Left Termination of the query pattern permute_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:



Prolog
  ↳ PrologToPiTRSProof

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).

Queries:

permute(g,a).

We use the technique of [30]. 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



↳ Prolog
  ↳ PrologToPiTRSProof
PiTRS
      ↳ DependencyPairsProof

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)


Using Dependency Pairs [1,30] 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)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)
PERMUTE_IN_GA(x1, x2)  =  PERMUTE_IN_GA(x1)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x5)

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:

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)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)
PERMUTE_IN_GA(x1, x2)  =  PERMUTE_IN_GA(x1)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 3 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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.
By using the subterm criterion [20] together with the size-change analysis [32] 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:



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
PiDP
                ↳ UsableRulesProof

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))
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)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof

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))
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:

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ RuleRemovalProof

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.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

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)

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x1, x2)) = 1 + x1 + x2   
POL(PERMUTE_IN_GA(x1)) = 2 + 2·x1   
POL(U1_GA(x1)) = x1   
POL(U3_aga(x1, x2)) = 2 + 2·x1 + x2   
POL(delete_in_aga(x1)) = 1 + 2·x1   
POL(delete_out_aga(x1, x2)) = 2 + x1 + 2·x2   



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ RuleRemovalProof
QDP
                            ↳ DependencyGraphProof

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

U1_GA(delete_out_aga(U, W)) → PERMUTE_IN_GA(W)

The TRS R consists of the following rules:

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.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.