Termination w.r.t. Q proof of /tmp/tmp04fzh6/gcd2.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
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → if3(gt(x, 0), x, y)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
if3(true, x, y) → gcd(x, minus(y, x))
if3(false, x, y) → y
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

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
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → if3(gt(x, 0), x, y)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
if3(true, x, y) → gcd(x, minus(y, x))
if3(false, x, y) → y
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

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
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → if3(gt(x, 0), x, y)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
if3(true, x, y) → gcd(x, minus(y, x))
if3(false, x, y) → y
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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)
GCD(x, y) → IF1(ge(x, y), x, y)
GCD(x, y) → GE(x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF1(true, x, y) → GT(y, 0)
IF1(false, x, y) → IF3(gt(x, 0), x, y)
IF1(false, x, y) → GT(x, 0)
IF2(true, x, y) → GCD(minus(x, y), y)
IF2(true, x, y) → MINUS(x, y)
IF3(true, x, y) → GCD(x, minus(y, x))
IF3(true, x, y) → MINUS(y, x)
GT(s(x), s(y)) → GT(x, y)
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
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → if3(gt(x, 0), x, y)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
if3(true, x, y) → gcd(x, minus(y, x))
if3(false, x, y) → y
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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)
GCD(x, y) → IF1(ge(x, y), x, y)
GCD(x, y) → GE(x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF1(true, x, y) → GT(y, 0)
IF1(false, x, y) → IF3(gt(x, 0), x, y)
IF1(false, x, y) → GT(x, 0)
IF2(true, x, y) → GCD(minus(x, y), y)
IF2(true, x, y) → MINUS(x, y)
IF3(true, x, y) → GCD(x, minus(y, x))
IF3(true, x, y) → MINUS(y, x)
GT(s(x), s(y)) → GT(x, y)
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
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → if3(gt(x, 0), x, y)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
if3(true, x, y) → gcd(x, minus(y, x))
if3(false, x, y) → y
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 6 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:

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
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → if3(gt(x, 0), x, y)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
if3(true, x, y) → gcd(x, minus(y, x))
if3(false, x, y) → y
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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:

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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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:

GE(s(x), s(y)) → GE(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:

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

                ↳ UsableRulesProof

              ↳ 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
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → if3(gt(x, 0), x, y)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
if3(true, x, y) → gcd(x, minus(y, x))
if3(false, x, y) → y
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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:

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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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:

GT(s(x), s(y)) → GT(x, y)

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

              ↳ 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
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → if3(gt(x, 0), x, y)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
if3(true, x, y) → gcd(x, minus(y, x))
if3(false, x, y) → y
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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) → GCD(minus(x, y), y)
GCD(x, y) → IF1(ge(x, y), x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF1(false, x, y) → IF3(gt(x, 0), x, y)
IF3(true, x, y) → GCD(x, minus(y, x))

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
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → if3(gt(x, 0), x, y)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
if3(true, x, y) → gcd(x, minus(y, x))
if3(false, x, y) → y
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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) → GCD(minus(x, y), y)
GCD(x, y) → IF1(ge(x, y), x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF1(false, x, y) → IF3(gt(x, 0), x, y)
IF3(true, x, y) → GCD(x, minus(y, x))

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)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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].

gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
if3(true, x0, x1)
if3(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) → GCD(minus(x, y), y)
GCD(x, y) → IF1(ge(x, y), x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF1(false, x, y) → IF3(gt(x, 0), x, y)
IF3(true, x, y) → GCD(x, minus(y, x))

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

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

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

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

IF2(true, x, y) → GCD(minus(x, y), y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF1(false, x, y) → IF3(gt(x, 0), x, y)
IF3(true, x, y) → GCD(x, minus(y, x))
GCD(0, s(x0)) → IF1(false, 0, s(x0))
GCD(x0, 0) → IF1(true, x0, 0)
GCD(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)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

