Left Termination proof of /tmp/tmpBqUT-z/permutation.pl

Left Termination of the query pattern perm_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

app1(.(X0, X), Y, .(X0, Z)) :- app1(X, Y, Z).
app1([], Y, Y).
app2(.(X0, X), Y, .(X0, Z)) :- app2(X, Y, Z).
app2([], Y, Y).
perm(X, .(X0, Y)) :- ','(app1(X1, .(X0, X2), X), ','(app2(X1, X2, Z), perm(Z, Y))).
perm([], []).

Queries:

perm(g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm_in: (b,f)
app1_in: (f,f,b)
app2_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, 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:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

PERM_IN_GA(X, .(X0, Y)) → U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
PERM_IN_GA(X, .(X0, Y)) → APP1_IN_AAG(X1, .(X0, X2), X)
APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → U1_AAG(X0, X, Y, Z, app1_in_aag(X, Y, Z))
APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → APP1_IN_AAG(X, Y, Z)
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → APP2_IN_GGA(X1, X2, Z)
APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → U2_GGA(X0, X, Y, Z, app2_in_gga(X, Y, Z))
APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → APP2_IN_GGA(X, Y, Z)
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_GA(X, X0, Y, perm_in_ga(Z, Y))
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → PERM_IN_GA(Z, Y)

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
app1_in_aag(x1, x2, x3) = app1_in_aag(x3)
.(x1, x2) = .(x1, x2)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5)
app1_out_aag(x1, x2, x3) = app1_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x2, x6)
app2_in_gga(x1, x2, x3) = app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x5)
[] = []
app2_out_gga(x1, x2, x3) = app2_out_gga(x3)
U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4)
perm_out_ga(x1, x2) = perm_out_ga(x2)
U5_GA(x1, x2, x3, x4) = U5_GA(x2, x4)
U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x1, x5)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x1, x5)
APP1_IN_AAG(x1, x2, x3) = APP1_IN_AAG(x3)
APP2_IN_GGA(x1, x2, x3) = APP2_IN_GGA(x1, x2)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x2, x6)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
PERM_IN_GA(x1, x2) = PERM_IN_GA(x1)

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:

PERM_IN_GA(X, .(X0, Y)) → U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
PERM_IN_GA(X, .(X0, Y)) → APP1_IN_AAG(X1, .(X0, X2), X)
APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → U1_AAG(X0, X, Y, Z, app1_in_aag(X, Y, Z))
APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → APP1_IN_AAG(X, Y, Z)
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → APP2_IN_GGA(X1, X2, Z)
APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → U2_GGA(X0, X, Y, Z, app2_in_gga(X, Y, Z))
APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → APP2_IN_GGA(X, Y, Z)
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_GA(X, X0, Y, perm_in_ga(Z, Y))
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → PERM_IN_GA(Z, Y)

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → APP2_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
app1_in_aag(x1, x2, x3) = app1_in_aag(x3)
.(x1, x2) = .(x1, x2)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5)
app1_out_aag(x1, x2, x3) = app1_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x2, x6)
app2_in_gga(x1, x2, x3) = app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x5)
[] = []
app2_out_gga(x1, x2, x3) = app2_out_gga(x3)
U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4)
perm_out_ga(x1, x2) = perm_out_ga(x2)
APP2_IN_GGA(x1, x2, x3) = APP2_IN_GGA(x1, 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

              ↳ PiDP

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

APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → APP2_IN_GGA(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP2_IN_GGA(x1, x2, x3) = APP2_IN_GGA(x1, 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

              ↳ PiDP

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

APP2_IN_GGA(.(X0, X), Y) → APP2_IN_GGA(X, 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:

APP2_IN_GGA(.(X0, X), Y) → APP2_IN_GGA(X, Y)
The graph contains the following edges 1 > 1, 2 >= 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → APP1_IN_AAG(X, Y, Z)

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
app1_in_aag(x1, x2, x3) = app1_in_aag(x3)
.(x1, x2) = .(x1, x2)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5)
app1_out_aag(x1, x2, x3) = app1_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x2, x6)
app2_in_gga(x1, x2, x3) = app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x5)
[] = []
app2_out_gga(x1, x2, x3) = app2_out_gga(x3)
U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4)
perm_out_ga(x1, x2) = perm_out_ga(x2)
APP1_IN_AAG(x1, x2, x3) = APP1_IN_AAG(x3)

