Termination w.r.t. Q proof of ../tpdb/TRS/AG01/#4.30c.trs

YES Termination w.r.t. Q proof of ../tpdb/TRS/AG01/#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(true, x, y) → GCD(minus(x, y), y)
IF_GCD(true, x, y) → MINUS(x, y)
MINUS(x, s(y)) → PRED(minus(x, y))
LE(s(x), s(y)) → LE(x, y)
IF_GCD(false, x, y) → MINUS(y, x)
GCD(s(x), s(y)) → LE(y, x)
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) → GCD(minus(y, x), x)

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

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

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

The TRS R consists of the following rules:

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

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

The TRS R consists of the following rules:

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

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(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:
(1)    (IF_GCD(le(x3, x2), s(x2), s(x3))=IF_GCD(true, x4, x5) ⇒ IF_GCD(true, x4, x5)≥GCD(minus(x4, x5), x5))

We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
(2)    (le(x3, x2)=true ⇒ IF_GCD(true, s(x2), s(x3))≥GCD(minus(s(x2), s(x3)), s(x3)))

We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on le(x3, x2)=true which results in the following new constraints:
(3)    (true=true ⇒ IF_GCD(true, s(x26), s(0))≥GCD(minus(s(x26), s(0)), s(0)))

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

We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
(5)    (IF_GCD(true, s(x26), s(0))≥GCD(minus(s(x26), s(0)), s(0)))

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

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(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:
(7)    (GCD(minus(x8, x9), x9)=GCD(s(x10), s(x11)) ⇒ GCD(s(x10), s(x11))≥IF_GCD(le(x11, x10), s(x10), s(x11)))

We simplified constraint (7) using rules (I), (II) which results in the following new constraint:
(8)    (minus(x8, x9)=s(x10)∧x9=s(x11) ⇒ GCD(s(x10), s(x11))≥IF_GCD(le(x11, x10), s(x10), s(x11)))

We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on minus(x8, x9)=s(x10) which results in the following new constraints:
(9)    (x30=s(x10)∧0=s(x11) ⇒ GCD(s(x10), s(x11))≥IF_GCD(le(x11, x10), s(x10), s(x11)))

(10)    (pred(minus(x31, x32))=s(x10)∧s(x32)=s(x11)∧(∀x33,x34:minus(x31, x32)=s(x33)∧x32=s(x34) ⇒ GCD(s(x33), s(x34))≥IF_GCD(le(x34, x33), s(x33), s(x34))) ⇒ GCD(s(x10), s(x11))≥IF_GCD(le(x11, x10), s(x10), s(x11)))

We solved constraint (9) using rules (I), (II).We simplified constraint (10) using rules (I), (II), (III), (VII) which results in the following new constraint:
(11)    (minus(x31, x32)=x35∧pred(x35)=s(x10)∧(∀x33,x34:minus(x31, x32)=s(x33)∧x32=s(x34) ⇒ GCD(s(x33), s(x34))≥IF_GCD(le(x34, x33), s(x33), s(x34))) ⇒ GCD(s(x10), s(x32))≥IF_GCD(le(x32, x10), s(x10), s(x32)))

We simplified constraint (11) using rule (V) (with possible (I) afterwards) using induction on pred(x35)=s(x10) which results in the following new constraint:
(12)    (x36=s(x10)∧minus(x31, x32)=s(x36)∧(∀x33,x34:minus(x31, x32)=s(x33)∧x32=s(x34) ⇒ GCD(s(x33), s(x34))≥IF_GCD(le(x34, x33), s(x33), s(x34))) ⇒ GCD(s(x10), s(x32))≥IF_GCD(le(x32, x10), s(x10), s(x32)))

We simplified constraint (12) using rules (III), (IV) which results in the following new constraint:
(13)    (minus(x31, x32)=s(s(x10)) ⇒ GCD(s(x10), s(x32))≥IF_GCD(le(x32, x10), s(x10), s(x32)))

We simplified constraint (13) using rule (V) (with possible (I) afterwards) using induction on minus(x31, x32)=s(s(x10)) which results in the following new constraints:
(14)    (x37=s(s(x10)) ⇒ GCD(s(x10), s(0))≥IF_GCD(le(0, x10), s(x10), s(0)))

(15)    (pred(minus(x38, x39))=s(s(x10))∧(∀x40:minus(x38, x39)=s(s(x40)) ⇒ GCD(s(x40), s(x39))≥IF_GCD(le(x39, x40), s(x40), s(x39))) ⇒ GCD(s(x10), s(s(x39)))≥IF_GCD(le(s(x39), x10), s(x10), s(s(x39))))

We simplified constraint (14) using rule (III) which results in the following new constraint:
(16)    (GCD(s(x10), s(0))≥IF_GCD(le(0, x10), s(x10), s(0)))

