Left Termination proof of /tmp/tmpa307Ry/NJ6.pl

Left Termination of the query pattern f_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

f(A, [], RES) :- g(A, [], RES).
f(.(A, As), .(B, Bs), RES) :- f(.(B, .(A, As)), Bs, RES).
g([], RES, RES).
g(.(C, Cs), D, RES) :- g(Cs, .(C, D), RES).

Queries:

f(g,g,a).

We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

f_in(.(A, As), .(B, Bs), RES) → U2(A, As, B, Bs, RES, f_in(.(B, .(A, As)), Bs, RES))
f_in(A, [], RES) → U1(A, RES, g_in(A, [], RES))
g_in(.(C, Cs), D, RES) → U3(C, Cs, D, RES, g_in(Cs, .(C, D), RES))
g_in([], RES, RES) → g_out([], RES, RES)
U3(C, Cs, D, RES, g_out(Cs, .(C, D), RES)) → g_out(.(C, Cs), D, RES)
U1(A, RES, g_out(A, [], RES)) → f_out(A, [], RES)
U2(A, As, B, Bs, RES, f_out(.(B, .(A, As)), Bs, RES)) → f_out(.(A, As), .(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_in(.(A, As), .(B, Bs), RES) → U2(A, As, B, Bs, RES, f_in(.(B, .(A, As)), Bs, RES))
f_in(A, [], RES) → U1(A, RES, g_in(A, [], RES))
g_in(.(C, Cs), D, RES) → U3(C, Cs, D, RES, g_in(Cs, .(C, D), RES))
g_in([], RES, RES) → g_out([], RES, RES)
U3(C, Cs, D, RES, g_out(Cs, .(C, D), RES)) → g_out(.(C, Cs), D, RES)
U1(A, RES, g_out(A, [], RES)) → f_out(A, [], RES)
U2(A, As, B, Bs, RES, f_out(.(B, .(A, As)), Bs, RES)) → f_out(.(A, As), .(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_in(x1, x2, x3) = f_in(x1, x2)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5, x6) = U2(x6)
[] = []
U1(x1, x2, x3) = U1(x3)
g_in(x1, x2, x3) = g_in(x1, x2)
U3(x1, x2, x3, x4, x5) = U3(x5)
g_out(x1, x2, x3) = g_out(x3)
f_out(x1, x2, x3) = f_out(x3)

Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

F_IN(.(A, As), .(B, Bs), RES) → U2¹(A, As, B, Bs, RES, f_in(.(B, .(A, As)), Bs, RES))
F_IN(.(A, As), .(B, Bs), RES) → F_IN(.(B, .(A, As)), Bs, RES)
F_IN(A, [], RES) → U1¹(A, RES, g_in(A, [], RES))
F_IN(A, [], RES) → G_IN(A, [], RES)
G_IN(.(C, Cs), D, RES) → U3¹(C, Cs, D, RES, g_in(Cs, .(C, D), RES))
G_IN(.(C, Cs), D, RES) → G_IN(Cs, .(C, D), RES)

The TRS R consists of the following rules:

f_in(.(A, As), .(B, Bs), RES) → U2(A, As, B, Bs, RES, f_in(.(B, .(A, As)), Bs, RES))
f_in(A, [], RES) → U1(A, RES, g_in(A, [], RES))
g_in(.(C, Cs), D, RES) → U3(C, Cs, D, RES, g_in(Cs, .(C, D), RES))
g_in([], RES, RES) → g_out([], RES, RES)
U3(C, Cs, D, RES, g_out(Cs, .(C, D), RES)) → g_out(.(C, Cs), D, RES)
U1(A, RES, g_out(A, [], RES)) → f_out(A, [], RES)
U2(A, As, B, Bs, RES, f_out(.(B, .(A, As)), Bs, RES)) → f_out(.(A, As), .(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_in(x1, x2, x3) = f_in(x1, x2)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5, x6) = U2(x6)
[] = []
U1(x1, x2, x3) = U1(x3)
g_in(x1, x2, x3) = g_in(x1, x2)
U3(x1, x2, x3, x4, x5) = U3(x5)
g_out(x1, x2, x3) = g_out(x3)
f_out(x1, x2, x3) = f_out(x3)
G_IN(x1, x2, x3) = G_IN(x1, x2)
F_IN(x1, x2, x3) = F_IN(x1, x2)
U3¹(x1, x2, x3, x4, x5) = U3¹(x5)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x6)
U1¹(x1, x2, x3) = U1¹(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_IN(.(A, As), .(B, Bs), RES) → U2¹(A, As, B, Bs, RES, f_in(.(B, .(A, As)), Bs, RES))
F_IN(.(A, As), .(B, Bs), RES) → F_IN(.(B, .(A, As)), Bs, RES)
F_IN(A, [], RES) → U1¹(A, RES, g_in(A, [], RES))
F_IN(A, [], RES) → G_IN(A, [], RES)
G_IN(.(C, Cs), D, RES) → U3¹(C, Cs, D, RES, g_in(Cs, .(C, D), RES))
G_IN(.(C, Cs), D, RES) → G_IN(Cs, .(C, D), RES)

