ag(x1, x2)  =  is_out_ag
1  =  1
U12_ggg(x1, x2, x3, x4, x5)  =  U12_ggg(x5)
noattack_in_gga(x1, x2, x3)  =  noattack_in_gga(x1, x2)
noattack_out_gga(x1, x2, x3)  =  noattack_out_gga
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
=\=_in_ga(x1, x2)  =  =\=_in_ga(x1)
=\=_out_ga(x1, x2)  =  =\=_out_ga
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x3, x5)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x3, x5)
is_in_aa(x1, x2)  =  is_in_aa
is_out_aa(x1, x2)  =  is_out_aa
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U7_g(x1, x2, x3)  =  U7_g(x3)
queens_out_ga(x1, x2)  =  queens_out_ga(x2)
QUEENS_IN_GA(x1, x2)  =  QUEENS_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
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_GA(x1, x2, x3)  =  U2_GA(x2, x3)
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)
=\=_IN_GG(x1, x2)  =  =\=_IN_GG(x1, x2)
U9_GGG(x1, x2, x3, x4, x5)  =  U9_GGG(x1, x2, x3, x4, x5)
U10_GGG(x1, x2, x3, x4, x5)  =  U10_GGG(x1, x3, x4, x5)
U11_GGG(x1, x2, x3, x4, x5)  =  U11_GGG(x1, x3, x5)
IS_IN_AG(x1, x2)  =  IS_IN_AG(x2)
U12_GGG(x1, x2, x3, x4, x5)  =  U12_GGG(x5)
NOATTACK_IN_GGA(x1, x2, x3)  =  NOATTACK_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x2, x3, x5)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)
=\=_IN_GA(x1, x2)  =  =\=_IN_GA(x1)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x3, x5)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x1, x3, x5)
IS_IN_AA(x1, x2)  =  IS_IN_AA
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x5)
U7_G(x1, x2, x3)  =  U7_G(x3)

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

(6) Obligation:

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

QUEENS_IN_GA(X, Y) → U1_GA(X, Y, perm_in_ga(X, Y))
QUEENS_IN_GA(X, Y) → PERM_IN_GA(X, 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_GA(X, Y, perm_out_ga(X, Y)) → U2_GA(X, Y, safe_in_g(Y))
U1_GA(X, Y, perm_out_ga(X, Y)) → SAFE_IN_G(Y)
SAFE_IN_G(.(X, Y)) → U6_G(X, Y, noattack_in_ggg(X, Y, 1))
SAFE_IN_G(.(X, Y)) → NOATTACK_IN_GGG(X, Y, 1)
NOATTACK_IN_GGG(X, .(F, T), N) → U8_GGG(X, F, T, N, =\=_in_gg(X, F))
NOATTACK_IN_GGG(X, .(F, T), N) → =\=_IN_GG(X, F)
U8_GGG(X, F, T, N, =\=_out_gg(X, F)) → U9_GGG(X, F, T, N, =\=_in_gg(X, +(F, N)))
U8_GGG(X, F, T, N, =\=_out_gg(X, F)) → =\=_IN_GG(X, +(F, N))
U9_GGG(X, F, T, N, =\=_out_gg(X, +(F, N))) → U10_GGG(X, F, T, N, =\=_in_gg(F, +(X, N)))
U9_GGG(X, F, T, N, =\=_out_gg(X, +(F, N))) → =\=_IN_GG(F, +(X, N))
U10_GGG(X, F, T, N, =\=_out_gg(F, +(X, N))) → U11_GGG(X, F, T, N, is_in_ag(N1, +(N, 1)))
U10_GGG(X, F, T, N, =\=_out_gg(F, +(X, N))) → IS_IN_AG(N1, +(N, 1))
U11_GGG(X, F, T, N, is_out_ag(N1, +(N, 1))) → U12_GGG(X, F, T, N, noattack_in_gga(X, T, N1))
U11_GGG(X, F, T, N, is_out_ag(N1, +(N, 1))) → NOATTACK_IN_GGA(X, T, N1)
NOATTACK_IN_GGA(X, .(F, T), N) → U8_GGA(X, F, T, N, =\=_in_gg(X, F))
NOATTACK_IN_GGA(X, .(F, T), N) → =\=_IN_GG(X, F)
U8_GGA(X, F, T, N, =\=_out_gg(X, F)) → U9_GGA(X, F, T, N, =\=_in_ga(X, +(F, N)))
U8_GGA(X, F, T, N, =\=_out_gg(X, F)) → =\=_IN_GA(X, +(F, N))
U9_GGA(X, F, T, N, =\=_out_ga(X, +(F, N))) → U10_GGA(X, F, T, N, =\=_in_ga(F, +(X, N)))
U9_GGA(X, F, T, N, =\=_out_ga(X, +(F, N))) → =\=_IN_GA(F, +(X, N))
U10_GGA(X, F, T, N, =\=_out_ga(F, +(X, N))) → U11_GGA(X, F, T, N, is_in_aa(N1, +(N, 1)))
U10_GGA(X, F, T, N, =\=_out_ga(F, +(X, N))) → IS_IN_AA(N1, +(N, 1))
U11_GGA(X, F, T, N, is_out_aa(N1, +(N, 1))) → U12_GGA(X, F, T, N, noattack_in_gga(X, T, N1))
U11_GGA(X, F, T, N, is_out_aa(N1, +(N, 1))) → NOATTACK_IN_GGA(X, T, N1)
U6_G(X, Y, noattack_out_ggg(X, Y, 1)) → U7_G(X, Y, safe_in_g(Y))
U6_G(X, Y, noattack_out_ggg(X, Y, 1)) → SAFE_IN_G(Y)

