Left Termination proof of /tmp/tmpJSHYW7/pl8.3.1a.pl

Left Termination of the query pattern minsort_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

minsort([], []).
minsort(L, .(X, L1)) :- ','(min1(X, L), ','(remove(X, L, L2), minsort(L2, L1))).
min1(M, .(X, L)) :- min2(X, M, L).
min2(X, X, []).
min2(X, A, .(M, L)) :- ','(min(X, M, B), min2(B, A, L)).
min(X, Y, X) :- le(X, Y).
min(X, Y, Y) :- gt(X, Y).
remove(N, .(N, L), L).
remove(N, .(M, L), .(M, L1)) :- ','(notEq(N, M), remove(N, L, L1)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).
notEq(s(X), s(Y)) :- notEq(X, Y).
notEq(s(X), 0).
notEq(0, s(X)).

Queries:

minsort(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:

minsort_in(L, .(X, L1)) → U1(L, X, L1, min1_in(X, L))
min1_in(M, .(X, L)) → U4(M, X, L, min2_in(X, M, L))
min2_in(X, A, .(M, L)) → U5(X, A, M, L, min_in(X, M, B))
min_in(X, Y, Y) → U8(X, Y, gt_in(X, Y))
gt_in(s(X), 0) → gt_out(s(X), 0)
gt_in(s(X), s(Y)) → U11(X, Y, gt_in(X, Y))
U11(X, Y, gt_out(X, Y)) → gt_out(s(X), s(Y))
U8(X, Y, gt_out(X, Y)) → min_out(X, Y, Y)
min_in(X, Y, X) → U7(X, Y, le_in(X, Y))
le_in(0, 0) → le_out(0, 0)
le_in(0, s(Y)) → le_out(0, s(Y))
le_in(s(X), s(Y)) → U12(X, Y, le_in(X, Y))
U12(X, Y, le_out(X, Y)) → le_out(s(X), s(Y))
U7(X, Y, le_out(X, Y)) → min_out(X, Y, X)
U5(X, A, M, L, min_out(X, M, B)) → U6(X, A, M, L, min2_in(B, A, L))
min2_in(X, X, []) → min2_out(X, X, [])
U6(X, A, M, L, min2_out(B, A, L)) → min2_out(X, A, .(M, L))
U4(M, X, L, min2_out(X, M, L)) → min1_out(M, .(X, L))
U1(L, X, L1, min1_out(X, L)) → U2(L, X, L1, remove_in(X, L, L2))
remove_in(N, .(M, L), .(M, L1)) → U9(N, M, L, L1, notEq_in(N, M))
notEq_in(0, s(X)) → notEq_out(0, s(X))
notEq_in(s(X), 0) → notEq_out(s(X), 0)
notEq_in(s(X), s(Y)) → U13(X, Y, notEq_in(X, Y))
U13(X, Y, notEq_out(X, Y)) → notEq_out(s(X), s(Y))
U9(N, M, L, L1, notEq_out(N, M)) → U10(N, M, L, L1, remove_in(N, L, L1))
remove_in(N, .(N, L), L) → remove_out(N, .(N, L), L)
U10(N, M, L, L1, remove_out(N, L, L1)) → remove_out(N, .(M, L), .(M, L1))
U2(L, X, L1, remove_out(X, L, L2)) → U3(L, X, L1, minsort_in(L2, L1))
minsort_in([], []) → minsort_out([], [])
U3(L, X, L1, minsort_out(L2, L1)) → minsort_out(L, .(X, L1))

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:

minsort_in(L, .(X, L1)) → U1(L, X, L1, min1_in(X, L))
min1_in(M, .(X, L)) → U4(M, X, L, min2_in(X, M, L))
min2_in(X, A, .(M, L)) → U5(X, A, M, L, min_in(X, M, B))
min_in(X, Y, Y) → U8(X, Y, gt_in(X, Y))
gt_in(s(X), 0) → gt_out(s(X), 0)
gt_in(s(X), s(Y)) → U11(X, Y, gt_in(X, Y))
U11(X, Y, gt_out(X, Y)) → gt_out(s(X), s(Y))
U8(X, Y, gt_out(X, Y)) → min_out(X, Y, Y)
min_in(X, Y, X) → U7(X, Y, le_in(X, Y))
le_in(0, 0) → le_out(0, 0)
le_in(0, s(Y)) → le_out(0, s(Y))
le_in(s(X), s(Y)) → U12(X, Y, le_in(X, Y))
U12(X, Y, le_out(X, Y)) → le_out(s(X), s(Y))
U7(X, Y, le_out(X, Y)) → min_out(X, Y, X)
U5(X, A, M, L, min_out(X, M, B)) → U6(X, A, M, L, min2_in(B, A, L))
min2_in(X, X, []) → min2_out(X, X, [])
U6(X, A, M, L, min2_out(B, A, L)) → min2_out(X, A, .(M, L))
U4(M, X, L, min2_out(X, M, L)) → min1_out(M, .(X, L))
U1(L, X, L1, min1_out(X, L)) → U2(L, X, L1, remove_in(X, L, L2))
remove_in(N, .(M, L), .(M, L1)) → U9(N, M, L, L1, notEq_in(N, M))
notEq_in(0, s(X)) → notEq_out(0, s(X))
notEq_in(s(X), 0) → notEq_out(s(X), 0)
notEq_in(s(X), s(Y)) → U13(X, Y, notEq_in(X, Y))
U13(X, Y, notEq_out(X, Y)) → notEq_out(s(X), s(Y))
U9(N, M, L, L1, notEq_out(N, M)) → U10(N, M, L, L1, remove_in(N, L, L1))
remove_in(N, .(N, L), L) → remove_out(N, .(N, L), L)
U10(N, M, L, L1, remove_out(N, L, L1)) → remove_out(N, .(M, L), .(M, L1))
U2(L, X, L1, remove_out(X, L, L2)) → U3(L, X, L1, minsort_in(L2, L1))
minsort_in([], []) → minsort_out([], [])
U3(L, X, L1, minsort_out(L2, L1)) → minsort_out(L, .(X, L1))

