Left Termination proof of /tmp/tmpnHWR5G/g.pl

Left Termination of the query pattern g_in_1(a) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

g(W) :- ','(eq(X, .(.(a, []), .(.(R, []), []))), ','(eq(Y, .(.(S, .(c, [])), .([], []))), ','(app_1(X, Y, Z), ','(eq(Z, .(U, V)), app_2(U, U, W))))).
app_1([], X, X).
app_1(.(X, Xs), Ys, .(X, Zs)) :- app_1(Xs, Ys, Zs).
app_2([], X, X).
app_2(.(X, Xs), Ys, .(X, Zs)) :- app_2(Xs, Ys, Zs).
eq(X, X).

Queries:

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:

g_in(W) → U1(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
eq_in(X, X) → eq_out(X, X)
U1(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U2(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3(W, app_1_in(X, Y, Z))
app_1_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
app_1_in([], X, X) → app_1_out([], X, X)
U6(X, Xs, Ys, Zs, app_1_out(Xs, Ys, Zs)) → app_1_out(.(X, Xs), Ys, .(X, Zs))
U3(W, app_1_out(X, Y, Z)) → U4(W, eq_in(Z, .(U, V)))
U4(W, eq_out(Z, .(U, V))) → U5(W, app_2_in(U, U, W))
app_2_in(.(X, Xs), Ys, .(X, Zs)) → U7(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
app_2_in([], X, X) → app_2_out([], X, X)
U7(X, Xs, Ys, Zs, app_2_out(Xs, Ys, Zs)) → app_2_out(.(X, Xs), Ys, .(X, Zs))
U5(W, app_2_out(U, U, W)) → g_out(W)

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:

g_in(W) → U1(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
eq_in(X, X) → eq_out(X, X)
U1(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U2(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3(W, app_1_in(X, Y, Z))
app_1_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
app_1_in([], X, X) → app_1_out([], X, X)
U6(X, Xs, Ys, Zs, app_1_out(Xs, Ys, Zs)) → app_1_out(.(X, Xs), Ys, .(X, Zs))
U3(W, app_1_out(X, Y, Z)) → U4(W, eq_in(Z, .(U, V)))
U4(W, eq_out(Z, .(U, V))) → U5(W, app_2_in(U, U, W))
app_2_in(.(X, Xs), Ys, .(X, Zs)) → U7(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
app_2_in([], X, X) → app_2_out([], X, X)
U7(X, Xs, Ys, Zs, app_2_out(Xs, Ys, Zs)) → app_2_out(.(X, Xs), Ys, .(X, Zs))
U5(W, app_2_out(U, U, W)) → g_out(W)

G_IN(W) → U1¹(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
G_IN(W) → EQ_IN(X, .(.(a, []), .(.(R, []), [])))
U1¹(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2¹(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U1¹(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → EQ_IN(Y, .(.(S, .(c, [])), .([], [])))
U2¹(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3¹(W, app_1_in(X, Y, Z))
U2¹(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → APP_1_IN(X, Y, Z)
APP_1_IN(.(X, Xs), Ys, .(X, Zs)) → U6¹(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
APP_1_IN(.(X, Xs), Ys, .(X, Zs)) → APP_1_IN(Xs, Ys, Zs)
U3¹(W, app_1_out(X, Y, Z)) → U4¹(W, eq_in(Z, .(U, V)))
U3¹(W, app_1_out(X, Y, Z)) → EQ_IN(Z, .(U, V))
U4¹(W, eq_out(Z, .(U, V))) → U5¹(W, app_2_in(U, U, W))
U4¹(W, eq_out(Z, .(U, V))) → APP_2_IN(U, U, W)
APP_2_IN(.(X, Xs), Ys, .(X, Zs)) → U7¹(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
APP_2_IN(.(X, Xs), Ys, .(X, Zs)) → APP_2_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

