Termination w.r.t. Q proof of /tmp/tmpEh-KFT/#4.30c.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:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
gcd(0, y) → y
gcd(s(x), 0) → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) → gcd(minus(x, y), y)
if_gcd(false, x, y) → gcd(minus(y, x), x)

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
gcd(0, y) → y
gcd(s(x), 0) → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) → gcd(minus(x, y), y)
if_gcd(false, x, y) → gcd(minus(y, x), x)

Q is empty.

The TRS is overlay and locally confluent. By [19] we can switch to innermost.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

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

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
gcd(0, y) → y
gcd(s(x), 0) → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) → gcd(minus(x, y), y)
if_gcd(false, x, y) → gcd(minus(y, x), x)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if_gcd(true, x0, x1)
if_gcd(false, x0, x1)

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

IF_GCD(false, x, y) → GCD(minus(y, x), x)
IF_GCD(true, x, y) → GCD(minus(x, y), y)
GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y))
MINUS(x, s(y)) → MINUS(x, y)
IF_GCD(false, x, y) → MINUS(y, x)
IF_GCD(true, x, y) → MINUS(x, y)
LE(s(x), s(y)) → LE(x, y)
GCD(s(x), s(y)) → LE(y, x)
MINUS(x, s(y)) → PRED(minus(x, y))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
gcd(0, y) → y
gcd(s(x), 0) → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) → gcd(minus(x, y), y)
if_gcd(false, x, y) → gcd(minus(y, x), x)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if_gcd(true, x0, x1)
if_gcd(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:

IF_GCD(false, x, y) → GCD(minus(y, x), x)
IF_GCD(true, x, y) → GCD(minus(x, y), y)
GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y))
MINUS(x, s(y)) → MINUS(x, y)
IF_GCD(false, x, y) → MINUS(y, x)
IF_GCD(true, x, y) → MINUS(x, y)
LE(s(x), s(y)) → LE(x, y)
GCD(s(x), s(y)) → LE(y, x)
MINUS(x, s(y)) → PRED(minus(x, y))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
gcd(0, y) → y
gcd(s(x), 0) → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) → gcd(minus(x, y), y)
if_gcd(false, x, y) → gcd(minus(y, x), x)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if_gcd(true, x0, x1)
if_gcd(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 4 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

MINUS(x, s(y)) → MINUS(x, y)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
gcd(0, y) → y
gcd(s(x), 0) → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) → gcd(minus(x, y), y)
if_gcd(false, x, y) → gcd(minus(y, x), x)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if_gcd(true, x0, x1)
if_gcd(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 [15] we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

MINUS(x, s(y)) → MINUS(x, y)

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

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if_gcd(true, x0, x1)
if_gcd(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.

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if_gcd(true, x0, x1)
if_gcd(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

MINUS(x, s(y)) → MINUS(x, y)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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

MINUS(x, s(y)) → MINUS(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

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

LE(s(x), s(y)) → LE(x, y)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
gcd(0, y) → y
gcd(s(x), 0) → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) → gcd(minus(x, y), y)
if_gcd(false, x, y) → gcd(minus(y, x), x)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if_gcd(true, x0, x1)
if_gcd(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

LE(s(x), s(y)) → LE(x, y)

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

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if_gcd(true, x0, x1)
if_gcd(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.

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if_gcd(true, x0, x1)
if_gcd(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

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

LE(s(x), s(y)) → LE(x, y)

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

LE(s(x), s(y)) → LE(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

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

IF_GCD(false, x, y) → GCD(minus(y, x), x)
IF_GCD(true, x, y) → GCD(minus(x, y), y)
GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
gcd(0, y) → y
gcd(s(x), 0) → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) → gcd(minus(x, y), y)
if_gcd(false, x, y) → gcd(minus(y, x), x)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if_gcd(true, x0, x1)
if_gcd(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

IF_GCD(false, x, y) → GCD(minus(y, x), x)
IF_GCD(true, x, y) → GCD(minus(x, y), y)
GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if_gcd(true, x0, x1)
if_gcd(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.

gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if_gcd(true, x0, x1)
if_gcd(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

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

IF_GCD(false, x, y) → GCD(minus(y, x), x)
IF_GCD(true, x, y) → GCD(minus(x, y), y)
GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
The DP Problem is simplified using the Induction Calculus [18] with the following steps:
Note that final constraints are written in bold face.

