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:



↳ 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)

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
              ↳ 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)

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:



↳ 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)

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
              ↳ 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:




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




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




To summarize, we get the following constraints P for the following pairs.



The constraints for P> respective Pbound 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 Pbound.
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 Pbound:

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:

truele(0, y)
falsele(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))

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
              ↳ 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))

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
              ↳ 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))

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
              ↳ 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))

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:



↳ 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/
+
/11\
\01/
·x1+
/00\
\00/
·x2

M( le(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( true ) =
/0\
\0/

M( pred(x1) ) =
/0\
\0/
+
/01\
\01/
·x1

M( false ) =
/0\
\0/

M( s(x1) ) =
/1\
\0/
+
/11\
\11/
·x1

M( 0 ) =
/0\
\0/

Tuple symbols:
M( IF_GCD(x1, ..., x3) ) = 0+
[0,0]
·x1+
[1,0]
·x2+
[0,1]
·x3

M( GCD(x1, x2) ) = 1+
[1,0]
·x1+
[0,1]
·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.