Left Termination proof of /tmp/tmp5

Left Termination of the query pattern div_in_3(g, g, a) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

le(0, Y, true).
le(s(X), 0, false).
le(s(X), s(Y), B) :- le(X, Y, B).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
div(X, s(Y), Z) :- ','(le(s(Y), X, B), if(B, X, s(Y), Z)).
if(false, X, s(Y), 0).
if(true, X, s(Y), s(Z)) :- ','(minus(X, Y, U), div(U, s(Y), Z)).

Queries:

div(g,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:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

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:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

DIV_IN(X, s(Y), Z) → U3¹(X, Y, Z, le_in(s(Y), X, B))
DIV_IN(X, s(Y), Z) → LE_IN(s(Y), X, B)
LE_IN(s(X), s(Y), B) → U1¹(X, Y, B, le_in(X, Y, B))
LE_IN(s(X), s(Y), B) → LE_IN(X, Y, B)
U3¹(X, Y, Z, le_out(s(Y), X, B)) → U4¹(X, Y, Z, if_in(B, X, s(Y), Z))
U3¹(X, Y, Z, le_out(s(Y), X, B)) → IF_IN(B, X, s(Y), Z)
IF_IN(true, X, s(Y), s(Z)) → U5¹(X, Y, Z, minus_in(X, Y, U))
IF_IN(true, X, s(Y), s(Z)) → MINUS_IN(X, Y, U)
MINUS_IN(s(X), s(Y), Z) → U2¹(X, Y, Z, minus_in(X, Y, Z))
MINUS_IN(s(X), s(Y), Z) → MINUS_IN(X, Y, Z)
U5¹(X, Y, Z, minus_out(X, Y, U)) → U6¹(X, Y, Z, div_in(U, s(Y), Z))
U5¹(X, Y, Z, minus_out(X, Y, U)) → DIV_IN(U, s(Y), Z)

The TRS R consists of the following rules:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

The argument filtering Pi contains the following mapping:
div_in(x1, x2, x3) = div_in(x1, x2)
s(x1) = s(x1)
U3(x1, x2, x3, x4) = U3(x1, x2, x4)
le_in(x1, x2, x3) = le_in(x1, x2)
U1(x1, x2, x3, x4) = U1(x4)
0 = 0
false = false
le_out(x1, x2, x3) = le_out(x3)
true = true
U4(x1, x2, x3, x4) = U4(x4)
if_in(x1, x2, x3, x4) = if_in(x1, x2, x3)
U5(x1, x2, x3, x4) = U5(x2, x4)
minus_in(x1, x2, x3) = minus_in(x1, x2)
U2(x1, x2, x3, x4) = U2(x4)
minus_out(x1, x2, x3) = minus_out(x3)
U6(x1, x2, x3, x4) = U6(x4)
div_out(x1, x2, x3) = div_out(x3)
if_out(x1, x2, x3, x4) = if_out(x4)
U5¹(x1, x2, x3, x4) = U5¹(x2, x4)
LE_IN(x1, x2, x3) = LE_IN(x1, x2)
U3¹(x1, x2, x3, x4) = U3¹(x1, x2, x4)
DIV_IN(x1, x2, x3) = DIV_IN(x1, x2)
MINUS_IN(x1, x2, x3) = MINUS_IN(x1, x2)
U4¹(x1, x2, x3, x4) = U4¹(x4)
U2¹(x1, x2, x3, x4) = U2¹(x4)
IF_IN(x1, x2, x3, x4) = IF_IN(x1, x2, x3)
U6¹(x1, x2, x3, x4) = U6¹(x4)
U1¹(x1, x2, x3, x4) = U1¹(x4)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

DIV_IN(X, s(Y), Z) → U3¹(X, Y, Z, le_in(s(Y), X, B))
DIV_IN(X, s(Y), Z) → LE_IN(s(Y), X, B)
LE_IN(s(X), s(Y), B) → U1¹(X, Y, B, le_in(X, Y, B))
LE_IN(s(X), s(Y), B) → LE_IN(X, Y, B)
U3¹(X, Y, Z, le_out(s(Y), X, B)) → U4¹(X, Y, Z, if_in(B, X, s(Y), Z))
U3¹(X, Y, Z, le_out(s(Y), X, B)) → IF_IN(B, X, s(Y), Z)
IF_IN(true, X, s(Y), s(Z)) → U5¹(X, Y, Z, minus_in(X, Y, U))
IF_IN(true, X, s(Y), s(Z)) → MINUS_IN(X, Y, U)
MINUS_IN(s(X), s(Y), Z) → U2¹(X, Y, Z, minus_in(X, Y, Z))
MINUS_IN(s(X), s(Y), Z) → MINUS_IN(X, Y, Z)
U5¹(X, Y, Z, minus_out(X, Y, U)) → U6¹(X, Y, Z, div_in(U, s(Y), Z))
U5¹(X, Y, Z, minus_out(X, Y, U)) → DIV_IN(U, s(Y), Z)