g_in(W) → U1(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
eq_in(X, X) → eq_out(X, X)
U1(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U2(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3(W, app_1_in(X, Y, Z))
app_1_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
app_1_in([], X, X) → app_1_out([], X, X)
U6(X, Xs, Ys, Zs, app_1_out(Xs, Ys, Zs)) → app_1_out(.(X, Xs), Ys, .(X, Zs))
U3(W, app_1_out(X, Y, Z)) → U4(W, eq_in(Z, .(U, V)))
U4(W, eq_out(Z, .(U, V))) → U5(W, app_2_in(U, U, W))
app_2_in(.(X, Xs), Ys, .(X, Zs)) → U7(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
app_2_in([], X, X) → app_2_out([], X, X)
U7(X, Xs, Ys, Zs, app_2_out(Xs, Ys, Zs)) → app_2_out(.(X, Xs), Ys, .(X, Zs))
U5(W, app_2_out(U, U, W)) → g_out(W)

The argument filtering Pi contains the following mapping:
g_in(x1) = g_in
U1(x1, x2) = U1(x2)
eq_in(x1, x2) = eq_in
.(x1, x2) = .(x1, x2)
a = a
[] = []
eq_out(x1, x2) = eq_out
U2(x1, x2, x3) = U2(x3)
c = c
U3(x1, x2) = U3(x2)
app_1_in(x1, x2, x3) = app_1_in
U6(x1, x2, x3, x4, x5) = U6(x5)
app_1_out(x1, x2, x3) = app_1_out
U4(x1, x2) = U4(x2)
U5(x1, x2) = U5(x2)
app_2_in(x1, x2, x3) = app_2_in
U7(x1, x2, x3, x4, x5) = U7(x5)
app_2_out(x1, x2, x3) = app_2_out
g_out(x1) = g_out
U5¹(x1, x2) = U5¹(x2)
G_IN(x1) = G_IN
U4¹(x1, x2) = U4¹(x2)
APP_2_IN(x1, x2, x3) = APP_2_IN
U1¹(x1, x2) = U1¹(x2)
U3¹(x1, x2) = U3¹(x2)
U7¹(x1, x2, x3, x4, x5) = U7¹(x5)
U6¹(x1, x2, x3, x4, x5) = U6¹(x5)
U2¹(x1, x2, x3) = U2¹(x3)
APP_1_IN(x1, x2, x3) = APP_1_IN
EQ_IN(x1, x2) = EQ_IN

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:

G_IN(W) → U1¹(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
G_IN(W) → EQ_IN(X, .(.(a, []), .(.(R, []), [])))
U1¹(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2¹(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U1¹(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → EQ_IN(Y, .(.(S, .(c, [])), .([], [])))
U2¹(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3¹(W, app_1_in(X, Y, Z))
U2¹(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → APP_1_IN(X, Y, Z)
APP_1_IN(.(X, Xs), Ys, .(X, Zs)) → U6¹(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
APP_1_IN(.(X, Xs), Ys, .(X, Zs)) → APP_1_IN(Xs, Ys, Zs)
U3¹(W, app_1_out(X, Y, Z)) → U4¹(W, eq_in(Z, .(U, V)))
U3¹(W, app_1_out(X, Y, Z)) → EQ_IN(Z, .(U, V))
U4¹(W, eq_out(Z, .(U, V))) → U5¹(W, app_2_in(U, U, W))
U4¹(W, eq_out(Z, .(U, V))) → APP_2_IN(U, U, W)
APP_2_IN(.(X, Xs), Ys, .(X, Zs)) → U7¹(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
APP_2_IN(.(X, Xs), Ys, .(X, Zs)) → APP_2_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

g_in(W) → U1(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
eq_in(X, X) → eq_out(X, X)
U1(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U2(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3(W, app_1_in(X, Y, Z))
app_1_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
app_1_in([], X, X) → app_1_out([], X, X)
U6(X, Xs, Ys, Zs, app_1_out(Xs, Ys, Zs)) → app_1_out(.(X, Xs), Ys, .(X, Zs))
U3(W, app_1_out(X, Y, Z)) → U4(W, eq_in(Z, .(U, V)))
U4(W, eq_out(Z, .(U, V))) → U5(W, app_2_in(U, U, W))
app_2_in(.(X, Xs), Ys, .(X, Zs)) → U7(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
app_2_in([], X, X) → app_2_out([], X, X)
U7(X, Xs, Ys, Zs, app_2_out(Xs, Ys, Zs)) → app_2_out(.(X, Xs), Ys, .(X, Zs))
U5(W, app_2_out(U, U, W)) → g_out(W)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_2_IN(.(X, Xs), Ys, .(X, Zs)) → APP_2_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