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

IF2(true, x, y) → GCD(minus(x, y), y)
IF1(false, x, y) → IF3(gt(x, 0), x, y)
IF3(true, x, y) → GCD(x, minus(y, x))
GCD(0, s(x0)) → IF1(false, 0, s(x0))
GCD(x0, 0) → IF1(true, x0, 0)
GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
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)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

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

GCD(0, s(x0)) → IF1(false, 0, s(x0))
IF1(false, x, y) → IF3(gt(x, 0), x, y)
IF3(true, x, y) → GCD(x, minus(y, x))
GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, y0, s(x0)) → IF2(true, y0, s(x0))
IF2(true, x, y) → GCD(minus(x, y), 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)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule IF1(false, x, y) → IF3(gt(x, 0), x, y) at position [0] we obtained the following new rules [LPAR04]:

IF1(false, 0, y1) → IF3(false, 0, y1)
IF1(false, s(x0), y1) → IF3(true, s(x0), y1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

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

GCD(0, s(x0)) → IF1(false, 0, s(x0))
IF3(true, x, y) → GCD(x, minus(y, x))
GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, y0, s(x0)) → IF2(true, y0, s(x0))
IF2(true, x, y) → GCD(minus(x, y), y)
IF1(false, 0, y1) → IF3(false, 0, y1)
IF1(false, s(x0), y1) → IF3(true, s(x0), y1)

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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, y0, s(x0)) → IF2(true, y0, s(x0))
IF2(true, x, y) → GCD(minus(x, y), y)
IF1(false, s(x0), y1) → IF3(true, s(x0), y1)
IF3(true, x, y) → GCD(x, minus(y, x))

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

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

IF2(true, s(x0), x1) → GCD(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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, y0, s(x0)) → IF2(true, y0, s(x0))
IF1(false, s(x0), y1) → IF3(true, s(x0), y1)
IF3(true, x, y) → GCD(x, minus(y, x))
IF2(true, s(x0), x1) → GCD(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)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule IF3(true, x, y) → GCD(x, minus(y, x)) at position [1] we obtained the following new rules [LPAR04]:

IF3(true, x1, s(x0)) → GCD(x1, if(gt(s(x0), x1), x0, x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, y0, s(x0)) → IF2(true, y0, s(x0))
IF1(false, s(x0), y1) → IF3(true, s(x0), y1)
IF2(true, s(x0), x1) → GCD(if(gt(s(x0), x1), x0, x1), x1)
IF3(true, x1, s(x0)) → GCD(x1, if(gt(s(x0), x1), x0, 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)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(false, s(x0), y1) → IF3(true, s(x0), y1)
IF2(true, s(x0), x1) → GCD(if(gt(s(x0), x1), x0, x1), x1)
IF3(true, x1, s(x0)) → GCD(x1, if(gt(s(x0), x1), x0, 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)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule IF1(false, s(x0), y1) → IF3(true, s(x0), y1) we obtained the following new rules [LPAR04]:

IF1(false, s(z0), s(z1)) → IF3(true, s(z0), s(z1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF2(true, s(x0), x1) → GCD(if(gt(s(x0), x1), x0, x1), x1)
IF3(true, x1, s(x0)) → GCD(x1, if(gt(s(x0), x1), x0, x1))
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))
IF1(false, s(z0), s(z1)) → IF3(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)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

IF2(true, s(z0), s(z1)) → GCD(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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Rewriting

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF3(true, x1, s(x0)) → GCD(x1, if(gt(s(x0), x1), x0, x1))
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))
IF1(false, s(z0), s(z1)) → IF3(true, s(z0), s(z1))
IF2(true, s(z0), s(z1)) → GCD(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)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(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)) → GCD(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)) → GCD(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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF3(true, x1, s(x0)) → GCD(x1, if(gt(s(x0), x1), x0, x1))
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))
IF1(false, s(z0), s(z1)) → IF3(true, s(z0), s(z1))
IF2(true, s(z0), s(z1)) → GCD(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)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule IF3(true, x1, s(x0)) → GCD(x1, if(gt(s(x0), x1), x0, x1)) we obtained the following new rules [LPAR04]:

IF3(true, s(z0), s(z1)) → GCD(s(z0), if(gt(s(z1), s(z0)), z1, s(z0)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Rewriting

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))
IF1(false, s(z0), s(z1)) → IF3(true, s(z0), s(z1))
IF2(true, s(z0), s(z1)) → GCD(if(gt(z0, z1), z0, s(z1)), s(z1))
IF3(true, s(z0), s(z1)) → GCD(s(z0), if(gt(s(z1), s(z0)), z1, s(z0)))

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule IF3(true, s(z0), s(z1)) → GCD(s(z0), if(gt(s(z1), s(z0)), z1, s(z0))) at position [1,0] we obtained the following new rules [LPAR04]:

IF3(true, s(z0), s(z1)) → GCD(s(z0), if(gt(z1, z0), z1, s(z0)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Induction-Processor

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))
IF1(false, s(z0), s(z1)) → IF3(true, s(z0), s(z1))
IF2(true, s(z0), s(z1)) → GCD(if(gt(z0, z1), z0, s(z1)), s(z1))
IF3(true, s(z0), s(z1)) → GCD(s(z0), if(gt(z1, z0), z1, s(z0)))

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)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(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(z00), s(z10)) → GCD(if(gt(z00, z10), z00, s(z10)), s(z10))

This order was computed:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(GCD(x₁, x₂)) = x₁
POL(IF1(x₁, x₂, x₃)) = x₂
POL(IF2(x₁, x₂, x₃)) = x₂
POL(IF3(x₁, x₂, x₃)) = x₂
POL(false) = 1
POL(ge(x₁, x₂)) = 1 + x₁
POL(gt(x₁, x₂)) = 1
POL(if(x₁, x₂, x₃)) = x₁ + x₂
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = 1 + x₁
POL(true) = 1

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

The following formula is valid:
z00:sort[a27],z10:sort[a27].if'(gt(z00 , z10 ), z00 , s(z10 ))=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[a27](0, 0) → true
equal_sort[a27](0, s(x0)) → false
equal_sort[a27](s(x0), 0) → false
equal_sort[a27](s(x0), s(x1)) → equal_sort[a27](x0, x1)
equal_sort[a44](witness_sort[a44], witness_sort[a44]) → true

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Induction-Processor

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                ↳ QTRS

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))
IF1(false, s(z0), s(z1)) → IF3(true, s(z0), s(z1))
IF3(true, s(z0), s(z1)) → GCD(s(z0), if(gt(z1, z0), z1, s(z0)))

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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Induction-Processor

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Induction-Processor

                                                                                ↳ QTRS

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

IF1(false, s(z0), s(z1)) → IF3(true, s(z0), s(z1))
IF3(true, s(z0), s(z1)) → GCD(s(z0), if(gt(z1, z0), z1, s(z0)))
GCD(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)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

This DP could be deleted by the Induction-Processor:
IF3(true, s(z0''), s(z1'')) → GCD(s(z0''), if(gt(z1'', z0''), z1'', s(z0'')))

This order was computed:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(GCD(x₁, x₂)) = x₂
POL(IF1(x₁, x₂, x₃)) = x₃
POL(IF3(x₁, x₂, x₃)) = x₃
POL(false) = 0
POL(ge(x₁, x₂)) = x₁
POL(gt(x₁, x₂)) = 0
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:
z1'':sort[a23],z0'':sort[a23].if'(gt(z1'' , z0'' ), z1'' , s(z0'' ))=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[a41](witness_sort[a41], witness_sort[a41]) → true

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Induction-Processor

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Induction-Processor

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                          ↳ QTRS

                                                                                ↳ QTRS

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

IF1(false, s(z0), s(z1)) → IF3(true, s(z0), s(z1))
GCD(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)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Induction-Processor

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Induction-Processor

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                          ↳ QTRS

                                                                                            ↳ QTRSRRRProof

                                                                                ↳ QTRS

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[a41](witness_sort[a41], witness_sort[a41]) → 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[a41](witness_sort[a41], witness_sort[a41]) → true