MINSORT_IN(L, .(X, L1)) → U1¹(L, X, L1, min1_in(X, L))
MINSORT_IN(L, .(X, L1)) → MIN1_IN(X, L)
MIN1_IN(M, .(X, L)) → U4¹(M, X, L, min2_in(X, M, L))
MIN1_IN(M, .(X, L)) → MIN2_IN(X, M, L)
MIN2_IN(X, A, .(M, L)) → U5¹(X, A, M, L, min_in(X, M, B))
MIN2_IN(X, A, .(M, L)) → MIN_IN(X, M, B)
MIN_IN(X, Y, Y) → U8¹(X, Y, gt_in(X, Y))
MIN_IN(X, Y, Y) → GT_IN(X, Y)
GT_IN(s(X), s(Y)) → U11¹(X, Y, gt_in(X, Y))
GT_IN(s(X), s(Y)) → GT_IN(X, Y)
MIN_IN(X, Y, X) → U7¹(X, Y, le_in(X, Y))
MIN_IN(X, Y, X) → LE_IN(X, Y)
LE_IN(s(X), s(Y)) → U12¹(X, Y, le_in(X, Y))
LE_IN(s(X), s(Y)) → LE_IN(X, Y)
U5¹(X, A, M, L, min_out(X, M, B)) → U6¹(X, A, M, L, min2_in(B, A, L))
U5¹(X, A, M, L, min_out(X, M, B)) → MIN2_IN(B, A, L)
U1¹(L, X, L1, min1_out(X, L)) → U2¹(L, X, L1, remove_in(X, L, L2))
U1¹(L, X, L1, min1_out(X, L)) → REMOVE_IN(X, L, L2)
REMOVE_IN(N, .(M, L), .(M, L1)) → U9¹(N, M, L, L1, notEq_in(N, M))
REMOVE_IN(N, .(M, L), .(M, L1)) → NOTEQ_IN(N, M)
NOTEQ_IN(s(X), s(Y)) → U13¹(X, Y, notEq_in(X, Y))
NOTEQ_IN(s(X), s(Y)) → NOTEQ_IN(X, Y)
U9¹(N, M, L, L1, notEq_out(N, M)) → U10¹(N, M, L, L1, remove_in(N, L, L1))
U9¹(N, M, L, L1, notEq_out(N, M)) → REMOVE_IN(N, L, L1)
U2¹(L, X, L1, remove_out(X, L, L2)) → U3¹(L, X, L1, minsort_in(L2, L1))
U2¹(L, X, L1, remove_out(X, L, L2)) → MINSORT_IN(L2, L1)

The TRS R consists of the following rules:

minsort_in(L, .(X, L1)) → U1(L, X, L1, min1_in(X, L))
min1_in(M, .(X, L)) → U4(M, X, L, min2_in(X, M, L))
min2_in(X, A, .(M, L)) → U5(X, A, M, L, min_in(X, M, B))
min_in(X, Y, Y) → U8(X, Y, gt_in(X, Y))
gt_in(s(X), 0) → gt_out(s(X), 0)
gt_in(s(X), s(Y)) → U11(X, Y, gt_in(X, Y))
U11(X, Y, gt_out(X, Y)) → gt_out(s(X), s(Y))
U8(X, Y, gt_out(X, Y)) → min_out(X, Y, Y)
min_in(X, Y, X) → U7(X, Y, le_in(X, Y))
le_in(0, 0) → le_out(0, 0)
le_in(0, s(Y)) → le_out(0, s(Y))
le_in(s(X), s(Y)) → U12(X, Y, le_in(X, Y))
U12(X, Y, le_out(X, Y)) → le_out(s(X), s(Y))
U7(X, Y, le_out(X, Y)) → min_out(X, Y, X)
U5(X, A, M, L, min_out(X, M, B)) → U6(X, A, M, L, min2_in(B, A, L))
min2_in(X, X, []) → min2_out(X, X, [])
U6(X, A, M, L, min2_out(B, A, L)) → min2_out(X, A, .(M, L))
U4(M, X, L, min2_out(X, M, L)) → min1_out(M, .(X, L))
U1(L, X, L1, min1_out(X, L)) → U2(L, X, L1, remove_in(X, L, L2))
remove_in(N, .(M, L), .(M, L1)) → U9(N, M, L, L1, notEq_in(N, M))
notEq_in(0, s(X)) → notEq_out(0, s(X))
notEq_in(s(X), 0) → notEq_out(s(X), 0)
notEq_in(s(X), s(Y)) → U13(X, Y, notEq_in(X, Y))
U13(X, Y, notEq_out(X, Y)) → notEq_out(s(X), s(Y))
U9(N, M, L, L1, notEq_out(N, M)) → U10(N, M, L, L1, remove_in(N, L, L1))
remove_in(N, .(N, L), L) → remove_out(N, .(N, L), L)
U10(N, M, L, L1, remove_out(N, L, L1)) → remove_out(N, .(M, L), .(M, L1))
U2(L, X, L1, remove_out(X, L, L2)) → U3(L, X, L1, minsort_in(L2, L1))
minsort_in([], []) → minsort_out([], [])
U3(L, X, L1, minsort_out(L2, L1)) → minsort_out(L, .(X, L1))

The argument filtering Pi contains the following mapping:
minsort_in(x1, x2) = minsort_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4) = U1(x1, x4)
min1_in(x1, x2) = min1_in(x2)
U4(x1, x2, x3, x4) = U4(x4)
min2_in(x1, x2, x3) = min2_in(x1, x3)
U5(x1, x2, x3, x4, x5) = U5(x4, x5)
min_in(x1, x2, x3) = min_in(x1, x2)
U8(x1, x2, x3) = U8(x2, x3)
gt_in(x1, x2) = gt_in(x1, x2)
s(x1) = s(x1)
0 = 0
gt_out(x1, x2) = gt_out
U11(x1, x2, x3) = U11(x3)
min_out(x1, x2, x3) = min_out(x3)
U7(x1, x2, x3) = U7(x1, x3)
le_in(x1, x2) = le_in(x1, x2)
le_out(x1, x2) = le_out
U12(x1, x2, x3) = U12(x3)
U6(x1, x2, x3, x4, x5) = U6(x5)
[] = []
min2_out(x1, x2, x3) = min2_out(x2)
min1_out(x1, x2) = min1_out(x1)
U2(x1, x2, x3, x4) = U2(x2, x4)
remove_in(x1, x2, x3) = remove_in(x1, x2)
U9(x1, x2, x3, x4, x5) = U9(x1, x2, x3, x5)
notEq_in(x1, x2) = notEq_in(x1, x2)
notEq_out(x1, x2) = notEq_out
U13(x1, x2, x3) = U13(x3)
U10(x1, x2, x3, x4, x5) = U10(x2, x5)
remove_out(x1, x2, x3) = remove_out(x3)
U3(x1, x2, x3, x4) = U3(x2, x4)
minsort_out(x1, x2) = minsort_out(x2)
MINSORT_IN(x1, x2) = MINSORT_IN(x1)
MIN2_IN(x1, x2, x3) = MIN2_IN(x1, x3)
U11¹(x1, x2, x3) = U11¹(x3)
U7¹(x1, x2, x3) = U7¹(x1, x3)
U13¹(x1, x2, x3) = U13¹(x3)
U4¹(x1, x2, x3, x4) = U4¹(x4)
U8¹(x1, x2, x3) = U8¹(x2, x3)
U9¹(x1, x2, x3, x4, x5) = U9¹(x1, x2, x3, x5)
U2¹(x1, x2, x3, x4) = U2¹(x2, x4)
U6¹(x1, x2, x3, x4, x5) = U6¹(x5)
LE_IN(x1, x2) = LE_IN(x1, x2)
U10¹(x1, x2, x3, x4, x5) = U10¹(x2, x5)
U3¹(x1, x2, x3, x4) = U3¹(x2, x4)
U5¹(x1, x2, x3, x4, x5) = U5¹(x4, x5)
MIN_IN(x1, x2, x3) = MIN_IN(x1, x2)
REMOVE_IN(x1, x2, x3) = REMOVE_IN(x1, x2)
U12¹(x1, x2, x3) = U12¹(x3)
NOTEQ_IN(x1, x2) = NOTEQ_IN(x1, x2)
MIN1_IN(x1, x2) = MIN1_IN(x2)
GT_IN(x1, x2) = GT_IN(x1, x2)
U1¹(x1, x2, x3, x4) = U1¹(x1, x4)