The TRS R consists of the following rules:

queens_in_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, 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_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(X, 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, 1))
noattack_in_ggg(X, [], N) → noattack_out_ggg(X, [], N)
noattack_in_ggg(X, .(F, T), N) → U8_ggg(X, F, T, N, =\=_in_gg(X, F))
=\=_in_gg(X0, X1) → =\=_out_gg(X0, X1)
U8_ggg(X, F, T, N, =\=_out_gg(X, F)) → U9_ggg(X, F, T, N, =\=_in_gg(X, +(F, N)))
U9_ggg(X, F, T, N, =\=_out_gg(X, +(F, N))) → U10_ggg(X, F, T, N, =\=_in_gg(F, +(X, N)))
U10_ggg(X, F, T, N, =\=_out_gg(F, +(X, N))) → U11_ggg(X, F, T, N, is_in_ag(N1, +(N, 1)))
is_in_ag(X0, X1) → is_out_ag(X0, X1)
U11_ggg(X, F, T, N, is_out_ag(N1, +(N, 1))) → U12_ggg(X, F, T, N, noattack_in_gga(X, T, N1))
noattack_in_gga(X, [], N) → noattack_out_gga(X, [], N)
noattack_in_gga(X, .(F, T), N) → U8_gga(X, F, T, N, =\=_in_gg(X, F))
U8_gga(X, F, T, N, =\=_out_gg(X, F)) → U9_gga(X, F, T, N, =\=_in_ga(X, +(F, N)))
=\=_in_ga(X0, X1) → =\=_out_ga(X0, X1)
U9_gga(X, F, T, N, =\=_out_ga(X, +(F, N))) → U10_gga(X, F, T, N, =\=_in_ga(F, +(X, N)))
U10_gga(X, F, T, N, =\=_out_ga(F, +(X, N))) → U11_gga(X, F, T, N, is_in_aa(N1, +(N, 1)))
is_in_aa(X0, X1) → is_out_aa(X0, X1)
U11_gga(X, F, T, N, is_out_aa(N1, +(N, 1))) → U12_gga(X, F, T, N, noattack_in_gga(X, T, N1))
U12_gga(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_gga(X, .(F, T), N)
U12_ggg(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_ggg(X, .(F, T), N)
U6_g(X, Y, noattack_out_ggg(X, Y, 1)) → U7_g(X, Y, safe_in_g(Y))
U7_g(X, Y, safe_out_g(Y)) → safe_out_g(.(X, Y))
U2_ga(X, Y, safe_out_g(Y)) → queens_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
queens_in_ga(x1, x2)  =  queens_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
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)
U2_ga(x1, x2, x3)  =  U2_ga(x2, x3)
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)
=\=_in_gg(x1, x2)  =  =\=_in_gg(x1, x2)
=\=_out_gg(x1, x2)  =  =\=_out_gg
U9_ggg(x1, x2, x3, x4, x5)  =  U9_ggg(x1, x2, x3, x4, x5)
+(x1, x2)  =  +(x1, x2)
U10_ggg(x1, x2, x3, x4, x5)  =  U10_ggg(x1, x3, x4, x5)
U11_ggg(x1, x2, x3, x4, x5)  =  U11_ggg(x1, x3, x5)
is_in_ag(x1, x2)  =  is_in_ag(x2)
is_out_ag(x1, x2)  =  is_out_ag
1  =  1
U12_ggg(x1, x2, x3, x4, x5)  =  U12_ggg(x5)
noattack_in_gga(x1, x2, x3)  =  noattack_in_gga(x1, x2)
noattack_out_gga(x1, x2, x3)  =  noattack_out_gga
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
=\=_in_ga(x1, x2)  =  =\=_in_ga(x1)
=\=_out_ga(x1, x2)  =  =\=_out_ga
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x3, x5)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x3, x5)
is_in_aa(x1, x2)  =  is_in_aa
is_out_aa(x1, x2)  =  is_out_aa
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U7_g(x1, x2, x3)  =  U7_g(x3)
queens_out_ga(x1, x2)  =  queens_out_ga(x2)
QUEENS_IN_GA(x1, x2)  =  QUEENS_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
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_GA(x1, x2, x3)  =  U2_GA(x2, x3)
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)
=\=_IN_GG(x1, x2)  =  =\=_IN_GG(x1, x2)
U9_GGG(x1, x2, x3, x4, x5)  =  U9_GGG(x1, x2, x3, x4, x5)
U10_GGG(x1, x2, x3, x4, x5)  =  U10_GGG(x1, x3, x4, x5)
U11_GGG(x1, x2, x3, x4, x5)  =  U11_GGG(x1, x3, x5)
IS_IN_AG(x1, x2)  =  IS_IN_AG(x2)
U12_GGG(x1, x2, x3, x4, x5)  =  U12_GGG(x5)
NOATTACK_IN_GGA(x1, x2, x3)  =  NOATTACK_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x2, x3, x5)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)
=\=_IN_GA(x1, x2)  =  =\=_IN_GA(x1)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x3, x5)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x1, x3, x5)
IS_IN_AA(x1, x2)  =  IS_IN_AA
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x5)
U7_G(x1, x2, x3)  =  U7_G(x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 24 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