For Pair IF_GCD(false, x, y) → GCD(minus(y, x), x) the following chains were created:

We consider the chain GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y)), IF_GCD(false, x, y) → GCD(minus(y, x), x) which results in the following constraint:
(1)    (IF_GCD(le(x5, x4), s(x4), s(x5))=IF_GCD(false, x6, x7) ⇒ IF_GCD(false, x6, x7)≥GCD(minus(x7, x6), x6))

We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
(2)    (le(x5, x4)=false ⇒ IF_GCD(false, s(x4), s(x5))≥GCD(minus(s(x5), s(x4)), s(x4)))

We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on le(x5, x4)=false which results in the following new constraints:
(3)    (false=false ⇒ IF_GCD(false, s(0), s(s(x27)))≥GCD(minus(s(s(x27)), s(0)), s(0)))

(4)    (le(x28, x29)=false∧(le(x28, x29)=false ⇒ IF_GCD(false, s(x29), s(x28))≥GCD(minus(s(x28), s(x29)), s(x29))) ⇒ IF_GCD(false, s(s(x29)), s(s(x28)))≥GCD(minus(s(s(x28)), s(s(x29))), s(s(x29))))

We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
(5)    (IF_GCD(false, s(0), s(s(x27)))≥GCD(minus(s(s(x27)), s(0)), s(0)))

We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (le(x28, x29)=false ⇒ IF_GCD(false, s(x29), s(x28))≥GCD(minus(s(x28), s(x29)), s(x29))) with σ = [ ] which results in the following new constraint:
(6)    (IF_GCD(false, s(x29), s(x28))≥GCD(minus(s(x28), s(x29)), s(x29)) ⇒ IF_GCD(false, s(s(x29)), s(s(x28)))≥GCD(minus(s(s(x28)), s(s(x29))), s(s(x29))))

For Pair IF_GCD(true, x, y) → GCD(minus(x, y), y) the following chains were created:

We consider the chain GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y)), IF_GCD(true, x, y) → GCD(minus(x, y), y) which results in the following constraint:
(7)    (IF_GCD(le(x13, x12), s(x12), s(x13))=IF_GCD(true, x14, x15) ⇒ IF_GCD(true, x14, x15)≥GCD(minus(x14, x15), x15))

We simplified constraint (7) using rules (I), (II), (III) which results in the following new constraint:
(8)    (le(x13, x12)=true ⇒ IF_GCD(true, s(x12), s(x13))≥GCD(minus(s(x12), s(x13)), s(x13)))

We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on le(x13, x12)=true which results in the following new constraints:
(9)    (true=true ⇒ IF_GCD(true, s(x30), s(0))≥GCD(minus(s(x30), s(0)), s(0)))

(10)    (le(x32, x33)=true∧(le(x32, x33)=true ⇒ IF_GCD(true, s(x33), s(x32))≥GCD(minus(s(x33), s(x32)), s(x32))) ⇒ IF_GCD(true, s(s(x33)), s(s(x32)))≥GCD(minus(s(s(x33)), s(s(x32))), s(s(x32))))

We simplified constraint (9) using rules (I), (II) which results in the following new constraint:
(11)    (IF_GCD(true, s(x30), s(0))≥GCD(minus(s(x30), s(0)), s(0)))

We simplified constraint (10) using rule (VI) where we applied the induction hypothesis (le(x32, x33)=true ⇒ IF_GCD(true, s(x33), s(x32))≥GCD(minus(s(x33), s(x32)), s(x32))) with σ = [ ] which results in the following new constraint:
(12)    (IF_GCD(true, s(x33), s(x32))≥GCD(minus(s(x33), s(x32)), s(x32)) ⇒ IF_GCD(true, s(s(x33)), s(s(x32)))≥GCD(minus(s(s(x33)), s(s(x32))), s(s(x32))))