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

              ↳ PiDP

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

APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → APP1_IN_AAG(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP1_IN_AAG(x1, x2, x3) = APP1_IN_AAG(x3)

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

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

APP1_IN_AAG(.(X0, Z)) → APP1_IN_AAG(Z)

From the DPs we obtained the following set of size-change graphs:

APP1_IN_AAG(.(X0, Z)) → APP1_IN_AAG(Z)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

PERM_IN_GA(X, .(X0, Y)) → U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → PERM_IN_GA(Z, Y)

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
app1_in_aag(x1, x2, x3) = app1_in_aag(x3)
.(x1, x2) = .(x1, x2)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5)
app1_out_aag(x1, x2, x3) = app1_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x2, x6)
app2_in_gga(x1, x2, x3) = app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x5)
[] = []
app2_out_gga(x1, x2, x3) = app2_out_gga(x3)
U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4)
perm_out_ga(x1, x2) = perm_out_ga(x2)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x2, x6)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
PERM_IN_GA(x1, x2) = PERM_IN_GA(x1)

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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

PERM_IN_GA(X, .(X0, Y)) → U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → PERM_IN_GA(Z, Y)

The TRS R consists of the following rules:

app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))

The argument filtering Pi contains the following mapping:
app1_in_aag(x1, x2, x3) = app1_in_aag(x3)
.(x1, x2) = .(x1, x2)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5)
app1_out_aag(x1, x2, x3) = app1_out_aag(x1, x2)
app2_in_gga(x1, x2, x3) = app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x1, x5)
[] = []
app2_out_gga(x1, x2, x3) = app2_out_gga(x3)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x2, x6)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
PERM_IN_GA(x1, x2) = PERM_IN_GA(x1)

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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

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

U4_GA(X0, app2_out_gga(Z)) → PERM_IN_GA(Z)
U3_GA(app1_out_aag(X1, .(X0, X2))) → U4_GA(X0, app2_in_gga(X1, X2))
PERM_IN_GA(X) → U3_GA(app1_in_aag(X))

The TRS R consists of the following rules:

app1_in_aag(.(X0, Z)) → U1_aag(X0, app1_in_aag(Z))
app1_in_aag(Y) → app1_out_aag([], Y)
app2_in_gga(.(X0, X), Y) → U2_gga(X0, app2_in_gga(X, Y))
app2_in_gga([], Y) → app2_out_gga(Y)
U1_aag(X0, app1_out_aag(X, Y)) → app1_out_aag(.(X0, X), Y)
U2_gga(X0, app2_out_gga(Z)) → app2_out_gga(.(X0, Z))

The set Q consists of the following terms:

app1_in_aag(x0)
app2_in_gga(x0, x1)
U1_aag(x0, x1)
U2_gga(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:

U3_GA(app1_out_aag(X1, .(X0, X2))) → U4_GA(X0, app2_in_gga(X1, X2))

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 1 + x₁ + x₂
POL(PERM_IN_GA(x₁)) = 2·x₁
POL(U1_aag(x₁, x₂)) = 1 + x₁ + x₂
POL(U2_gga(x₁, x₂)) = 1 + x₁ + x₂
POL(U3_GA(x₁)) = 2·x₁
POL(U4_GA(x₁, x₂)) = x₁ + 2·x₂
POL([]) = 0
POL(app1_in_aag(x₁)) = x₁
POL(app1_out_aag(x₁, x₂)) = x₁ + x₂
POL(app2_in_gga(x₁, x₂)) = x₁ + x₂
POL(app2_out_gga(x₁)) = x₁

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

U4_GA(X0, app2_out_gga(Z)) → PERM_IN_GA(Z)
PERM_IN_GA(X) → U3_GA(app1_in_aag(X))

The TRS R consists of the following rules:

app1_in_aag(.(X0, Z)) → U1_aag(X0, app1_in_aag(Z))
app1_in_aag(Y) → app1_out_aag([], Y)
app2_in_gga(.(X0, X), Y) → U2_gga(X0, app2_in_gga(X, Y))
app2_in_gga([], Y) → app2_out_gga(Y)
U1_aag(X0, app1_out_aag(X, Y)) → app1_out_aag(.(X0, X), Y)
U2_gga(X0, app2_out_gga(Z)) → app2_out_gga(.(X0, Z))

The set Q consists of the following terms:

app1_in_aag(x0)
app2_in_gga(x0, x1)
U1_aag(x0, x1)
U2_gga(x0, x1)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.