Termination w.r.t. Q proof of /tmp/tmp0kpXyN/div.trs

Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → 0
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

Q restricted rewrite system:
The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → 0
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0

Q is empty.

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → 0
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

MINUS(s(x), y) → IF(gt(s(x), y), x, y)
MINUS(s(x), y) → GT(s(x), y)
IF(true, x, y) → MINUS(x, y)
GE(s(x), s(y)) → GE(x, y)
GT(s(x), s(y)) → GT(x, y)
DIV(x, y) → IF1(ge(x, y), x, y)
DIV(x, y) → GE(x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF1(true, x, y) → GT(y, 0)
IF2(true, x, y) → DIV(minus(x, y), y)
IF2(true, x, y) → MINUS(x, y)

The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → 0
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

MINUS(s(x), y) → IF(gt(s(x), y), x, y)
MINUS(s(x), y) → GT(s(x), y)
IF(true, x, y) → MINUS(x, y)
GE(s(x), s(y)) → GE(x, y)
GT(s(x), s(y)) → GT(x, y)
DIV(x, y) → IF1(ge(x, y), x, y)
DIV(x, y) → GE(x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF1(true, x, y) → GT(y, 0)
IF2(true, x, y) → DIV(minus(x, y), y)
IF2(true, x, y) → MINUS(x, y)

The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → 0
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 4 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GT(s(x), s(y)) → GT(x, y)

The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → 0
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

We have to consider all minimal (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 [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GT(s(x), s(y)) → GT(x, y)

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

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

We have to consider all minimal (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.[THIEMANN].

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GT(s(x), s(y)) → GT(x, y)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

GT(s(x), s(y)) → GT(x, y)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

GE(s(x), s(y)) → GE(x, y)

The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → 0
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

GE(s(x), s(y)) → GE(x, y)

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

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

We have to consider all minimal (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.[THIEMANN].

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

GE(s(x), s(y)) → GE(x, y)

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

GE(s(x), s(y)) → GE(x, y)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

IF(true, x, y) → MINUS(x, y)
MINUS(s(x), y) → IF(gt(s(x), y), x, y)

The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → 0
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

IF(true, x, y) → MINUS(x, y)
MINUS(s(x), y) → IF(gt(s(x), y), x, y)

The TRS R consists of the following rules:

gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
gt(0, y) → false

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

We have to consider all minimal (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.[THIEMANN].

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

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

IF(true, x, y) → MINUS(x, y)
MINUS(s(x), y) → IF(gt(s(x), y), x, y)

The TRS R consists of the following rules:

gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
gt(0, y) → false

The set Q consists of the following terms:

gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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(s(x), y) → IF(gt(s(x), y), x, y)
The graph contains the following edges 1 > 2, 2 >= 3
IF(true, x, y) → MINUS(x, y)
The graph contains the following edges 2 >= 1, 3 >= 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

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

IF2(true, x, y) → DIV(minus(x, y), y)
DIV(x, y) → IF1(ge(x, y), x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)

The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → 0
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

IF2(true, x, y) → DIV(minus(x, y), y)
DIV(x, y) → IF1(ge(x, y), x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

We have to consider all minimal (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.[THIEMANN].

div(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

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

IF2(true, x, y) → DIV(minus(x, y), y)
DIV(x, y) → IF1(ge(x, y), x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule DIV(x, y) → IF1(ge(x, y), x, y) at position [0] we obtained the following new rules [LPAR04]:

DIV(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
DIV(0, s(x0)) → IF1(false, 0, s(x0))
DIV(x0, 0) → IF1(true, x0, 0)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

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

IF2(true, x, y) → DIV(minus(x, y), y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
DIV(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
DIV(0, s(x0)) → IF1(false, 0, s(x0))
DIV(x0, 0) → IF1(true, x0, 0)

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

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

DIV(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF2(true, x, y) → DIV(minus(x, y), y)
DIV(x0, 0) → IF1(true, x0, 0)

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule IF1(true, x, y) → IF2(gt(y, 0), x, y) at position [0] we obtained the following new rules [LPAR04]:

IF1(true, y0, s(x0)) → IF2(true, y0, s(x0))
IF1(true, y0, 0) → IF2(false, y0, 0)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

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

DIV(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF2(true, x, y) → DIV(minus(x, y), y)
DIV(x0, 0) → IF1(true, x0, 0)
IF1(true, y0, s(x0)) → IF2(true, y0, s(x0))
IF1(true, y0, 0) → IF2(false, y0, 0)

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

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

IF1(true, y0, s(x0)) → IF2(true, y0, s(x0))
IF2(true, x, y) → DIV(minus(x, y), y)
DIV(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule IF2(true, x, y) → DIV(minus(x, y), y) at position [0] we obtained the following new rules [LPAR04]:

IF2(true, s(x0), x1) → DIV(if(gt(s(x0), x1), x0, x1), x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Instantiation

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

IF1(true, y0, s(x0)) → IF2(true, y0, s(x0))
DIV(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF2(true, s(x0), x1) → DIV(if(gt(s(x0), x1), x0, x1), x1)

