Left Termination proof of /tmp/tmpYXgMdA/slowsort-fb.pl

Left Termination of the query pattern ss_in_2(a, g) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

ss(Xs, Ys) :- ','(perm(Xs, Ys), ordered(Ys)).
perm([], []).
perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
ordered([]).
ordered(.(X, [])).
ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs))).
less(0, s(X)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

ss(a,g).

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:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, 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:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

SS_IN(Xs, Ys) → U1¹(Xs, Ys, perm_in(Xs, Ys))
SS_IN(Xs, Ys) → PERM_IN(Xs, Ys)
PERM_IN(Xs, .(X, Ys)) → U3¹(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)) → U6¹(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
U3¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4¹(Xs, X, Ys, app_in(X1s, X2s, Zs))
U3¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → APP_IN(X1s, X2s, Zs)
U4¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5¹(Xs, X, Ys, perm_in(Zs, Ys))
U4¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → PERM_IN(Zs, Ys)
U1¹(Xs, Ys, perm_out(Xs, Ys)) → U2¹(Xs, Ys, ordered_in(Ys))
U1¹(Xs, Ys, perm_out(Xs, Ys)) → ORDERED_IN(Ys)
ORDERED_IN(.(X, .(Y, Xs))) → U7¹(X, Y, Xs, less_in(X, s(Y)))
ORDERED_IN(.(X, .(Y, Xs))) → LESS_IN(X, s(Y))
LESS_IN(s(X), s(Y)) → U9¹(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U7¹(X, Y, Xs, less_out(X, s(Y))) → U8¹(X, Y, Xs, ordered_in(.(Y, Xs)))
U7¹(X, Y, Xs, less_out(X, s(Y))) → ORDERED_IN(.(Y, Xs))

The TRS R consists of the following rules:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in(x1, x2) = ss_in(x2)
U1(x1, x2, x3) = U1(x2, x3)
perm_in(x1, x2) = perm_in(x2)
.(x1, x2) = .(x1, x2)
U3(x1, x2, x3, x4) = U3(x3, x4)
app_in(x1, x2, x3) = app_in
U6(x1, x2, x3, x4, x5) = U6(x5)
[] = []
app_out(x1, x2, x3) = app_out
U4(x1, x2, x3, x4) = U4(x3, x4)
U5(x1, x2, x3, x4) = U5(x4)
perm_out(x1, x2) = perm_out
U2(x1, x2, x3) = U2(x3)
ordered_in(x1) = ordered_in(x1)
U7(x1, x2, x3, x4) = U7(x2, x3, x4)
less_in(x1, x2) = less_in(x1, x2)
s(x1) = s(x1)
U9(x1, x2, x3) = U9(x3)
0 = 0
less_out(x1, x2) = less_out
U8(x1, x2, x3, x4) = U8(x4)
ordered_out(x1) = ordered_out
ss_out(x1, x2) = ss_out
U4¹(x1, x2, x3, x4) = U4¹(x3, x4)
ORDERED_IN(x1) = ORDERED_IN(x1)
PERM_IN(x1, x2) = PERM_IN(x2)
U9¹(x1, x2, x3) = U9¹(x3)
U6¹(x1, x2, x3, x4, x5) = U6¹(x5)
U7¹(x1, x2, x3, x4) = U7¹(x2, x3, x4)
U1¹(x1, x2, x3) = U1¹(x2, x3)
APP_IN(x1, x2, x3) = APP_IN
SS_IN(x1, x2) = SS_IN(x2)
LESS_IN(x1, x2) = LESS_IN(x1, x2)
U5¹(x1, x2, x3, x4) = U5¹(x4)
U3¹(x1, x2, x3, x4) = U3¹(x3, x4)
U8¹(x1, x2, x3, x4) = U8¹(x4)
U2¹(x1, x2, x3) = U2¹(x3)

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:

SS_IN(Xs, Ys) → U1¹(Xs, Ys, perm_in(Xs, Ys))
SS_IN(Xs, Ys) → PERM_IN(Xs, Ys)
PERM_IN(Xs, .(X, Ys)) → U3¹(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)) → U6¹(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
U3¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4¹(Xs, X, Ys, app_in(X1s, X2s, Zs))
U3¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → APP_IN(X1s, X2s, Zs)
U4¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5¹(Xs, X, Ys, perm_in(Zs, Ys))
U4¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → PERM_IN(Zs, Ys)
U1¹(Xs, Ys, perm_out(Xs, Ys)) → U2¹(Xs, Ys, ordered_in(Ys))
U1¹(Xs, Ys, perm_out(Xs, Ys)) → ORDERED_IN(Ys)
ORDERED_IN(.(X, .(Y, Xs))) → U7¹(X, Y, Xs, less_in(X, s(Y)))
ORDERED_IN(.(X, .(Y, Xs))) → LESS_IN(X, s(Y))
LESS_IN(s(X), s(Y)) → U9¹(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U7¹(X, Y, Xs, less_out(X, s(Y))) → U8¹(X, Y, Xs, ordered_in(.(Y, Xs)))
U7¹(X, Y, Xs, less_out(X, s(Y))) → ORDERED_IN(.(Y, Xs))