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

[true, false, 0, minus₂, if₃, ge₂, not₁, equal_sort[a41]₂] > s₁ > [minus'₂, if'₃] > gt₂
equal_bool₂ > gt₂
and₂ > gt₂
or₂ > gt₂
isa_true₁ > gt₂
isa_false₁ > gt₂
equal_sort[a23]₂ > gt₂
witness_sort[a41] > gt₂

Status:

equal_sort[a23]₂: multiset
minus₂: [2,1]
minus'₂: [1,2]
true: multiset
or₂: multiset
equal_sort[a41]₂: [1,2]
and₂: multiset
gt₂: multiset
0: multiset
equal_bool₂: multiset
if₃: [3,2,1]
if'₃: [2,3,1]
witness_sort[a41]: multiset
not₁: multiset
isa_false₁: [1]
false: multiset
s₁: multiset
ge₂: multiset
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
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[a41](witness_sort[a41], witness_sort[a41]) → true

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Induction-Processor

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Induction-Processor

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                          ↳ QTRS

                                                                                            ↳ QTRSRRRProof

                                                                                              ↳ QTRS

                                                                                                ↳ RisEmptyProof

                                                                                                ↳ RisEmptyProof

                                                                                                ↳ RisEmptyProof

                                                                                ↳ QTRS

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.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ 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[a27](0, 0) → true
equal_sort[a27](0, s(x0)) → false
equal_sort[a27](s(x0), 0) → false
equal_sort[a27](s(x0), s(x1)) → equal_sort[a27](x0, x1)
equal_sort[a44](witness_sort[a44], witness_sort[a44]) → 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[a27](0, 0) → true
equal_sort[a27](0, s(x0)) → false
equal_sort[a27](s(x0), 0) → false
equal_sort[a27](s(x0), s(x1)) → equal_sort[a27](x0, x1)
equal_sort[a44](witness_sort[a44], witness_sort[a44]) → true

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

[minus'₂, if'₃] > s₁ > [false, ge₂, equal_bool₂] > [true, witness_sort[a44]] > gt₂
[minus₂, if₃] > s₁ > [false, ge₂, equal_bool₂] > [true, witness_sort[a44]] > gt₂
[minus₂, if₃] > 0 > [false, ge₂, equal_bool₂] > [true, witness_sort[a44]] > gt₂
and₂ > [false, ge₂, equal_bool₂] > [true, witness_sort[a44]] > gt₂
or₂ > [true, witness_sort[a44]] > gt₂
not₁ > [false, ge₂, equal_bool₂] > [true, witness_sort[a44]] > gt₂
isa_true₁ > [false, ge₂, equal_bool₂] > [true, witness_sort[a44]] > gt₂
isa_false₁ > [false, ge₂, equal_bool₂] > [true, witness_sort[a44]] > gt₂
equal_sort[a27]₂ > [false, ge₂, equal_bool₂] > [true, witness_sort[a44]] > gt₂
equal_sort[a44]₂ > [true, witness_sort[a44]] > gt₂

Status:

minus₂: [2,1]
minus'₂: [2,1]
true: multiset
or₂: multiset
and₂: [2,1]
gt₂: multiset
0: multiset
equal_sort[a44]₂: multiset
witness_sort[a44]: multiset
equal_bool₂: [2,1]
if₃: [3,2,1]
if'₃: [3,2,1]
equal_sort[a27]₂: multiset
not₁: multiset
isa_false₁: [1]
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
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a27](0, 0) → true
equal_sort[a27](0, s(x0)) → false
equal_sort[a27](s(x0), 0) → false
equal_sort[a27](s(x0), s(x1)) → equal_sort[a27](x0, x1)
equal_sort[a44](witness_sort[a44], witness_sort[a44]) → true

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Induction-Processor

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                ↳ QTRS

                                                                                  ↳ QTRSRRRProof

                                                                                    ↳ QTRS

                                                                                      ↳ RisEmptyProof

                                                                                      ↳ RisEmptyProof

                                                                                      ↳ RisEmptyProof