g_in(W) → U1(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
eq_in(X, X) → eq_out(X, X)
U1(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U2(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3(W, app_1_in(X, Y, Z))
app_1_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
app_1_in([], X, X) → app_1_out([], X, X)
U6(X, Xs, Ys, Zs, app_1_out(Xs, Ys, Zs)) → app_1_out(.(X, Xs), Ys, .(X, Zs))
U3(W, app_1_out(X, Y, Z)) → U4(W, eq_in(Z, .(U, V)))
U4(W, eq_out(Z, .(U, V))) → U5(W, app_2_in(U, U, W))
app_2_in(.(X, Xs), Ys, .(X, Zs)) → U7(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
app_2_in([], X, X) → app_2_out([], X, X)
U7(X, Xs, Ys, Zs, app_2_out(Xs, Ys, Zs)) → app_2_out(.(X, Xs), Ys, .(X, Zs))
U5(W, app_2_out(U, U, W)) → g_out(W)

The argument filtering Pi contains the following mapping:
g_in(x1) = g_in
U1(x1, x2) = U1(x2)
eq_in(x1, x2) = eq_in
.(x1, x2) = .(x1, x2)
a = a
[] = []
eq_out(x1, x2) = eq_out
U2(x1, x2, x3) = U2(x3)
c = c
U3(x1, x2) = U3(x2)
app_1_in(x1, x2, x3) = app_1_in
U6(x1, x2, x3, x4, x5) = U6(x5)
app_1_out(x1, x2, x3) = app_1_out
U4(x1, x2) = U4(x2)
U5(x1, x2) = U5(x2)
app_2_in(x1, x2, x3) = app_2_in
U7(x1, x2, x3, x4, x5) = U7(x5)
app_2_out(x1, x2, x3) = app_2_out
g_out(x1) = g_out
APP_2_IN(x1, x2, x3) = APP_2_IN

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:

APP_2_IN(.(X, Xs), Ys, .(X, Zs)) → APP_2_IN(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP_2_IN(x1, x2, x3) = APP_2_IN

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:

APP_2_IN → APP_2_IN

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:

APP_2_IN → APP_2_IN

The TRS R consists of the following rules:none

s = APP_2_IN evaluates to t =APP_2_IN

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 APP_2_IN to APP_2_IN.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

APP_1_IN(.(X, Xs), Ys, .(X, Zs)) → APP_1_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

g_in(W) → U1(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
eq_in(X, X) → eq_out(X, X)
U1(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U2(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3(W, app_1_in(X, Y, Z))
app_1_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
app_1_in([], X, X) → app_1_out([], X, X)
U6(X, Xs, Ys, Zs, app_1_out(Xs, Ys, Zs)) → app_1_out(.(X, Xs), Ys, .(X, Zs))
U3(W, app_1_out(X, Y, Z)) → U4(W, eq_in(Z, .(U, V)))
U4(W, eq_out(Z, .(U, V))) → U5(W, app_2_in(U, U, W))
app_2_in(.(X, Xs), Ys, .(X, Zs)) → U7(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
app_2_in([], X, X) → app_2_out([], X, X)
U7(X, Xs, Ys, Zs, app_2_out(Xs, Ys, Zs)) → app_2_out(.(X, Xs), Ys, .(X, Zs))
U5(W, app_2_out(U, U, W)) → g_out(W)