U8_GGA(X, F, T, N, =\=_out_gg(X, F)) → U9_GGA(X, F, T, N, =\=_in_ga(X, +(F, N)))
U9_GGA(X, F, T, N, =\=_out_ga(X, +(F, N))) → U10_GGA(X, F, T, N, =\=_in_ga(F, +(X, N)))
U10_GGA(X, F, T, N, =\=_out_ga(F, +(X, N))) → U11_GGA(X, F, T, N, is_in_aa(N1, +(N, 1)))
U11_GGA(X, F, T, N, is_out_aa(N1, +(N, 1))) → NOATTACK_IN_GGA(X, T, N1)
NOATTACK_IN_GGA(X, .(F, T), N) → U8_GGA(X, F, T, N, =\=_in_gg(X, F))

The TRS R consists of the following rules:

queens_in_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, 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_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(X, 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, 1))
noattack_in_ggg(X, [], N) → noattack_out_ggg(X, [], N)
noattack_in_ggg(X, .(F, T), N) → U8_ggg(X, F, T, N, =\=_in_gg(X, F))
=\=_in_gg(X0, X1) → =\=_out_gg(X0, X1)
U8_ggg(X, F, T, N, =\=_out_gg(X, F)) → U9_ggg(X, F, T, N, =\=_in_gg(X, +(F, N)))
U9_ggg(X, F, T, N, =\=_out_gg(X, +(F, N))) → U10_ggg(X, F, T, N, =\=_in_gg(F, +(X, N)))
U10_ggg(X, F, T, N, =\=_out_gg(F, +(X, N))) → U11_ggg(X, F, T, N, is_in_ag(N1, +(N, 1)))
is_in_ag(X0, X1) → is_out_ag(X0, X1)
U11_ggg(X, F, T, N, is_out_ag(N1, +(N, 1))) → U12_ggg(X, F, T, N, noattack_in_gga(X, T, N1))
noattack_in_gga(X, [], N) → noattack_out_gga(X, [], N)
noattack_in_gga(X, .(F, T), N) → U8_gga(X, F, T, N, =\=_in_gg(X, F))
U8_gga(X, F, T, N, =\=_out_gg(X, F)) → U9_gga(X, F, T, N, =\=_in_ga(X, +(F, N)))
=\=_in_ga(X0, X1) → =\=_out_ga(X0, X1)
U9_gga(X, F, T, N, =\=_out_ga(X, +(F, N))) → U10_gga(X, F, T, N, =\=_in_ga(F, +(X, N)))
U10_gga(X, F, T, N, =\=_out_ga(F, +(X, N))) → U11_gga(X, F, T, N, is_in_aa(N1, +(N, 1)))
is_in_aa(X0, X1) → is_out_aa(X0, X1)
U11_gga(X, F, T, N, is_out_aa(N1, +(N, 1))) → U12_gga(X, F, T, N, noattack_in_gga(X, T, N1))
U12_gga(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_gga(X, .(F, T), N)
U12_ggg(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_ggg(X, .(F, T), N)
U6_g(X, Y, noattack_out_ggg(X, Y, 1)) → U7_g(X, Y, safe_in_g(Y))
U7_g(X, Y, safe_out_g(Y)) → safe_out_g(.(X, Y))
U2_ga(X, Y, safe_out_g(Y)) → queens_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
queens_in_ga(x1, x2)  =  queens_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
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)
U2_ga(x1, x2, x3)  =  U2_ga(x2, x3)
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)
=\=_in_gg(x1, x2)  =  =\=_in_gg(x1, x2)
=\=_out_gg(x1, x2)  =  =\=_out_gg
U9_ggg(x1, x2, x3, x4, x5)  =  U9_ggg(x1, x2, x3, x4, x5)
+(x1, x2)  =  +(x1, x2)
U10_ggg(x1, x2, x3, x4, x5)  =  U10_ggg(x1, x3, x4, x5)
U11_ggg(x1, x2, x3, x4, x5)  =  U11_ggg(x1, x3, x5)
is_in_ag(x1, x2)  =  is_in_ag(x2)
is_out_ag(x1, x2)  =  is_out_ag
1  =  1
U12_ggg(x1, x2, x3, x4, x5)  =  U12_ggg(x5)
noattack_in_gga(x1, x2, x3)  =  noattack_in_gga(x1, x2)
noattack_out_gga(x1, x2, x3)  =  noattack_out_gga
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
=\=_in_ga(x1, x2)  =  =\=_in_ga(x1)
=\=_out_ga(x1, x2)  =  =\=_out_ga
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x3, x5)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x3, x5)
is_in_aa(x1, x2)  =  is_in_aa
is_out_aa(x1, x2)  =  is_out_aa
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U7_g(x1, x2, x3)  =  U7_g(x3)
queens_out_ga(x1, x2)  =  queens_out_ga(x2)
NOATTACK_IN_GGA(x1, x2, x3)  =  NOATTACK_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x2, x3, x5)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x3, x5)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x1, x3, x5)

