Left Termination proof of /tmp/tmp65G

Left Termination of the query pattern fl(b,b,f) w.r.t. the given Prolog program could successfully be proven:

↳ PROLOG

  ↳ PrologToPiTRSProof

fl₃({}₀, {}₀, 0₀).
fl₃(.₂(E, X), R, s₁(Z)) :- append₃(E, Y, R), fl₃(X, Y, Z).
append₃({}₀, X, X).
append₃(.₂(X, Xs), Ys, .₂(X, Zs)) :- append₃(Xs, Ys, Zs).

With regard to the inferred argument filtering the predicates were used in the following modes:
fl₃: (b,b,f)
append₃: (b,f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

fl_3_in_gga₃([]_0, []_0, 0_0) -> fl_3_out_gga₃([]_0, []_0, 0_0)
fl_3_in_gga₃(._2₂(E, X), R, s_1₁(Z)) -> if_fl_3_in_1_gga₅(E, X, R, Z, append_3_in_gag₃(E, Y, R))
append_3_in_gag₃([]_0, X, X) -> append_3_out_gag₃([]_0, X, X)
append_3_in_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_in_gag₃(Xs, Ys, Zs))
if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_out_gag₃(Xs, Ys, Zs)) -> append_3_out_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_fl_3_in_1_gga₅(E, X, R, Z, append_3_out_gag₃(E, Y, R)) -> if_fl_3_in_2_gga₆(E, X, R, Z, Y, fl_3_in_gga₃(X, Y, Z))
if_fl_3_in_2_gga₆(E, X, R, Z, Y, fl_3_out_gga₃(X, Y, Z)) -> fl_3_out_gga₃(._2₂(E, X), R, s_1₁(Z))

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:

fl_3_in_gga₃([]_0, []_0, 0_0) -> fl_3_out_gga₃([]_0, []_0, 0_0)
fl_3_in_gga₃(._2₂(E, X), R, s_1₁(Z)) -> if_fl_3_in_1_gga₅(E, X, R, Z, append_3_in_gag₃(E, Y, R))
append_3_in_gag₃([]_0, X, X) -> append_3_out_gag₃([]_0, X, X)
append_3_in_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_in_gag₃(Xs, Ys, Zs))
if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_out_gag₃(Xs, Ys, Zs)) -> append_3_out_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_fl_3_in_1_gga₅(E, X, R, Z, append_3_out_gag₃(E, Y, R)) -> if_fl_3_in_2_gga₆(E, X, R, Z, Y, fl_3_in_gga₃(X, Y, Z))
if_fl_3_in_2_gga₆(E, X, R, Z, Y, fl_3_out_gga₃(X, Y, Z)) -> fl_3_out_gga₃(._2₂(E, X), R, s_1₁(Z))

FL_3_IN_GGA₃(._2₂(E, X), R, s_1₁(Z)) -> IF_FL_3_IN_1_GGA₅(E, X, R, Z, append_3_in_gag₃(E, Y, R))
FL_3_IN_GGA₃(._2₂(E, X), R, s_1₁(Z)) -> APPEND_3_IN_GAG₃(E, Y, R)
APPEND_3_IN_GAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APPEND_3_IN_1_GAG₅(X, Xs, Ys, Zs, append_3_in_gag₃(Xs, Ys, Zs))
APPEND_3_IN_GAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APPEND_3_IN_GAG₃(Xs, Ys, Zs)
IF_FL_3_IN_1_GGA₅(E, X, R, Z, append_3_out_gag₃(E, Y, R)) -> IF_FL_3_IN_2_GGA₆(E, X, R, Z, Y, fl_3_in_gga₃(X, Y, Z))
IF_FL_3_IN_1_GGA₅(E, X, R, Z, append_3_out_gag₃(E, Y, R)) -> FL_3_IN_GGA₃(X, Y, Z)