We simplified constraint (15) using rules (IV), (VII) which results in the following new constraint:
(17)    (GCD(s(x10), s(s(x39)))≥IF_GCD(le(s(x39), x10), s(x10), s(s(x39))))
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:
(18)    (GCD(minus(x15, x14), x14)=GCD(s(x16), s(x17)) ⇒ GCD(s(x16), s(x17))≥IF_GCD(le(x17, x16), s(x16), s(x17)))

We simplified constraint (18) using rules (I), (II) which results in the following new constraint:
(19)    (minus(x15, x14)=s(x16)∧x14=s(x17) ⇒ GCD(s(x16), s(x17))≥IF_GCD(le(x17, x16), s(x16), s(x17)))

We simplified constraint (19) using rule (V) (with possible (I) afterwards) using induction on minus(x15, x14)=s(x16) which results in the following new constraints:
(20)    (x42=s(x16)∧0=s(x17) ⇒ GCD(s(x16), s(x17))≥IF_GCD(le(x17, x16), s(x16), s(x17)))

(21)    (pred(minus(x43, x44))=s(x16)∧s(x44)=s(x17)∧(∀x45,x46:minus(x43, x44)=s(x45)∧x44=s(x46) ⇒ GCD(s(x45), s(x46))≥IF_GCD(le(x46, x45), s(x45), s(x46))) ⇒ GCD(s(x16), s(x17))≥IF_GCD(le(x17, x16), s(x16), s(x17)))

We solved constraint (20) using rules (I), (II).We simplified constraint (21) using rules (I), (II), (III), (VII) which results in the following new constraint:
(22)    (minus(x43, x44)=x47∧pred(x47)=s(x16)∧(∀x45,x46:minus(x43, x44)=s(x45)∧x44=s(x46) ⇒ GCD(s(x45), s(x46))≥IF_GCD(le(x46, x45), s(x45), s(x46))) ⇒ GCD(s(x16), s(x44))≥IF_GCD(le(x44, x16), s(x16), s(x44)))

We simplified constraint (22) using rule (V) (with possible (I) afterwards) using induction on pred(x47)=s(x16) which results in the following new constraint:
(23)    (x48=s(x16)∧minus(x43, x44)=s(x48)∧(∀x45,x46:minus(x43, x44)=s(x45)∧x44=s(x46) ⇒ GCD(s(x45), s(x46))≥IF_GCD(le(x46, x45), s(x45), s(x46))) ⇒ GCD(s(x16), s(x44))≥IF_GCD(le(x44, x16), s(x16), s(x44)))

We simplified constraint (23) using rules (III), (IV) which results in the following new constraint:
(24)    (minus(x43, x44)=s(s(x16)) ⇒ GCD(s(x16), s(x44))≥IF_GCD(le(x44, x16), s(x16), s(x44)))

We simplified constraint (24) using rule (V) (with possible (I) afterwards) using induction on minus(x43, x44)=s(s(x16)) which results in the following new constraints:
(25)    (x49=s(s(x16)) ⇒ GCD(s(x16), s(0))≥IF_GCD(le(0, x16), s(x16), s(0)))

(26)    (pred(minus(x50, x51))=s(s(x16))∧(∀x52:minus(x50, x51)=s(s(x52)) ⇒ GCD(s(x52), s(x51))≥IF_GCD(le(x51, x52), s(x52), s(x51))) ⇒ GCD(s(x16), s(s(x51)))≥IF_GCD(le(s(x51), x16), s(x16), s(s(x51))))

We simplified constraint (25) using rule (III) which results in the following new constraint:
(27)    (GCD(s(x16), s(0))≥IF_GCD(le(0, x16), s(x16), s(0)))

We simplified constraint (26) using rules (IV), (VII) which results in the following new constraint:
(28)    (GCD(s(x16), s(s(x51)))≥IF_GCD(le(s(x51), x16), s(x16), s(s(x51))))

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:
(29)    (IF_GCD(le(x21, x20), s(x20), s(x21))=IF_GCD(false, x22, x23) ⇒ IF_GCD(false, x22, x23)≥GCD(minus(x23, x22), x22))

We simplified constraint (29) using rules (I), (II), (III) which results in the following new constraint:
(30)    (le(x21, x20)=false ⇒ IF_GCD(false, s(x20), s(x21))≥GCD(minus(s(x21), s(x20)), s(x20)))

We simplified constraint (30) using rule (V) (with possible (I) afterwards) using induction on le(x21, x20)=false which results in the following new constraints:
(31)    (false=false ⇒ IF_GCD(false, s(0), s(s(x55)))≥GCD(minus(s(s(x55)), s(0)), s(0)))

(32)    (le(x56, x57)=false∧(le(x56, x57)=false ⇒ IF_GCD(false, s(x57), s(x56))≥GCD(minus(s(x56), s(x57)), s(x57))) ⇒ IF_GCD(false, s(s(x57)), s(s(x56)))≥GCD(minus(s(s(x56)), s(s(x57))), s(s(x57))))

We simplified constraint (31) using rules (I), (II) which results in the following new constraint:
(33)    (IF_GCD(false, s(0), s(s(x55)))≥GCD(minus(s(s(x55)), s(0)), s(0)))