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

(10) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(11) Obligation:

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

U8_GGA(X, F, T, N, =\=_out_gg(X, F)) → U9_GGA(X, F, T, N, =\=_in_ga(X, +(F, N)))
U9_GGA(X, F, T, N, =\=_out_ga(X, +(F, N))) → U10_GGA(X, F, T, N, =\=_in_ga(F, +(X, N)))
U10_GGA(X, F, T, N, =\=_out_ga(F, +(X, N))) → U11_GGA(X, F, T, N, is_in_aa(N1, +(N, 1)))
U11_GGA(X, F, T, N, is_out_aa(N1, +(N, 1))) → NOATTACK_IN_GGA(X, T, N1)
NOATTACK_IN_GGA(X, .(F, T), N) → U8_GGA(X, F, T, N, =\=_in_gg(X, F))

The TRS R consists of the following rules:

=\=_in_ga(X0, X1) → =\=_out_ga(X0, X1)
is_in_aa(X0, X1) → is_out_aa(X0, X1)
=\=_in_gg(X0, X1) → =\=_out_gg(X0, X1)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
=\=_in_gg(x1, x2)  =  =\=_in_gg(x1, x2)
=\=_out_gg(x1, x2)  =  =\=_out_gg
+(x1, x2)  =  +(x1, x2)
1  =  1
=\=_in_ga(x1, x2)  =  =\=_in_ga(x1)
=\=_out_ga(x1, x2)  =  =\=_out_ga
is_in_aa(x1, x2)  =  is_in_aa
is_out_aa(x1, x2)  =  is_out_aa
NOATTACK_IN_GGA(x1, x2, x3)  =  NOATTACK_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x2, x3, x5)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x3, x5)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x1, x3, x5)

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

(12) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(13) Obligation:

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

U8_GGA(X, F, T, =\=_out_gg) → U9_GGA(X, F, T, =\=_in_ga(X))
U9_GGA(X, F, T, =\=_out_ga) → U10_GGA(X, T, =\=_in_ga(F))
U10_GGA(X, T, =\=_out_ga) → U11_GGA(X, T, is_in_aa)
U11_GGA(X, T, is_out_aa) → NOATTACK_IN_GGA(X, T)
NOATTACK_IN_GGA(X, .(F, T)) → U8_GGA(X, F, T, =\=_in_gg(X, F))

The TRS R consists of the following rules:

=\=_in_ga(X0) → =\=_out_ga
is_in_aa → is_out_aa
=\=_in_gg(X0, X1) → =\=_out_gg

The set Q consists of the following terms:

=\=_in_ga(x0)
is_in_aa
=\=_in_gg(x0, x1)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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_GGA(X, F, T, =\=_out_ga) → U10_GGA(X, T, =\=_in_ga(F))
    The graph contains the following edges 1 >= 1, 3 >= 2

  • NOATTACK_IN_GGA(X, .(F, T)) → U8_GGA(X, F, T, =\=_in_gg(X, F))
    The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

  • U10_GGA(X, T, =\=_out_ga) → U11_GGA(X, T, is_in_aa)
    The graph contains the following edges 1 >= 1, 2 >= 2

  • U8_GGA(X, F, T, =\=_out_gg) → U9_GGA(X, F, T, =\=_in_ga(X))
    The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3