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:

MINSORT_IN(L, .(X, L1)) → U1¹(L, X, L1, min1_in(X, L))
MINSORT_IN(L, .(X, L1)) → MIN1_IN(X, L)
MIN1_IN(M, .(X, L)) → U4¹(M, X, L, min2_in(X, M, L))
MIN1_IN(M, .(X, L)) → MIN2_IN(X, M, L)
MIN2_IN(X, A, .(M, L)) → U5¹(X, A, M, L, min_in(X, M, B))
MIN2_IN(X, A, .(M, L)) → MIN_IN(X, M, B)
MIN_IN(X, Y, Y) → U8¹(X, Y, gt_in(X, Y))
MIN_IN(X, Y, Y) → GT_IN(X, Y)
GT_IN(s(X), s(Y)) → U11¹(X, Y, gt_in(X, Y))
GT_IN(s(X), s(Y)) → GT_IN(X, Y)
MIN_IN(X, Y, X) → U7¹(X, Y, le_in(X, Y))
MIN_IN(X, Y, X) → LE_IN(X, Y)
LE_IN(s(X), s(Y)) → U12¹(X, Y, le_in(X, Y))
LE_IN(s(X), s(Y)) → LE_IN(X, Y)
U5¹(X, A, M, L, min_out(X, M, B)) → U6¹(X, A, M, L, min2_in(B, A, L))
U5¹(X, A, M, L, min_out(X, M, B)) → MIN2_IN(B, A, L)
U1¹(L, X, L1, min1_out(X, L)) → U2¹(L, X, L1, remove_in(X, L, L2))
U1¹(L, X, L1, min1_out(X, L)) → REMOVE_IN(X, L, L2)
REMOVE_IN(N, .(M, L), .(M, L1)) → U9¹(N, M, L, L1, notEq_in(N, M))
REMOVE_IN(N, .(M, L), .(M, L1)) → NOTEQ_IN(N, M)
NOTEQ_IN(s(X), s(Y)) → U13¹(X, Y, notEq_in(X, Y))
NOTEQ_IN(s(X), s(Y)) → NOTEQ_IN(X, Y)
U9¹(N, M, L, L1, notEq_out(N, M)) → U10¹(N, M, L, L1, remove_in(N, L, L1))
U9¹(N, M, L, L1, notEq_out(N, M)) → REMOVE_IN(N, L, L1)
U2¹(L, X, L1, remove_out(X, L, L2)) → U3¹(L, X, L1, minsort_in(L2, L1))
U2¹(L, X, L1, remove_out(X, L, L2)) → MINSORT_IN(L2, L1)

The TRS R consists of the following rules:

minsort_in(L, .(X, L1)) → U1(L, X, L1, min1_in(X, L))
min1_in(M, .(X, L)) → U4(M, X, L, min2_in(X, M, L))
min2_in(X, A, .(M, L)) → U5(X, A, M, L, min_in(X, M, B))
min_in(X, Y, Y) → U8(X, Y, gt_in(X, Y))
gt_in(s(X), 0) → gt_out(s(X), 0)
gt_in(s(X), s(Y)) → U11(X, Y, gt_in(X, Y))
U11(X, Y, gt_out(X, Y)) → gt_out(s(X), s(Y))
U8(X, Y, gt_out(X, Y)) → min_out(X, Y, Y)
min_in(X, Y, X) → U7(X, Y, le_in(X, Y))
le_in(0, 0) → le_out(0, 0)
le_in(0, s(Y)) → le_out(0, s(Y))
le_in(s(X), s(Y)) → U12(X, Y, le_in(X, Y))
U12(X, Y, le_out(X, Y)) → le_out(s(X), s(Y))
U7(X, Y, le_out(X, Y)) → min_out(X, Y, X)
U5(X, A, M, L, min_out(X, M, B)) → U6(X, A, M, L, min2_in(B, A, L))
min2_in(X, X, []) → min2_out(X, X, [])
U6(X, A, M, L, min2_out(B, A, L)) → min2_out(X, A, .(M, L))
U4(M, X, L, min2_out(X, M, L)) → min1_out(M, .(X, L))
U1(L, X, L1, min1_out(X, L)) → U2(L, X, L1, remove_in(X, L, L2))
remove_in(N, .(M, L), .(M, L1)) → U9(N, M, L, L1, notEq_in(N, M))
notEq_in(0, s(X)) → notEq_out(0, s(X))
notEq_in(s(X), 0) → notEq_out(s(X), 0)
notEq_in(s(X), s(Y)) → U13(X, Y, notEq_in(X, Y))
U13(X, Y, notEq_out(X, Y)) → notEq_out(s(X), s(Y))
U9(N, M, L, L1, notEq_out(N, M)) → U10(N, M, L, L1, remove_in(N, L, L1))
remove_in(N, .(N, L), L) → remove_out(N, .(N, L), L)
U10(N, M, L, L1, remove_out(N, L, L1)) → remove_out(N, .(M, L), .(M, L1))
U2(L, X, L1, remove_out(X, L, L2)) → U3(L, X, L1, minsort_in(L2, L1))
minsort_in([], []) → minsort_out([], [])
U3(L, X, L1, minsort_out(L2, L1)) → minsort_out(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