For Pair GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y)) the following chains were created:

We consider the chain IF_GCD(false, x, y) → GCD(minus(y, x), x), GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y)) which results in the following constraint:
(13)    (GCD(minus(x17, x16), x16)=GCD(s(x18), s(x19)) ⇒ GCD(s(x18), s(x19))≥IF_GCD(le(x19, x18), s(x18), s(x19)))

We simplified constraint (13) using rules (I), (II) which results in the following new constraint:
(14)    (minus(x17, x16)=s(x18)∧x16=s(x19) ⇒ GCD(s(x18), s(x19))≥IF_GCD(le(x19, x18), s(x18), s(x19)))

We simplified constraint (14) using rule (V) (with possible (I) afterwards) using induction on minus(x17, x16)=s(x18) which results in the following new constraints:
(15)    (x34=s(x18)∧0=s(x19) ⇒ GCD(s(x18), s(x19))≥IF_GCD(le(x19, x18), s(x18), s(x19)))

(16)    (pred(minus(x35, x36))=s(x18)∧s(x36)=s(x19)∧(∀x37,x38:minus(x35, x36)=s(x37)∧x36=s(x38) ⇒ GCD(s(x37), s(x38))≥IF_GCD(le(x38, x37), s(x37), s(x38))) ⇒ GCD(s(x18), s(x19))≥IF_GCD(le(x19, x18), s(x18), s(x19)))

We solved constraint (15) using rules (I), (II).We simplified constraint (16) using rules (I), (II), (III), (VII) which results in the following new constraint:
(17)    (minus(x35, x36)=x39∧pred(x39)=s(x18)∧(∀x37,x38:minus(x35, x36)=s(x37)∧x36=s(x38) ⇒ GCD(s(x37), s(x38))≥IF_GCD(le(x38, x37), s(x37), s(x38))) ⇒ GCD(s(x18), s(x36))≥IF_GCD(le(x36, x18), s(x18), s(x36)))

We simplified constraint (17) using rule (V) (with possible (I) afterwards) using induction on pred(x39)=s(x18) which results in the following new constraint:
(18)    (x40=s(x18)∧minus(x35, x36)=s(x40)∧(∀x37,x38:minus(x35, x36)=s(x37)∧x36=s(x38) ⇒ GCD(s(x37), s(x38))≥IF_GCD(le(x38, x37), s(x37), s(x38))) ⇒ GCD(s(x18), s(x36))≥IF_GCD(le(x36, x18), s(x18), s(x36)))

We simplified constraint (18) using rules (III), (IV) which results in the following new constraint:
(19)    (minus(x35, x36)=s(s(x18)) ⇒ GCD(s(x18), s(x36))≥IF_GCD(le(x36, x18), s(x18), s(x36)))

We simplified constraint (19) using rule (V) (with possible (I) afterwards) using induction on minus(x35, x36)=s(s(x18)) which results in the following new constraints:
(20)    (x41=s(s(x18)) ⇒ GCD(s(x18), s(0))≥IF_GCD(le(0, x18), s(x18), s(0)))

(21)    (pred(minus(x42, x43))=s(s(x18))∧(∀x44:minus(x42, x43)=s(s(x44)) ⇒ GCD(s(x44), s(x43))≥IF_GCD(le(x43, x44), s(x44), s(x43))) ⇒ GCD(s(x18), s(s(x43)))≥IF_GCD(le(s(x43), x18), s(x18), s(s(x43))))

We simplified constraint (20) using rule (III) which results in the following new constraint:
(22)    (GCD(s(x18), s(0))≥IF_GCD(le(0, x18), s(x18), s(0)))