  • U11_GGA(X, T, is_out_aa) → NOATTACK_IN_GGA(X, T)
    The graph contains the following edges 1 >= 1, 2 >= 2

(15) YES

(16) Obligation:

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

U6_G(X, Y, noattack_out_ggg(X, Y, 1)) → SAFE_IN_G(Y)
SAFE_IN_G(.(X, Y)) → U6_G(X, Y, noattack_in_ggg(X, Y, 1))

The TRS R consists of the following rules:

queens_in_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, 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_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(X, 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, 1))
noattack_in_ggg(X, [], N) → noattack_out_ggg(X, [], N)
noattack_in_ggg(X, .(F, T), N) → U8_ggg(X, F, T, N, =\=_in_gg(X, F))
=\=_in_gg(X0, X1) → =\=_out_gg(X0, X1)
U8_ggg(X, F, T, N, =\=_out_gg(X, F)) → U9_ggg(X, F, T, N, =\=_in_gg(X, +(F, N)))
U9_ggg(X, F, T, N, =\=_out_gg(X, +(F, N))) → U10_ggg(X, F, T, N, =\=_in_gg(F, +(X, N)))
U10_ggg(X, F, T, N, =\=_out_gg(F, +(X, N))) → U11_ggg(X, F, T, N, is_in_ag(N1, +(N, 1)))
is_in_ag(X0, X1) → is_out_ag(X0, X1)
U11_ggg(X, F, T, N, is_out_ag(N1, +(N, 1))) → U12_ggg(X, F, T, N, noattack_in_gga(X, T, N1))
noattack_in_gga(X, [], N) → noattack_out_gga(X, [], N)
noattack_in_gga(X, .(F, T), N) → U8_gga(X, F, T, N, =\=_in_gg(X, F))
U8_gga(X, F, T, N, =\=_out_gg(X, F)) → U9_gga(X, F, T, N, =\=_in_ga(X, +(F, N)))
=\=_in_ga(X0, X1) → =\=_out_ga(X0, X1)
U9_gga(X, F, T, N, =\=_out_ga(X, +(F, N))) → U10_gga(X, F, T, N, =\=_in_ga(F, +(X, N)))
U10_gga(X, F, T, N, =\=_out_ga(F, +(X, N))) → U11_gga(X, F, T, N, is_in_aa(N1, +(N, 1)))
is_in_aa(X0, X1) → is_out_aa(X0, X1)
U11_gga(X, F, T, N, is_out_aa(N1, +(N, 1))) → U12_gga(X, F, T, N, noattack_in_gga(X, T, N1))
U12_gga(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_gga(X, .(F, T), N)
U12_ggg(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_ggg(X, .(F, T), N)
U6_g(X, Y, noattack_out_ggg(X, Y, 1)) → U7_g(X, Y, safe_in_g(Y))
U7_g(X, Y, safe_out_g(Y)) → safe_out_g(.(X, Y))
U2_ga(X, Y, safe_out_g(Y)) → queens_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
queens_in_ga(x1, x2)  =  queens_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
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)
U2_ga(x1, x2, x3)  =  U2_ga(x2, x3)
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)
=\=_in_gg(x1, x2)  =  =\=_in_gg(x1, x2)
=\=_out_gg(x1, x2)  =  =\=_out_gg
U9_ggg(x1, x2, x3, x4, x5)  =  U9_ggg(x1, x2, x3, x4, x5)
+(x1, x2)  =  +(x1, x2)
U10_ggg(x1, x2, x3, x4, x5)  =  U10_ggg(x1, x3, x4, x5)
U11_ggg(x1, x2, x3, x4, x5)  =  U11_ggg(x1, x3, x5)
is_in_ag(x1, x2)  =  is_in_ag(x2)
is_out_ag(x1, x2)  =  is_out_ag
1  =  1
U12_ggg(x1, x2, x3, x4, x5)  =  U12_ggg(x5)
noattack_in_gga(x1, x2, x3)  =  noattack_in_gga(x1, x2)
noattack_out_gga(x1, x2, x3)  =  noattack_out_gga
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
=\=_in_ga(x1, x2)  =  =\=_in_ga(x1)
=\=_out_ga(x1, x2)  =  =\=_out_ga
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x3, x5)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x3, x5)
is_in_aa(x1, x2)  =  is_in_aa
is_out_aa(x1, x2)  =  is_out_aa
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U7_g(x1, x2, x3)  =  U7_g(x3)
queens_out_ga(x1, x2)  =  queens_out_ga(x2)
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

(17) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(18) Obligation:

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

U6_G(X, Y, noattack_out_ggg(X, Y, 1)) → SAFE_IN_G(Y)
SAFE_IN_G(.(X, Y)) → U6_G(X, Y, noattack_in_ggg(X, Y, 1))