NOTEQ_IN(s(X), s(Y)) → NOTEQ_IN(X, Y)

The TRS R consists of the following rules:

minsort_in(L, .(X, L1)) → U1(L, X, L1, min1_in(X, L))
min1_in(M, .(X, L)) → U4(M, X, L, min2_in(X, M, L))
min2_in(X, A, .(M, L)) → U5(X, A, M, L, min_in(X, M, B))
min_in(X, Y, Y) → U8(X, Y, gt_in(X, Y))
gt_in(s(X), 0) → gt_out(s(X), 0)
gt_in(s(X), s(Y)) → U11(X, Y, gt_in(X, Y))
U11(X, Y, gt_out(X, Y)) → gt_out(s(X), s(Y))
U8(X, Y, gt_out(X, Y)) → min_out(X, Y, Y)
min_in(X, Y, X) → U7(X, Y, le_in(X, Y))
le_in(0, 0) → le_out(0, 0)
le_in(0, s(Y)) → le_out(0, s(Y))
le_in(s(X), s(Y)) → U12(X, Y, le_in(X, Y))
U12(X, Y, le_out(X, Y)) → le_out(s(X), s(Y))
U7(X, Y, le_out(X, Y)) → min_out(X, Y, X)
U5(X, A, M, L, min_out(X, M, B)) → U6(X, A, M, L, min2_in(B, A, L))
min2_in(X, X, []) → min2_out(X, X, [])
U6(X, A, M, L, min2_out(B, A, L)) → min2_out(X, A, .(M, L))
U4(M, X, L, min2_out(X, M, L)) → min1_out(M, .(X, L))
U1(L, X, L1, min1_out(X, L)) → U2(L, X, L1, remove_in(X, L, L2))
remove_in(N, .(M, L), .(M, L1)) → U9(N, M, L, L1, notEq_in(N, M))
notEq_in(0, s(X)) → notEq_out(0, s(X))
notEq_in(s(X), 0) → notEq_out(s(X), 0)
notEq_in(s(X), s(Y)) → U13(X, Y, notEq_in(X, Y))
U13(X, Y, notEq_out(X, Y)) → notEq_out(s(X), s(Y))
U9(N, M, L, L1, notEq_out(N, M)) → U10(N, M, L, L1, remove_in(N, L, L1))
remove_in(N, .(N, L), L) → remove_out(N, .(N, L), L)
U10(N, M, L, L1, remove_out(N, L, L1)) → remove_out(N, .(M, L), .(M, L1))
U2(L, X, L1, remove_out(X, L, L2)) → U3(L, X, L1, minsort_in(L2, L1))
minsort_in([], []) → minsort_out([], [])
U3(L, X, L1, minsort_out(L2, L1)) → minsort_out(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

NOTEQ_IN(s(X), s(Y)) → NOTEQ_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

              ↳ PiDP

              ↳ PiDP

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

NOTEQ_IN(s(X), s(Y)) → NOTEQ_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:

NOTEQ_IN(s(X), s(Y)) → NOTEQ_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

              ↳ PiDP

              ↳ PiDP

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

REMOVE_IN(N, .(M, L), .(M, L1)) → U9¹(N, M, L, L1, notEq_in(N, M))
U9¹(N, M, L, L1, notEq_out(N, M)) → REMOVE_IN(N, L, L1)

The TRS R consists of the following rules:

minsort_in(L, .(X, L1)) → U1(L, X, L1, min1_in(X, L))
min1_in(M, .(X, L)) → U4(M, X, L, min2_in(X, M, L))
min2_in(X, A, .(M, L)) → U5(X, A, M, L, min_in(X, M, B))
min_in(X, Y, Y) → U8(X, Y, gt_in(X, Y))
gt_in(s(X), 0) → gt_out(s(X), 0)
gt_in(s(X), s(Y)) → U11(X, Y, gt_in(X, Y))
U11(X, Y, gt_out(X, Y)) → gt_out(s(X), s(Y))
U8(X, Y, gt_out(X, Y)) → min_out(X, Y, Y)
min_in(X, Y, X) → U7(X, Y, le_in(X, Y))
le_in(0, 0) → le_out(0, 0)
le_in(0, s(Y)) → le_out(0, s(Y))
le_in(s(X), s(Y)) → U12(X, Y, le_in(X, Y))
U12(X, Y, le_out(X, Y)) → le_out(s(X), s(Y))
U7(X, Y, le_out(X, Y)) → min_out(X, Y, X)
U5(X, A, M, L, min_out(X, M, B)) → U6(X, A, M, L, min2_in(B, A, L))
min2_in(X, X, []) → min2_out(X, X, [])
U6(X, A, M, L, min2_out(B, A, L)) → min2_out(X, A, .(M, L))
U4(M, X, L, min2_out(X, M, L)) → min1_out(M, .(X, L))
U1(L, X, L1, min1_out(X, L)) → U2(L, X, L1, remove_in(X, L, L2))
remove_in(N, .(M, L), .(M, L1)) → U9(N, M, L, L1, notEq_in(N, M))
notEq_in(0, s(X)) → notEq_out(0, s(X))
notEq_in(s(X), 0) → notEq_out(s(X), 0)
notEq_in(s(X), s(Y)) → U13(X, Y, notEq_in(X, Y))
U13(X, Y, notEq_out(X, Y)) → notEq_out(s(X), s(Y))
U9(N, M, L, L1, notEq_out(N, M)) → U10(N, M, L, L1, remove_in(N, L, L1))
remove_in(N, .(N, L), L) → remove_out(N, .(N, L), L)
U10(N, M, L, L1, remove_out(N, L, L1)) → remove_out(N, .(M, L), .(M, L1))
U2(L, X, L1, remove_out(X, L, L2)) → U3(L, X, L1, minsort_in(L2, L1))
minsort_in([], []) → minsort_out([], [])
U3(L, X, L1, minsort_out(L2, L1)) → minsort_out(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

REMOVE_IN(N, .(M, L), .(M, L1)) → U9¹(N, M, L, L1, notEq_in(N, M))
U9¹(N, M, L, L1, notEq_out(N, M)) → REMOVE_IN(N, L, L1)

