Left Termination proof of /tmp/tmpkfpDG0/permutation1.pl

Left Termination of the query pattern perm_in_2(g, a) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).
perm([], []).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
app([], Ys, Ys).

Queries:

perm(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:

perm_in([], []) → perm_out([], [])
perm_in(Xs, .(X, Ys)) → U1(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U1(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2(Xs, X, Ys, app_in(X1s, X2s, Zs))
U2(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3(Xs, X, Ys, perm_in(Zs, Ys))
U3(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))

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:

perm_in([], []) → perm_out([], [])
perm_in(Xs, .(X, Ys)) → U1(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U1(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2(Xs, X, Ys, app_in(X1s, X2s, Zs))
U2(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3(Xs, X, Ys, perm_in(Zs, Ys))
U3(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in(x1, x2) = perm_in
[] = []
perm_out(x1, x2) = perm_out(x2)
.(x1, x2) = .(x2)
U1(x1, x2, x3, x4) = U1(x4)
app_in(x1, x2, x3) = app_in
app_out(x1, x2, x3) = app_out(x1)
U4(x1, x2, x3, x4, x5) = U4(x5)
U2(x1, x2, x3, x4) = U2(x4)
U3(x1, x2, x3, x4) = U3(x4)

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

PERM_IN(Xs, .(X, Ys)) → U1¹(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
PERM_IN(Xs, .(X, Ys)) → APP_IN(X1s, .(X, X2s), Xs)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U4¹(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
U1¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2¹(Xs, X, Ys, app_in(X1s, X2s, Zs))
U1¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → APP_IN(X1s, X2s, Zs)
U2¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3¹(Xs, X, Ys, perm_in(Zs, Ys))
U2¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → PERM_IN(Zs, Ys)

The TRS R consists of the following rules:

perm_in([], []) → perm_out([], [])
perm_in(Xs, .(X, Ys)) → U1(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U1(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2(Xs, X, Ys, app_in(X1s, X2s, Zs))
U2(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3(Xs, X, Ys, perm_in(Zs, Ys))
U3(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in(x1, x2) = perm_in
[] = []
perm_out(x1, x2) = perm_out(x2)
.(x1, x2) = .(x2)
U1(x1, x2, x3, x4) = U1(x4)
app_in(x1, x2, x3) = app_in
app_out(x1, x2, x3) = app_out(x1)
U4(x1, x2, x3, x4, x5) = U4(x5)
U2(x1, x2, x3, x4) = U2(x4)
U3(x1, x2, x3, x4) = U3(x4)
U3¹(x1, x2, x3, x4) = U3¹(x4)
U4¹(x1, x2, x3, x4, x5) = U4¹(x5)
U2¹(x1, x2, x3, x4) = U2¹(x4)
PERM_IN(x1, x2) = PERM_IN
APP_IN(x1, x2, x3) = APP_IN
U1¹(x1, x2, x3, x4) = U1¹(x4)

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:

PERM_IN(Xs, .(X, Ys)) → U1¹(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
PERM_IN(Xs, .(X, Ys)) → APP_IN(X1s, .(X, X2s), Xs)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U4¹(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
U1¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2¹(Xs, X, Ys, app_in(X1s, X2s, Zs))
U1¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → APP_IN(X1s, X2s, Zs)
U2¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3¹(Xs, X, Ys, perm_in(Zs, Ys))
U2¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → PERM_IN(Zs, Ys)

The TRS R consists of the following rules:

perm_in([], []) → perm_out([], [])
perm_in(Xs, .(X, Ys)) → U1(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U1(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2(Xs, X, Ys, app_in(X1s, X2s, Zs))
U2(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3(Xs, X, Ys, perm_in(Zs, Ys))
U3(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in(x1, x2) = perm_in
[] = []
perm_out(x1, x2) = perm_out(x2)
.(x1, x2) = .(x2)
U1(x1, x2, x3, x4) = U1(x4)
app_in(x1, x2, x3) = app_in
app_out(x1, x2, x3) = app_out(x1)
U4(x1, x2, x3, x4, x5) = U4(x5)
U2(x1, x2, x3, x4) = U2(x4)
U3(x1, x2, x3, x4) = U3(x4)
U3¹(x1, x2, x3, x4) = U3¹(x4)
U4¹(x1, x2, x3, x4, x5) = U4¹(x5)
U2¹(x1, x2, x3, x4) = U2¹(x4)
PERM_IN(x1, x2) = PERM_IN
APP_IN(x1, x2, x3) = APP_IN
U1¹(x1, x2, x3, x4) = U1¹(x4)