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, =\=_in_gg(X, F))
U8_ggg(X, F, T, N, =\=_out_gg(X, F)) → U9_ggg(X, F, T, N, =\=_in_gg(X, +(F, N)))
=\=_in_gg(X0, X1) → =\=_out_gg(X0, X1)
U9_ggg(X, F, T, N, =\=_out_gg(X, +(F, N))) → U10_ggg(X, F, T, N, =\=_in_gg(F, +(X, N)))
U10_ggg(X, F, T, N, =\=_out_gg(F, +(X, N))) → U11_ggg(X, F, T, N, is_in_ag(N1, +(N, 1)))
U11_ggg(X, F, T, N, is_out_ag(N1, +(N, 1))) → U12_ggg(X, F, T, N, noattack_in_gga(X, T, N1))
is_in_ag(X0, X1) → is_out_ag(X0, X1)
U12_ggg(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_ggg(X, .(F, T), N)
noattack_in_gga(X, [], N) → noattack_out_gga(X, [], N)
noattack_in_gga(X, .(F, T), N) → U8_gga(X, F, T, N, =\=_in_gg(X, F))
U8_gga(X, F, T, N, =\=_out_gg(X, F)) → U9_gga(X, F, T, N, =\=_in_ga(X, +(F, N)))
U9_gga(X, F, T, N, =\=_out_ga(X, +(F, N))) → U10_gga(X, F, T, N, =\=_in_ga(F, +(X, N)))
=\=_in_ga(X0, X1) → =\=_out_ga(X0, X1)
U10_gga(X, F, T, N, =\=_out_ga(F, +(X, N))) → U11_gga(X, F, T, N, is_in_aa(N1, +(N, 1)))
U11_gga(X, F, T, N, is_out_aa(N1, +(N, 1))) → U12_gga(X, F, T, N, noattack_in_gga(X, T, N1))
is_in_aa(X0, X1) → is_out_aa(X0, X1)
U12_gga(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_gga(X, .(F, T), N)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
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)
=\=_in_gg(x1, x2)  =  =\=_in_gg(x1, x2)
=\=_out_gg(x1, x2)  =  =\=_out_gg
U9_ggg(x1, x2, x3, x4, x5)  =  U9_ggg(x1, x2, x3, x4, x5)
+(x1, x2)  =  +(x1, x2)
U10_ggg(x1, x2, x3, x4, x5)  =  U10_ggg(x1, x3, x4, x5)
U11_ggg(x1, x2, x3, x4, x5)  =  U11_ggg(x1, x3, x5)
is_in_ag(x1, x2)  =  is_in_ag(x2)
is_out_ag(x1, x2)  =  is_out_ag
1  =  1
U12_ggg(x1, x2, x3, x4, x5)  =  U12_ggg(x5)
noattack_in_gga(x1, x2, x3)  =  noattack_in_gga(x1, x2)
noattack_out_gga(x1, x2, x3)  =  noattack_out_gga
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
=\=_in_ga(x1, x2)  =  =\=_in_ga(x1)
=\=_out_ga(x1, x2)  =  =\=_out_ga
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x3, x5)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x3, x5)
is_in_aa(x1, x2)  =  is_in_aa
is_out_aa(x1, x2)  =  is_out_aa
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(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

(19) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(20) 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, 1))

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, =\=_in_gg(X, F))
U8_ggg(X, F, T, N, =\=_out_gg) → U9_ggg(X, F, T, N, =\=_in_gg(X, +(F, N)))
=\=_in_gg(X0, X1) → =\=_out_gg
U9_ggg(X, F, T, N, =\=_out_gg) → U10_ggg(X, T, N, =\=_in_gg(F, +(X, N)))
U10_ggg(X, T, N, =\=_out_gg) → U11_ggg(X, T, is_in_ag(+(N, 1)))
U11_ggg(X, T, is_out_ag) → U12_ggg(noattack_in_gga(X, T))
is_in_ag(X1) → is_out_ag
U12_ggg(noattack_out_gga) → noattack_out_ggg
noattack_in_gga(X, []) → noattack_out_gga
noattack_in_gga(X, .(F, T)) → U8_gga(X, F, T, =\=_in_gg(X, F))
U8_gga(X, F, T, =\=_out_gg) → U9_gga(X, F, T, =\=_in_ga(X))
U9_gga(X, F, T, =\=_out_ga) → U10_gga(X, T, =\=_in_ga(F))
=\=_in_ga(X0) → =\=_out_ga
U10_gga(X, T, =\=_out_ga) → U11_gga(X, T, is_in_aa)
U11_gga(X, T, is_out_aa) → U12_gga(noattack_in_gga(X, T))
is_in_aa → is_out_aa
U12_gga(noattack_out_gga) → noattack_out_gga