The TRS R consists of the following rules:

notEq_in(0, s(X)) → notEq_out(0, s(X))
notEq_in(s(X), 0) → notEq_out(s(X), 0)
notEq_in(s(X), s(Y)) → U13(X, Y, notEq_in(X, Y))
U13(X, Y, notEq_out(X, Y)) → notEq_out(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
0 = 0
notEq_in(x1, x2) = notEq_in(x1, x2)
notEq_out(x1, x2) = notEq_out
U13(x1, x2, x3) = U13(x3)
U9¹(x1, x2, x3, x4, x5) = U9¹(x1, x2, x3, x5)
REMOVE_IN(x1, x2, x3) = REMOVE_IN(x1, x2)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

REMOVE_IN(N, .(M, L)) → U9¹(N, M, L, notEq_in(N, M))
U9¹(N, M, L, notEq_out) → REMOVE_IN(N, L)

The TRS R consists of the following rules:

notEq_in(0, s(X)) → notEq_out
notEq_in(s(X), 0) → notEq_out
notEq_in(s(X), s(Y)) → U13(notEq_in(X, Y))
U13(notEq_out) → notEq_out

The set Q consists of the following terms:

notEq_in(x0, x1)
U13(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:

U9¹(N, M, L, notEq_out) → REMOVE_IN(N, L)
The graph contains the following edges 1 >= 1, 3 >= 2
REMOVE_IN(N, .(M, L)) → U9¹(N, M, L, notEq_in(N, M))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

LE_IN(s(X), s(Y)) → LE_IN(X, Y)

The TRS R consists of the following rules:

minsort_in(L, .(X, L1)) → U1(L, X, L1, min1_in(X, L))
min1_in(M, .(X, L)) → U4(M, X, L, min2_in(X, M, L))
min2_in(X, A, .(M, L)) → U5(X, A, M, L, min_in(X, M, B))
min_in(X, Y, Y) → U8(X, Y, gt_in(X, Y))
gt_in(s(X), 0) → gt_out(s(X), 0)
gt_in(s(X), s(Y)) → U11(X, Y, gt_in(X, Y))
U11(X, Y, gt_out(X, Y)) → gt_out(s(X), s(Y))
U8(X, Y, gt_out(X, Y)) → min_out(X, Y, Y)
min_in(X, Y, X) → U7(X, Y, le_in(X, Y))
le_in(0, 0) → le_out(0, 0)
le_in(0, s(Y)) → le_out(0, s(Y))
le_in(s(X), s(Y)) → U12(X, Y, le_in(X, Y))
U12(X, Y, le_out(X, Y)) → le_out(s(X), s(Y))
U7(X, Y, le_out(X, Y)) → min_out(X, Y, X)
U5(X, A, M, L, min_out(X, M, B)) → U6(X, A, M, L, min2_in(B, A, L))
min2_in(X, X, []) → min2_out(X, X, [])
U6(X, A, M, L, min2_out(B, A, L)) → min2_out(X, A, .(M, L))
U4(M, X, L, min2_out(X, M, L)) → min1_out(M, .(X, L))
U1(L, X, L1, min1_out(X, L)) → U2(L, X, L1, remove_in(X, L, L2))
remove_in(N, .(M, L), .(M, L1)) → U9(N, M, L, L1, notEq_in(N, M))
notEq_in(0, s(X)) → notEq_out(0, s(X))
notEq_in(s(X), 0) → notEq_out(s(X), 0)
notEq_in(s(X), s(Y)) → U13(X, Y, notEq_in(X, Y))
U13(X, Y, notEq_out(X, Y)) → notEq_out(s(X), s(Y))
U9(N, M, L, L1, notEq_out(N, M)) → U10(N, M, L, L1, remove_in(N, L, L1))
remove_in(N, .(N, L), L) → remove_out(N, .(N, L), L)
U10(N, M, L, L1, remove_out(N, L, L1)) → remove_out(N, .(M, L), .(M, L1))
U2(L, X, L1, remove_out(X, L, L2)) → U3(L, X, L1, minsort_in(L2, L1))
minsort_in([], []) → minsort_out([], [])
U3(L, X, L1, minsort_out(L2, L1)) → minsort_out(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

LE_IN(s(X), s(Y)) → LE_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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

LE_IN(s(X), s(Y)) → LE_IN(X, Y)

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

GT_IN(s(X), s(Y)) → GT_IN(X, Y)

The TRS R consists of the following rules:

minsort_in(L, .(X, L1)) → U1(L, X, L1, min1_in(X, L))
min1_in(M, .(X, L)) → U4(M, X, L, min2_in(X, M, L))
min2_in(X, A, .(M, L)) → U5(X, A, M, L, min_in(X, M, B))
min_in(X, Y, Y) → U8(X, Y, gt_in(X, Y))
gt_in(s(X), 0) → gt_out(s(X), 0)
gt_in(s(X), s(Y)) → U11(X, Y, gt_in(X, Y))
U11(X, Y, gt_out(X, Y)) → gt_out(s(X), s(Y))
U8(X, Y, gt_out(X, Y)) → min_out(X, Y, Y)
min_in(X, Y, X) → U7(X, Y, le_in(X, Y))
le_in(0, 0) → le_out(0, 0)
le_in(0, s(Y)) → le_out(0, s(Y))
le_in(s(X), s(Y)) → U12(X, Y, le_in(X, Y))
U12(X, Y, le_out(X, Y)) → le_out(s(X), s(Y))
U7(X, Y, le_out(X, Y)) → min_out(X, Y, X)
U5(X, A, M, L, min_out(X, M, B)) → U6(X, A, M, L, min2_in(B, A, L))
min2_in(X, X, []) → min2_out(X, X, [])
U6(X, A, M, L, min2_out(B, A, L)) → min2_out(X, A, .(M, L))
U4(M, X, L, min2_out(X, M, L)) → min1_out(M, .(X, L))
U1(L, X, L1, min1_out(X, L)) → U2(L, X, L1, remove_in(X, L, L2))
remove_in(N, .(M, L), .(M, L1)) → U9(N, M, L, L1, notEq_in(N, M))
notEq_in(0, s(X)) → notEq_out(0, s(X))
notEq_in(s(X), 0) → notEq_out(s(X), 0)
notEq_in(s(X), s(Y)) → U13(X, Y, notEq_in(X, Y))
U13(X, Y, notEq_out(X, Y)) → notEq_out(s(X), s(Y))
U9(N, M, L, L1, notEq_out(N, M)) → U10(N, M, L, L1, remove_in(N, L, L1))
remove_in(N, .(N, L), L) → remove_out(N, .(N, L), L)
U10(N, M, L, L1, remove_out(N, L, L1)) → remove_out(N, .(M, L), .(M, L1))
U2(L, X, L1, remove_out(X, L, L2)) → U3(L, X, L1, minsort_in(L2, L1))
minsort_in([], []) → minsort_out([], [])
U3(L, X, L1, minsort_out(L2, L1)) → minsort_out(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

