(0) Obligation:

Clauses:

permute([], []).
permute(.(X, Y), .(U, V)) :- ','(delete(U, .(X, Y), W), permute(W, V)).
delete(X, .(X, Y), Y).
delete(U, .(X, Y), .(X, Z)) :- delete(U, Y, Z).

Query: permute(g,a)

(1) PrologToPiTRSProof (SOUND transformation)

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

permute_in_ga([], []) → permute_out_ga([], [])
permute_in_ga(.(X, Y), .(U, V)) → U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))
U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_ga(X, Y, U, V, permute_in_ga(W, V))
U2_ga(X, Y, U, V, permute_out_ga(W, V)) → permute_out_ga(.(X, Y), .(U, V))

The argument filtering Pi contains the following mapping:
permute_in_ga(x1, x2)  =  permute_in_ga(x1)
[]  =  []
permute_out_ga(x1, x2)  =  permute_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

permute_in_ga([], []) → permute_out_ga([], [])
permute_in_ga(.(X, Y), .(U, V)) → U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))
U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_ga(X, Y, U, V, permute_in_ga(W, V))
U2_ga(X, Y, U, V, permute_out_ga(W, V)) → permute_out_ga(.(X, Y), .(U, V))

The argument filtering Pi contains the following mapping:
permute_in_ga(x1, x2)  =  permute_in_ga(x1)
[]  =  []
permute_out_ga(x1, x2)  =  permute_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)

(3) 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:

PERMUTE_IN_GA(.(X, Y), .(U, V)) → U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
PERMUTE_IN_GA(.(X, Y), .(U, V)) → DELETE_IN_AGA(U, .(X, Y), W)
DELETE_IN_AGA(U, .(X, Y), .(X, Z)) → U3_AGA(U, X, Y, Z, delete_in_aga(U, Y, Z))
DELETE_IN_AGA(U, .(X, Y), .(X, Z)) → DELETE_IN_AGA(U, Y, Z)
U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_GA(X, Y, U, V, permute_in_ga(W, V))
U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → PERMUTE_IN_GA(W, V)

The TRS R consists of the following rules:

permute_in_ga([], []) → permute_out_ga([], [])
permute_in_ga(.(X, Y), .(U, V)) → U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))
U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_ga(X, Y, U, V, permute_in_ga(W, V))
U2_ga(X, Y, U, V, permute_out_ga(W, V)) → permute_out_ga(.(X, Y), .(U, V))

The argument filtering Pi contains the following mapping:
permute_in_ga(x1, x2)  =  permute_in_ga(x1)
[]  =  []
permute_out_ga(x1, x2)  =  permute_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
PERMUTE_IN_GA(x1, x2)  =  PERMUTE_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)

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

(4) Obligation:

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

PERMUTE_IN_GA(.(X, Y), .(U, V)) → U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
PERMUTE_IN_GA(.(X, Y), .(U, V)) → DELETE_IN_AGA(U, .(X, Y), W)
DELETE_IN_AGA(U, .(X, Y), .(X, Z)) → U3_AGA(U, X, Y, Z, delete_in_aga(U, Y, Z))
DELETE_IN_AGA(U, .(X, Y), .(X, Z)) → DELETE_IN_AGA(U, Y, Z)
U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_GA(X, Y, U, V, permute_in_ga(W, V))
U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → PERMUTE_IN_GA(W, V)

The TRS R consists of the following rules:

permute_in_ga([], []) → permute_out_ga([], [])
permute_in_ga(.(X, Y), .(U, V)) → U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))
U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_ga(X, Y, U, V, permute_in_ga(W, V))
U2_ga(X, Y, U, V, permute_out_ga(W, V)) → permute_out_ga(.(X, Y), .(U, V))

The argument filtering Pi contains the following mapping:
permute_in_ga(x1, x2)  =  permute_in_ga(x1)
[]  =  []
permute_out_ga(x1, x2)  =  permute_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
PERMUTE_IN_GA(x1, x2)  =  PERMUTE_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

DELETE_IN_AGA(U, .(X, Y), .(X, Z)) → DELETE_IN_AGA(U, Y, Z)

The TRS R consists of the following rules:

permute_in_ga([], []) → permute_out_ga([], [])
permute_in_ga(.(X, Y), .(U, V)) → U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))
U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_ga(X, Y, U, V, permute_in_ga(W, V))
U2_ga(X, Y, U, V, permute_out_ga(W, V)) → permute_out_ga(.(X, Y), .(U, V))

The argument filtering Pi contains the following mapping:
permute_in_ga(x1, x2)  =  permute_in_ga(x1)
[]  =  []
permute_out_ga(x1, x2)  =  permute_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

DELETE_IN_AGA(U, .(X, Y), .(X, Z)) → DELETE_IN_AGA(U, Y, Z)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

DELETE_IN_AGA(.(X, Y)) → DELETE_IN_AGA(Y)

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

(12) 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(.(X, Y)) → DELETE_IN_AGA(Y)
    The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → PERMUTE_IN_GA(W, V)
PERMUTE_IN_GA(.(X, Y), .(U, V)) → U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W))

The TRS R consists of the following rules:

permute_in_ga([], []) → permute_out_ga([], [])
permute_in_ga(.(X, Y), .(U, V)) → U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))
U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_ga(X, Y, U, V, permute_in_ga(W, V))
U2_ga(X, Y, U, V, permute_out_ga(W, V)) → permute_out_ga(.(X, Y), .(U, V))

The argument filtering Pi contains the following mapping:
permute_in_ga(x1, x2)  =  permute_in_ga(x1)
[]  =  []
permute_out_ga(x1, x2)  =  permute_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
PERMUTE_IN_GA(x1, x2)  =  PERMUTE_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → PERMUTE_IN_GA(W, V)
PERMUTE_IN_GA(.(X, Y), .(U, V)) → U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W))

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(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))

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)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
PERMUTE_IN_GA(x1, x2)  =  PERMUTE_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

U1_GA(delete_out_aga(U, W)) → PERMUTE_IN_GA(W)
PERMUTE_IN_GA(.(X, Y)) → U1_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(.(X, Y)) → U3_aga(X, delete_in_aga(Y))
U3_aga(X, delete_out_aga(U, Z)) → delete_out_aga(U, .(X, Z))

The set Q consists of the following terms:

delete_in_aga(x0)
U3_aga(x0, x1)

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

(19) 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:

U1_GA(delete_out_aga(U, W)) → PERMUTE_IN_GA(W)
PERMUTE_IN_GA(.(X, Y)) → U1_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(.(X, Y)) → U3_aga(X, delete_in_aga(Y))
U3_aga(X, delete_out_aga(U, Z)) → delete_out_aga(U, .(X, Z))

Used ordering: Knuth-Bendix order [KBO] with precedence:
.2 > deleteinaga1 > U3aga2 > U1GA1 > PERMUTEINGA1 > deleteoutaga2

and weight map:

delete_in_aga_1=1
U1_GA_1=1
PERMUTE_IN_GA_1=3
._2=0
delete_out_aga_2=1
U3_aga_2=0

The variable weight is 1

(20) Obligation:

Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:

delete_in_aga(x0)
U3_aga(x0, x1)

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

(21) PisEmptyProof (EQUIVALENT transformation)

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

(22) YES