Left Termination proof of /tmp/tmp0N39W4/permutation-bf.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:

perm([], []).
perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

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)
app_in: (f,f,b) (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([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

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([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP_IN_AAG(X1s, .(X, X2s), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → APP_IN_GGA(X1s, X2s, Zs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
[] = []
perm_out_ga(x1, x2) = perm_out_ga(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x4)
app_in_aag(x1, x2, x3) = app_in_aag(x3)
app_out_aag(x1, x2, x3) = app_out_aag(x1, x2)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x5)
U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5)
U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4)
APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2)
U4_AAG(x1, x2, x3, x4, x5) = U4_AAG(x1, x5)
APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x5)
U3_GA(x1, x2, x3, x4) = U3_GA(x2, x4)
U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4)
U1_GA(x1, x2, x3, x4) = U1_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(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP_IN_AAG(X1s, .(X, X2s), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → APP_IN_GGA(X1s, X2s, Zs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
[] = []
perm_out_ga(x1, x2) = perm_out_ga(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x4)
app_in_aag(x1, x2, x3) = app_in_aag(x3)
app_out_aag(x1, x2, x3) = app_out_aag(x1, x2)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x5)
U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5)
U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4)
APP_IN_GGA(x1, x2, x3) = APP_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:

APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

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

APP_IN_GGA(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)

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:

APP_IN_GGA(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)
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:

APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
[] = []
perm_out_ga(x1, x2) = perm_out_ga(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x4)
app_in_aag(x1, x2, x3) = app_in_aag(x3)
app_out_aag(x1, x2, x3) = app_out_aag(x1, x2)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x5)
U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5)
U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4)
APP_IN_AAG(x1, x2, x3) = APP_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:

APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)
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:

U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
[] = []
perm_out_ga(x1, x2) = perm_out_ga(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x4)
app_in_aag(x1, x2, x3) = app_in_aag(x3)
app_out_aag(x1, x2, x3) = app_out_aag(x1, x2)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x5)
U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5)
U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4)
U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4)
U1_GA(x1, x2, x3, x4) = U1_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:

U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[] = []
app_in_aag(x1, x2, x3) = app_in_aag(x3)
app_out_aag(x1, x2, x3) = app_out_aag(x1, x2)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x5)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5)
U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4)
U1_GA(x1, x2, x3, x4) = U1_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:

U1_GA(app_out_aag(X1s, .(X, X2s))) → U2_GA(X, app_in_gga(X1s, X2s))
PERM_IN_GA(Xs) → U1_GA(app_in_aag(Xs))
U2_GA(X, app_out_gga(Zs)) → PERM_IN_GA(Zs)

The TRS R consists of the following rules:

app_in_gga([], X) → app_out_gga(X)
app_in_gga(.(X, Xs), Ys) → U4_gga(X, app_in_gga(Xs, Ys))
app_in_aag(X) → app_out_aag([], X)
app_in_aag(.(X, Zs)) → U4_aag(X, app_in_aag(Zs))
U4_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
U4_aag(X, app_out_aag(Xs, Ys)) → app_out_aag(.(X, Xs), Ys)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_aag(x0)
U4_gga(x0, x1)
U4_aag(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:

U1_GA(app_out_aag(X1s, .(X, X2s))) → U2_GA(X, app_in_gga(X1s, X2s))

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(PERM_IN_GA(x₁)) = 2·x₁
POL(U1_GA(x₁)) = 2·x₁
POL(U2_GA(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U4_aag(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(U4_gga(x₁, x₂)) = 2 + 2·x₁ + x₂
POL([]) = 0
POL(app_in_aag(x₁)) = x₁
POL(app_in_gga(x₁, x₂)) = x₁ + x₂
POL(app_out_aag(x₁, x₂)) = x₁ + x₂
POL(app_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:

PERM_IN_GA(Xs) → U1_GA(app_in_aag(Xs))
U2_GA(X, app_out_gga(Zs)) → PERM_IN_GA(Zs)

The TRS R consists of the following rules:

app_in_gga([], X) → app_out_gga(X)
app_in_gga(.(X, Xs), Ys) → U4_gga(X, app_in_gga(Xs, Ys))
app_in_aag(X) → app_out_aag([], X)
app_in_aag(.(X, Zs)) → U4_aag(X, app_in_aag(Zs))
U4_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
U4_aag(X, app_out_aag(Xs, Ys)) → app_out_aag(.(X, Xs), Ys)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_aag(x0)
U4_gga(x0, x1)
U4_aag(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.