Left Termination proof of /tmp/tmpcBf2Qh/quicksort-bf.pl

Left Termination of the query pattern qs_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

qs([], []).
qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))).
part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(less(X, Y), part(X, Xs, Ls, Bs)).
part(X, .(Y, Xs), Ls, .(Y, Bs)) :- part(X, Xs, Ls, Bs).
part(X, [], [], []).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
less(0, s(X)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

qs(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:

qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, 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))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, 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))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)

QS_IN(.(X, Xs), Ys) → U1¹(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
QS_IN(.(X, Xs), Ys) → PART_IN(X, Xs, Littles, Bigs)
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → U7¹(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN(X, Xs, Ls, Bs)
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → U5¹(X, Y, Xs, Ls, Bs, less_in(X, Y))
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN(X, Y)
LESS_IN(s(X), s(Y)) → U9¹(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U5¹(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6¹(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U5¹(X, Y, Xs, Ls, Bs, less_out(X, Y)) → PART_IN(X, Xs, Ls, Bs)
U1¹(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2¹(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
U1¹(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → QS_IN(Littles, Ls)
U2¹(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3¹(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U2¹(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → QS_IN(Bigs, Bs)
U3¹(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4¹(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
U3¹(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → APP_IN(Ls, .(X, Bs), Ys)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U8¹(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, 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))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in(x1, x2) = qs_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4) = U1(x1, x4)
part_in(x1, x2, x3, x4) = part_in(x1, x2)
[] = []
part_out(x1, x2, x3, x4) = part_out(x3, x4)
U7(x1, x2, x3, x4, x5, x6) = U7(x2, x6)
U5(x1, x2, x3, x4, x5, x6) = U5(x1, x2, x3, x6)
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
U6(x1, x2, x3, x4, x5, x6) = U6(x2, x6)
U2(x1, x2, x3, x4, x5) = U2(x1, x4, x5)
qs_out(x1, x2) = qs_out(x2)
U3(x1, x2, x3, x4, x5) = U3(x1, x4, x5)
U4(x1, x2, x3, x4) = U4(x4)
app_in(x1, x2, x3) = app_in(x1, x2)
U8(x1, x2, x3, x4, x5) = U8(x1, x5)
app_out(x1, x2, x3) = app_out(x3)
U5¹(x1, x2, x3, x4, x5, x6) = U5¹(x1, x2, x3, x6)
U4¹(x1, x2, x3, x4) = U4¹(x4)
U3¹(x1, x2, x3, x4, x5) = U3¹(x1, x4, x5)
PART_IN(x1, x2, x3, x4) = PART_IN(x1, x2)
U9¹(x1, x2, x3) = U9¹(x3)
QS_IN(x1, x2) = QS_IN(x1)
U7¹(x1, x2, x3, x4, x5, x6) = U7¹(x2, x6)
APP_IN(x1, x2, x3) = APP_IN(x1, x2)
LESS_IN(x1, x2) = LESS_IN(x1, x2)
U8¹(x1, x2, x3, x4, x5) = U8¹(x1, x5)
U2¹(x1, x2, x3, x4, x5) = U2¹(x1, x4, x5)
U1¹(x1, x2, x3, x4) = U1¹(x1, x4)
U6¹(x1, x2, x3, x4, x5, x6) = U6¹(x2, x6)

We have to consider all (P,R,Pi)-chains

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

QS_IN(.(X, Xs), Ys) → U1¹(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
QS_IN(.(X, Xs), Ys) → PART_IN(X, Xs, Littles, Bigs)
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → U7¹(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN(X, Xs, Ls, Bs)
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → U5¹(X, Y, Xs, Ls, Bs, less_in(X, Y))
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN(X, Y)
LESS_IN(s(X), s(Y)) → U9¹(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U5¹(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6¹(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U5¹(X, Y, Xs, Ls, Bs, less_out(X, Y)) → PART_IN(X, Xs, Ls, Bs)
U1¹(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2¹(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
U1¹(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → QS_IN(Littles, Ls)
U2¹(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3¹(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U2¹(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → QS_IN(Bigs, Bs)
U3¹(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4¹(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
U3¹(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → APP_IN(Ls, .(X, Bs), Ys)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U8¹(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, 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))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ 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:

qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, 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))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

APP_IN(.(X, Xs), Ys) → APP_IN(Xs, Ys)

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:

APP_IN(.(X, Xs), Ys) → APP_IN(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ 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:

qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, 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))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ 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

