Left Termination proof of /tmp/tmpkera8y/NJ6.pl

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

↳ PROLOG

  ↳ PrologToPiTRSProof

f₃(A, {}₀, RES) :- g₃(A, {}₀, RES).
f₃(.₂(A, As), .₂(B, Bs), RES) :- f₃(.₂(B, .₂(A, As)), Bs, RES).
g₃({}₀, RES, RES).
g₃(.₂(C, Cs), D, RES) :- g₃(Cs, .₂(C, D), RES).

With regard to the inferred argument filtering the predicates were used in the following modes:
f₃: (b,b,f)
g₃: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

f_3_in_gga₃(A, []_0, RES) -> if_f_3_in_1_gga₃(A, RES, g_3_in_gga₃(A, []_0, RES))
g_3_in_gga₃([]_0, RES, RES) -> g_3_out_gga₃([]_0, RES, RES)
g_3_in_gga₃(._2₂(C, Cs), D, RES) -> if_g_3_in_1_gga₅(C, Cs, D, RES, g_3_in_gga₃(Cs, ._2₂(C, D), RES))
if_g_3_in_1_gga₅(C, Cs, D, RES, g_3_out_gga₃(Cs, ._2₂(C, D), RES)) -> g_3_out_gga₃(._2₂(C, Cs), D, RES)
if_f_3_in_1_gga₃(A, RES, g_3_out_gga₃(A, []_0, RES)) -> f_3_out_gga₃(A, []_0, RES)
f_3_in_gga₃(._2₂(A, As), ._2₂(B, Bs), RES) -> if_f_3_in_2_gga₆(A, As, B, Bs, RES, f_3_in_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES))
if_f_3_in_2_gga₆(A, As, B, Bs, RES, f_3_out_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES)) -> f_3_out_gga₃(._2₂(A, As), ._2₂(B, Bs), RES)

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:

f_3_in_gga₃(A, []_0, RES) -> if_f_3_in_1_gga₃(A, RES, g_3_in_gga₃(A, []_0, RES))
g_3_in_gga₃([]_0, RES, RES) -> g_3_out_gga₃([]_0, RES, RES)
g_3_in_gga₃(._2₂(C, Cs), D, RES) -> if_g_3_in_1_gga₅(C, Cs, D, RES, g_3_in_gga₃(Cs, ._2₂(C, D), RES))
if_g_3_in_1_gga₅(C, Cs, D, RES, g_3_out_gga₃(Cs, ._2₂(C, D), RES)) -> g_3_out_gga₃(._2₂(C, Cs), D, RES)
if_f_3_in_1_gga₃(A, RES, g_3_out_gga₃(A, []_0, RES)) -> f_3_out_gga₃(A, []_0, RES)
f_3_in_gga₃(._2₂(A, As), ._2₂(B, Bs), RES) -> if_f_3_in_2_gga₆(A, As, B, Bs, RES, f_3_in_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES))
if_f_3_in_2_gga₆(A, As, B, Bs, RES, f_3_out_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES)) -> f_3_out_gga₃(._2₂(A, As), ._2₂(B, Bs), RES)

F_3_IN_GGA₃(A, []_0, RES) -> IF_F_3_IN_1_GGA₃(A, RES, g_3_in_gga₃(A, []_0, RES))
F_3_IN_GGA₃(A, []_0, RES) -> G_3_IN_GGA₃(A, []_0, RES)
G_3_IN_GGA₃(._2₂(C, Cs), D, RES) -> IF_G_3_IN_1_GGA₅(C, Cs, D, RES, g_3_in_gga₃(Cs, ._2₂(C, D), RES))
G_3_IN_GGA₃(._2₂(C, Cs), D, RES) -> G_3_IN_GGA₃(Cs, ._2₂(C, D), RES)
F_3_IN_GGA₃(._2₂(A, As), ._2₂(B, Bs), RES) -> IF_F_3_IN_2_GGA₆(A, As, B, Bs, RES, f_3_in_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES))
F_3_IN_GGA₃(._2₂(A, As), ._2₂(B, Bs), RES) -> F_3_IN_GGA₃(._2₂(B, ._2₂(A, As)), Bs, RES)

The TRS R consists of the following rules:

f_3_in_gga₃(A, []_0, RES) -> if_f_3_in_1_gga₃(A, RES, g_3_in_gga₃(A, []_0, RES))
g_3_in_gga₃([]_0, RES, RES) -> g_3_out_gga₃([]_0, RES, RES)
g_3_in_gga₃(._2₂(C, Cs), D, RES) -> if_g_3_in_1_gga₅(C, Cs, D, RES, g_3_in_gga₃(Cs, ._2₂(C, D), RES))
if_g_3_in_1_gga₅(C, Cs, D, RES, g_3_out_gga₃(Cs, ._2₂(C, D), RES)) -> g_3_out_gga₃(._2₂(C, Cs), D, RES)
if_f_3_in_1_gga₃(A, RES, g_3_out_gga₃(A, []_0, RES)) -> f_3_out_gga₃(A, []_0, RES)
f_3_in_gga₃(._2₂(A, As), ._2₂(B, Bs), RES) -> if_f_3_in_2_gga₆(A, As, B, Bs, RES, f_3_in_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES))
if_f_3_in_2_gga₆(A, As, B, Bs, RES, f_3_out_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES)) -> f_3_out_gga₃(._2₂(A, As), ._2₂(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_3_in_gga₃(x1, x2, x3) = f_3_in_gga₂(x1, x2)
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
if_f_3_in_1_gga₃(x1, x2, x3) = if_f_3_in_1_gga₁(x3)
g_3_in_gga₃(x1, x2, x3) = g_3_in_gga₂(x1, x2)
g_3_out_gga₃(x1, x2, x3) = g_3_out_gga₁(x3)
if_g_3_in_1_gga₅(x1, x2, x3, x4, x5) = if_g_3_in_1_gga₁(x5)
f_3_out_gga₃(x1, x2, x3) = f_3_out_gga₁(x3)
if_f_3_in_2_gga₆(x1, x2, x3, x4, x5, x6) = if_f_3_in_2_gga₁(x6)
IF_G_3_IN_1_GGA₅(x1, x2, x3, x4, x5) = IF_G_3_IN_1_GGA₁(x5)
G_3_IN_GGA₃(x1, x2, x3) = G_3_IN_GGA₂(x1, x2)
F_3_IN_GGA₃(x1, x2, x3) = F_3_IN_GGA₂(x1, x2)
IF_F_3_IN_2_GGA₆(x1, x2, x3, x4, x5, x6) = IF_F_3_IN_2_GGA₁(x6)
IF_F_3_IN_1_GGA₃(x1, x2, x3) = IF_F_3_IN_1_GGA₁(x3)

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:

F_3_IN_GGA₃(A, []_0, RES) -> IF_F_3_IN_1_GGA₃(A, RES, g_3_in_gga₃(A, []_0, RES))
F_3_IN_GGA₃(A, []_0, RES) -> G_3_IN_GGA₃(A, []_0, RES)
G_3_IN_GGA₃(._2₂(C, Cs), D, RES) -> IF_G_3_IN_1_GGA₅(C, Cs, D, RES, g_3_in_gga₃(Cs, ._2₂(C, D), RES))
G_3_IN_GGA₃(._2₂(C, Cs), D, RES) -> G_3_IN_GGA₃(Cs, ._2₂(C, D), RES)
F_3_IN_GGA₃(._2₂(A, As), ._2₂(B, Bs), RES) -> IF_F_3_IN_2_GGA₆(A, As, B, Bs, RES, f_3_in_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES))
F_3_IN_GGA₃(._2₂(A, As), ._2₂(B, Bs), RES) -> F_3_IN_GGA₃(._2₂(B, ._2₂(A, As)), Bs, RES)

The TRS R consists of the following rules:

f_3_in_gga₃(A, []_0, RES) -> if_f_3_in_1_gga₃(A, RES, g_3_in_gga₃(A, []_0, RES))
g_3_in_gga₃([]_0, RES, RES) -> g_3_out_gga₃([]_0, RES, RES)
g_3_in_gga₃(._2₂(C, Cs), D, RES) -> if_g_3_in_1_gga₅(C, Cs, D, RES, g_3_in_gga₃(Cs, ._2₂(C, D), RES))
if_g_3_in_1_gga₅(C, Cs, D, RES, g_3_out_gga₃(Cs, ._2₂(C, D), RES)) -> g_3_out_gga₃(._2₂(C, Cs), D, RES)
if_f_3_in_1_gga₃(A, RES, g_3_out_gga₃(A, []_0, RES)) -> f_3_out_gga₃(A, []_0, RES)
f_3_in_gga₃(._2₂(A, As), ._2₂(B, Bs), RES) -> if_f_3_in_2_gga₆(A, As, B, Bs, RES, f_3_in_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES))
if_f_3_in_2_gga₆(A, As, B, Bs, RES, f_3_out_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES)) -> f_3_out_gga₃(._2₂(A, As), ._2₂(B, Bs), RES)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

G_3_IN_GGA₃(._2₂(C, Cs), D, RES) -> G_3_IN_GGA₃(Cs, ._2₂(C, D), RES)