We simplified constraint (21) using rules (IV), (VII) which results in the following new constraint:
(23)    (GCD(s(x18), s(s(x43)))≥IF_GCD(le(s(x43), x18), s(x18), s(s(x43))))
We consider the chain IF_GCD(true, x, y) → GCD(minus(x, y), y), GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y)) which results in the following constraint:
(24)    (GCD(minus(x20, x21), x21)=GCD(s(x22), s(x23)) ⇒ GCD(s(x22), s(x23))≥IF_GCD(le(x23, x22), s(x22), s(x23)))

We simplified constraint (24) using rules (I), (II) which results in the following new constraint:
(25)    (minus(x20, x21)=s(x22)∧x21=s(x23) ⇒ GCD(s(x22), s(x23))≥IF_GCD(le(x23, x22), s(x22), s(x23)))

We simplified constraint (25) using rule (V) (with possible (I) afterwards) using induction on minus(x20, x21)=s(x22) which results in the following new constraints:
(26)    (x46=s(x22)∧0=s(x23) ⇒ GCD(s(x22), s(x23))≥IF_GCD(le(x23, x22), s(x22), s(x23)))

(27)    (pred(minus(x47, x48))=s(x22)∧s(x48)=s(x23)∧(∀x49,x50:minus(x47, x48)=s(x49)∧x48=s(x50) ⇒ GCD(s(x49), s(x50))≥IF_GCD(le(x50, x49), s(x49), s(x50))) ⇒ GCD(s(x22), s(x23))≥IF_GCD(le(x23, x22), s(x22), s(x23)))

We solved constraint (26) using rules (I), (II).We simplified constraint (27) using rules (I), (II), (III), (VII) which results in the following new constraint:
(28)    (minus(x47, x48)=x51∧pred(x51)=s(x22)∧(∀x49,x50:minus(x47, x48)=s(x49)∧x48=s(x50) ⇒ GCD(s(x49), s(x50))≥IF_GCD(le(x50, x49), s(x49), s(x50))) ⇒ GCD(s(x22), s(x48))≥IF_GCD(le(x48, x22), s(x22), s(x48)))

We simplified constraint (28) using rule (V) (with possible (I) afterwards) using induction on pred(x51)=s(x22) which results in the following new constraint:
(29)    (x52=s(x22)∧minus(x47, x48)=s(x52)∧(∀x49,x50:minus(x47, x48)=s(x49)∧x48=s(x50) ⇒ GCD(s(x49), s(x50))≥IF_GCD(le(x50, x49), s(x49), s(x50))) ⇒ GCD(s(x22), s(x48))≥IF_GCD(le(x48, x22), s(x22), s(x48)))

We simplified constraint (29) using rules (III), (IV) which results in the following new constraint:
(30)    (minus(x47, x48)=s(s(x22)) ⇒ GCD(s(x22), s(x48))≥IF_GCD(le(x48, x22), s(x22), s(x48)))

We simplified constraint (30) using rule (V) (with possible (I) afterwards) using induction on minus(x47, x48)=s(s(x22)) which results in the following new constraints:
(31)    (x53=s(s(x22)) ⇒ GCD(s(x22), s(0))≥IF_GCD(le(0, x22), s(x22), s(0)))

(32)    (pred(minus(x54, x55))=s(s(x22))∧(∀x56:minus(x54, x55)=s(s(x56)) ⇒ GCD(s(x56), s(x55))≥IF_GCD(le(x55, x56), s(x56), s(x55))) ⇒ GCD(s(x22), s(s(x55)))≥IF_GCD(le(s(x55), x22), s(x22), s(s(x55))))

We simplified constraint (31) using rule (III) which results in the following new constraint:
(33)    (GCD(s(x22), s(0))≥IF_GCD(le(0, x22), s(x22), s(0)))

We simplified constraint (32) using rules (IV), (VII) which results in the following new constraint:
(34)    (GCD(s(x22), s(s(x55)))≥IF_GCD(le(s(x55), x22), s(x22), s(s(x55))))

To summarize, we get the following constraints P_≥ for the following pairs.