The TRS R consists of the following rules:

f_in(.(A, As), .(B, Bs), RES) → U2(A, As, B, Bs, RES, f_in(.(B, .(A, As)), Bs, RES))
f_in(A, [], RES) → U1(A, RES, g_in(A, [], RES))
g_in(.(C, Cs), D, RES) → U3(C, Cs, D, RES, g_in(Cs, .(C, D), RES))
g_in([], RES, RES) → g_out([], RES, RES)
U3(C, Cs, D, RES, g_out(Cs, .(C, D), RES)) → g_out(.(C, Cs), D, RES)
U1(A, RES, g_out(A, [], RES)) → f_out(A, [], RES)
U2(A, As, B, Bs, RES, f_out(.(B, .(A, As)), Bs, RES)) → f_out(.(A, As), .(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_in(x1, x2, x3) = f_in(x1, x2)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5, x6) = U2(x6)
[] = []
U1(x1, x2, x3) = U1(x3)
g_in(x1, x2, x3) = g_in(x1, x2)
U3(x1, x2, x3, x4, x5) = U3(x5)
g_out(x1, x2, x3) = g_out(x3)
f_out(x1, x2, x3) = f_out(x3)
G_IN(x1, x2, x3) = G_IN(x1, x2)
F_IN(x1, x2, x3) = F_IN(x1, x2)
U3¹(x1, x2, x3, x4, x5) = U3¹(x5)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x6)
U1¹(x1, x2, x3) = U1¹(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 4 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

G_IN(.(C, Cs), D, RES) → G_IN(Cs, .(C, D), RES)

The TRS R consists of the following rules:

f_in(.(A, As), .(B, Bs), RES) → U2(A, As, B, Bs, RES, f_in(.(B, .(A, As)), Bs, RES))
f_in(A, [], RES) → U1(A, RES, g_in(A, [], RES))
g_in(.(C, Cs), D, RES) → U3(C, Cs, D, RES, g_in(Cs, .(C, D), RES))
g_in([], RES, RES) → g_out([], RES, RES)
U3(C, Cs, D, RES, g_out(Cs, .(C, D), RES)) → g_out(.(C, Cs), D, RES)
U1(A, RES, g_out(A, [], RES)) → f_out(A, [], RES)
U2(A, As, B, Bs, RES, f_out(.(B, .(A, As)), Bs, RES)) → f_out(.(A, As), .(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_in(x1, x2, x3) = f_in(x1, x2)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5, x6) = U2(x6)
[] = []
U1(x1, x2, x3) = U1(x3)
g_in(x1, x2, x3) = g_in(x1, x2)
U3(x1, x2, x3, x4, x5) = U3(x5)
g_out(x1, x2, x3) = g_out(x3)
f_out(x1, x2, x3) = f_out(x3)
G_IN(x1, x2, x3) = G_IN(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

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

G_IN(.(C, Cs), D, RES) → G_IN(Cs, .(C, D), RES)

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

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

G_IN(.(C, Cs), D) → G_IN(Cs, .(C, D))

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:

G_IN(.(C, Cs), D) → G_IN(Cs, .(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_IN(.(A, As), .(B, Bs), RES) → F_IN(.(B, .(A, As)), Bs, RES)

The TRS R consists of the following rules:

f_in(.(A, As), .(B, Bs), RES) → U2(A, As, B, Bs, RES, f_in(.(B, .(A, As)), Bs, RES))
f_in(A, [], RES) → U1(A, RES, g_in(A, [], RES))
g_in(.(C, Cs), D, RES) → U3(C, Cs, D, RES, g_in(Cs, .(C, D), RES))
g_in([], RES, RES) → g_out([], RES, RES)
U3(C, Cs, D, RES, g_out(Cs, .(C, D), RES)) → g_out(.(C, Cs), D, RES)
U1(A, RES, g_out(A, [], RES)) → f_out(A, [], RES)
U2(A, As, B, Bs, RES, f_out(.(B, .(A, As)), Bs, RES)) → f_out(.(A, As), .(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_in(x1, x2, x3) = f_in(x1, x2)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5, x6) = U2(x6)
[] = []
U1(x1, x2, x3) = U1(x3)
g_in(x1, x2, x3) = g_in(x1, x2)
U3(x1, x2, x3, x4, x5) = U3(x5)
g_out(x1, x2, x3) = g_out(x3)
f_out(x1, x2, x3) = f_out(x3)
F_IN(x1, x2, x3) = F_IN(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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

F_IN(.(A, As), .(B, Bs), RES) → F_IN(.(B, .(A, As)), Bs, RES)

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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

F_IN(.(A, As), .(B, Bs)) → F_IN(.(B, .(A, As)), Bs)

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

F_IN(.(A, As), .(B, Bs)) → F_IN(.(B, .(A, As)), Bs)
The graph contains the following edges 2 > 2