The argument filtering Pi contains the following mapping:
g_in(x1) = g_in
U1(x1, x2) = U1(x2)
eq_in(x1, x2) = eq_in
.(x1, x2) = .(x1, x2)
a = a
[] = []
eq_out(x1, x2) = eq_out
U2(x1, x2, x3) = U2(x3)
c = c
U3(x1, x2) = U3(x2)
app_1_in(x1, x2, x3) = app_1_in
U6(x1, x2, x3, x4, x5) = U6(x5)
app_1_out(x1, x2, x3) = app_1_out
U4(x1, x2) = U4(x2)
U5(x1, x2) = U5(x2)
app_2_in(x1, x2, x3) = app_2_in
U7(x1, x2, x3, x4, x5) = U7(x5)
app_2_out(x1, x2, x3) = app_2_out
g_out(x1) = g_out
APP_1_IN(x1, x2, x3) = APP_1_IN

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:

APP_1_IN(.(X, Xs), Ys, .(X, Zs)) → APP_1_IN(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP_1_IN(x1, x2, x3) = APP_1_IN

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

                        ↳ NonTerminationProof

  ↳ PrologToPiTRSProof

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

APP_1_IN → APP_1_IN

The TRS R consists of the following rules:none

s = APP_1_IN evaluates to t =APP_1_IN

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 APP_1_IN to APP_1_IN.

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:

g_in(W) → U1(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
eq_in(X, X) → eq_out(X, X)
U1(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U2(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3(W, app_1_in(X, Y, Z))
app_1_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
app_1_in([], X, X) → app_1_out([], X, X)
U6(X, Xs, Ys, Zs, app_1_out(Xs, Ys, Zs)) → app_1_out(.(X, Xs), Ys, .(X, Zs))
U3(W, app_1_out(X, Y, Z)) → U4(W, eq_in(Z, .(U, V)))
U4(W, eq_out(Z, .(U, V))) → U5(W, app_2_in(U, U, W))
app_2_in(.(X, Xs), Ys, .(X, Zs)) → U7(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
app_2_in([], X, X) → app_2_out([], X, X)
U7(X, Xs, Ys, Zs, app_2_out(Xs, Ys, Zs)) → app_2_out(.(X, Xs), Ys, .(X, Zs))
U5(W, app_2_out(U, U, W)) → g_out(W)

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:

g_in(W) → U1(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
eq_in(X, X) → eq_out(X, X)
U1(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U2(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3(W, app_1_in(X, Y, Z))
app_1_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
app_1_in([], X, X) → app_1_out([], X, X)
U6(X, Xs, Ys, Zs, app_1_out(Xs, Ys, Zs)) → app_1_out(.(X, Xs), Ys, .(X, Zs))
U3(W, app_1_out(X, Y, Z)) → U4(W, eq_in(Z, .(U, V)))
U4(W, eq_out(Z, .(U, V))) → U5(W, app_2_in(U, U, W))
app_2_in(.(X, Xs), Ys, .(X, Zs)) → U7(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
app_2_in([], X, X) → app_2_out([], X, X)
U7(X, Xs, Ys, Zs, app_2_out(Xs, Ys, Zs)) → app_2_out(.(X, Xs), Ys, .(X, Zs))
U5(W, app_2_out(U, U, W)) → g_out(W)

G_IN(W) → U1¹(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
G_IN(W) → EQ_IN(X, .(.(a, []), .(.(R, []), [])))
U1¹(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2¹(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U1¹(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → EQ_IN(Y, .(.(S, .(c, [])), .([], [])))
U2¹(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3¹(W, app_1_in(X, Y, Z))
U2¹(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → APP_1_IN(X, Y, Z)
APP_1_IN(.(X, Xs), Ys, .(X, Zs)) → U6¹(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
APP_1_IN(.(X, Xs), Ys, .(X, Zs)) → APP_1_IN(Xs, Ys, Zs)
U3¹(W, app_1_out(X, Y, Z)) → U4¹(W, eq_in(Z, .(U, V)))
U3¹(W, app_1_out(X, Y, Z)) → EQ_IN(Z, .(U, V))
U4¹(W, eq_out(Z, .(U, V))) → U5¹(W, app_2_in(U, U, W))
U4¹(W, eq_out(Z, .(U, V))) → APP_2_IN(U, U, W)
APP_2_IN(.(X, Xs), Ys, .(X, Zs)) → U7¹(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
APP_2_IN(.(X, Xs), Ys, .(X, Zs)) → APP_2_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

