Left Termination proof of /tmp/tmpZ5qKZR/flatlength-ffb.pl

Left Termination of the query pattern fl_in_3(a, a, g) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

fl([], [], 0).
fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z)).
append([], X, X).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).

Queries:

fl(a,a,g).

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

fl_in_aag([], [], 0) → fl_out_aag([], [], 0)
fl_in_aag(.(E, X), R, s(Z)) → U1_aag(E, X, R, Z, append_in_aaa(E, Y, R))
append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z))
U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) → fl_out_aag(.(E, X), R, s(Z))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

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

fl_in_aag([], [], 0) → fl_out_aag([], [], 0)
fl_in_aag(.(E, X), R, s(Z)) → U1_aag(E, X, R, Z, append_in_aaa(E, Y, R))
append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z))
U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) → fl_out_aag(.(E, X), R, s(Z))

The argument filtering Pi contains the following mapping:
fl_in_aag(x1, x2, x3) = fl_in_aag(x3)
0 = 0
fl_out_aag(x1, x2, x3) = fl_out_aag
s(x1) = s(x1)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(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:

FL_IN_AAG(.(E, X), R, s(Z)) → U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R))
FL_IN_AAG(.(E, X), R, s(Z)) → APPEND_IN_AAA(E, Y, R)
APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U3_AAA(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN_AAA(Xs, Ys, Zs)
U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_AAG(E, X, R, Z, fl_in_aag(X, Y, Z))
U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) → FL_IN_AAG(X, Y, Z)

The TRS R consists of the following rules:

fl_in_aag([], [], 0) → fl_out_aag([], [], 0)
fl_in_aag(.(E, X), R, s(Z)) → U1_aag(E, X, R, Z, append_in_aaa(E, Y, R))
append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z))
U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) → fl_out_aag(.(E, X), R, s(Z))

The argument filtering Pi contains the following mapping:
fl_in_aag(x1, x2, x3) = fl_in_aag(x3)
0 = 0
fl_out_aag(x1, x2, x3) = fl_out_aag
s(x1) = s(x1)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5)
U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x5)
APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA
FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3)
U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(x5)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

FL_IN_AAG(.(E, X), R, s(Z)) → U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R))
FL_IN_AAG(.(E, X), R, s(Z)) → APPEND_IN_AAA(E, Y, R)
APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U3_AAA(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN_AAA(Xs, Ys, Zs)
U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_AAG(E, X, R, Z, fl_in_aag(X, Y, Z))
U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) → FL_IN_AAG(X, Y, Z)

The TRS R consists of the following rules:

fl_in_aag([], [], 0) → fl_out_aag([], [], 0)
fl_in_aag(.(E, X), R, s(Z)) → U1_aag(E, X, R, Z, append_in_aaa(E, Y, R))
append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z))
U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) → fl_out_aag(.(E, X), R, s(Z))

The argument filtering Pi contains the following mapping:
fl_in_aag(x1, x2, x3) = fl_in_aag(x3)
0 = 0
fl_out_aag(x1, x2, x3) = fl_out_aag
s(x1) = s(x1)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5)
U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x5)
APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA
FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3)
U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(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

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

fl_in_aag([], [], 0) → fl_out_aag([], [], 0)
fl_in_aag(.(E, X), R, s(Z)) → U1_aag(E, X, R, Z, append_in_aaa(E, Y, R))
append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z))
U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) → fl_out_aag(.(E, X), R, s(Z))

The argument filtering Pi contains the following mapping:
fl_in_aag(x1, x2, x3) = fl_in_aag(x3)
0 = 0
fl_out_aag(x1, x2, x3) = fl_out_aag
s(x1) = s(x1)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5)
APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA

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

  ↳ PrologToPiTRSProof

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

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

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

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

                        ↳ NonTerminationProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APPEND_IN_AAA → APPEND_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

APPEND_IN_AAA → APPEND_IN_AAA

The TRS R consists of the following rules:none

s = APPEND_IN_AAA evaluates to t =APPEND_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND_IN_AAA to APPEND_IN_AAA.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

FL_IN_AAG(.(E, X), R, s(Z)) → U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R))
U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) → FL_IN_AAG(X, Y, Z)