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ 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

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN(X, Xs, Ls, Bs)
U5¹(X, Y, Xs, Ls, Bs, less_out(X, Y)) → PART_IN(X, Xs, Ls, Bs)
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → U5¹(X, Y, Xs, Ls, Bs, less_in(X, Y))

The TRS R consists of the following rules:

qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, 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))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN(X, Xs, Ls, Bs)
U5¹(X, Y, Xs, Ls, Bs, less_out(X, Y)) → PART_IN(X, Xs, Ls, Bs)
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → U5¹(X, Y, Xs, Ls, Bs, less_in(X, 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
U5¹(x1, x2, x3, x4, x5, x6) = U5¹(x1, x2, x3, x6)
PART_IN(x1, x2, x3, x4) = PART_IN(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

PART_IN(X, .(Y, Xs)) → PART_IN(X, Xs)
PART_IN(X, .(Y, Xs)) → U5¹(X, Y, Xs, less_in(X, Y))
U5¹(X, Y, Xs, less_out) → PART_IN(X, Xs)

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 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:

PART_IN(X, .(Y, Xs)) → PART_IN(X, Xs)
The graph contains the following edges 1 >= 1, 2 > 2
PART_IN(X, .(Y, Xs)) → U5¹(X, Y, Xs, less_in(X, Y))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
U5¹(X, Y, Xs, less_out) → PART_IN(X, Xs)
The graph contains the following edges 1 >= 1, 3 >= 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

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

U1¹(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2¹(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
U1¹(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → QS_IN(Littles, Ls)
QS_IN(.(X, Xs), Ys) → U1¹(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
U2¹(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → QS_IN(Bigs, Bs)

The TRS R consists of the following rules:

qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, 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))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in(x1, x2) = qs_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4) = U1(x1, x4)
part_in(x1, x2, x3, x4) = part_in(x1, x2)
[] = []
part_out(x1, x2, x3, x4) = part_out(x3, x4)
U7(x1, x2, x3, x4, x5, x6) = U7(x2, x6)
U5(x1, x2, x3, x4, x5, x6) = U5(x1, x2, x3, x6)
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
U6(x1, x2, x3, x4, x5, x6) = U6(x2, x6)
U2(x1, x2, x3, x4, x5) = U2(x1, x4, x5)
qs_out(x1, x2) = qs_out(x2)
U3(x1, x2, x3, x4, x5) = U3(x1, x4, x5)
U4(x1, x2, x3, x4) = U4(x4)
app_in(x1, x2, x3) = app_in(x1, x2)
U8(x1, x2, x3, x4, x5) = U8(x1, x5)
app_out(x1, x2, x3) = app_out(x3)
QS_IN(x1, x2) = QS_IN(x1)
U2¹(x1, x2, x3, x4, x5) = U2¹(x1, x4, x5)
U1¹(x1, x2, x3, x4) = U1¹(x1, 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

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPOrderProof

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

U2¹(X, Bigs, qs_out(Ls)) → QS_IN(Bigs)
QS_IN(.(X, Xs)) → U1¹(X, part_in(X, Xs))
U1¹(X, part_out(Littles, Bigs)) → QS_IN(Littles)
U1¹(X, part_out(Littles, Bigs)) → U2¹(X, Bigs, qs_in(Littles))

The TRS R consists of the following rules:

qs_in(.(X, Xs)) → U1(X, part_in(X, Xs))
part_in(X, []) → part_out([], [])
part_in(X, .(Y, Xs)) → U7(Y, part_in(X, Xs))
part_in(X, .(Y, Xs)) → U5(X, Y, Xs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(less_in(X, Y))
less_in(0, s(X)) → less_out
U9(less_out) → less_out
U5(X, Y, Xs, less_out) → U6(Y, part_in(X, Xs))
U6(Y, part_out(Ls, Bs)) → part_out(.(Y, Ls), Bs)
U7(Y, part_out(Ls, Bs)) → part_out(Ls, .(Y, Bs))
U1(X, part_out(Littles, Bigs)) → U2(X, Bigs, qs_in(Littles))
qs_in([]) → qs_out([])
U2(X, Bigs, qs_out(Ls)) → U3(X, Ls, qs_in(Bigs))
U3(X, Ls, qs_out(Bs)) → U4(app_in(Ls, .(X, Bs)))
app_in(.(X, Xs), Ys) → U8(X, app_in(Xs, Ys))
app_in([], X) → app_out(X)
U8(X, app_out(Zs)) → app_out(.(X, Zs))
U4(app_out(Ys)) → qs_out(Ys)

The set Q consists of the following terms:

qs_in(x0)
part_in(x0, x1)
less_in(x0, x1)
U9(x0)
U5(x0, x1, x2, x3)
U6(x0, x1)
U7(x0, x1)
U1(x0, x1)
U2(x0, x1, x2)
U3(x0, x1, x2)
app_in(x0, x1)
U8(x0, x1)
U4(x0)

We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

QS_IN(.(X, Xs)) → U1¹(X, part_in(X, Xs))

The remaining pairs can at least be oriented weakly.