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

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

perm_in([], []) → perm_out([], [])
perm_in(Xs, .(X, Ys)) → U1(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U1(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2(Xs, X, Ys, app_in(X1s, X2s, Zs))
U2(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3(Xs, X, Ys, perm_in(Zs, Ys))
U3(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in(x1, x2) = perm_in
[] = []
perm_out(x1, x2) = perm_out(x2)
.(x1, x2) = .(x2)
U1(x1, x2, x3, x4) = U1(x4)
app_in(x1, x2, x3) = app_in
app_out(x1, x2, x3) = app_out(x1)
U4(x1, x2, x3, x4, x5) = U4(x5)
U2(x1, x2, x3, x4) = U2(x4)
U3(x1, x2, x3, x4) = U3(x4)
APP_IN(x1, x2, x3) = APP_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_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

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

The TRS R consists of the following rules:none

s = APP_IN evaluates to t =APP_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_IN to APP_IN.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

U2¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → PERM_IN(Zs, Ys)
U1¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2¹(Xs, X, Ys, app_in(X1s, X2s, Zs))
PERM_IN(Xs, .(X, Ys)) → U1¹(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

perm_in([], []) → perm_out([], [])
perm_in(Xs, .(X, Ys)) → U1(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U1(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2(Xs, X, Ys, app_in(X1s, X2s, Zs))
U2(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3(Xs, X, Ys, perm_in(Zs, Ys))
U3(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in(x1, x2) = perm_in
[] = []
perm_out(x1, x2) = perm_out(x2)
.(x1, x2) = .(x2)
U1(x1, x2, x3, x4) = U1(x4)
app_in(x1, x2, x3) = app_in
app_out(x1, x2, x3) = app_out(x1)
U4(x1, x2, x3, x4, x5) = U4(x5)
U2(x1, x2, x3, x4) = U2(x4)
U3(x1, x2, x3, x4) = U3(x4)
U2¹(x1, x2, x3, x4) = U2¹(x4)
PERM_IN(x1, x2) = PERM_IN
U1¹(x1, x2, x3, x4) = U1¹(x4)

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:

U2¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → PERM_IN(Zs, Ys)
U1¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2¹(Xs, X, Ys, app_in(X1s, X2s, Zs))
PERM_IN(Xs, .(X, Ys)) → U1¹(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x2)
app_in(x1, x2, x3) = app_in
app_out(x1, x2, x3) = app_out(x1)
U4(x1, x2, x3, x4, x5) = U4(x5)
U2¹(x1, x2, x3, x4) = U2¹(x4)
PERM_IN(x1, x2) = PERM_IN
U1¹(x1, x2, x3, x4) = U1¹(x4)

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

                        ↳ Narrowing

  ↳ PrologToPiTRSProof

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

U2¹(app_out(X1s)) → PERM_IN
U1¹(app_out(X1s)) → U2¹(app_in)
PERM_IN → U1¹(app_in)

The TRS R consists of the following rules:

app_in → app_out([])
app_in → U4(app_in)
U4(app_out(Xs)) → app_out(.(Xs))

The set Q consists of the following terms:

app_in
U4(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1¹(app_out(X1s)) → U2¹(app_in) at position [0] we obtained the following new rules:

U1¹(app_out(y0)) → U2¹(app_out([]))
U1¹(app_out(y0)) → U2¹(U4(app_in))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

  ↳ PrologToPiTRSProof

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

U2¹(app_out(X1s)) → PERM_IN
U1¹(app_out(y0)) → U2¹(app_out([]))
U1¹(app_out(y0)) → U2¹(U4(app_in))
PERM_IN → U1¹(app_in)

The TRS R consists of the following rules:

app_in → app_out([])
app_in → U4(app_in)
U4(app_out(Xs)) → app_out(.(Xs))

The set Q consists of the following terms:

app_in
U4(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PERM_IN → U1¹(app_in) at position [0] we obtained the following new rules:

PERM_IN → U1¹(app_out([]))
PERM_IN → U1¹(U4(app_in))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ NonTerminationProof

  ↳ PrologToPiTRSProof

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

PERM_IN → U1¹(app_out([]))
U2¹(app_out(X1s)) → PERM_IN
PERM_IN → U1¹(U4(app_in))
U1¹(app_out(y0)) → U2¹(app_out([]))
U1¹(app_out(y0)) → U2¹(U4(app_in))

The TRS R consists of the following rules:

app_in → app_out([])
app_in → U4(app_in)
U4(app_out(Xs)) → app_out(.(Xs))

The set Q consists of the following terms:

app_in
U4(x0)

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 narrowing to the left:

The TRS P consists of the following rules:

PERM_IN → U1¹(app_out([]))
U2¹(app_out(X1s)) → PERM_IN
PERM_IN → U1¹(U4(app_in))
U1¹(app_out(y0)) → U2¹(app_out([]))
U1¹(app_out(y0)) → U2¹(U4(app_in))