The TRS R consists of the following rules:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

The TRS R consists of the following rules:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

R is empty.
Pi is empty.
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

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

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:

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U7¹(X, Y, Xs, less_out(X, s(Y))) → ORDERED_IN(.(Y, Xs))
ORDERED_IN(.(X, .(Y, Xs))) → U7¹(X, Y, Xs, less_in(X, s(Y)))

The TRS R consists of the following rules:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U7¹(X, Y, Xs, less_out(X, s(Y))) → ORDERED_IN(.(Y, Xs))
ORDERED_IN(.(X, .(Y, Xs))) → U7¹(X, Y, Xs, less_in(X, s(Y)))

The TRS R consists of the following rules:

less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
less_in(x1, x2) = less_in(x1, x2)
s(x1) = s(x1)
U9(x1, x2, x3) = U9(x3)
0 = 0
less_out(x1, x2) = less_out
ORDERED_IN(x1) = ORDERED_IN(x1)
U7¹(x1, x2, x3, x4) = U7¹(x2, x3, 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

                        ↳ UsableRulesReductionPairsProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U7¹(Y, Xs, less_out) → ORDERED_IN(.(Y, Xs))
ORDERED_IN(.(X, .(Y, Xs))) → U7¹(Y, Xs, less_in(X, s(Y)))

The TRS R consists of the following rules:

less_in(s(X), s(Y)) → U9(less_in(X, Y))
less_in(0, s(X)) → less_out
U9(less_out) → less_out

The set Q consists of the following terms:

less_in(x0, x1)
U9(x0)

We have to consider all (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

U7¹(Y, Xs, less_out) → ORDERED_IN(.(Y, Xs))

The following rules are removed from R:

less_in(0, s(X)) → less_out

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁, x₂)) = x₁ + 2·x₂
POL(0) = 2
POL(ORDERED_IN(x₁)) = x₁
POL(U7¹(x₁, x₂, x₃)) = x₁ + 2·x₂ + x₃
POL(U9(x₁)) = x₁
POL(less_in(x₁, x₂)) = x₁ + x₂
POL(less_out) = 2
POL(s(x₁)) = x₁

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ DependencyGraphProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

ORDERED_IN(.(X, .(Y, Xs))) → U7¹(Y, Xs, less_in(X, s(Y)))

The TRS R consists of the following rules:

less_in(s(X), s(Y)) → U9(less_in(X, Y))
U9(less_out) → less_out

The set Q consists of the following terms:

less_in(x0, x1)
U9(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ 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:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ 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) = .(x1, 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

              ↳ PiDP

              ↳ 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

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(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) = .(x1, x2)
app_in(x1, x2, x3) = app_in
U6(x1, x2, x3, x4, x5) = U6(x5)
[] = []
app_out(x1, x2, x3) = app_out
U4¹(x1, x2, x3, x4) = U4¹(x3, x4)
PERM_IN(x1, x2) = PERM_IN(x2)
U3¹(x1, x2, x3, x4) = U3¹(x3, 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

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

  ↳ PrologToPiTRSProof

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

U3¹(Ys, app_out) → U4¹(Ys, app_in)
U4¹(Ys, app_out) → PERM_IN(Ys)
PERM_IN(.(X, Ys)) → U3¹(Ys, app_in)

The TRS R consists of the following rules:

app_in → U6(app_in)
app_in → app_out
U6(app_out) → app_out

The set Q consists of the following terms:

app_in
U6(x0)

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:

U4¹(Ys, app_out) → PERM_IN(Ys)
The graph contains the following edges 1 >= 1
PERM_IN(.(X, Ys)) → U3¹(Ys, app_in)
The graph contains the following edges 1 > 1
U3¹(Ys, app_out) → U4¹(Ys, app_in)
The graph contains the following edges 1 >= 1

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:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, 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:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