U2¹(X, Bigs, qs_out(Ls)) → QS_IN(Bigs)
U1¹(X, part_out(Littles, Bigs)) → QS_IN(Littles)
U1¹(X, part_out(Littles, Bigs)) → U2¹(X, Bigs, qs_in(Littles))

Used ordering: Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(QS_IN(x₁)) = x₁
POL(U1(x₁, x₂)) = 0
POL(U1¹(x₁, x₂)) = x₂
POL(U2(x₁, x₂, x₃)) = 0
POL(U2¹(x₁, x₂, x₃)) = x₂
POL(U3(x₁, x₂, x₃)) = 0
POL(U4(x₁)) = 0
POL(U5(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U6(x₁, x₂)) = 1 + x₂
POL(U7(x₁, x₂)) = 1 + x₂
POL(U8(x₁, x₂)) = 0
POL(U9(x₁)) = 0
POL([]) = 0
POL(app_in(x₁, x₂)) = 1
POL(app_out(x₁)) = 0
POL(less_in(x₁, x₂)) = 0
POL(less_out) = 0
POL(part_in(x₁, x₂)) = x₂
POL(part_out(x₁, x₂)) = x₁ + x₂
POL(qs_in(x₁)) = 0
POL(qs_out(x₁)) = 0
POL(s(x₁)) = 0

The following usable rules [17] were oriented:

U7(Y, part_out(Ls, Bs)) → part_out(Ls, .(Y, Bs))
U6(Y, part_out(Ls, Bs)) → part_out(.(Y, Ls), Bs)
part_in(X, .(Y, Xs)) → U5(X, Y, Xs, less_in(X, Y))
U5(X, Y, Xs, less_out) → U6(Y, part_in(X, Xs))
part_in(X, .(Y, Xs)) → U7(Y, part_in(X, Xs))
part_in(X, []) → part_out([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

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

U2¹(X, Bigs, qs_out(Ls)) → QS_IN(Bigs)
U1¹(X, part_out(Littles, Bigs)) → U2¹(X, Bigs, qs_in(Littles))
U1¹(X, part_out(Littles, Bigs)) → QS_IN(Littles)

The TRS R consists of the following rules:

qs_in(.(X, Xs)) → U1(X, part_in(X, Xs))
part_in(X, []) → part_out([], [])
part_in(X, .(Y, Xs)) → U7(Y, part_in(X, Xs))
part_in(X, .(Y, Xs)) → U5(X, Y, Xs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(less_in(X, Y))
less_in(0, s(X)) → less_out
U9(less_out) → less_out
U5(X, Y, Xs, less_out) → U6(Y, part_in(X, Xs))
U6(Y, part_out(Ls, Bs)) → part_out(.(Y, Ls), Bs)
U7(Y, part_out(Ls, Bs)) → part_out(Ls, .(Y, Bs))
U1(X, part_out(Littles, Bigs)) → U2(X, Bigs, qs_in(Littles))
qs_in([]) → qs_out([])
U2(X, Bigs, qs_out(Ls)) → U3(X, Ls, qs_in(Bigs))
U3(X, Ls, qs_out(Bs)) → U4(app_in(Ls, .(X, Bs)))
app_in(.(X, Xs), Ys) → U8(X, app_in(Xs, Ys))
app_in([], X) → app_out(X)
U8(X, app_out(Zs)) → app_out(.(X, Zs))
U4(app_out(Ys)) → qs_out(Ys)

The set Q consists of the following terms:

qs_in(x0)
part_in(x0, x1)
less_in(x0, x1)
U9(x0)
U5(x0, x1, x2, x3)
U6(x0, x1)
U7(x0, x1)
U1(x0, x1)
U2(x0, x1, x2)
U3(x0, x1, x2)
app_in(x0, x1)
U8(x0, x1)
U4(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.