GT_IN(s(X), s(Y)) → GT_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

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

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

GT_IN(s(X), s(Y)) → GT_IN(X, Y)

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

MIN2_IN(X, A, .(M, L)) → U5¹(X, A, M, L, min_in(X, M, B))
U5¹(X, A, M, L, min_out(X, M, B)) → MIN2_IN(B, A, L)

The TRS R consists of the following rules:

minsort_in(L, .(X, L1)) → U1(L, X, L1, min1_in(X, L))
min1_in(M, .(X, L)) → U4(M, X, L, min2_in(X, M, L))
min2_in(X, A, .(M, L)) → U5(X, A, M, L, min_in(X, M, B))
min_in(X, Y, Y) → U8(X, Y, gt_in(X, Y))
gt_in(s(X), 0) → gt_out(s(X), 0)
gt_in(s(X), s(Y)) → U11(X, Y, gt_in(X, Y))
U11(X, Y, gt_out(X, Y)) → gt_out(s(X), s(Y))
U8(X, Y, gt_out(X, Y)) → min_out(X, Y, Y)
min_in(X, Y, X) → U7(X, Y, le_in(X, Y))
le_in(0, 0) → le_out(0, 0)
le_in(0, s(Y)) → le_out(0, s(Y))
le_in(s(X), s(Y)) → U12(X, Y, le_in(X, Y))
U12(X, Y, le_out(X, Y)) → le_out(s(X), s(Y))
U7(X, Y, le_out(X, Y)) → min_out(X, Y, X)
U5(X, A, M, L, min_out(X, M, B)) → U6(X, A, M, L, min2_in(B, A, L))
min2_in(X, X, []) → min2_out(X, X, [])
U6(X, A, M, L, min2_out(B, A, L)) → min2_out(X, A, .(M, L))
U4(M, X, L, min2_out(X, M, L)) → min1_out(M, .(X, L))
U1(L, X, L1, min1_out(X, L)) → U2(L, X, L1, remove_in(X, L, L2))
remove_in(N, .(M, L), .(M, L1)) → U9(N, M, L, L1, notEq_in(N, M))
notEq_in(0, s(X)) → notEq_out(0, s(X))
notEq_in(s(X), 0) → notEq_out(s(X), 0)
notEq_in(s(X), s(Y)) → U13(X, Y, notEq_in(X, Y))
U13(X, Y, notEq_out(X, Y)) → notEq_out(s(X), s(Y))
U9(N, M, L, L1, notEq_out(N, M)) → U10(N, M, L, L1, remove_in(N, L, L1))
remove_in(N, .(N, L), L) → remove_out(N, .(N, L), L)
U10(N, M, L, L1, remove_out(N, L, L1)) → remove_out(N, .(M, L), .(M, L1))
U2(L, X, L1, remove_out(X, L, L2)) → U3(L, X, L1, minsort_in(L2, L1))
minsort_in([], []) → minsort_out([], [])
U3(L, X, L1, minsort_out(L2, L1)) → minsort_out(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

MIN2_IN(X, A, .(M, L)) → U5¹(X, A, M, L, min_in(X, M, B))
U5¹(X, A, M, L, min_out(X, M, B)) → MIN2_IN(B, A, L)

The TRS R consists of the following rules:

min_in(X, Y, Y) → U8(X, Y, gt_in(X, Y))
min_in(X, Y, X) → U7(X, Y, le_in(X, Y))
U8(X, Y, gt_out(X, Y)) → min_out(X, Y, Y)
U7(X, Y, le_out(X, Y)) → min_out(X, Y, X)
gt_in(s(X), 0) → gt_out(s(X), 0)
gt_in(s(X), s(Y)) → U11(X, Y, gt_in(X, Y))
le_in(0, 0) → le_out(0, 0)
le_in(0, s(Y)) → le_out(0, s(Y))
le_in(s(X), s(Y)) → U12(X, Y, le_in(X, Y))
U11(X, Y, gt_out(X, Y)) → gt_out(s(X), s(Y))
U12(X, Y, le_out(X, Y)) → le_out(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
min_in(x1, x2, x3) = min_in(x1, x2)
U8(x1, x2, x3) = U8(x2, x3)
gt_in(x1, x2) = gt_in(x1, x2)
s(x1) = s(x1)
0 = 0
gt_out(x1, x2) = gt_out
U11(x1, x2, x3) = U11(x3)
min_out(x1, x2, x3) = min_out(x3)
U7(x1, x2, x3) = U7(x1, x3)
le_in(x1, x2) = le_in(x1, x2)
le_out(x1, x2) = le_out
U12(x1, x2, x3) = U12(x3)
MIN2_IN(x1, x2, x3) = MIN2_IN(x1, x3)
U5¹(x1, x2, x3, x4, x5) = U5¹(x4, x5)

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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

U5¹(L, min_out(B)) → MIN2_IN(B, L)
MIN2_IN(X, .(M, L)) → U5¹(L, min_in(X, M))

The TRS R consists of the following rules:

min_in(X, Y) → U8(Y, gt_in(X, Y))
min_in(X, Y) → U7(X, le_in(X, Y))
U8(Y, gt_out) → min_out(Y)
U7(X, le_out) → min_out(X)
gt_in(s(X), 0) → gt_out
gt_in(s(X), s(Y)) → U11(gt_in(X, Y))
le_in(0, 0) → le_out
le_in(0, s(Y)) → le_out
le_in(s(X), s(Y)) → U12(le_in(X, Y))
U11(gt_out) → gt_out
U12(le_out) → le_out

The set Q consists of the following terms:

min_in(x0, x1)
U8(x0, x1)
U7(x0, x1)
gt_in(x0, x1)
le_in(x0, x1)
U11(x0)
U12(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:

U5¹(L, min_out(B)) → MIN2_IN(B, L)
The graph contains the following edges 2 > 1, 1 >= 2
MIN2_IN(X, .(M, L)) → U5¹(L, min_in(X, M))
The graph contains the following edges 2 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

U2¹(L, X, L1, remove_out(X, L, L2)) → MINSORT_IN(L2, L1)
MINSORT_IN(L, .(X, L1)) → U1¹(L, X, L1, min1_in(X, L))
U1¹(L, X, L1, min1_out(X, L)) → U2¹(L, X, L1, remove_in(X, L, L2))