g_in(W) → U1(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
eq_in(X, X) → eq_out(X, X)
U1(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U2(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3(W, app_1_in(X, Y, Z))
app_1_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
app_1_in([], X, X) → app_1_out([], X, X)
U6(X, Xs, Ys, Zs, app_1_out(Xs, Ys, Zs)) → app_1_out(.(X, Xs), Ys, .(X, Zs))
U3(W, app_1_out(X, Y, Z)) → U4(W, eq_in(Z, .(U, V)))
U4(W, eq_out(Z, .(U, V))) → U5(W, app_2_in(U, U, W))
app_2_in(.(X, Xs), Ys, .(X, Zs)) → U7(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
app_2_in([], X, X) → app_2_out([], X, X)
U7(X, Xs, Ys, Zs, app_2_out(Xs, Ys, Zs)) → app_2_out(.(X, Xs), Ys, .(X, Zs))
U5(W, app_2_out(U, U, W)) → g_out(W)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

G_IN(W) → U1¹(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
G_IN(W) → EQ_IN(X, .(.(a, []), .(.(R, []), [])))
U1¹(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2¹(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U1¹(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → EQ_IN(Y, .(.(S, .(c, [])), .([], [])))
U2¹(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3¹(W, app_1_in(X, Y, Z))
U2¹(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → APP_1_IN(X, Y, Z)
APP_1_IN(.(X, Xs), Ys, .(X, Zs)) → U6¹(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
APP_1_IN(.(X, Xs), Ys, .(X, Zs)) → APP_1_IN(Xs, Ys, Zs)
U3¹(W, app_1_out(X, Y, Z)) → U4¹(W, eq_in(Z, .(U, V)))
U3¹(W, app_1_out(X, Y, Z)) → EQ_IN(Z, .(U, V))
U4¹(W, eq_out(Z, .(U, V))) → U5¹(W, app_2_in(U, U, W))
U4¹(W, eq_out(Z, .(U, V))) → APP_2_IN(U, U, W)
APP_2_IN(.(X, Xs), Ys, .(X, Zs)) → U7¹(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
APP_2_IN(.(X, Xs), Ys, .(X, Zs)) → APP_2_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

g_in(W) → U1(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
eq_in(X, X) → eq_out(X, X)
U1(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U2(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3(W, app_1_in(X, Y, Z))
app_1_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
app_1_in([], X, X) → app_1_out([], X, X)
U6(X, Xs, Ys, Zs, app_1_out(Xs, Ys, Zs)) → app_1_out(.(X, Xs), Ys, .(X, Zs))
U3(W, app_1_out(X, Y, Z)) → U4(W, eq_in(Z, .(U, V)))
U4(W, eq_out(Z, .(U, V))) → U5(W, app_2_in(U, U, W))
app_2_in(.(X, Xs), Ys, .(X, Zs)) → U7(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
app_2_in([], X, X) → app_2_out([], X, X)
U7(X, Xs, Ys, Zs, app_2_out(Xs, Ys, Zs)) → app_2_out(.(X, Xs), Ys, .(X, Zs))
U5(W, app_2_out(U, U, W)) → g_out(W)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

APP_2_IN(.(X, Xs), Ys, .(X, Zs)) → APP_2_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

g_in(W) → U1(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
eq_in(X, X) → eq_out(X, X)
U1(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U2(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3(W, app_1_in(X, Y, Z))
app_1_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
app_1_in([], X, X) → app_1_out([], X, X)
U6(X, Xs, Ys, Zs, app_1_out(Xs, Ys, Zs)) → app_1_out(.(X, Xs), Ys, .(X, Zs))
U3(W, app_1_out(X, Y, Z)) → U4(W, eq_in(Z, .(U, V)))
U4(W, eq_out(Z, .(U, V))) → U5(W, app_2_in(U, U, W))
app_2_in(.(X, Xs), Ys, .(X, Zs)) → U7(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
app_2_in([], X, X) → app_2_out([], X, X)
U7(X, Xs, Ys, Zs, app_2_out(Xs, Ys, Zs)) → app_2_out(.(X, Xs), Ys, .(X, Zs))
U5(W, app_2_out(U, U, W)) → g_out(W)