The TRS R consists of the following rules:

fl_3_in_gga₃([]_0, []_0, 0_0) -> fl_3_out_gga₃([]_0, []_0, 0_0)
fl_3_in_gga₃(._2₂(E, X), R, s_1₁(Z)) -> if_fl_3_in_1_gga₅(E, X, R, Z, append_3_in_gag₃(E, Y, R))
append_3_in_gag₃([]_0, X, X) -> append_3_out_gag₃([]_0, X, X)
append_3_in_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_in_gag₃(Xs, Ys, Zs))
if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_out_gag₃(Xs, Ys, Zs)) -> append_3_out_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_fl_3_in_1_gga₅(E, X, R, Z, append_3_out_gag₃(E, Y, R)) -> if_fl_3_in_2_gga₆(E, X, R, Z, Y, fl_3_in_gga₃(X, Y, Z))
if_fl_3_in_2_gga₆(E, X, R, Z, Y, fl_3_out_gga₃(X, Y, Z)) -> fl_3_out_gga₃(._2₂(E, X), R, s_1₁(Z))

The argument filtering Pi contains the following mapping:
fl_3_in_gga₃(x1, x2, x3) = fl_3_in_gga₂(x1, x2)
[]_0 = []_0
0_0 = 0_0
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
fl_3_out_gga₃(x1, x2, x3) = fl_3_out_gga₁(x3)
if_fl_3_in_1_gga₅(x1, x2, x3, x4, x5) = if_fl_3_in_1_gga₂(x2, x5)
append_3_in_gag₃(x1, x2, x3) = append_3_in_gag₂(x1, x3)
append_3_out_gag₃(x1, x2, x3) = append_3_out_gag₁(x2)
if_append_3_in_1_gag₅(x1, x2, x3, x4, x5) = if_append_3_in_1_gag₁(x5)
if_fl_3_in_2_gga₆(x1, x2, x3, x4, x5, x6) = if_fl_3_in_2_gga₁(x6)
IF_APPEND_3_IN_1_GAG₅(x1, x2, x3, x4, x5) = IF_APPEND_3_IN_1_GAG₁(x5)
APPEND_3_IN_GAG₃(x1, x2, x3) = APPEND_3_IN_GAG₂(x1, x3)
IF_FL_3_IN_2_GGA₆(x1, x2, x3, x4, x5, x6) = IF_FL_3_IN_2_GGA₁(x6)
IF_FL_3_IN_1_GGA₅(x1, x2, x3, x4, x5) = IF_FL_3_IN_1_GGA₂(x2, x5)
FL_3_IN_GGA₃(x1, x2, x3) = FL_3_IN_GGA₂(x1, x2)

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:

FL_3_IN_GGA₃(._2₂(E, X), R, s_1₁(Z)) -> IF_FL_3_IN_1_GGA₅(E, X, R, Z, append_3_in_gag₃(E, Y, R))
FL_3_IN_GGA₃(._2₂(E, X), R, s_1₁(Z)) -> APPEND_3_IN_GAG₃(E, Y, R)
APPEND_3_IN_GAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APPEND_3_IN_1_GAG₅(X, Xs, Ys, Zs, append_3_in_gag₃(Xs, Ys, Zs))
APPEND_3_IN_GAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APPEND_3_IN_GAG₃(Xs, Ys, Zs)
IF_FL_3_IN_1_GGA₅(E, X, R, Z, append_3_out_gag₃(E, Y, R)) -> IF_FL_3_IN_2_GGA₆(E, X, R, Z, Y, fl_3_in_gga₃(X, Y, Z))
IF_FL_3_IN_1_GGA₅(E, X, R, Z, append_3_out_gag₃(E, Y, R)) -> FL_3_IN_GGA₃(X, Y, Z)

The TRS R consists of the following rules:

fl_3_in_gga₃([]_0, []_0, 0_0) -> fl_3_out_gga₃([]_0, []_0, 0_0)
fl_3_in_gga₃(._2₂(E, X), R, s_1₁(Z)) -> if_fl_3_in_1_gga₅(E, X, R, Z, append_3_in_gag₃(E, Y, R))
append_3_in_gag₃([]_0, X, X) -> append_3_out_gag₃([]_0, X, X)
append_3_in_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_in_gag₃(Xs, Ys, Zs))
if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_out_gag₃(Xs, Ys, Zs)) -> append_3_out_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_fl_3_in_1_gga₅(E, X, R, Z, append_3_out_gag₃(E, Y, R)) -> if_fl_3_in_2_gga₆(E, X, R, Z, Y, fl_3_in_gga₃(X, Y, Z))
if_fl_3_in_2_gga₆(E, X, R, Z, Y, fl_3_out_gga₃(X, Y, Z)) -> fl_3_out_gga₃(._2₂(E, X), R, s_1₁(Z))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