SS_IN(Xs, Ys) → U1¹(Xs, Ys, perm_in(Xs, Ys))
SS_IN(Xs, Ys) → PERM_IN(Xs, Ys)
PERM_IN(Xs, .(X, Ys)) → U3¹(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)) → U6¹(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
U3¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4¹(Xs, X, Ys, app_in(X1s, X2s, Zs))
U3¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → APP_IN(X1s, X2s, Zs)
U4¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5¹(Xs, X, Ys, perm_in(Zs, Ys))
U4¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → PERM_IN(Zs, Ys)
U1¹(Xs, Ys, perm_out(Xs, Ys)) → U2¹(Xs, Ys, ordered_in(Ys))
U1¹(Xs, Ys, perm_out(Xs, Ys)) → ORDERED_IN(Ys)
ORDERED_IN(.(X, .(Y, Xs))) → U7¹(X, Y, Xs, less_in(X, s(Y)))
ORDERED_IN(.(X, .(Y, Xs))) → LESS_IN(X, s(Y))
LESS_IN(s(X), s(Y)) → U9¹(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U7¹(X, Y, Xs, less_out(X, s(Y))) → U8¹(X, Y, Xs, ordered_in(.(Y, Xs)))
U7¹(X, Y, Xs, less_out(X, s(Y))) → ORDERED_IN(.(Y, Xs))

The TRS R consists of the following rules:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in(x1, x2) = ss_in(x2)
U1(x1, x2, x3) = U1(x2, x3)
perm_in(x1, x2) = perm_in(x2)
.(x1, x2) = .(x1, x2)
U3(x1, x2, x3, x4) = U3(x2, x3, x4)
app_in(x1, x2, x3) = app_in
U6(x1, x2, x3, x4, x5) = U6(x5)
[] = []
app_out(x1, x2, x3) = app_out
U4(x1, x2, x3, x4) = U4(x2, x3, x4)
U5(x1, x2, x3, x4) = U5(x2, x3, x4)
perm_out(x1, x2) = perm_out(x2)
U2(x1, x2, x3) = U2(x2, x3)
ordered_in(x1) = ordered_in(x1)
U7(x1, x2, x3, x4) = U7(x1, x2, x3, x4)
less_in(x1, x2) = less_in(x1, x2)
s(x1) = s(x1)
U9(x1, x2, x3) = U9(x1, x2, x3)
0 = 0
less_out(x1, x2) = less_out(x1, x2)
U8(x1, x2, x3, x4) = U8(x1, x2, x3, x4)
ordered_out(x1) = ordered_out(x1)
ss_out(x1, x2) = ss_out(x2)
U4¹(x1, x2, x3, x4) = U4¹(x2, x3, x4)
ORDERED_IN(x1) = ORDERED_IN(x1)
PERM_IN(x1, x2) = PERM_IN(x2)
U9¹(x1, x2, x3) = U9¹(x1, x2, x3)
U6¹(x1, x2, x3, x4, x5) = U6¹(x5)
U7¹(x1, x2, x3, x4) = U7¹(x1, x2, x3, x4)
U1¹(x1, x2, x3) = U1¹(x2, x3)
APP_IN(x1, x2, x3) = APP_IN
SS_IN(x1, x2) = SS_IN(x2)
LESS_IN(x1, x2) = LESS_IN(x1, x2)
U5¹(x1, x2, x3, x4) = U5¹(x2, x3, x4)
U3¹(x1, x2, x3, x4) = U3¹(x2, x3, x4)
U8¹(x1, x2, x3, x4) = U8¹(x1, x2, x3, x4)
U2¹(x1, x2, x3) = U2¹(x2, x3)

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:

SS_IN(Xs, Ys) → U1¹(Xs, Ys, perm_in(Xs, Ys))
SS_IN(Xs, Ys) → PERM_IN(Xs, Ys)
PERM_IN(Xs, .(X, Ys)) → U3¹(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)) → U6¹(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
U3¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4¹(Xs, X, Ys, app_in(X1s, X2s, Zs))
U3¹(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → APP_IN(X1s, X2s, Zs)
U4¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5¹(Xs, X, Ys, perm_in(Zs, Ys))
U4¹(Xs, X, Ys, app_out(X1s, X2s, Zs)) → PERM_IN(Zs, Ys)
U1¹(Xs, Ys, perm_out(Xs, Ys)) → U2¹(Xs, Ys, ordered_in(Ys))
U1¹(Xs, Ys, perm_out(Xs, Ys)) → ORDERED_IN(Ys)
ORDERED_IN(.(X, .(Y, Xs))) → U7¹(X, Y, Xs, less_in(X, s(Y)))
ORDERED_IN(.(X, .(Y, Xs))) → LESS_IN(X, s(Y))
LESS_IN(s(X), s(Y)) → U9¹(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U7¹(X, Y, Xs, less_out(X, s(Y))) → U8¹(X, Y, Xs, ordered_in(.(Y, Xs)))
U7¹(X, Y, Xs, less_out(X, s(Y))) → ORDERED_IN(.(Y, Xs))