The TRS R consists of the following rules:

f_3_in_gga₃(A, []_0, RES) -> if_f_3_in_1_gga₃(A, RES, g_3_in_gga₃(A, []_0, RES))
g_3_in_gga₃([]_0, RES, RES) -> g_3_out_gga₃([]_0, RES, RES)
g_3_in_gga₃(._2₂(C, Cs), D, RES) -> if_g_3_in_1_gga₅(C, Cs, D, RES, g_3_in_gga₃(Cs, ._2₂(C, D), RES))
if_g_3_in_1_gga₅(C, Cs, D, RES, g_3_out_gga₃(Cs, ._2₂(C, D), RES)) -> g_3_out_gga₃(._2₂(C, Cs), D, RES)
if_f_3_in_1_gga₃(A, RES, g_3_out_gga₃(A, []_0, RES)) -> f_3_out_gga₃(A, []_0, RES)
f_3_in_gga₃(._2₂(A, As), ._2₂(B, Bs), RES) -> if_f_3_in_2_gga₆(A, As, B, Bs, RES, f_3_in_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES))
if_f_3_in_2_gga₆(A, As, B, Bs, RES, f_3_out_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES)) -> f_3_out_gga₃(._2₂(A, As), ._2₂(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_3_in_gga₃(x1, x2, x3) = f_3_in_gga₂(x1, x2)
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
if_f_3_in_1_gga₃(x1, x2, x3) = if_f_3_in_1_gga₁(x3)
g_3_in_gga₃(x1, x2, x3) = g_3_in_gga₂(x1, x2)
g_3_out_gga₃(x1, x2, x3) = g_3_out_gga₁(x3)
if_g_3_in_1_gga₅(x1, x2, x3, x4, x5) = if_g_3_in_1_gga₁(x5)
f_3_out_gga₃(x1, x2, x3) = f_3_out_gga₁(x3)
if_f_3_in_2_gga₆(x1, x2, x3, x4, x5, x6) = if_f_3_in_2_gga₁(x6)
G_3_IN_GGA₃(x1, x2, x3) = G_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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

G_3_IN_GGA₃(._2₂(C, Cs), D, RES) -> G_3_IN_GGA₃(Cs, ._2₂(C, D), RES)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
G_3_IN_GGA₃(x1, x2, x3) = G_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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

G_3_IN_GGA₂(._2₂(C, Cs), D) -> G_3_IN_GGA₂(Cs, ._2₂(C, D))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {G_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:

G_3_IN_GGA₂(._2₂(C, Cs), D) -> G_3_IN_GGA₂(Cs, ._2₂(C, D))
The graph contains the following edges 1 > 1

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

F_3_IN_GGA₃(._2₂(A, As), ._2₂(B, Bs), RES) -> F_3_IN_GGA₃(._2₂(B, ._2₂(A, As)), Bs, RES)

The TRS R consists of the following rules:

f_3_in_gga₃(A, []_0, RES) -> if_f_3_in_1_gga₃(A, RES, g_3_in_gga₃(A, []_0, RES))
g_3_in_gga₃([]_0, RES, RES) -> g_3_out_gga₃([]_0, RES, RES)
g_3_in_gga₃(._2₂(C, Cs), D, RES) -> if_g_3_in_1_gga₅(C, Cs, D, RES, g_3_in_gga₃(Cs, ._2₂(C, D), RES))
if_g_3_in_1_gga₅(C, Cs, D, RES, g_3_out_gga₃(Cs, ._2₂(C, D), RES)) -> g_3_out_gga₃(._2₂(C, Cs), D, RES)
if_f_3_in_1_gga₃(A, RES, g_3_out_gga₃(A, []_0, RES)) -> f_3_out_gga₃(A, []_0, RES)
f_3_in_gga₃(._2₂(A, As), ._2₂(B, Bs), RES) -> if_f_3_in_2_gga₆(A, As, B, Bs, RES, f_3_in_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES))
if_f_3_in_2_gga₆(A, As, B, Bs, RES, f_3_out_gga₃(._2₂(B, ._2₂(A, As)), Bs, RES)) -> f_3_out_gga₃(._2₂(A, As), ._2₂(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_3_in_gga₃(x1, x2, x3) = f_3_in_gga₂(x1, x2)
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
if_f_3_in_1_gga₃(x1, x2, x3) = if_f_3_in_1_gga₁(x3)
g_3_in_gga₃(x1, x2, x3) = g_3_in_gga₂(x1, x2)
g_3_out_gga₃(x1, x2, x3) = g_3_out_gga₁(x3)
if_g_3_in_1_gga₅(x1, x2, x3, x4, x5) = if_g_3_in_1_gga₁(x5)
f_3_out_gga₃(x1, x2, x3) = f_3_out_gga₁(x3)
if_f_3_in_2_gga₆(x1, x2, x3, x4, x5, x6) = if_f_3_in_2_gga₁(x6)
F_3_IN_GGA₃(x1, x2, x3) = F_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:

F_3_IN_GGA₃(._2₂(A, As), ._2₂(B, Bs), RES) -> F_3_IN_GGA₃(._2₂(B, ._2₂(A, As)), Bs, RES)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
F_3_IN_GGA₃(x1, x2, x3) = F_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:

F_3_IN_GGA₂(._2₂(A, As), ._2₂(B, Bs)) -> F_3_IN_GGA₂(._2₂(B, ._2₂(A, As)), Bs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {F_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:

F_3_IN_GGA₂(._2₂(A, As), ._2₂(B, Bs)) -> F_3_IN_GGA₂(._2₂(B, ._2₂(A, As)), Bs)
The graph contains the following edges 2 > 2