The TRS R consists of the following rules:

minsort_in(L, .(X, L1)) → U1(L, X, L1, min1_in(X, L))
min1_in(M, .(X, L)) → U4(M, X, L, min2_in(X, M, L))
min2_in(X, A, .(M, L)) → U5(X, A, M, L, min_in(X, M, B))
min_in(X, Y, Y) → U8(X, Y, gt_in(X, Y))
gt_in(s(X), 0) → gt_out(s(X), 0)
gt_in(s(X), s(Y)) → U11(X, Y, gt_in(X, Y))
U11(X, Y, gt_out(X, Y)) → gt_out(s(X), s(Y))
U8(X, Y, gt_out(X, Y)) → min_out(X, Y, Y)
min_in(X, Y, X) → U7(X, Y, le_in(X, Y))
le_in(0, 0) → le_out(0, 0)
le_in(0, s(Y)) → le_out(0, s(Y))
le_in(s(X), s(Y)) → U12(X, Y, le_in(X, Y))
U12(X, Y, le_out(X, Y)) → le_out(s(X), s(Y))
U7(X, Y, le_out(X, Y)) → min_out(X, Y, X)
U5(X, A, M, L, min_out(X, M, B)) → U6(X, A, M, L, min2_in(B, A, L))
min2_in(X, X, []) → min2_out(X, X, [])
U6(X, A, M, L, min2_out(B, A, L)) → min2_out(X, A, .(M, L))
U4(M, X, L, min2_out(X, M, L)) → min1_out(M, .(X, L))
U1(L, X, L1, min1_out(X, L)) → U2(L, X, L1, remove_in(X, L, L2))
remove_in(N, .(M, L), .(M, L1)) → U9(N, M, L, L1, notEq_in(N, M))
notEq_in(0, s(X)) → notEq_out(0, s(X))
notEq_in(s(X), 0) → notEq_out(s(X), 0)
notEq_in(s(X), s(Y)) → U13(X, Y, notEq_in(X, Y))
U13(X, Y, notEq_out(X, Y)) → notEq_out(s(X), s(Y))
U9(N, M, L, L1, notEq_out(N, M)) → U10(N, M, L, L1, remove_in(N, L, L1))
remove_in(N, .(N, L), L) → remove_out(N, .(N, L), L)
U10(N, M, L, L1, remove_out(N, L, L1)) → remove_out(N, .(M, L), .(M, L1))
U2(L, X, L1, remove_out(X, L, L2)) → U3(L, X, L1, minsort_in(L2, L1))
minsort_in([], []) → minsort_out([], [])
U3(L, X, L1, minsort_out(L2, L1)) → minsort_out(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

U2¹(L, X, L1, remove_out(X, L, L2)) → MINSORT_IN(L2, L1)
MINSORT_IN(L, .(X, L1)) → U1¹(L, X, L1, min1_in(X, L))
U1¹(L, X, L1, min1_out(X, L)) → U2¹(L, X, L1, remove_in(X, L, L2))

The TRS R consists of the following rules:

min1_in(M, .(X, L)) → U4(M, X, L, min2_in(X, M, L))
remove_in(N, .(M, L), .(M, L1)) → U9(N, M, L, L1, notEq_in(N, M))
remove_in(N, .(N, L), L) → remove_out(N, .(N, L), L)
U4(M, X, L, min2_out(X, M, L)) → min1_out(M, .(X, L))
U9(N, M, L, L1, notEq_out(N, M)) → U10(N, M, L, L1, remove_in(N, L, L1))
min2_in(X, A, .(M, L)) → U5(X, A, M, L, min_in(X, M, B))
min2_in(X, X, []) → min2_out(X, X, [])
notEq_in(0, s(X)) → notEq_out(0, s(X))
notEq_in(s(X), 0) → notEq_out(s(X), 0)
notEq_in(s(X), s(Y)) → U13(X, Y, notEq_in(X, Y))
U10(N, M, L, L1, remove_out(N, L, L1)) → remove_out(N, .(M, L), .(M, L1))
U5(X, A, M, L, min_out(X, M, B)) → U6(X, A, M, L, min2_in(B, A, L))
U13(X, Y, notEq_out(X, Y)) → notEq_out(s(X), s(Y))
min_in(X, Y, Y) → U8(X, Y, gt_in(X, Y))
min_in(X, Y, X) → U7(X, Y, le_in(X, Y))
U6(X, A, M, L, min2_out(B, A, L)) → min2_out(X, A, .(M, L))
U8(X, Y, gt_out(X, Y)) → min_out(X, Y, Y)
U7(X, Y, le_out(X, Y)) → min_out(X, Y, X)
gt_in(s(X), 0) → gt_out(s(X), 0)
gt_in(s(X), s(Y)) → U11(X, Y, gt_in(X, Y))
le_in(0, 0) → le_out(0, 0)
le_in(0, s(Y)) → le_out(0, s(Y))
le_in(s(X), s(Y)) → U12(X, Y, le_in(X, Y))
U11(X, Y, gt_out(X, Y)) → gt_out(s(X), s(Y))
U12(X, Y, le_out(X, Y)) → le_out(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
min1_in(x1, x2) = min1_in(x2)
U4(x1, x2, x3, x4) = U4(x4)
min2_in(x1, x2, x3) = min2_in(x1, x3)
U5(x1, x2, x3, x4, x5) = U5(x4, x5)
min_in(x1, x2, x3) = min_in(x1, x2)
U8(x1, x2, x3) = U8(x2, x3)
gt_in(x1, x2) = gt_in(x1, x2)
s(x1) = s(x1)
0 = 0
gt_out(x1, x2) = gt_out
U11(x1, x2, x3) = U11(x3)
min_out(x1, x2, x3) = min_out(x3)
U7(x1, x2, x3) = U7(x1, x3)
le_in(x1, x2) = le_in(x1, x2)
le_out(x1, x2) = le_out
U12(x1, x2, x3) = U12(x3)
U6(x1, x2, x3, x4, x5) = U6(x5)
[] = []
min2_out(x1, x2, x3) = min2_out(x2)
min1_out(x1, x2) = min1_out(x1)
remove_in(x1, x2, x3) = remove_in(x1, x2)
U9(x1, x2, x3, x4, x5) = U9(x1, x2, x3, x5)
notEq_in(x1, x2) = notEq_in(x1, x2)
notEq_out(x1, x2) = notEq_out
U13(x1, x2, x3) = U13(x3)
U10(x1, x2, x3, x4, x5) = U10(x2, x5)
remove_out(x1, x2, x3) = remove_out(x3)
MINSORT_IN(x1, x2) = MINSORT_IN(x1)
U2¹(x1, x2, x3, x4) = U2¹(x2, x4)
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