The TRS R consists of the following rules:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in(x1, x2) = ss_in(x2)
U1(x1, x2, x3) = U1(x2, x3)
perm_in(x1, x2) = perm_in(x2)
.(x1, x2) = .(x1, x2)
U3(x1, x2, x3, x4) = U3(x2, x3, x4)
app_in(x1, x2, x3) = app_in
U6(x1, x2, x3, x4, x5) = U6(x5)
[] = []
app_out(x1, x2, x3) = app_out
U4(x1, x2, x3, x4) = U4(x2, x3, x4)
U5(x1, x2, x3, x4) = U5(x2, x3, x4)
perm_out(x1, x2) = perm_out(x2)
U2(x1, x2, x3) = U2(x2, x3)
ordered_in(x1) = ordered_in(x1)
U7(x1, x2, x3, x4) = U7(x1, x2, x3, x4)
less_in(x1, x2) = less_in(x1, x2)
s(x1) = s(x1)
U9(x1, x2, x3) = U9(x1, x2, x3)
0 = 0
less_out(x1, x2) = less_out(x1, x2)
U8(x1, x2, x3, x4) = U8(x1, x2, x3, x4)
ordered_out(x1) = ordered_out(x1)
ss_out(x1, x2) = ss_out(x2)
U4¹(x1, x2, x3, x4) = U4¹(x2, x3, x4)
ORDERED_IN(x1) = ORDERED_IN(x1)
PERM_IN(x1, x2) = PERM_IN(x2)
U9¹(x1, x2, x3) = U9¹(x1, x2, x3)
U6¹(x1, x2, x3, x4, x5) = U6¹(x5)
U7¹(x1, x2, x3, x4) = U7¹(x1, x2, x3, x4)
U1¹(x1, x2, x3) = U1¹(x2, x3)
APP_IN(x1, x2, x3) = APP_IN
SS_IN(x1, x2) = SS_IN(x2)
LESS_IN(x1, x2) = LESS_IN(x1, x2)
U5¹(x1, x2, x3, x4) = U5¹(x2, x3, x4)
U3¹(x1, x2, x3, x4) = U3¹(x2, x3, x4)
U8¹(x1, x2, x3, x4) = U8¹(x1, x2, x3, x4)
U2¹(x1, x2, x3) = U2¹(x2, x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 4 SCCs with 11 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

The TRS R consists of the following rules:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

R is empty.
Pi is empty.
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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

U7¹(X, Y, Xs, less_out(X, s(Y))) → ORDERED_IN(.(Y, Xs))
ORDERED_IN(.(X, .(Y, Xs))) → U7¹(X, Y, Xs, less_in(X, s(Y)))

The TRS R consists of the following rules:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

U7¹(X, Y, Xs, less_out(X, s(Y))) → ORDERED_IN(.(Y, Xs))
ORDERED_IN(.(X, .(Y, Xs))) → U7¹(X, Y, Xs, less_in(X, s(Y)))

The TRS R consists of the following rules:

less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))

Pi is empty.
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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

              ↳ PiDP

              ↳ PiDP

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

U7¹(X, Y, Xs, less_out(X, s(Y))) → ORDERED_IN(.(Y, Xs))
ORDERED_IN(.(X, .(Y, Xs))) → U7¹(X, Y, Xs, less_in(X, s(Y)))

The TRS R consists of the following rules:

less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))

The set Q consists of the following terms:

less_in(x0, x1)
U9(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

ORDERED_IN(.(X, .(Y, Xs))) → U7¹(X, Y, Xs, less_in(X, s(Y)))

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 2·x₁ + x₂
POL(0) = 2
POL(ORDERED_IN(x₁)) = 2 + 2·x₁
POL(U7¹(x₁, x₂, x₃, x₄)) = x₁ + 2·x₂ + 2·x₃ + x₄
POL(U9(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃
POL(less_in(x₁, x₂)) = 2·x₁ + x₂
POL(less_out(x₁, x₂)) = 2 + x₁ + x₂
POL(s(x₁)) = 2·x₁