The set Q consists of the following terms:

noattack_in_ggg(x0, x1, x2)
U8_ggg(x0, x1, x2, x3, x4)
=\=_in_gg(x0, x1)
U9_ggg(x0, x1, x2, x3, x4)
U10_ggg(x0, x1, x2, x3)
U11_ggg(x0, x1, x2)
is_in_ag(x0)
U12_ggg(x0)
noattack_in_gga(x0, x1)
U8_gga(x0, x1, x2, x3)
U9_gga(x0, x1, x2, x3)
=\=_in_ga(x0)
U10_gga(x0, x1, x2)
U11_gga(x0, x1, x2)
is_in_aa
U12_gga(x0)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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, 1))
    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

(22) YES

(23) 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_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, 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_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(X, 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, 1))
noattack_in_ggg(X, [], N) → noattack_out_ggg(X, [], N)
noattack_in_ggg(X, .(F, T), N) → U8_ggg(X, F, T, N, =\=_in_gg(X, F))
=\=_in_gg(X0, X1) → =\=_out_gg(X0, X1)
U8_ggg(X, F, T, N, =\=_out_gg(X, F)) → U9_ggg(X, F, T, N, =\=_in_gg(X, +(F, N)))
U9_ggg(X, F, T, N, =\=_out_gg(X, +(F, N))) → U10_ggg(X, F, T, N, =\=_in_gg(F, +(X, N)))
U10_ggg(X, F, T, N, =\=_out_gg(F, +(X, N))) → U11_ggg(X, F, T, N, is_in_ag(N1, +(N, 1)))
is_in_ag(X0, X1) → is_out_ag(X0, X1)
U11_ggg(X, F, T, N, is_out_ag(N1, +(N, 1))) → U12_ggg(X, F, T, N, noattack_in_gga(X, T, N1))
noattack_in_gga(X, [], N) → noattack_out_gga(X, [], N)
noattack_in_gga(X, .(F, T), N) → U8_gga(X, F, T, N, =\=_in_gg(X, F))
U8_gga(X, F, T, N, =\=_out_gg(X, F)) → U9_gga(X, F, T, N, =\=_in_ga(X, +(F, N)))
=\=_in_ga(X0, X1) → =\=_out_ga(X0, X1)
U9_gga(X, F, T, N, =\=_out_ga(X, +(F, N))) → U10_gga(X, F, T, N, =\=_in_ga(F, +(X, N)))
U10_gga(X, F, T, N, =\=_out_ga(F, +(X, N))) → U11_gga(X, F, T, N, is_in_aa(N1, +(N, 1)))
is_in_aa(X0, X1) → is_out_aa(X0, X1)
U11_gga(X, F, T, N, is_out_aa(N1, +(N, 1))) → U12_gga(X, F, T, N, noattack_in_gga(X, T, N1))
U12_gga(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_gga(X, .(F, T), N)
U12_ggg(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_ggg(X, .(F, T), N)
U6_g(X, Y, noattack_out_ggg(X, Y, 1)) → U7_g(X, Y, safe_in_g(Y))
U7_g(X, Y, safe_out_g(Y)) → safe_out_g(.(X, Y))
U2_ga(X, Y, safe_out_g(Y)) → queens_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
queens_in_ga(x1, x2)  =  queens_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
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)
U2_ga(x1, x2, x3)  =  U2_ga(x2, x3)
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)
=\=_in_gg(x1, x2)  =  =\=_in_gg(x1, x2)
=\=_out_gg(x1, x2)  =  =\=_out_gg
U9_ggg(x1, x2, x3, x4, x5)  =  U9_ggg(x1, x2, x3, x4, x5)
+(x1, x2)  =  +(x1, x2)
U10_ggg(x1, x2, x3, x4, x5)  =  U10_ggg(x1, x3, x4, x5)
U11_ggg(x1, x2, x3, x4, x5)  =  U11_ggg(x1, x3, x5)
is_in_ag(x1, x2)  =  is_in_ag(x2)
is_out_ag(x1, x2)  =  is_out_ag
1  =  1
U12_ggg(x1, x2, x3, x4, x5)  =  U12_ggg(x5)
noattack_in_gga(x1, x2, x3)  =  noattack_in_gga(x1, x2)
noattack_out_gga(x1, x2, x3)  =  noattack_out_gga
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
=\=_in_ga(x1, x2)  =  =\=_in_ga(x1)
=\=_out_ga(x1, x2)  =  =\=_out_ga
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x3, x5)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x3, x5)
is_in_aa(x1, x2)  =  is_in_aa
is_out_aa(x1, x2)  =  is_out_aa
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U7_g(x1, x2, x3)  =  U7_g(x3)
queens_out_ga(x1, x2)  =  queens_out_ga(x2)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(25) 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

(26) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(27) 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.