The TRS R consists of the following rules:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MINUS_IN(s(X), s(Y), Z) → MINUS_IN(X, Y, Z)

The TRS R consists of the following rules:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MINUS_IN(s(X), s(Y), Z) → MINUS_IN(X, Y, Z)

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

  ↳ PrologToPiTRSProof

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

MINUS_IN(s(X), s(Y)) → MINUS_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:

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LE_IN(s(X), s(Y), B) → LE_IN(X, Y, B)

The TRS R consists of the following rules:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LE_IN(s(X), s(Y), B) → LE_IN(X, Y, B)

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

  ↳ PrologToPiTRSProof

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

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

IF_IN(true, X, s(Y), s(Z)) → U5¹(X, Y, Z, minus_in(X, Y, U))
U3¹(X, Y, Z, le_out(s(Y), X, B)) → IF_IN(B, X, s(Y), Z)
U5¹(X, Y, Z, minus_out(X, Y, U)) → DIV_IN(U, s(Y), Z)
DIV_IN(X, s(Y), Z) → U3¹(X, Y, Z, le_in(s(Y), X, B))

The TRS R consists of the following rules:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

The argument filtering Pi contains the following mapping:
div_in(x1, x2, x3) = div_in(x1, x2)
s(x1) = s(x1)
U3(x1, x2, x3, x4) = U3(x1, x2, x4)
le_in(x1, x2, x3) = le_in(x1, x2)
U1(x1, x2, x3, x4) = U1(x4)
0 = 0
false = false
le_out(x1, x2, x3) = le_out(x3)
true = true
U4(x1, x2, x3, x4) = U4(x4)
if_in(x1, x2, x3, x4) = if_in(x1, x2, x3)
U5(x1, x2, x3, x4) = U5(x2, x4)
minus_in(x1, x2, x3) = minus_in(x1, x2)
U2(x1, x2, x3, x4) = U2(x4)
minus_out(x1, x2, x3) = minus_out(x3)
U6(x1, x2, x3, x4) = U6(x4)
div_out(x1, x2, x3) = div_out(x3)
if_out(x1, x2, x3, x4) = if_out(x4)
U5¹(x1, x2, x3, x4) = U5¹(x2, x4)
U3¹(x1, x2, x3, x4) = U3¹(x1, x2, x4)
DIV_IN(x1, x2, x3) = DIV_IN(x1, x2)
IF_IN(x1, x2, x3, x4) = IF_IN(x1, x2, x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

IF_IN(true, X, s(Y), s(Z)) → U5¹(X, Y, Z, minus_in(X, Y, U))
U3¹(X, Y, Z, le_out(s(Y), X, B)) → IF_IN(B, X, s(Y), Z)
U5¹(X, Y, Z, minus_out(X, Y, U)) → DIV_IN(U, s(Y), Z)
DIV_IN(X, s(Y), Z) → U3¹(X, Y, Z, le_in(s(Y), X, B))

The TRS R consists of the following rules:

minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
le_in(0, Y, true) → le_out(0, Y, true)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
le_in(x1, x2, x3) = le_in(x1, x2)
U1(x1, x2, x3, x4) = U1(x4)
0 = 0
false = false
le_out(x1, x2, x3) = le_out(x3)
true = true
minus_in(x1, x2, x3) = minus_in(x1, x2)
U2(x1, x2, x3, x4) = U2(x4)
minus_out(x1, x2, x3) = minus_out(x3)
U5¹(x1, x2, x3, x4) = U5¹(x2, x4)
U3¹(x1, x2, x3, x4) = U3¹(x1, x2, x4)
DIV_IN(x1, x2, x3) = DIV_IN(x1, x2)
IF_IN(x1, x2, x3, x4) = IF_IN(x1, x2, x3)

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

                        ↳ Narrowing

  ↳ PrologToPiTRSProof

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

DIV_IN(X, s(Y)) → U3¹(X, Y, le_in(s(Y), X))
U5¹(Y, minus_out(U)) → DIV_IN(U, s(Y))
U3¹(X, Y, le_out(B)) → IF_IN(B, X, s(Y))
IF_IN(true, X, s(Y)) → U5¹(Y, minus_in(X, Y))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule IF_IN(true, X, s(Y)) → U5¹(Y, minus_in(X, Y)) at position [1] we obtained the following new rules:

IF_IN(true, x0, s(0)) → U5¹(0, minus_out(x0))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

  ↳ PrologToPiTRSProof

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

DIV_IN(X, s(Y)) → U3¹(X, Y, le_in(s(Y), X))
U5¹(Y, minus_out(U)) → DIV_IN(U, s(Y))
U3¹(X, Y, le_out(B)) → IF_IN(B, X, s(Y))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))
IF_IN(true, x0, s(0)) → U5¹(0, minus_out(x0))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule DIV_IN(X, s(Y)) → U3¹(X, Y, le_in(s(Y), X)) at position [2] we obtained the following new rules:

DIV_IN(s(x1), s(x0)) → U3¹(s(x1), x0, U1(le_in(x0, x1)))
DIV_IN(0, s(x0)) → U3¹(0, x0, le_out(false))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

  ↳ PrologToPiTRSProof

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

DIV_IN(s(x1), s(x0)) → U3¹(s(x1), x0, U1(le_in(x0, x1)))
U5¹(Y, minus_out(U)) → DIV_IN(U, s(Y))
U3¹(X, Y, le_out(B)) → IF_IN(B, X, s(Y))
DIV_IN(0, s(x0)) → U3¹(0, x0, le_out(false))
IF_IN(true, x0, s(0)) → U5¹(0, minus_out(x0))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U3¹(X, Y, le_out(B)) → IF_IN(B, X, s(Y)) we obtained the following new rules:

U3¹(s(z0), z1, le_out(x2)) → IF_IN(x2, s(z0), s(z1))
U3¹(0, z0, le_out(false)) → IF_IN(false, 0, s(z0))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

U3¹(s(z0), z1, le_out(x2)) → IF_IN(x2, s(z0), s(z1))
DIV_IN(s(x1), s(x0)) → U3¹(s(x1), x0, U1(le_in(x0, x1)))
U5¹(Y, minus_out(U)) → DIV_IN(U, s(Y))
U3¹(0, z0, le_out(false)) → IF_IN(false, 0, s(z0))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))
IF_IN(true, x0, s(0)) → U5¹(0, minus_out(x0))
DIV_IN(0, s(x0)) → U3¹(0, x0, le_out(false))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

  ↳ PrologToPiTRSProof

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

U3¹(s(z0), z1, le_out(x2)) → IF_IN(x2, s(z0), s(z1))
DIV_IN(s(x1), s(x0)) → U3¹(s(x1), x0, U1(le_in(x0, x1)))
U5¹(Y, minus_out(U)) → DIV_IN(U, s(Y))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))
IF_IN(true, x0, s(0)) → U5¹(0, minus_out(x0))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule IF_IN(true, x0, s(0)) → U5¹(0, minus_out(x0)) we obtained the following new rules:

IF_IN(true, s(z0), s(0)) → U5¹(0, minus_out(s(z0)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

  ↳ PrologToPiTRSProof

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

DIV_IN(s(x1), s(x0)) → U3¹(s(x1), x0, U1(le_in(x0, x1)))
U3¹(s(z0), z1, le_out(x2)) → IF_IN(x2, s(z0), s(z1))
U5¹(Y, minus_out(U)) → DIV_IN(U, s(Y))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))
IF_IN(true, s(z0), s(0)) → U5¹(0, minus_out(s(z0)))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U5¹(Y, minus_out(U)) → DIV_IN(U, s(Y)) we obtained the following new rules:

U5¹(0, minus_out(s(z0))) → DIV_IN(s(z0), s(0))
U5¹(s(z1), minus_out(x1)) → DIV_IN(x1, s(s(z1)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

  ↳ PrologToPiTRSProof

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

U3¹(s(z0), z1, le_out(x2)) → IF_IN(x2, s(z0), s(z1))
DIV_IN(s(x1), s(x0)) → U3¹(s(x1), x0, U1(le_in(x0, x1)))
U5¹(0, minus_out(s(z0))) → DIV_IN(s(z0), s(0))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))
U5¹(s(z1), minus_out(x1)) → DIV_IN(x1, s(s(z1)))
IF_IN(true, s(z0), s(0)) → U5¹(0, minus_out(s(z0)))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule DIV_IN(s(x1), s(x0)) → U3¹(s(x1), x0, U1(le_in(x0, x1))) we obtained the following new rules:

DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, U1(le_in(0, z0)))
DIV_IN(s(x0), s(s(z0))) → U3¹(s(x0), s(z0), U1(le_in(s(z0), x0)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

  ↳ PrologToPiTRSProof

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

DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, U1(le_in(0, z0)))
U3¹(s(z0), z1, le_out(x2)) → IF_IN(x2, s(z0), s(z1))
U5¹(0, minus_out(s(z0))) → DIV_IN(s(z0), s(0))
DIV_IN(s(x0), s(s(z0))) → U3¹(s(x0), s(z0), U1(le_in(s(z0), x0)))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))
IF_IN(true, s(z0), s(0)) → U5¹(0, minus_out(s(z0)))
U5¹(s(z1), minus_out(x1)) → DIV_IN(x1, s(s(z1)))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, U1(le_in(0, z0))) at position [2,0] we obtained the following new rules:

DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, U1(le_out(true)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