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ DependencyGraphProof

              ↳ PiDP

              ↳ PiDP

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

U7¹(X, Y, Xs, less_out(X, s(Y))) → ORDERED_IN(.(Y, Xs))

The TRS R consists of the following rules:

less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))

The set Q consists of the following terms:

less_in(x0, x1)
U9(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ 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:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ 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

              ↳ PiDP

              ↳ 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

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

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

The TRS R consists of the following rules:

ss_in(Xs, Ys) → U1(Xs, Ys, perm_in(Xs, Ys))
perm_in(Xs, .(X, Ys)) → U3(Xs, X, Ys, app_in(X1s, .(X, X2s), Xs))
app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U3(Xs, X, Ys, app_out(X1s, .(X, X2s), Xs)) → U4(Xs, X, Ys, app_in(X1s, X2s, Zs))
U4(Xs, X, Ys, app_out(X1s, X2s, Zs)) → U5(Xs, X, Ys, perm_in(Zs, Ys))
perm_in([], []) → perm_out([], [])
U5(Xs, X, Ys, perm_out(Zs, Ys)) → perm_out(Xs, .(X, Ys))
U1(Xs, Ys, perm_out(Xs, Ys)) → U2(Xs, Ys, ordered_in(Ys))
ordered_in(.(X, .(Y, Xs))) → U7(X, Y, Xs, less_in(X, s(Y)))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U7(X, Y, Xs, less_out(X, s(Y))) → U8(X, Y, Xs, ordered_in(.(Y, Xs)))
ordered_in(.(X, [])) → ordered_out(.(X, []))
ordered_in([]) → ordered_out([])
U8(X, Y, Xs, ordered_out(.(Y, Xs))) → ordered_out(.(X, .(Y, Xs)))
U2(Xs, Ys, ordered_out(Ys)) → ss_out(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in(x1, x2) = ss_in(x2)
U1(x1, x2, x3) = U1(x2, x3)
perm_in(x1, x2) = perm_in(x2)
.(x1, x2) = .(x1, x2)
U3(x1, x2, x3, x4) = U3(x2, x3, x4)
app_in(x1, x2, x3) = app_in
U6(x1, x2, x3, x4, x5) = U6(x5)
[] = []
app_out(x1, x2, x3) = app_out
U4(x1, x2, x3, x4) = U4(x2, x3, x4)
U5(x1, x2, x3, x4) = U5(x2, x3, x4)
perm_out(x1, x2) = perm_out(x2)
U2(x1, x2, x3) = U2(x2, x3)
ordered_in(x1) = ordered_in(x1)
U7(x1, x2, x3, x4) = U7(x1, x2, x3, x4)
less_in(x1, x2) = less_in(x1, x2)
s(x1) = s(x1)
U9(x1, x2, x3) = U9(x1, x2, x3)
0 = 0
less_out(x1, x2) = less_out(x1, x2)
U8(x1, x2, x3, x4) = U8(x1, x2, x3, x4)
ordered_out(x1) = ordered_out(x1)
ss_out(x1, x2) = ss_out(x2)
U4¹(x1, x2, x3, x4) = U4¹(x2, x3, x4)
PERM_IN(x1, x2) = PERM_IN(x2)
U3¹(x1, x2, x3, x4) = U3¹(x2, x3, 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

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

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

The TRS R consists of the following rules:

app_in(.(X, Xs), Ys, .(X, Zs)) → U6(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U6(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) = .(x1, x2)
app_in(x1, x2, x3) = app_in
U6(x1, x2, x3, x4, x5) = U6(x5)
[] = []
app_out(x1, x2, x3) = app_out
U4¹(x1, x2, x3, x4) = U4¹(x2, x3, x4)
PERM_IN(x1, x2) = PERM_IN(x2)
U3¹(x1, x2, x3, x4) = U3¹(x2, x3, 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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

PERM_IN(.(X, Ys)) → U3¹(X, Ys, app_in)
U3¹(X, Ys, app_out) → U4¹(X, Ys, app_in)
U4¹(X, Ys, app_out) → PERM_IN(Ys)

The TRS R consists of the following rules:

app_in → U6(app_in)
app_in → app_out
U6(app_out) → app_out

The set Q consists of the following terms:

app_in
U6(x0)

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:

U4¹(X, Ys, app_out) → PERM_IN(Ys)
The graph contains the following edges 2 >= 1
PERM_IN(.(X, Ys)) → U3¹(X, Ys, app_in)
The graph contains the following edges 1 > 1, 1 > 2
U3¹(X, Ys, app_out) → U4¹(X, Ys, app_in)
The graph contains the following edges 1 >= 1, 2 >= 2