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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

(29) YES

(30) 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_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, 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_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(X, 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, 1))
noattack_in_ggg(X, [], N) → noattack_out_ggg(X, [], N)
noattack_in_ggg(X, .(F, T), N) → U8_ggg(X, F, T, N, =\=_in_gg(X, F))
=\=_in_gg(X0, X1) → =\=_out_gg(X0, X1)
U8_ggg(X, F, T, N, =\=_out_gg(X, F)) → U9_ggg(X, F, T, N, =\=_in_gg(X, +(F, N)))
U9_ggg(X, F, T, N, =\=_out_gg(X, +(F, N))) → U10_ggg(X, F, T, N, =\=_in_gg(F, +(X, N)))
U10_ggg(X, F, T, N, =\=_out_gg(F, +(X, N))) → U11_ggg(X, F, T, N, is_in_ag(N1, +(N, 1)))
is_in_ag(X0, X1) → is_out_ag(X0, X1)
U11_ggg(X, F, T, N, is_out_ag(N1, +(N, 1))) → U12_ggg(X, F, T, N, noattack_in_gga(X, T, N1))
noattack_in_gga(X, [], N) → noattack_out_gga(X, [], N)
noattack_in_gga(X, .(F, T), N) → U8_gga(X, F, T, N, =\=_in_gg(X, F))
U8_gga(X, F, T, N, =\=_out_gg(X, F)) → U9_gga(X, F, T, N, =\=_in_ga(X, +(F, N)))
=\=_in_ga(X0, X1) → =\=_out_ga(X0, X1)
U9_gga(X, F, T, N, =\=_out_ga(X, +(F, N))) → U10_gga(X, F, T, N, =\=_in_ga(F, +(X, N)))
U10_gga(X, F, T, N, =\=_out_ga(F, +(X, N))) → U11_gga(X, F, T, N, is_in_aa(N1, +(N, 1)))
is_in_aa(X0, X1) → is_out_aa(X0, X1)
U11_gga(X, F, T, N, is_out_aa(N1, +(N, 1))) → U12_gga(X, F, T, N, noattack_in_gga(X, T, N1))
U12_gga(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_gga(X, .(F, T), N)
U12_ggg(X, F, T, N, noattack_out_gga(X, T, N1)) → noattack_out_ggg(X, .(F, T), N)
U6_g(X, Y, noattack_out_ggg(X, Y, 1)) → U7_g(X, Y, safe_in_g(Y))
U7_g(X, Y, safe_out_g(Y)) → safe_out_g(.(X, Y))
U2_ga(X, Y, safe_out_g(Y)) → queens_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
queens_in_ga(x1, x2)  =  queens_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
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)
U2_ga(x1, x2, x3)  =  U2_ga(x2, x3)
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)
=\=_in_gg(x1, x2)  =  =\=_in_gg(x1, x2)
=\=_out_gg(x1, x2)  =  =\=_out_gg
U9_ggg(x1, x2, x3, x4, x5)  =  U9_ggg(x1, x2, x3, x4, x5)
+(x1, x2)  =  +(x1, x2)
U10_ggg(x1, x2, x3, x4, x5)  =  U10_ggg(x1, x3, x4, x5)
U11_ggg(x1, x2, x3, x4, x5)  =  U11_ggg(x1, x3, x5)
is_in_ag(x1, x2)  =  is_in_ag(x2)
is_out_ag(x1, x2)  =  is_out_ag
1  =  1
U12_ggg(x1, x2, x3, x4, x5)  =  U12_ggg(x5)
noattack_in_gga(x1, x2, x3)  =  noattack_in_gga(x1, x2)
noattack_out_gga(x1, x2, x3)  =  noattack_out_gga
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
=\=_in_ga(x1, x2)  =  =\=_in_ga(x1)
=\=_out_ga(x1, x2)  =  =\=_out_ga
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x3, x5)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x3, x5)
is_in_aa(x1, x2)  =  is_in_aa
is_out_aa(x1, x2)  =  is_out_aa
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x5)
U7_g(x1, x2, x3)  =  U7_g(x3)
queens_out_ga(x1, x2)  =  queens_out_ga(x2)
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

(31) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(32) 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

(33) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(34) 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.

(35) MRRProof (EQUIVALENT transformation)

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)
PERM_IN_GA(.(X, Y)) → U3_GA(delete_in_aga(.(X, Y)))

Strictly oriented rules of the TRS R:

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))

Used ordering: Knuth-Bendix order [KBO] with precedence:
.2 > deleteinaga1 > U5aga2 > U3GA1 > PERMINGA1 > deleteoutaga2

and weight map:

delete_in_aga_1=1
U3_GA_1=1
PERM_IN_GA_1=3
._2=0
delete_out_aga_2=1
U5_aga_2=0

The variable weight is 1

(36) Obligation:

Q DP problem:
P is empty.
R is empty.
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.

(37) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(38) YES