Left Termination proof of /tmp/tmpzHPJR1/permute1.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(.(X, Xs), Ys, .(X, Zs)) :- app1(Xs, Ys, Zs).
app1([], Ys, Ys).
app2(.(X, Xs), Ys, .(X, Zs)) :- app2(Xs, Ys, Zs).
app2([], Ys, Ys).
perm(Xs, .(X, Ys)) :- ','(app2(X1s, .(X, X2s), Xs), ','(app1(X1s, X2s, Zs), perm(Zs, Ys))).
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)
app2_in: (f,f,b)
app1_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(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_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(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

PERM_IN_GA(Xs, .(X, Ys)) → U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP2_IN_AAG(X1s, .(X, X2s), Xs)
APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U2_AAG(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AAG(Xs, Ys, Zs)
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → APP1_IN_GGA(X1s, X2s, Zs)
APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U1_GGA(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP1_IN_GGA(Xs, Ys, Zs)
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_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)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
app2_in_aag(x1, x2, x3) = app2_in_aag(x3)
.(x1, x2) = .(x2)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5)
app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6)
app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5)
[] = []
app1_out_gga(x1, x2, x3) = app1_out_gga(x3)
U5_ga(x1, x2, x3, x4) = U5_ga(x4)
perm_out_ga(x1, x2) = perm_out_ga(x2)
U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x5)
U5_GA(x1, x2, x3, x4) = U5_GA(x4)
APP1_IN_GGA(x1, x2, x3) = APP1_IN_GGA(x1, x2)
APP2_IN_AAG(x1, x2, x3) = APP2_IN_AAG(x3)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6)
U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x5)
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(Xs, .(X, Ys)) → U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP2_IN_AAG(X1s, .(X, X2s), Xs)
APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U2_AAG(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AAG(Xs, Ys, Zs)
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → APP1_IN_GGA(X1s, X2s, Zs)
APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U1_GGA(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP1_IN_GGA(Xs, Ys, Zs)
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_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:

APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP1_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_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)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
app2_in_aag(x1, x2, x3) = app2_in_aag(x3)
.(x1, x2) = .(x2)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5)
app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6)
app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5)
[] = []
app1_out_gga(x1, x2, x3) = app1_out_gga(x3)
U5_ga(x1, x2, x3, x4) = U5_ga(x4)
perm_out_ga(x1, x2) = perm_out_ga(x2)
APP1_IN_GGA(x1, x2, x3) = APP1_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:

APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP1_IN_GGA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APP1_IN_GGA(x1, x2, x3) = APP1_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:

APP1_IN_GGA(.(Xs), Ys) → APP1_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:

APP1_IN_GGA(.(Xs), Ys) → APP1_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:

APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_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)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
app2_in_aag(x1, x2, x3) = app2_in_aag(x3)
.(x1, x2) = .(x2)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5)
app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6)
app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5)
[] = []
app1_out_gga(x1, x2, x3) = app1_out_gga(x3)
U5_ga(x1, x2, x3, x4) = U5_ga(x4)
perm_out_ga(x1, x2) = perm_out_ga(x2)
APP2_IN_AAG(x1, x2, x3) = APP2_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:

APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AAG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APP2_IN_AAG(x1, x2, x3) = APP2_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:

APP2_IN_AAG(.(Zs)) → APP2_IN_AAG(Zs)

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

APP2_IN_AAG(.(Zs)) → APP2_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:

PERM_IN_GA(Xs, .(X, Ys)) → U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_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)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
app2_in_aag(x1, x2, x3) = app2_in_aag(x3)
.(x1, x2) = .(x2)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5)
app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6)
app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5)
[] = []
app1_out_gga(x1, x2, x3) = app1_out_gga(x3)
U5_ga(x1, x2, x3, x4) = U5_ga(x4)
perm_out_ga(x1, x2) = perm_out_ga(x2)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(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(Xs, .(X, Ys)) → U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))

The TRS R consists of the following rules:

app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_aag(x1, x2, x3) = app2_in_aag(x3)
.(x1, x2) = .(x2)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5)
app2_out_aag(x1, x2, x3) = app2_out_aag(x1, x2)
app1_in_gga(x1, x2, x3) = app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5)
[] = []
app1_out_gga(x1, x2, x3) = app1_out_gga(x3)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(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:

PERM_IN_GA(Xs) → U3_GA(app2_in_aag(Xs))
U3_GA(app2_out_aag(X1s, .(X2s))) → U4_GA(app1_in_gga(X1s, X2s))
U4_GA(app1_out_gga(Zs)) → PERM_IN_GA(Zs)

The TRS R consists of the following rules:

app2_in_aag(.(Zs)) → U2_aag(app2_in_aag(Zs))
app2_in_aag(Ys) → app2_out_aag([], Ys)
app1_in_gga(.(Xs), Ys) → U1_gga(app1_in_gga(Xs, Ys))
app1_in_gga([], Ys) → app1_out_gga(Ys)
U2_aag(app2_out_aag(Xs, Ys)) → app2_out_aag(.(Xs), Ys)
U1_gga(app1_out_gga(Zs)) → app1_out_gga(.(Zs))

The set Q consists of the following terms:

app2_in_aag(x0)
app1_in_gga(x0, x1)
U2_aag(x0)
U1_gga(x0)

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(app2_out_aag(X1s, .(X2s))) → U4_GA(app1_in_gga(X1s, X2s))

Strictly oriented rules of the TRS R:

app2_in_aag(Ys) → app2_out_aag([], Ys)

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁)) = 2 + x₁
POL(PERM_IN_GA(x₁)) = 2 + 2·x₁
POL(U1_gga(x₁)) = 2 + x₁
POL(U2_aag(x₁)) = 2 + x₁
POL(U3_GA(x₁)) = 2·x₁
POL(U4_GA(x₁)) = 2 + 2·x₁
POL([]) = 0
POL(app1_in_gga(x₁, x₂)) = x₁ + x₂
POL(app1_out_gga(x₁)) = x₁
POL(app2_in_aag(x₁)) = 1 + x₁
POL(app2_out_aag(x₁, x₂)) = x₁ + x₂

↳ 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) → U3_GA(app2_in_aag(Xs))
U4_GA(app1_out_gga(Zs)) → PERM_IN_GA(Zs)

The TRS R consists of the following rules:

app2_in_aag(.(Zs)) → U2_aag(app2_in_aag(Zs))
app1_in_gga(.(Xs), Ys) → U1_gga(app1_in_gga(Xs, Ys))
app1_in_gga([], Ys) → app1_out_gga(Ys)
U2_aag(app2_out_aag(Xs, Ys)) → app2_out_aag(.(Xs), Ys)
U1_gga(app1_out_gga(Zs)) → app1_out_gga(.(Zs))

The set Q consists of the following terms:

app2_in_aag(x0)
app1_in_gga(x0, x1)
U2_aag(x0)
U1_gga(x0)

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.