The TRS R consists of the following rules:

fl_in_aag([], [], 0) → fl_out_aag([], [], 0)
fl_in_aag(.(E, X), R, s(Z)) → U1_aag(E, X, R, Z, append_in_aaa(E, Y, R))
append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z))
U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) → fl_out_aag(.(E, X), R, s(Z))

The argument filtering Pi contains the following mapping:
fl_in_aag(x1, x2, x3) = fl_in_aag(x3)
0 = 0
fl_out_aag(x1, x2, x3) = fl_out_aag
s(x1) = s(x1)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x5)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5)
FL_IN_AAG(x1, x2, x3) = FL_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

  ↳ PrologToPiTRSProof

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

FL_IN_AAG(.(E, X), R, s(Z)) → U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R))
U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) → FL_IN_AAG(X, Y, Z)

The TRS R consists of the following rules:

append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5)
FL_IN_AAG(x1, x2, x3) = FL_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

  ↳ PrologToPiTRSProof

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

U1_AAG(Z, append_out_aaa) → FL_IN_AAG(Z)
FL_IN_AAG(s(Z)) → U1_AAG(Z, append_in_aaa)

The TRS R consists of the following rules:

append_in_aaa → append_out_aaa
append_in_aaa → U3_aaa(append_in_aaa)
U3_aaa(append_out_aaa) → append_out_aaa

The set Q consists of the following terms:

append_in_aaa
U3_aaa(x0)

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:

U1_AAG(Z, append_out_aaa) → FL_IN_AAG(Z)
The graph contains the following edges 1 >= 1
FL_IN_AAG(s(Z)) → U1_AAG(Z, append_in_aaa)
The graph contains the following edges 1 > 1

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

fl_in_aag([], [], 0) → fl_out_aag([], [], 0)
fl_in_aag(.(E, X), R, s(Z)) → U1_aag(E, X, R, Z, append_in_aaa(E, Y, R))
append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z))
U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) → fl_out_aag(.(E, X), R, s(Z))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

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

fl_in_aag([], [], 0) → fl_out_aag([], [], 0)
fl_in_aag(.(E, X), R, s(Z)) → U1_aag(E, X, R, Z, append_in_aaa(E, Y, R))
append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z))
U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) → fl_out_aag(.(E, X), R, s(Z))

The argument filtering Pi contains the following mapping:
fl_in_aag(x1, x2, x3) = fl_in_aag(x3)
0 = 0
fl_out_aag(x1, x2, x3) = fl_out_aag(x3)
s(x1) = s(x1)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, 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:

FL_IN_AAG(.(E, X), R, s(Z)) → U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R))
FL_IN_AAG(.(E, X), R, s(Z)) → APPEND_IN_AAA(E, Y, R)
APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U3_AAA(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN_AAA(Xs, Ys, Zs)
U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_AAG(E, X, R, Z, fl_in_aag(X, Y, Z))
U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) → FL_IN_AAG(X, Y, Z)

The TRS R consists of the following rules:

fl_in_aag([], [], 0) → fl_out_aag([], [], 0)
fl_in_aag(.(E, X), R, s(Z)) → U1_aag(E, X, R, Z, append_in_aaa(E, Y, R))
append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z))
U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) → fl_out_aag(.(E, X), R, s(Z))

The argument filtering Pi contains the following mapping:
fl_in_aag(x1, x2, x3) = fl_in_aag(x3)
0 = 0
fl_out_aag(x1, x2, x3) = fl_out_aag(x3)
s(x1) = s(x1)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5)
U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x4, x5)
APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA
FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3)
U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(x5)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

FL_IN_AAG(.(E, X), R, s(Z)) → U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R))
FL_IN_AAG(.(E, X), R, s(Z)) → APPEND_IN_AAA(E, Y, R)
APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U3_AAA(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
APPEND_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN_AAA(Xs, Ys, Zs)
U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_AAG(E, X, R, Z, fl_in_aag(X, Y, Z))
U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) → FL_IN_AAG(X, Y, Z)