The TRS R consists of the following rules:

app_in → app_out([])
app_in → U4(app_in)
U4(app_out(Xs)) → app_out(.(Xs))

s = U1¹(app_out(y0)) evaluates to t =U1¹(app_out([]))

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

Matcher: [y0 / []]
Semiunifier: [ ]

Rewriting sequence

U1¹(app_out(y0)) → U2¹(app_out([]))
with rule U1¹(app_out(y0')) → U2¹(app_out([])) at position [] and matcher [y0' / y0]

U2¹(app_out([])) → PERM_IN
with rule U2¹(app_out(X1s)) → PERM_IN at position [] and matcher [X1s / []]

PERM_IN → U1¹(app_out([]))
with rule PERM_IN → U1¹(app_out([]))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

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:

perm_in([], []) → perm_out([], [])
perm_in(Xs, .(X, Ys)) → U1(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U1(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2(Xs, X, Ys, app_in(X1s, X2s, Zs))
U2(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3(Xs, X, Ys, perm_in(Zs, Ys))
U3(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))

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:

perm_in([], []) → perm_out([], [])
perm_in(Xs, .(X, Ys)) → U1(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U1(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2(Xs, X, Ys, app_in(X1s, X2s, Zs))
U2(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3(Xs, X, Ys, perm_in(Zs, Ys))
U3(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))

PERM_IN(Xs, .(X, Ys)) → U1¹(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
PERM_IN(Xs, .(X, Ys)) → APP_IN(X1s, .(X, X2s), Xs)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U4¹(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
U1¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2¹(Xs, X, Ys, app_in(X1s, X2s, Zs))
U1¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → APP_IN(X1s, X2s, Zs)
U2¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3¹(Xs, X, Ys, perm_in(Zs, Ys))
U2¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → PERM_IN(Zs, Ys)

The TRS R consists of the following rules:

perm_in([], []) → perm_out([], [])
perm_in(Xs, .(X, Ys)) → U1(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U1(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2(Xs, X, Ys, app_in(X1s, X2s, Zs))
U2(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3(Xs, X, Ys, perm_in(Zs, Ys))
U3(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in(x1, x2) = perm_in
[] = []
perm_out(x1, x2) = perm_out(x2)
.(x1, x2) = .(x2)
U1(x1, x2, x3, x4) = U1(x4)
app_in(x1, x2, x3) = app_in
app_out(x1, x2, x3) = app_out(x1)
U4(x1, x2, x3, x4, x5) = U4(x5)
U2(x1, x2, x3, x4) = U2(x4)
U3(x1, x2, x3, x4) = U3(x4)
U3¹(x1, x2, x3, x4) = U3¹(x4)
U4¹(x1, x2, x3, x4, x5) = U4¹(x5)
U2¹(x1, x2, x3, x4) = U2¹(x4)
PERM_IN(x1, x2) = PERM_IN
APP_IN(x1, x2, x3) = APP_IN
U1¹(x1, x2, x3, x4) = U1¹(x4)

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:

PERM_IN(Xs, .(X, Ys)) → U1¹(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
PERM_IN(Xs, .(X, Ys)) → APP_IN(X1s, .(X, X2s), Xs)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U4¹(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
U1¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2¹(Xs, X, Ys, app_in(X1s, X2s, Zs))
U1¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → APP_IN(X1s, X2s, Zs)
U2¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3¹(Xs, X, Ys, perm_in(Zs, Ys))
U2¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → PERM_IN(Zs, Ys)

The TRS R consists of the following rules:

perm_in([], []) → perm_out([], [])
perm_in(Xs, .(X, Ys)) → U1(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U1(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2(Xs, X, Ys, app_in(X1s, X2s, Zs))
U2(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3(Xs, X, Ys, perm_in(Zs, Ys))
U3(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in(x1, x2) = perm_in
[] = []
perm_out(x1, x2) = perm_out(x2)
.(x1, x2) = .(x2)
U1(x1, x2, x3, x4) = U1(x4)
app_in(x1, x2, x3) = app_in
app_out(x1, x2, x3) = app_out(x1)
U4(x1, x2, x3, x4, x5) = U4(x5)
U2(x1, x2, x3, x4) = U2(x4)
U3(x1, x2, x3, x4) = U3(x4)
U3¹(x1, x2, x3, x4) = U3¹(x4)
U4¹(x1, x2, x3, x4, x5) = U4¹(x5)
U2¹(x1, x2, x3, x4) = U2¹(x4)
PERM_IN(x1, x2) = PERM_IN
APP_IN(x1, x2, x3) = APP_IN
U1¹(x1, x2, x3, x4) = U1¹(x4)

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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