  ↳ PrologToPiTRSProof

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

U3¹(s(z0), z1, le_out(x2)) → IF_IN(x2, s(z0), s(z1))
DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, U1(le_out(true)))
U5¹(0, minus_out(s(z0))) → DIV_IN(s(z0), s(0))
DIV_IN(s(x0), s(s(z0))) → U3¹(s(x0), s(z0), U1(le_in(s(z0), x0)))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))
U5¹(s(z1), minus_out(x1)) → DIV_IN(x1, s(s(z1)))
IF_IN(true, s(z0), s(0)) → U5¹(0, minus_out(s(z0)))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, U1(le_out(true))) at position [2] we obtained the following new rules:

DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, le_out(true))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Instantiation

  ↳ PrologToPiTRSProof

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

U3¹(s(z0), z1, le_out(x2)) → IF_IN(x2, s(z0), s(z1))
DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, le_out(true))
U5¹(0, minus_out(s(z0))) → DIV_IN(s(z0), s(0))
DIV_IN(s(x0), s(s(z0))) → U3¹(s(x0), s(z0), U1(le_in(s(z0), x0)))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))
IF_IN(true, s(z0), s(0)) → U5¹(0, minus_out(s(z0)))
U5¹(s(z1), minus_out(x1)) → DIV_IN(x1, s(s(z1)))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U3¹(s(z0), z1, le_out(x2)) → IF_IN(x2, s(z0), s(z1)) we obtained the following new rules:

U3¹(s(z0), s(z1), le_out(x2)) → IF_IN(x2, s(z0), s(s(z1)))
U3¹(s(z0), 0, le_out(true)) → IF_IN(true, s(z0), s(0))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

U3¹(s(z0), s(z1), le_out(x2)) → IF_IN(x2, s(z0), s(s(z1)))
DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, le_out(true))
U3¹(s(z0), 0, le_out(true)) → IF_IN(true, s(z0), s(0))
U5¹(0, minus_out(s(z0))) → DIV_IN(s(z0), s(0))
DIV_IN(s(x0), s(s(z0))) → U3¹(s(x0), s(z0), U1(le_in(s(z0), x0)))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))
U5¹(s(z1), minus_out(x1)) → DIV_IN(x1, s(s(z1)))
IF_IN(true, s(z0), s(0)) → U5¹(0, minus_out(s(z0)))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ AND

                                                                    ↳ QDP

                                                                      ↳ UsableRulesProof

                                                                    ↳ QDP

  ↳ PrologToPiTRSProof

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

DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, le_out(true))
U3¹(s(z0), 0, le_out(true)) → IF_IN(true, s(z0), s(0))
U5¹(0, minus_out(s(z0))) → DIV_IN(s(z0), s(0))
IF_IN(true, s(z0), s(0)) → U5¹(0, minus_out(s(z0)))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ AND

                                                                    ↳ QDP

                                                                      ↳ UsableRulesProof

                                                                        ↳ QDP

                                                                          ↳ QReductionProof

                                                                    ↳ QDP

  ↳ PrologToPiTRSProof

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

DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, le_out(true))
U3¹(s(z0), 0, le_out(true)) → IF_IN(true, s(z0), s(0))
U5¹(0, minus_out(s(z0))) → DIV_IN(s(z0), s(0))
IF_IN(true, s(z0), s(0)) → U5¹(0, minus_out(s(z0)))

R is empty.
The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ AND

                                                                    ↳ QDP

                                                                      ↳ UsableRulesProof

                                                                        ↳ QDP

                                                                          ↳ QReductionProof

                                                                            ↳ QDP

                                                                              ↳ NonTerminationProof

                                                                    ↳ QDP

  ↳ PrologToPiTRSProof

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

DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, le_out(true))
U3¹(s(z0), 0, le_out(true)) → IF_IN(true, s(z0), s(0))
U5¹(0, minus_out(s(z0))) → DIV_IN(s(z0), s(0))
IF_IN(true, s(z0), s(0)) → U5¹(0, minus_out(s(z0)))

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 narrowing to the left:

The TRS P consists of the following rules:

DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, le_out(true))
U3¹(s(z0), 0, le_out(true)) → IF_IN(true, s(z0), s(0))
U5¹(0, minus_out(s(z0))) → DIV_IN(s(z0), s(0))
IF_IN(true, s(z0), s(0)) → U5¹(0, minus_out(s(z0)))

The TRS R consists of the following rules:none

s = U3¹(s(z0'), 0, le_out(true)) evaluates to t =U3¹(s(z0'), 0, le_out(true))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U3¹(s(z0'), 0, le_out(true)) → IF_IN(true, s(z0'), s(0))
with rule U3¹(s(z0''), 0, le_out(true)) → IF_IN(true, s(z0''), s(0)) at position [] and matcher [z0'' / z0']