IF_GCD(false, x, y) → GCD(minus(y, x), x)
- (IF_GCD(false, s(0), s(s(x27)))≥GCD(minus(s(s(x27)), s(0)), s(0)))
- (IF_GCD(false, s(x29), s(x28))≥GCD(minus(s(x28), s(x29)), s(x29)) ⇒ IF_GCD(false, s(s(x29)), s(s(x28)))≥GCD(minus(s(s(x28)), s(s(x29))), s(s(x29))))
IF_GCD(true, x, y) → GCD(minus(x, y), y)
- (IF_GCD(true, s(x30), s(0))≥GCD(minus(s(x30), s(0)), s(0)))
- (IF_GCD(true, s(x33), s(x32))≥GCD(minus(s(x33), s(x32)), s(x32)) ⇒ IF_GCD(true, s(s(x33)), s(s(x32)))≥GCD(minus(s(s(x33)), s(s(x32))), s(s(x32))))
GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y))
- (GCD(s(x18), s(0))≥IF_GCD(le(0, x18), s(x18), s(0)))
- (GCD(s(x18), s(s(x43)))≥IF_GCD(le(s(x43), x18), s(x18), s(s(x43))))
- (GCD(s(x22), s(0))≥IF_GCD(le(0, x22), s(x22), s(0)))
- (GCD(s(x22), s(s(x55)))≥IF_GCD(le(s(x55), x22), s(x22), s(s(x55))))

The constraints for P_> respective P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation [18]:

POL(0) = 0
POL(GCD(x₁, x₂)) = 1 + x₂
POL(IF_GCD(x₁, x₂, x₃)) = 1 - x₁ + x₃
POL(c) = -1
POL(false) = 0
POL(le(x₁, x₂)) = 0
POL(minus(x₁, x₂)) = 0
POL(pred(x₁)) = 1
POL(s(x₁)) = 1 + x₁
POL(true) = 0

The following pairs are in P_>:

IF_GCD(false, x, y) → GCD(minus(y, x), x)

The following pairs are in P_bound:

IF_GCD(false, x, y) → GCD(minus(y, x), x)
IF_GCD(true, x, y) → GCD(minus(x, y), y)
GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y))

The following rules are usable:

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

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

IF_GCD(true, x, y) → GCD(minus(x, y), y)
GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule IF_GCD(true, x, y) → GCD(minus(x, y), y) at position [0] we obtained the following new rules:

IF_GCD(true, x0, 0) → GCD(x0, 0)
IF_GCD(true, x0, s(x1)) → GCD(pred(minus(x0, x1)), s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

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

IF_GCD(true, x0, 0) → GCD(x0, 0)
IF_GCD(true, x0, s(x1)) → GCD(pred(minus(x0, x1)), s(x1))
GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

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

IF_GCD(true, x0, s(x1)) → GCD(pred(minus(x0, x1)), s(x1))
GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y)) at position [0] we obtained the following new rules:

GCD(s(s(x1)), s(s(x0))) → IF_GCD(le(x0, x1), s(s(x1)), s(s(x0)))
GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
GCD(s(0), s(s(x0))) → IF_GCD(false, s(0), s(s(x0)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

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

GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
GCD(s(s(x1)), s(s(x0))) → IF_GCD(le(x0, x1), s(s(x1)), s(s(x0)))
IF_GCD(true, x0, s(x1)) → GCD(pred(minus(x0, x1)), s(x1))
GCD(s(0), s(s(x0))) → IF_GCD(false, s(0), s(s(x0)))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

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

GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
GCD(s(s(x1)), s(s(x0))) → IF_GCD(le(x0, x1), s(s(x1)), s(s(x0)))
IF_GCD(true, x0, s(x1)) → GCD(pred(minus(x0, x1)), s(x1))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IF_GCD(true, x0, s(x1)) → GCD(pred(minus(x0, x1)), s(x1)) we obtained the following new rules:

IF_GCD(true, s(z0), s(0)) → GCD(pred(minus(s(z0), 0)), s(0))
IF_GCD(true, s(s(z0)), s(s(z1))) → GCD(pred(minus(s(s(z0)), s(z1))), s(s(z1)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

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

GCD(s(s(x1)), s(s(x0))) → IF_GCD(le(x0, x1), s(s(x1)), s(s(x0)))
GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
IF_GCD(true, s(z0), s(0)) → GCD(pred(minus(s(z0), 0)), s(0))
IF_GCD(true, s(s(z0)), s(s(z1))) → GCD(pred(minus(s(s(z0)), s(z1))), s(s(z1)))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                    ↳ QDP

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

GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
IF_GCD(true, s(z0), s(0)) → GCD(pred(minus(s(z0), 0)), s(0))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                    ↳ QDP

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

GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
IF_GCD(true, s(z0), s(0)) → GCD(pred(minus(s(z0), 0)), s(0))

The TRS R consists of the following rules:

minus(x, 0) → x
pred(s(x)) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(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.

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                    ↳ QDP

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

GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
IF_GCD(true, s(z0), s(0)) → GCD(pred(minus(s(z0), 0)), s(0))

The TRS R consists of the following rules:

minus(x, 0) → x
pred(s(x)) → x

The set Q consists of the following terms:

pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IF_GCD(true, s(z0), s(0)) → GCD(pred(minus(s(z0), 0)), s(0)) at position [0,0] we obtained the following new rules:

IF_GCD(true, s(z0), s(0)) → GCD(pred(s(z0)), s(0))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                    ↳ QDP

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

GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
IF_GCD(true, s(z0), s(0)) → GCD(pred(s(z0)), s(0))

The TRS R consists of the following rules:

minus(x, 0) → x
pred(s(x)) → x

The set Q consists of the following terms:

pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                    ↳ QDP

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

GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
IF_GCD(true, s(z0), s(0)) → GCD(pred(s(z0)), s(0))

The TRS R consists of the following rules:

pred(s(x)) → x

The set Q consists of the following terms:

pred(s(x0))
minus(x0, 0)
minus(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.

minus(x0, 0)
minus(x0, s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Rewriting

                                                    ↳ QDP

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

GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
IF_GCD(true, s(z0), s(0)) → GCD(pred(s(z0)), s(0))

The TRS R consists of the following rules:

pred(s(x)) → x

The set Q consists of the following terms:

pred(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IF_GCD(true, s(z0), s(0)) → GCD(pred(s(z0)), s(0)) at position [0] we obtained the following new rules:

IF_GCD(true, s(z0), s(0)) → GCD(z0, s(0))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Rewriting

                                                                            ↳ QDP

                                                                              ↳ UsableRulesProof

                                                    ↳ QDP

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

GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
IF_GCD(true, s(z0), s(0)) → GCD(z0, s(0))

The TRS R consists of the following rules:

pred(s(x)) → x

The set Q consists of the following terms:

pred(s(x0))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Rewriting

                                                                            ↳ QDP

                                                                              ↳ UsableRulesProof

                                                                                ↳ QDP

                                                                                  ↳ QReductionProof

                                                    ↳ QDP

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

GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
IF_GCD(true, s(z0), s(0)) → GCD(z0, s(0))

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

pred(s(x0))

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.

pred(s(x0))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Rewriting

                                                                            ↳ QDP

                                                                              ↳ UsableRulesProof

                                                                                ↳ QDP

                                                                                  ↳ QReductionProof

                                                                                    ↳ QDP

                                                                                      ↳ ForwardInstantiation

                                                    ↳ QDP

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

GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
IF_GCD(true, s(z0), s(0)) → GCD(z0, s(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule IF_GCD(true, s(z0), s(0)) → GCD(z0, s(0)) we obtained the following new rules:

IF_GCD(true, s(s(y_0)), s(0)) → GCD(s(y_0), s(0))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Rewriting