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

The TRS R consists of the following rules:

perm_in([], []) → perm_out([], [])
perm_in(Xs, .(X, Ys)) → U1(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U1(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2(Xs, X, Ys, app_in(X1s, X2s, Zs))
U2(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3(Xs, X, Ys, perm_in(Zs, Ys))
U3(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in(x1, x2) = perm_in
[] = []
perm_out(x1, x2) = perm_out(x2)
.(x1, x2) = .(x2)
U1(x1, x2, x3, x4) = U1(x4)
app_in(x1, x2, x3) = app_in
app_out(x1, x2, x3) = app_out(x1)
U4(x1, x2, x3, x4, x5) = U4(x5)
U2(x1, x2, x3, x4) = U2(x4)
U3(x1, x2, x3, x4) = U3(x4)
APP_IN(x1, x2, x3) = APP_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_IN(.(X, Xs), Ys, .(X, Zs)) → APP_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_IN → APP_IN

The TRS R consists of the following rules:none

s = APP_IN evaluates to t =APP_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_IN to APP_IN.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

U2¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → PERM_IN(Zs, Ys)
U1¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2¹(Xs, X, Ys, app_in(X1s, X2s, Zs))
PERM_IN(Xs, .(X, Ys)) → U1¹(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

perm_in([], []) → perm_out([], [])
perm_in(Xs, .(X, Ys)) → U1(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U1(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2(Xs, X, Ys, app_in(X1s, X2s, Zs))
U2(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U3(Xs, X, Ys, perm_in(Zs, Ys))
U3(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in(x1, x2) = perm_in
[] = []
perm_out(x1, x2) = perm_out(x2)
.(x1, x2) = .(x2)
U1(x1, x2, x3, x4) = U1(x4)
app_in(x1, x2, x3) = app_in
app_out(x1, x2, x3) = app_out(x1)
U4(x1, x2, x3, x4, x5) = U4(x5)
U2(x1, x2, x3, x4) = U2(x4)
U3(x1, x2, x3, x4) = U3(x4)
U2¹(x1, x2, x3, x4) = U2¹(x4)
PERM_IN(x1, x2) = PERM_IN
U1¹(x1, x2, x3, x4) = U1¹(x4)

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:

U2¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → PERM_IN(Zs, Ys)
U1¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U2¹(Xs, X, Ys, app_in(X1s, X2s, Zs))
PERM_IN(Xs, .(X, Ys)) → U1¹(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

app_in([], Ys, Ys) → app_out([], Ys, Ys)
app_in(.(X, Xs), Ys, .(X, Zs)) → U4(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
U4(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

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

U2¹(app_out(X1s)) → PERM_IN
U1¹(app_out(X1s)) → U2¹(app_in)
PERM_IN → U1¹(app_in)

The TRS R consists of the following rules:

app_in → app_out([])
app_in → U4(app_in)
U4(app_out(Xs)) → app_out(.(Xs))

The set Q consists of the following terms:

app_in
U4(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1¹(app_out(X1s)) → U2¹(app_in) at position [0] we obtained the following new rules:

U1¹(app_out(y0)) → U2¹(app_out([]))
U1¹(app_out(y0)) → U2¹(U4(app_in))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

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

U2¹(app_out(X1s)) → PERM_IN
U1¹(app_out(y0)) → U2¹(app_out([]))
U1¹(app_out(y0)) → U2¹(U4(app_in))
PERM_IN → U1¹(app_in)

The TRS R consists of the following rules:

app_in → app_out([])
app_in → U4(app_in)
U4(app_out(Xs)) → app_out(.(Xs))

The set Q consists of the following terms:

app_in
U4(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PERM_IN → U1¹(app_in) at position [0] we obtained the following new rules:

PERM_IN → U1¹(app_out([]))
PERM_IN → U1¹(U4(app_in))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ NonTerminationProof

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

PERM_IN → U1¹(app_out([]))
U2¹(app_out(X1s)) → PERM_IN
PERM_IN → U1¹(U4(app_in))
U1¹(app_out(y0)) → U2¹(app_out([]))
U1¹(app_out(y0)) → U2¹(U4(app_in))

The TRS R consists of the following rules:

app_in → app_out([])
app_in → U4(app_in)
U4(app_out(Xs)) → app_out(.(Xs))

The set Q consists of the following terms:

app_in
U4(x0)

PERM_IN → U1¹(app_out([]))
U2¹(app_out(X1s)) → PERM_IN
PERM_IN → U1¹(U4(app_in))
U1¹(app_out(y0)) → U2¹(app_out([]))
U1¹(app_out(y0)) → U2¹(U4(app_in))

The TRS R consists of the following rules:

app_in → app_out([])
app_in → U4(app_in)
U4(app_out(Xs)) → app_out(.(Xs))

s = U1¹(app_out(y0)) evaluates to t =U1¹(app_out([]))

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

Matcher: [y0 / []]
Semiunifier: [ ]