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule IF1(true, y0, s(x0)) → IF2(true, y0, s(x0)) we obtained the following new rules [LPAR04]:

IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

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

DIV(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF2(true, s(x0), x1) → DIV(if(gt(s(x0), x1), x0, x1), x1)
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule IF2(true, s(x0), x1) → DIV(if(gt(s(x0), x1), x0, x1), x1) we obtained the following new rules [LPAR04]:

IF2(true, s(z0), s(z1)) → DIV(if(gt(s(z0), s(z1)), z0, s(z1)), s(z1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

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

DIV(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))
IF2(true, s(z0), s(z1)) → DIV(if(gt(s(z0), s(z1)), z0, s(z1)), s(z1))

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule IF2(true, s(z0), s(z1)) → DIV(if(gt(s(z0), s(z1)), z0, s(z1)), s(z1)) at position [0,0] we obtained the following new rules [LPAR04]:

IF2(true, s(z0), s(z1)) → DIV(if(gt(z0, z1), z0, s(z1)), s(z1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Induction-Processor

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

DIV(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))
IF2(true, s(z0), s(z1)) → DIV(if(gt(z0, z1), z0, s(z1)), s(z1))

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

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

This DP could be deleted by the Induction-Processor:
IF2(true, s(z0''), s(z1'')) → DIV(if(gt(z0'', z1''), z0'', s(z1'')), s(z1''))

This order was computed:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(DIV(x₁, x₂)) = x₁
POL(IF1(x₁, x₂, x₃)) = x₂
POL(IF2(x₁, x₂, x₃)) = x₂
POL(false) = 0
POL(ge(x₁, x₂)) = 1 + x₂
POL(gt(x₁, x₂)) = x₁
POL(if(x₁, x₂, x₃)) = 1 + x₂
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = 1 + x₁
POL(true) = 0

At least one of these decreasing rules is always used after the deleted DP:
if(false, x413, y313) → 0

The following formula is valid:
z0'':sort[a23],z1'':sort[a23].if'(gt(z0'' , z1'' ), z0'' , s(z1'' ))=true

The transformed set:
minus'(s(x14), y10) → if'(gt(s(x14), y10), x14, y10)
if'(true, x32, y24) → minus'(x32, y24)
if'(false, x41, y31) → true
minus'(0, x1) → false
gt(0, y') → false
gt(s(x5), 0) → true
minus(s(x14), y10) → if(gt(s(x14), y10), x14, y10)
gt(s(x23), s(y17)) → gt(x23, y17)
if(true, x32, y24) → s(minus(x32, y24))
if(false, x41, y31) → 0
ge(x50, 0) → true
ge(0, s(x59)) → false
ge(s(x68), s(y50)) → ge(x68, y50)
minus(0, x1) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a23](0, 0) → true
equal_sort[a23](0, s(x0)) → false
equal_sort[a23](s(x0), 0) → false
equal_sort[a23](s(x0), s(x1)) → equal_sort[a23](x0, x1)
equal_sort[a40](witness_sort[a40], witness_sort[a40]) → true

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Induction-Processor

                                                          ↳ AND

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                            ↳ QTRS

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

DIV(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Induction-Processor

                                                          ↳ AND

                                                            ↳ QDP

                                                            ↳ QTRS

                                                              ↳ QTRSRRRProof

Q restricted rewrite system:
The TRS R consists of the following rules:

minus'(s(x14), y10) → if'(gt(s(x14), y10), x14, y10)
if'(true, x32, y24) → minus'(x32, y24)
if'(false, x41, y31) → true
minus'(0, x1) → false
gt(0, y') → false
gt(s(x5), 0) → true
minus(s(x14), y10) → if(gt(s(x14), y10), x14, y10)
gt(s(x23), s(y17)) → gt(x23, y17)
if(true, x32, y24) → s(minus(x32, y24))
if(false, x41, y31) → 0
ge(x50, 0) → true
ge(0, s(x59)) → false
ge(s(x68), s(y50)) → ge(x68, y50)
minus(0, x1) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a23](0, 0) → true
equal_sort[a23](0, s(x0)) → false
equal_sort[a23](s(x0), 0) → false
equal_sort[a23](s(x0), s(x1)) → equal_sort[a23](x0, x1)
equal_sort[a40](witness_sort[a40], witness_sort[a40]) → true

Q is empty.