We simplified constraint (32) using rule (VI) where we applied the induction hypothesis (le(x56, x57)=false ⇒ IF_GCD(false, s(x57), s(x56))≥GCD(minus(s(x56), s(x57)), s(x57))) with σ = [ ] which results in the following new constraint:
(34)    (IF_GCD(false, s(x57), s(x56))≥GCD(minus(s(x56), s(x57)), s(x57)) ⇒ IF_GCD(false, s(s(x57)), s(s(x56)))≥GCD(minus(s(s(x56)), s(s(x57))), s(s(x57))))

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

IF_GCD(true, x, y) → GCD(minus(x, y), y)
- (IF_GCD(true, s(x26), s(0))≥GCD(minus(s(x26), s(0)), s(0)))
- (IF_GCD(true, s(x29), s(x28))≥GCD(minus(s(x29), s(x28)), s(x28)) ⇒ IF_GCD(true, s(s(x29)), s(s(x28)))≥GCD(minus(s(s(x29)), s(s(x28))), s(s(x28))))
GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y))
- (GCD(s(x10), s(0))≥IF_GCD(le(0, x10), s(x10), s(0)))
- (GCD(s(x10), s(s(x39)))≥IF_GCD(le(s(x39), x10), s(x10), s(s(x39))))
- (GCD(s(x16), s(0))≥IF_GCD(le(0, x16), s(x16), s(0)))
- (GCD(s(x16), s(s(x51)))≥IF_GCD(le(s(x51), x16), s(x16), s(s(x51))))
IF_GCD(false, x, y) → GCD(minus(y, x), x)
- (IF_GCD(false, s(0), s(s(x55)))≥GCD(minus(s(s(x55)), s(0)), s(0)))
- (IF_GCD(false, s(x57), s(x56))≥GCD(minus(s(x56), s(x57)), s(x57)) ⇒ IF_GCD(false, s(s(x57)), s(s(x56)))≥GCD(minus(s(s(x56)), s(s(x57))), s(s(x57))))

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)) = 1
POL(pred(x₁)) = 0
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(true, x, y) → GCD(minus(x, y), y)
GCD(s(x), s(y)) → IF_GCD(le(y, x), s(x), s(y))
IF_GCD(false, x, y) → GCD(minus(y, x), x)

The following rules are usable:

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

↳ 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:

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

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:

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

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:

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

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(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)))
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:

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

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:

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

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

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, s(s(z0)), s(s(z1))) → GCD(pred(minus(s(s(z0)), s(z1))), s(s(z1)))
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
minus(x, s(y)) → pred(minus(x, y))
pred(s(x)) → x
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)

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

                                                      ↳ Rewriting

                                                    ↳ QDP

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:

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

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

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                    ↳ QDP

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:

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

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( true ) =

/	0	\
\	0	/

M( minus(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	1	1	/

x₁

/	0	0	\
\	0	0	/

x₂

M( false ) =

/	0	\
\	0	/

M( 0 ) =

/	0	\
\	0	/

M( s(x₁) ) =

/	0	\
\	1	/

/	1	1	\
\	1	1	/

x₁

M( le(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( pred(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	0	/

x₁

Tuple symbols:

M( GCD(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( IF_GCD(x₁, ..., x₃) ) =

[

]

x₁

[

]

x₂

[

]

x₃

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))
minus(x, 0) → x
pred(s(x)) → 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

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                    ↳ QDP

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:

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

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.

↳ 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

                                                      ↳ UsableRulesProof

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

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

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

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

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

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

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

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

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

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

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Rewriting

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

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Rewriting

                                                                            ↳ QDP

                                                                              ↳ UsableRulesProof

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

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Rewriting

                                                                            ↳ QDP

                                                                              ↳ UsableRulesProof

                                                                                ↳ QDP

                                                                                  ↳ QReductionProof

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

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Rewriting

                                                                            ↳ QDP

                                                                              ↳ UsableRulesProof

                                                                                ↳ QDP

                                                                                  ↳ QReductionProof

                                                                                    ↳ QDP

                                                                                      ↳ ForwardInstantiation

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

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Rewriting

                                                                            ↳ QDP

                                                                              ↳ UsableRulesProof

                                                                                ↳ QDP

                                                                                  ↳ QReductionProof

                                                                                    ↳ QDP

                                                                                      ↳ ForwardInstantiation

                                                                                        ↳ QDP

                                                                                          ↳ ForwardInstantiation

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

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QReductionProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Rewriting

                                                                            ↳ QDP

                                                                              ↳ UsableRulesProof

                                                                                ↳ QDP

                                                                                  ↳ QReductionProof

                                                                                    ↳ QDP

                                                                                      ↳ ForwardInstantiation

                                                                                        ↳ QDP

                                                                                          ↳ ForwardInstantiation

                                                                                            ↳ QDP

                                                                                              ↳ QDPSizeChangeProof

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