The TRS R consists of the following rules:

fl_in_aag([], [], 0) → fl_out_aag([], [], 0)
fl_in_aag(.(E, X), R, s(Z)) → U1_aag(E, X, R, Z, append_in_aaa(E, Y, R))
append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z))
U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) → fl_out_aag(.(E, X), R, s(Z))

The argument filtering Pi contains the following mapping:
fl_in_aag(x1, x2, x3) = fl_in_aag(x3)
0 = 0
fl_out_aag(x1, x2, x3) = fl_out_aag(x3)
s(x1) = s(x1)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5)
U2_AAG(x1, x2, x3, x4, x5) = U2_AAG(x4, x5)
APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA
FL_IN_AAG(x1, x2, x3) = FL_IN_AAG(x3)
U3_AAA(x1, x2, x3, x4, x5) = U3_AAA(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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

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

The TRS R consists of the following rules:

fl_in_aag([], [], 0) → fl_out_aag([], [], 0)
fl_in_aag(.(E, X), R, s(Z)) → U1_aag(E, X, R, Z, append_in_aaa(E, Y, R))
append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z))
U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) → fl_out_aag(.(E, X), R, s(Z))

The argument filtering Pi contains the following mapping:
fl_in_aag(x1, x2, x3) = fl_in_aag(x3)
0 = 0
fl_out_aag(x1, x2, x3) = fl_out_aag(x3)
s(x1) = s(x1)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5)
APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA

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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

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

APPEND_IN_AAA → APPEND_IN_AAA

The TRS R consists of the following rules:none

s = APPEND_IN_AAA evaluates to t =APPEND_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND_IN_AAA to APPEND_IN_AAA.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

FL_IN_AAG(.(E, X), R, s(Z)) → U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R))
U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) → FL_IN_AAG(X, Y, Z)

The TRS R consists of the following rules:

fl_in_aag([], [], 0) → fl_out_aag([], [], 0)
fl_in_aag(.(E, X), R, s(Z)) → U1_aag(E, X, R, Z, append_in_aaa(E, Y, R))
append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))
U1_aag(E, X, R, Z, append_out_aaa(E, Y, R)) → U2_aag(E, X, R, Z, fl_in_aag(X, Y, Z))
U2_aag(E, X, R, Z, fl_out_aag(X, Y, Z)) → fl_out_aag(.(E, X), R, s(Z))

The argument filtering Pi contains the following mapping:
fl_in_aag(x1, x2, x3) = fl_in_aag(x3)
0 = 0
fl_out_aag(x1, x2, x3) = fl_out_aag(x3)
s(x1) = s(x1)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5)
append_in_aaa(x1, x2, x3) = append_in_aaa
append_out_aaa(x1, x2, x3) = append_out_aaa
U3_aaa(x1, x2, x3, x4, x5) = U3_aaa(x5)
U2_aag(x1, x2, x3, x4, x5) = U2_aag(x4, x5)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5)
FL_IN_AAG(x1, x2, x3) = FL_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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

FL_IN_AAG(.(E, X), R, s(Z)) → U1_AAG(E, X, R, Z, append_in_aaa(E, Y, R))
U1_AAG(E, X, R, Z, append_out_aaa(E, Y, R)) → FL_IN_AAG(X, Y, Z)

The TRS R consists of the following rules:

append_in_aaa([], X, X) → append_out_aaa([], X, X)
append_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U3_aaa(X, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
U3_aaa(X, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) → append_out_aaa(.(X, Xs), Ys, .(X, Zs))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

U1_AAG(Z, append_out_aaa) → FL_IN_AAG(Z)
FL_IN_AAG(s(Z)) → U1_AAG(Z, append_in_aaa)

The TRS R consists of the following rules:

append_in_aaa → append_out_aaa
append_in_aaa → U3_aaa(append_in_aaa)
U3_aaa(append_out_aaa) → append_out_aaa

The set Q consists of the following terms:

append_in_aaa
U3_aaa(x0)

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:

U1_AAG(Z, append_out_aaa) → FL_IN_AAG(Z)
The graph contains the following edges 1 >= 1
FL_IN_AAG(s(Z)) → U1_AAG(Z, append_in_aaa)
The graph contains the following edges 1 > 1