IF_IN(true, s(z0'), s(0)) → U5¹(0, minus_out(s(z0')))
with rule IF_IN(true, s(z0), s(0)) → U5¹(0, minus_out(s(z0))) at position [] and matcher [z0 / z0']

U5¹(0, minus_out(s(z0'))) → DIV_IN(s(z0'), s(0))
with rule U5¹(0, minus_out(s(z0''))) → DIV_IN(s(z0''), s(0)) at position [] and matcher [z0'' / z0']

DIV_IN(s(z0'), s(0)) → U3¹(s(z0'), 0, le_out(true))
with rule DIV_IN(s(z0), s(0)) → U3¹(s(z0), 0, le_out(true))

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ AND

                                                                    ↳ QDP

                                                                    ↳ QDP

                                                                      ↳ ForwardInstantiation

  ↳ PrologToPiTRSProof

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

U3¹(s(z0), s(z1), le_out(x2)) → IF_IN(x2, s(z0), s(s(z1)))
DIV_IN(s(x0), s(s(z0))) → U3¹(s(x0), s(z0), U1(le_in(s(z0), x0)))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))
U5¹(s(z1), minus_out(x1)) → DIV_IN(x1, s(s(z1)))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U5¹(s(z1), minus_out(x1)) → DIV_IN(x1, s(s(z1))) we obtained the following new rules:

U5¹(s(x0), minus_out(s(y_0))) → DIV_IN(s(y_0), s(s(x0)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ AND

                                                                    ↳ QDP

                                                                    ↳ QDP

                                                                      ↳ ForwardInstantiation

                                                                        ↳ QDP

                                                                          ↳ ForwardInstantiation

  ↳ PrologToPiTRSProof

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

U3¹(s(z0), s(z1), le_out(x2)) → IF_IN(x2, s(z0), s(s(z1)))
U5¹(s(x0), minus_out(s(y_0))) → DIV_IN(s(y_0), s(s(x0)))
DIV_IN(s(x0), s(s(z0))) → U3¹(s(x0), s(z0), U1(le_in(s(z0), x0)))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U3¹(s(z0), s(z1), le_out(x2)) → IF_IN(x2, s(z0), s(s(z1))) we obtained the following new rules:

U3¹(s(x0), s(x1), le_out(true)) → IF_IN(true, s(x0), s(s(x1)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ AND

                                                                    ↳ QDP

                                                                    ↳ QDP

                                                                      ↳ ForwardInstantiation

                                                                        ↳ QDP

                                                                          ↳ ForwardInstantiation

                                                                            ↳ QDP

                                                                              ↳ QDPOrderProof

  ↳ PrologToPiTRSProof

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

U3¹(s(x0), s(x1), le_out(true)) → IF_IN(true, s(x0), s(s(x1)))
U5¹(s(x0), minus_out(s(y_0))) → DIV_IN(s(y_0), s(s(x0)))
DIV_IN(s(x0), s(s(z0))) → U3¹(s(x0), s(z0), U1(le_in(s(z0), x0)))
IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(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.

IF_IN(true, s(x0), s(s(x1))) → U5¹(s(x1), U2(minus_in(x0, x1)))

The remaining pairs can at least be oriented weakly.

U3¹(s(x0), s(x1), le_out(true)) → IF_IN(true, s(x0), s(s(x1)))
U5¹(s(x0), minus_out(s(y_0))) → DIV_IN(s(y_0), s(s(x0)))
DIV_IN(s(x0), s(s(z0))) → U3¹(s(x0), s(z0), U1(le_in(s(z0), x0)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(DIV_IN(x₁, x₂)) = 1 + x₁ + x₂
POL(IF_IN(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U1(x₁)) = 1
POL(U2(x₁)) = x₁
POL(U3¹(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U5¹(x₁, x₂)) = 1 + x₁ + x₂
POL(false) = 0
POL(le_in(x₁, x₂)) = 0
POL(le_out(x₁)) = 1
POL(minus_in(x₁, x₂)) = 1 + x₁
POL(minus_out(x₁)) = 1 + x₁
POL(s(x₁)) = 1 + x₁
POL(true) = 0

The following usable rules [17] were oriented:

U2(minus_out(Z)) → minus_out(Z)
minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
U1(le_out(B)) → le_out(B)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ AND

                                                                    ↳ QDP

                                                                    ↳ QDP

                                                                      ↳ ForwardInstantiation

                                                                        ↳ QDP

                                                                          ↳ ForwardInstantiation

                                                                            ↳ QDP

                                                                              ↳ QDPOrderProof

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

U3¹(s(x0), s(x1), le_out(true)) → IF_IN(true, s(x0), s(s(x1)))
U5¹(s(x0), minus_out(s(y_0))) → DIV_IN(s(y_0), s(s(x0)))
DIV_IN(s(x0), s(s(z0))) → U3¹(s(x0), s(z0), U1(le_in(s(z0), x0)))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(minus_in(X, Y))
minus_in(X, 0) → minus_out(X)
le_in(s(X), s(Y)) → U1(le_in(X, Y))
le_in(s(X), 0) → le_out(false)
U2(minus_out(Z)) → minus_out(Z)
U1(le_out(B)) → le_out(B)
le_in(0, Y) → le_out(true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0)
U1(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.
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:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