The argument filtering Pi contains the following mapping:
g_in(x1) = g_in
U1(x1, x2) = U1(x2)
eq_in(x1, x2) = eq_in
.(x1, x2) = .(x1, x2)
a = a
[] = []
eq_out(x1, x2) = eq_out
U2(x1, x2, x3) = U2(x3)
c = c
U3(x1, x2) = U3(x2)
app_1_in(x1, x2, x3) = app_1_in
U6(x1, x2, x3, x4, x5) = U6(x5)
app_1_out(x1, x2, x3) = app_1_out
U4(x1, x2) = U4(x2)
U5(x1, x2) = U5(x2)
app_2_in(x1, x2, x3) = app_2_in
U7(x1, x2, x3, x4, x5) = U7(x5)
app_2_out(x1, x2, x3) = app_2_out
g_out(x1) = g_out
APP_2_IN(x1, x2, x3) = APP_2_IN

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:

APP_2_IN(.(X, Xs), Ys, .(X, Zs)) → APP_2_IN(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:

APP_2_IN → APP_2_IN

The TRS R consists of the following rules:none

s = APP_2_IN evaluates to t =APP_2_IN

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 APP_2_IN to APP_2_IN.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

APP_1_IN(.(X, Xs), Ys, .(X, Zs)) → APP_1_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

g_in(W) → U1(W, eq_in(X, .(.(a, []), .(.(R, []), []))))
eq_in(X, X) → eq_out(X, X)
U1(W, eq_out(X, .(.(a, []), .(.(R, []), [])))) → U2(W, X, eq_in(Y, .(.(S, .(c, [])), .([], []))))
U2(W, X, eq_out(Y, .(.(S, .(c, [])), .([], [])))) → U3(W, app_1_in(X, Y, Z))
app_1_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_1_in(Xs, Ys, Zs))
app_1_in([], X, X) → app_1_out([], X, X)
U6(X, Xs, Ys, Zs, app_1_out(Xs, Ys, Zs)) → app_1_out(.(X, Xs), Ys, .(X, Zs))
U3(W, app_1_out(X, Y, Z)) → U4(W, eq_in(Z, .(U, V)))
U4(W, eq_out(Z, .(U, V))) → U5(W, app_2_in(U, U, W))
app_2_in(.(X, Xs), Ys, .(X, Zs)) → U7(X, Xs, Ys, Zs, app_2_in(Xs, Ys, Zs))
app_2_in([], X, X) → app_2_out([], X, X)
U7(X, Xs, Ys, Zs, app_2_out(Xs, Ys, Zs)) → app_2_out(.(X, Xs), Ys, .(X, Zs))
U5(W, app_2_out(U, U, W)) → g_out(W)

The argument filtering Pi contains the following mapping:
g_in(x1) = g_in
U1(x1, x2) = U1(x2)
eq_in(x1, x2) = eq_in
.(x1, x2) = .(x1, x2)
a = a
[] = []
eq_out(x1, x2) = eq_out
U2(x1, x2, x3) = U2(x3)
c = c
U3(x1, x2) = U3(x2)
app_1_in(x1, x2, x3) = app_1_in
U6(x1, x2, x3, x4, x5) = U6(x5)
app_1_out(x1, x2, x3) = app_1_out
U4(x1, x2) = U4(x2)
U5(x1, x2) = U5(x2)
app_2_in(x1, x2, x3) = app_2_in
U7(x1, x2, x3, x4, x5) = U7(x5)
app_2_out(x1, x2, x3) = app_2_out
g_out(x1) = g_out
APP_1_IN(x1, x2, x3) = APP_1_IN

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:

APP_1_IN(.(X, Xs), Ys, .(X, Zs)) → APP_1_IN(Xs, Ys, Zs)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

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

APP_1_IN → APP_1_IN

The TRS R consists of the following rules:none

s = APP_1_IN evaluates to t =APP_1_IN

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 APP_1_IN to APP_1_IN.