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

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

U2¹(X, remove_out(L2)) → MINSORT_IN(L2)
U1¹(L, min1_out(X)) → U2¹(X, remove_in(X, L))
MINSORT_IN(L) → U1¹(L, min1_in(L))

The TRS R consists of the following rules:

min1_in(.(X, L)) → U4(min2_in(X, L))
remove_in(N, .(M, L)) → U9(N, M, L, notEq_in(N, M))
remove_in(N, .(N, L)) → remove_out(L)
U4(min2_out(M)) → min1_out(M)
U9(N, M, L, notEq_out) → U10(M, remove_in(N, L))
min2_in(X, .(M, L)) → U5(L, min_in(X, M))
min2_in(X, []) → min2_out(X)
notEq_in(0, s(X)) → notEq_out
notEq_in(s(X), 0) → notEq_out
notEq_in(s(X), s(Y)) → U13(notEq_in(X, Y))
U10(M, remove_out(L1)) → remove_out(.(M, L1))
U5(L, min_out(B)) → U6(min2_in(B, L))
U13(notEq_out) → notEq_out
min_in(X, Y) → U8(Y, gt_in(X, Y))
min_in(X, Y) → U7(X, le_in(X, Y))
U6(min2_out(A)) → min2_out(A)
U8(Y, gt_out) → min_out(Y)
U7(X, le_out) → min_out(X)
gt_in(s(X), 0) → gt_out
gt_in(s(X), s(Y)) → U11(gt_in(X, Y))
le_in(0, 0) → le_out
le_in(0, s(Y)) → le_out
le_in(s(X), s(Y)) → U12(le_in(X, Y))
U11(gt_out) → gt_out
U12(le_out) → le_out

The set Q consists of the following terms:

min1_in(x0)
remove_in(x0, x1)
U4(x0)
U9(x0, x1, x2, x3)
min2_in(x0, x1)
notEq_in(x0, x1)
U10(x0, x1)
U5(x0, x1)
U13(x0)
min_in(x0, x1)
U6(x0)
U8(x0, x1)
U7(x0, x1)
gt_in(x0, x1)
le_in(x0, x1)
U11(x0)
U12(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.

U2¹(X, remove_out(L2)) → MINSORT_IN(L2)

The remaining pairs can at least be oriented weakly.

U1¹(L, min1_out(X)) → U2¹(X, remove_in(X, L))
MINSORT_IN(L) → U1¹(L, min1_in(L))

Used ordering: Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 1 + x₁ + x₂
POL(0) = 1
POL(MINSORT_IN(x₁)) = 1 + x₁
POL(U10(x₁, x₂)) = 1 + x₁ + x₂
POL(U11(x₁)) = 0
POL(U12(x₁)) = x₁
POL(U13(x₁)) = 0
POL(U1¹(x₁, x₂)) = 1 + x₁
POL(U2¹(x₁, x₂)) = 1 + x₂
POL(U4(x₁)) = 0
POL(U5(x₁, x₂)) = 0
POL(U6(x₁)) = 0
POL(U7(x₁, x₂)) = 0
POL(U8(x₁, x₂)) = 1
POL(U9(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃
POL([]) = 0
POL(gt_in(x₁, x₂)) = 0
POL(gt_out) = 0
POL(le_in(x₁, x₂)) = x₁ + x₂
POL(le_out) = 1
POL(min1_in(x₁)) = 0
POL(min1_out(x₁)) = 0
POL(min2_in(x₁, x₂)) = 0
POL(min2_out(x₁)) = 0
POL(min_in(x₁, x₂)) = 1 + x₁ + x₂
POL(min_out(x₁)) = 0
POL(notEq_in(x₁, x₂)) = x₁
POL(notEq_out) = 0
POL(remove_in(x₁, x₂)) = x₂
POL(remove_out(x₁)) = 1 + x₁
POL(s(x₁)) = x₁

The following usable rules [17] were oriented:

U10(M, remove_out(L1)) → remove_out(.(M, L1))
remove_in(N, .(M, L)) → U9(N, M, L, notEq_in(N, M))
remove_in(N, .(N, L)) → remove_out(L)
U9(N, M, L, notEq_out) → U10(M, remove_in(N, L))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

U1¹(L, min1_out(X)) → U2¹(X, remove_in(X, L))
MINSORT_IN(L) → U1¹(L, min1_in(L))

The TRS R consists of the following rules:

min1_in(.(X, L)) → U4(min2_in(X, L))
remove_in(N, .(M, L)) → U9(N, M, L, notEq_in(N, M))
remove_in(N, .(N, L)) → remove_out(L)
U4(min2_out(M)) → min1_out(M)
U9(N, M, L, notEq_out) → U10(M, remove_in(N, L))
min2_in(X, .(M, L)) → U5(L, min_in(X, M))
min2_in(X, []) → min2_out(X)
notEq_in(0, s(X)) → notEq_out
notEq_in(s(X), 0) → notEq_out
notEq_in(s(X), s(Y)) → U13(notEq_in(X, Y))
U10(M, remove_out(L1)) → remove_out(.(M, L1))
U5(L, min_out(B)) → U6(min2_in(B, L))
U13(notEq_out) → notEq_out
min_in(X, Y) → U8(Y, gt_in(X, Y))
min_in(X, Y) → U7(X, le_in(X, Y))
U6(min2_out(A)) → min2_out(A)
U8(Y, gt_out) → min_out(Y)
U7(X, le_out) → min_out(X)
gt_in(s(X), 0) → gt_out
gt_in(s(X), s(Y)) → U11(gt_in(X, Y))
le_in(0, 0) → le_out
le_in(0, s(Y)) → le_out
le_in(s(X), s(Y)) → U12(le_in(X, Y))
U11(gt_out) → gt_out
U12(le_out) → le_out

The set Q consists of the following terms:

min1_in(x0)
remove_in(x0, x1)
U4(x0)
U9(x0, x1, x2, x3)
min2_in(x0, x1)
notEq_in(x0, x1)
U10(x0, x1)
U5(x0, x1)
U13(x0)
min_in(x0, x1)
U6(x0)
U8(x0, x1)
U7(x0, x1)
gt_in(x0, x1)
le_in(x0, x1)
U11(x0)
U12(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.