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:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

DIV_IN(X, s(Y), Z) → U3¹(X, Y, Z, le_in(s(Y), X, B))
DIV_IN(X, s(Y), Z) → LE_IN(s(Y), X, B)
LE_IN(s(X), s(Y), B) → U1¹(X, Y, B, le_in(X, Y, B))
LE_IN(s(X), s(Y), B) → LE_IN(X, Y, B)
U3¹(X, Y, Z, le_out(s(Y), X, B)) → U4¹(X, Y, Z, if_in(B, X, s(Y), Z))
U3¹(X, Y, Z, le_out(s(Y), X, B)) → IF_IN(B, X, s(Y), Z)
IF_IN(true, X, s(Y), s(Z)) → U5¹(X, Y, Z, minus_in(X, Y, U))
IF_IN(true, X, s(Y), s(Z)) → MINUS_IN(X, Y, U)
MINUS_IN(s(X), s(Y), Z) → U2¹(X, Y, Z, minus_in(X, Y, Z))
MINUS_IN(s(X), s(Y), Z) → MINUS_IN(X, Y, Z)
U5¹(X, Y, Z, minus_out(X, Y, U)) → U6¹(X, Y, Z, div_in(U, s(Y), Z))
U5¹(X, Y, Z, minus_out(X, Y, U)) → DIV_IN(U, s(Y), Z)

The TRS R consists of the following rules:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

The argument filtering Pi contains the following mapping:
div_in(x1, x2, x3) = div_in(x1, x2)
s(x1) = s(x1)
U3(x1, x2, x3, x4) = U3(x1, x2, x4)
le_in(x1, x2, x3) = le_in(x1, x2)
U1(x1, x2, x3, x4) = U1(x1, x2, x4)
0 = 0
false = false
le_out(x1, x2, x3) = le_out(x1, x2, x3)
true = true
U4(x1, x2, x3, x4) = U4(x1, x2, x4)
if_in(x1, x2, x3, x4) = if_in(x1, x2, x3)
U5(x1, x2, x3, x4) = U5(x1, x2, x4)
minus_in(x1, x2, x3) = minus_in(x1, x2)
U2(x1, x2, x3, x4) = U2(x1, x2, x4)
minus_out(x1, x2, x3) = minus_out(x1, x2, x3)
U6(x1, x2, x3, x4) = U6(x1, x2, x4)
div_out(x1, x2, x3) = div_out(x1, x2, x3)
if_out(x1, x2, x3, x4) = if_out(x1, x2, x3, x4)
U5¹(x1, x2, x3, x4) = U5¹(x1, x2, x4)
LE_IN(x1, x2, x3) = LE_IN(x1, x2)
U3¹(x1, x2, x3, x4) = U3¹(x1, x2, x4)
DIV_IN(x1, x2, x3) = DIV_IN(x1, x2)
MINUS_IN(x1, x2, x3) = MINUS_IN(x1, x2)
U4¹(x1, x2, x3, x4) = U4¹(x1, x2, x4)
U2¹(x1, x2, x3, x4) = U2¹(x1, x2, x4)
IF_IN(x1, x2, x3, x4) = IF_IN(x1, x2, x3)
U6¹(x1, x2, x3, x4) = U6¹(x1, x2, x4)
U1¹(x1, x2, x3, x4) = U1¹(x1, x2, x4)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

DIV_IN(X, s(Y), Z) → U3¹(X, Y, Z, le_in(s(Y), X, B))
DIV_IN(X, s(Y), Z) → LE_IN(s(Y), X, B)
LE_IN(s(X), s(Y), B) → U1¹(X, Y, B, le_in(X, Y, B))
LE_IN(s(X), s(Y), B) → LE_IN(X, Y, B)
U3¹(X, Y, Z, le_out(s(Y), X, B)) → U4¹(X, Y, Z, if_in(B, X, s(Y), Z))
U3¹(X, Y, Z, le_out(s(Y), X, B)) → IF_IN(B, X, s(Y), Z)
IF_IN(true, X, s(Y), s(Z)) → U5¹(X, Y, Z, minus_in(X, Y, U))
IF_IN(true, X, s(Y), s(Z)) → MINUS_IN(X, Y, U)
MINUS_IN(s(X), s(Y), Z) → U2¹(X, Y, Z, minus_in(X, Y, Z))
MINUS_IN(s(X), s(Y), Z) → MINUS_IN(X, Y, Z)
U5¹(X, Y, Z, minus_out(X, Y, U)) → U6¹(X, Y, Z, div_in(U, s(Y), Z))
U5¹(X, Y, Z, minus_out(X, Y, U)) → DIV_IN(U, s(Y), Z)