APPEND_3_IN_GAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APPEND_3_IN_GAG₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

fl_3_in_gga₃([]_0, []_0, 0_0) -> fl_3_out_gga₃([]_0, []_0, 0_0)
fl_3_in_gga₃(._2₂(E, X), R, s_1₁(Z)) -> if_fl_3_in_1_gga₅(E, X, R, Z, append_3_in_gag₃(E, Y, R))
append_3_in_gag₃([]_0, X, X) -> append_3_out_gag₃([]_0, X, X)
append_3_in_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_in_gag₃(Xs, Ys, Zs))
if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_out_gag₃(Xs, Ys, Zs)) -> append_3_out_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_fl_3_in_1_gga₅(E, X, R, Z, append_3_out_gag₃(E, Y, R)) -> if_fl_3_in_2_gga₆(E, X, R, Z, Y, fl_3_in_gga₃(X, Y, Z))
if_fl_3_in_2_gga₆(E, X, R, Z, Y, fl_3_out_gga₃(X, Y, Z)) -> fl_3_out_gga₃(._2₂(E, X), R, s_1₁(Z))

The argument filtering Pi contains the following mapping:
fl_3_in_gga₃(x1, x2, x3) = fl_3_in_gga₂(x1, x2)
[]_0 = []_0
0_0 = 0_0
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
fl_3_out_gga₃(x1, x2, x3) = fl_3_out_gga₁(x3)
if_fl_3_in_1_gga₅(x1, x2, x3, x4, x5) = if_fl_3_in_1_gga₂(x2, x5)
append_3_in_gag₃(x1, x2, x3) = append_3_in_gag₂(x1, x3)
append_3_out_gag₃(x1, x2, x3) = append_3_out_gag₁(x2)
if_append_3_in_1_gag₅(x1, x2, x3, x4, x5) = if_append_3_in_1_gag₁(x5)
if_fl_3_in_2_gga₆(x1, x2, x3, x4, x5, x6) = if_fl_3_in_2_gga₁(x6)
APPEND_3_IN_GAG₃(x1, x2, x3) = APPEND_3_IN_GAG₂(x1, x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting 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:

APPEND_3_IN_GAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APPEND_3_IN_GAG₃(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
APPEND_3_IN_GAG₃(x1, x2, x3) = APPEND_3_IN_GAG₂(x1, x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem 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:

APPEND_3_IN_GAG₂(._2₂(X, Xs), ._2₂(X, Zs)) -> APPEND_3_IN_GAG₂(Xs, Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {APPEND_3_IN_GAG₂}.
By using the subterm criterion together with the size-change analysis 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:

APPEND_3_IN_GAG₂(._2₂(X, Xs), ._2₂(X, Zs)) -> APPEND_3_IN_GAG₂(Xs, Zs)
The graph contains the following edges 1 > 1, 2 > 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

FL_3_IN_GGA₃(._2₂(E, X), R, s_1₁(Z)) -> IF_FL_3_IN_1_GGA₅(E, X, R, Z, append_3_in_gag₃(E, Y, R))
IF_FL_3_IN_1_GGA₅(E, X, R, Z, append_3_out_gag₃(E, Y, R)) -> FL_3_IN_GGA₃(X, Y, Z)

The TRS R consists of the following rules:

fl_3_in_gga₃([]_0, []_0, 0_0) -> fl_3_out_gga₃([]_0, []_0, 0_0)
fl_3_in_gga₃(._2₂(E, X), R, s_1₁(Z)) -> if_fl_3_in_1_gga₅(E, X, R, Z, append_3_in_gag₃(E, Y, R))
append_3_in_gag₃([]_0, X, X) -> append_3_out_gag₃([]_0, X, X)
append_3_in_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_in_gag₃(Xs, Ys, Zs))
if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_out_gag₃(Xs, Ys, Zs)) -> append_3_out_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_fl_3_in_1_gga₅(E, X, R, Z, append_3_out_gag₃(E, Y, R)) -> if_fl_3_in_2_gga₆(E, X, R, Z, Y, fl_3_in_gga₃(X, Y, Z))
if_fl_3_in_2_gga₆(E, X, R, Z, Y, fl_3_out_gga₃(X, Y, Z)) -> fl_3_out_gga₃(._2₂(E, X), R, s_1₁(Z))