                                                                            ↳ QDP

                                                                              ↳ UsableRulesProof

                                                                                ↳ QDP

                                                                                  ↳ QReductionProof

                                                                                    ↳ QDP

                                                                                      ↳ ForwardInstantiation

                                                                                        ↳ QDP

                                                                                          ↳ ForwardInstantiation

                                                    ↳ QDP

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

GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0))
IF_GCD(true, s(s(y_0)), s(0)) → GCD(s(y_0), s(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule GCD(s(x0), s(0)) → IF_GCD(true, s(x0), s(0)) we obtained the following new rules:

GCD(s(s(y_0)), s(0)) → IF_GCD(true, s(s(y_0)), s(0))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Rewriting

                                                                            ↳ QDP

                                                                              ↳ UsableRulesProof

                                                                                ↳ QDP

                                                                                  ↳ QReductionProof

                                                                                    ↳ QDP

                                                                                      ↳ ForwardInstantiation

                                                                                        ↳ QDP

                                                                                          ↳ ForwardInstantiation

                                                                                            ↳ QDP

                                                                                              ↳ QDPSizeChangeProof

                                                    ↳ QDP

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

GCD(s(s(y_0)), s(0)) → IF_GCD(true, s(s(y_0)), s(0))
IF_GCD(true, s(s(y_0)), s(0)) → GCD(s(y_0), s(0))

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

GCD(s(s(y_0)), s(0)) → IF_GCD(true, s(s(y_0)), s(0))
The graph contains the following edges 1 >= 2, 2 >= 3
IF_GCD(true, s(s(y_0)), s(0)) → GCD(s(y_0), s(0))
The graph contains the following edges 2 > 1, 3 >= 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Rewriting

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

GCD(s(s(x1)), s(s(x0))) → IF_GCD(le(x0, x1), s(s(x1)), s(s(x0)))
IF_GCD(true, s(s(z0)), s(s(z1))) → GCD(pred(minus(s(s(z0)), s(z1))), s(s(z1)))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IF_GCD(true, s(s(z0)), s(s(z1))) → GCD(pred(minus(s(s(z0)), s(z1))), s(s(z1))) at position [0,0] we obtained the following new rules:

IF_GCD(true, s(s(z0)), s(s(z1))) → GCD(pred(pred(minus(s(s(z0)), z1))), s(s(z1)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

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

GCD(s(s(x1)), s(s(x0))) → IF_GCD(le(x0, x1), s(s(x1)), s(s(x0)))
IF_GCD(true, s(s(z0)), s(s(z1))) → GCD(pred(pred(minus(s(s(z0)), z1))), s(s(z1)))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))

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

The following pairs can be oriented strictly and are deleted.

GCD(s(s(x1)), s(s(x0))) → IF_GCD(le(x0, x1), s(s(x1)), s(s(x0)))

The remaining pairs can at least be oriented weakly.

IF_GCD(true, s(s(z0)), s(s(z1))) → GCD(pred(pred(minus(s(s(z0)), z1))), s(s(z1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( minus(x₁, x₂) ) =

/	0	\
\	0	/

/	1	1	\
\	0	1	/

x₁

/	0	0	\
\	0	0	/

x₂

M( le(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( true ) =

/	0	\
\	0	/

M( pred(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

M( false ) =

/	0	\
\	0	/

M( s(x₁) ) =

/	1	\
\	0	/

/	1	1	\
\	1	1	/

x₁

M( 0 ) =

/	0	\
\	0	/

Tuple symbols:

M( IF_GCD(x₁, ..., x₃) ) =

[

]

x₁

[

]

x₂

[

]

x₃

M( GCD(x₁, x₂) ) =

[

]

x₁

[

]

x₂

Matrix type:
We used a basic matrix type which is not further parametrizeable.

As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x
minus(x, 0) → x

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ NonInfProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

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

IF_GCD(true, s(s(z0)), s(s(z1))) → GCD(pred(pred(minus(s(s(z0)), z1))), s(s(z1)))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
pred(s(x0))
minus(x0, 0)
minus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.