The TRS R consists of the following rules:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

The argument filtering Pi contains the following mapping:
div_in(x1, x2, x3) = div_in(x1, x2)
s(x1) = s(x1)
U3(x1, x2, x3, x4) = U3(x1, x2, x4)
le_in(x1, x2, x3) = le_in(x1, x2)
U1(x1, x2, x3, x4) = U1(x1, x2, x4)
0 = 0
false = false
le_out(x1, x2, x3) = le_out(x1, x2, x3)
true = true
U4(x1, x2, x3, x4) = U4(x1, x2, x4)
if_in(x1, x2, x3, x4) = if_in(x1, x2, x3)
U5(x1, x2, x3, x4) = U5(x1, x2, x4)
minus_in(x1, x2, x3) = minus_in(x1, x2)
U2(x1, x2, x3, x4) = U2(x1, x2, x4)
minus_out(x1, x2, x3) = minus_out(x1, x2, x3)
U6(x1, x2, x3, x4) = U6(x1, x2, x4)
div_out(x1, x2, x3) = div_out(x1, x2, x3)
if_out(x1, x2, x3, x4) = if_out(x1, x2, x3, x4)
U5¹(x1, x2, x3, x4) = U5¹(x1, x2, x4)
LE_IN(x1, x2, x3) = LE_IN(x1, x2)
U3¹(x1, x2, x3, x4) = U3¹(x1, x2, x4)
DIV_IN(x1, x2, x3) = DIV_IN(x1, x2)
MINUS_IN(x1, x2, x3) = MINUS_IN(x1, x2)
U4¹(x1, x2, x3, x4) = U4¹(x1, x2, x4)
U2¹(x1, x2, x3, x4) = U2¹(x1, x2, x4)
IF_IN(x1, x2, x3, x4) = IF_IN(x1, x2, x3)
U6¹(x1, x2, x3, x4) = U6¹(x1, x2, x4)
U1¹(x1, x2, x3, x4) = U1¹(x1, x2, x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 6 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

MINUS_IN(s(X), s(Y), Z) → MINUS_IN(X, Y, Z)

The TRS R consists of the following rules:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

The argument filtering Pi contains the following mapping:
div_in(x1, x2, x3) = div_in(x1, x2)
s(x1) = s(x1)
U3(x1, x2, x3, x4) = U3(x1, x2, x4)
le_in(x1, x2, x3) = le_in(x1, x2)
U1(x1, x2, x3, x4) = U1(x1, x2, x4)
0 = 0
false = false
le_out(x1, x2, x3) = le_out(x1, x2, x3)
true = true
U4(x1, x2, x3, x4) = U4(x1, x2, x4)
if_in(x1, x2, x3, x4) = if_in(x1, x2, x3)
U5(x1, x2, x3, x4) = U5(x1, x2, x4)
minus_in(x1, x2, x3) = minus_in(x1, x2)
U2(x1, x2, x3, x4) = U2(x1, x2, x4)
minus_out(x1, x2, x3) = minus_out(x1, x2, x3)
U6(x1, x2, x3, x4) = U6(x1, x2, x4)
div_out(x1, x2, x3) = div_out(x1, x2, x3)
if_out(x1, x2, x3, x4) = if_out(x1, x2, x3, x4)
MINUS_IN(x1, x2, x3) = MINUS_IN(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

MINUS_IN(s(X), s(Y), Z) → MINUS_IN(X, Y, Z)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

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

MINUS_IN(s(X), s(Y)) → MINUS_IN(X, Y)

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

MINUS_IN(s(X), s(Y)) → MINUS_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

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

LE_IN(s(X), s(Y), B) → LE_IN(X, Y, B)

The TRS R consists of the following rules:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

LE_IN(s(X), s(Y), B) → LE_IN(X, Y, B)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ 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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

IF_IN(true, X, s(Y), s(Z)) → U5¹(X, Y, Z, minus_in(X, Y, U))
U3¹(X, Y, Z, le_out(s(Y), X, B)) → IF_IN(B, X, s(Y), Z)
U5¹(X, Y, Z, minus_out(X, Y, U)) → DIV_IN(U, s(Y), Z)
DIV_IN(X, s(Y), Z) → U3¹(X, Y, Z, le_in(s(Y), X, B))