The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

minus'(s(x14), y10) → if'(gt(s(x14), y10), x14, y10)
if'(true, x32, y24) → minus'(x32, y24)
if'(false, x41, y31) → true
minus'(0, x1) → false
gt(0, y') → false
gt(s(x5), 0) → true
minus(s(x14), y10) → if(gt(s(x14), y10), x14, y10)
gt(s(x23), s(y17)) → gt(x23, y17)
if(true, x32, y24) → s(minus(x32, y24))
if(false, x41, y31) → 0
ge(x50, 0) → true
ge(0, s(x59)) → false
ge(s(x68), s(y50)) → ge(x68, y50)
minus(0, x1) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a23](0, 0) → true
equal_sort[a23](0, s(x0)) → false
equal_sort[a23](s(x0), 0) → false
equal_sort[a23](s(x0), s(x1)) → equal_sort[a23](x0, x1)
equal_sort[a40](witness_sort[a40], witness_sort[a40]) → true

Q is empty.
Used ordering:
Combined order from the following AFS and order.
minus'(x1, x2) = minus'(x1, x2)
s(x1) = s(x1)
if'(x1, x2, x3) = if'(x1, x2, x3)
gt(x1, x2) = gt(x1, x2)
true = true
false = false
0 = 0
minus(x1, x2) = minus(x1, x2)
if(x1, x2, x3) = if(x1, x2, x3)
ge(x1, x2) = ge(x1, x2)
equal_bool(x1, x2) = equal_bool(x1, x2)
and(x1, x2) = and(x1, x2)
or(x1, x2) = or(x1, x2)
not(x1) = x1
isa_true(x1) = isa_true(x1)
isa_false(x1) = isa_false(x1)
equal_sort[a23](x1, x2) = equal_sort[a23](x1, x2)
equal_sort[a40](x1, x2) = equal_sort[a40](x1, x2)
witness_sort[a40] = witness_sort[a40]

Recursive path order with status [RPO].
Quasi-Precedence:

[true, false, 0, minus₂, if₃, equal_sort[a23]₂, equal_sort[a40]₂] > s₁ > [minus'₂, if'₃] > gt₂
ge₂ > gt₂
equal_bool₂ > gt₂
and₂ > gt₂
or₂ > gt₂
isa_true₁ > gt₂
isa_false₁ > gt₂
witness_sort[a40] > gt₂

Status:

equal_sort[a23]₂: multiset
equal_sort[a40]₂: [1,2]
minus₂: [2,1]
minus'₂: [2,1]
true: multiset
or₂: multiset
and₂: multiset
gt₂: multiset
0: multiset
equal_bool₂: multiset
if₃: [3,2,1]
if'₃: [3,2,1]
isa_false₁: multiset
witness_sort[a40]: multiset
false: multiset
s₁: multiset
ge₂: [1,2]
isa_true₁: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

minus'(s(x14), y10) → if'(gt(s(x14), y10), x14, y10)
if'(true, x32, y24) → minus'(x32, y24)
if'(false, x41, y31) → true
minus'(0, x1) → false
gt(0, y') → false
gt(s(x5), 0) → true
minus(s(x14), y10) → if(gt(s(x14), y10), x14, y10)
gt(s(x23), s(y17)) → gt(x23, y17)
if(true, x32, y24) → s(minus(x32, y24))
if(false, x41, y31) → 0
ge(x50, 0) → true
ge(0, s(x59)) → false
ge(s(x68), s(y50)) → ge(x68, y50)
minus(0, x1) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a23](0, 0) → true
equal_sort[a23](0, s(x0)) → false
equal_sort[a23](s(x0), 0) → false
equal_sort[a23](s(x0), s(x1)) → equal_sort[a23](x0, x1)
equal_sort[a40](witness_sort[a40], witness_sort[a40]) → true

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Induction-Processor

                                                          ↳ AND

                                                            ↳ QDP

                                                            ↳ QTRS

                                                              ↳ QTRSRRRProof

                                                                ↳ QTRS

                                                                  ↳ QTRSRRRProof

                                                                  ↳ QTRSRRRProof

Q restricted rewrite system:
The TRS R consists of the following rules:

not(false) → true
not(true) → false

Q is empty.

The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

not(false) → true
not(true) → false

Q is empty.
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:

[not₁, false] > true

Status:

not₁: [1]
true: multiset
false: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

not(false) → true
not(true) → false

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Induction-Processor

                                                          ↳ AND

                                                            ↳ QDP

                                                            ↳ QTRS

                                                              ↳ QTRSRRRProof

                                                                ↳ QTRS

                                                                  ↳ QTRSRRRProof

                                                                    ↳ QTRS

                                                                      ↳ RisEmptyProof

                                                                      ↳ RisEmptyProof

                                                                      ↳ RisEmptyProof

                                                                  ↳ QTRSRRRProof

Q restricted rewrite system:
R is empty.
Q is empty.

The TRS R is empty. Hence, termination is trivially proven.
The TRS R is empty. Hence, termination is trivially proven.
The TRS R is empty. Hence, termination is trivially proven.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

not(false) → true
not(true) → false

Q is empty.
Used ordering:
Polynomial interpretation [POLO]:

POL(false) = 0
POL(not(x₁)) = 1 + x₁
POL(true) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

not(false) → true
not(true) → false

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Induction-Processor

                                                          ↳ AND

                                                            ↳ QDP

                                                            ↳ QTRS

                                                              ↳ QTRSRRRProof

                                                                ↳ QTRS

                                                                  ↳ QTRSRRRProof

                                                                  ↳ QTRSRRRProof

                                                                    ↳ QTRS

Q restricted rewrite system:
R is empty.
Q is empty.