The argument filtering Pi contains the following mapping:
fl_3_in_gga₃(x1, x2, x3) = fl_3_in_gga₂(x1, x2)
[]_0 = []_0
0_0 = 0_0
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
fl_3_out_gga₃(x1, x2, x3) = fl_3_out_gga₁(x3)
if_fl_3_in_1_gga₅(x1, x2, x3, x4, x5) = if_fl_3_in_1_gga₂(x2, x5)
append_3_in_gag₃(x1, x2, x3) = append_3_in_gag₂(x1, x3)
append_3_out_gag₃(x1, x2, x3) = append_3_out_gag₁(x2)
if_append_3_in_1_gag₅(x1, x2, x3, x4, x5) = if_append_3_in_1_gag₁(x5)
if_fl_3_in_2_gga₆(x1, x2, x3, x4, x5, x6) = if_fl_3_in_2_gga₁(x6)
IF_FL_3_IN_1_GGA₅(x1, x2, x3, x4, x5) = IF_FL_3_IN_1_GGA₂(x2, x5)
FL_3_IN_GGA₃(x1, x2, x3) = FL_3_IN_GGA₂(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting 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:

FL_3_IN_GGA₃(._2₂(E, X), R, s_1₁(Z)) -> IF_FL_3_IN_1_GGA₅(E, X, R, Z, append_3_in_gag₃(E, Y, R))
IF_FL_3_IN_1_GGA₅(E, X, R, Z, append_3_out_gag₃(E, Y, R)) -> FL_3_IN_GGA₃(X, Y, Z)

The TRS R consists of the following rules:

append_3_in_gag₃([]_0, X, X) -> append_3_out_gag₃([]_0, X, X)
append_3_in_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_in_gag₃(Xs, Ys, Zs))
if_append_3_in_1_gag₅(X, Xs, Ys, Zs, append_3_out_gag₃(Xs, Ys, Zs)) -> append_3_out_gag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))

The argument filtering Pi contains the following mapping:
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
append_3_in_gag₃(x1, x2, x3) = append_3_in_gag₂(x1, x3)
append_3_out_gag₃(x1, x2, x3) = append_3_out_gag₁(x2)
if_append_3_in_1_gag₅(x1, x2, x3, x4, x5) = if_append_3_in_1_gag₁(x5)
IF_FL_3_IN_1_GGA₅(x1, x2, x3, x4, x5) = IF_FL_3_IN_1_GGA₂(x2, x5)
FL_3_IN_GGA₃(x1, x2, x3) = FL_3_IN_GGA₂(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

FL_3_IN_GGA₂(._2₂(E, X), R) -> IF_FL_3_IN_1_GGA₂(X, append_3_in_gag₂(E, R))
IF_FL_3_IN_1_GGA₂(X, append_3_out_gag₁(Y)) -> FL_3_IN_GGA₂(X, Y)

The TRS R consists of the following rules:

append_3_in_gag₂([]_0, X) -> append_3_out_gag₁(X)
append_3_in_gag₂(._2₂(X, Xs), ._2₂(X, Zs)) -> if_append_3_in_1_gag₁(append_3_in_gag₂(Xs, Zs))
if_append_3_in_1_gag₁(append_3_out_gag₁(Ys)) -> append_3_out_gag₁(Ys)

The set Q consists of the following terms:

append_3_in_gag₂(x0, x1)
if_append_3_in_1_gag₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_FL_3_IN_1_GGA₂, FL_3_IN_GGA₂}.
By using the subterm criterion together with the size-change analysis 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:

IF_FL_3_IN_1_GGA₂(X, append_3_out_gag₁(Y)) -> FL_3_IN_GGA₂(X, Y)
The graph contains the following edges 1 >= 1, 2 > 2
FL_3_IN_GGA₂(._2₂(E, X), R) -> IF_FL_3_IN_1_GGA₂(X, append_3_in_gag₂(E, R))
The graph contains the following edges 1 > 1