The TRS R consists of the following rules:

div_in(X, s(Y), Z) → U3(X, Y, Z, le_in(s(Y), X, B))
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
le_in(0, Y, true) → le_out(0, Y, true)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
U3(X, Y, Z, le_out(s(Y), X, B)) → U4(X, Y, Z, if_in(B, X, s(Y), Z))
if_in(true, X, s(Y), s(Z)) → U5(X, Y, Z, minus_in(X, Y, U))
minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U5(X, Y, Z, minus_out(X, Y, U)) → U6(X, Y, Z, div_in(U, s(Y), Z))
U6(X, Y, Z, div_out(U, s(Y), Z)) → if_out(true, X, s(Y), s(Z))
if_in(false, X, s(Y), 0) → if_out(false, X, s(Y), 0)
U4(X, Y, Z, if_out(B, X, s(Y), Z)) → div_out(X, s(Y), Z)

The argument filtering Pi contains the following mapping:
div_in(x1, x2, x3) = div_in(x1, x2)
s(x1) = s(x1)
U3(x1, x2, x3, x4) = U3(x1, x2, x4)
le_in(x1, x2, x3) = le_in(x1, x2)
U1(x1, x2, x3, x4) = U1(x1, x2, x4)
0 = 0
false = false
le_out(x1, x2, x3) = le_out(x1, x2, x3)
true = true
U4(x1, x2, x3, x4) = U4(x1, x2, x4)
if_in(x1, x2, x3, x4) = if_in(x1, x2, x3)
U5(x1, x2, x3, x4) = U5(x1, x2, x4)
minus_in(x1, x2, x3) = minus_in(x1, x2)
U2(x1, x2, x3, x4) = U2(x1, x2, x4)
minus_out(x1, x2, x3) = minus_out(x1, x2, x3)
U6(x1, x2, x3, x4) = U6(x1, x2, x4)
div_out(x1, x2, x3) = div_out(x1, x2, x3)
if_out(x1, x2, x3, x4) = if_out(x1, x2, x3, x4)
U5¹(x1, x2, x3, x4) = U5¹(x1, x2, x4)
U3¹(x1, x2, x3, x4) = U3¹(x1, x2, x4)
DIV_IN(x1, x2, x3) = DIV_IN(x1, x2)
IF_IN(x1, x2, x3, x4) = IF_IN(x1, x2, x3)

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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

IF_IN(true, X, s(Y), s(Z)) → U5¹(X, Y, Z, minus_in(X, Y, U))
U3¹(X, Y, Z, le_out(s(Y), X, B)) → IF_IN(B, X, s(Y), Z)
U5¹(X, Y, Z, minus_out(X, Y, U)) → DIV_IN(U, s(Y), Z)
DIV_IN(X, s(Y), Z) → U3¹(X, Y, Z, le_in(s(Y), X, B))

The TRS R consists of the following rules:

minus_in(s(X), s(Y), Z) → U2(X, Y, Z, minus_in(X, Y, Z))
minus_in(X, 0, X) → minus_out(X, 0, X)
le_in(s(X), s(Y), B) → U1(X, Y, B, le_in(X, Y, B))
le_in(s(X), 0, false) → le_out(s(X), 0, false)
U2(X, Y, Z, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U1(X, Y, B, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
le_in(0, Y, true) → le_out(0, Y, true)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
le_in(x1, x2, x3) = le_in(x1, x2)
U1(x1, x2, x3, x4) = U1(x1, x2, x4)
0 = 0
false = false
le_out(x1, x2, x3) = le_out(x1, x2, x3)
true = true
minus_in(x1, x2, x3) = minus_in(x1, x2)
U2(x1, x2, x3, x4) = U2(x1, x2, x4)
minus_out(x1, x2, x3) = minus_out(x1, x2, x3)
U5¹(x1, x2, x3, x4) = U5¹(x1, x2, x4)
U3¹(x1, x2, x3, x4) = U3¹(x1, x2, x4)
DIV_IN(x1, x2, x3) = DIV_IN(x1, x2)
IF_IN(x1, x2, x3, x4) = IF_IN(x1, x2, x3)

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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

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

DIV_IN(X, s(Y)) → U3¹(X, Y, le_in(s(Y), X))
U5¹(X, Y, minus_out(X, Y, U)) → DIV_IN(U, s(Y))
IF_IN(true, X, s(Y)) → U5¹(X, Y, minus_in(X, Y))
U3¹(X, Y, le_out(s(Y), X, B)) → IF_IN(B, X, s(Y))

The TRS R consists of the following rules:

minus_in(s(X), s(Y)) → U2(X, Y, minus_in(X, Y))
minus_in(X, 0) → minus_out(X, 0, X)
le_in(s(X), s(Y)) → U1(X, Y, le_in(X, Y))
le_in(s(X), 0) → le_out(s(X), 0, false)
U2(X, Y, minus_out(X, Y, Z)) → minus_out(s(X), s(Y), Z)
U1(X, Y, le_out(X, Y, B)) → le_out(s(X), s(Y), B)
le_in(0, Y) → le_out(0, Y, true)

The set Q consists of the following terms:

minus_in(x0, x1)
le_in(x0, x1)
U2(x0, x1, x2)
U1(x0, x1, x2)

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