Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS
↳1 AAECC Innermost (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
  ↳7 QDP
    ↳8 QDPOrderProof (⇔)
    ↳9 QDP
    ↳10 PisEmptyProof (⇔)
    ↳11 TRUE
  ↳12 QDP
    ↳13 QDPOrderProof (⇔)
    ↳14 QDP
    ↳15 PisEmptyProof (⇔)
    ↳16 TRUE
  ↳17 QDP

(0) Obligation:

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

f(true, x, y) → f(gt(x, y), x, round(s(y)))
round(0) → 0
round(s(0)) → s(s(0))
round(s(s(x))) → s(s(round(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

Q is empty.

(1) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is

round(0) → 0
round(s(0)) → s(s(0))
round(s(s(x))) → s(s(round(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

The TRS R 2 is

f(true, x, y) → f(gt(x, y), x, round(s(y)))

The signature Sigma is {f}

(2) Obligation:

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

f(true, x, y) → f(gt(x, y), x, round(s(y)))
round(0) → 0
round(s(0)) → s(s(0))
round(s(s(x))) → s(s(round(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

The set Q consists of the following terms:

f(true, x0, x1)
round(0)
round(s(0))
round(s(s(x0)))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

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

F(true, x, y) → F(gt(x, y), x, round(s(y)))
F(true, x, y) → GT(x, y)
F(true, x, y) → ROUND(s(y))
ROUND(s(s(x))) → ROUND(x)
GT(s(u), s(v)) → GT(u, v)

The TRS R consists of the following rules:

f(true, x, y) → f(gt(x, y), x, round(s(y)))
round(0) → 0
round(s(0)) → s(s(0))
round(s(s(x))) → s(s(round(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

The set Q consists of the following terms:

f(true, x0, x1)
round(0)
round(s(0))
round(s(s(x0)))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 2 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GT(s(u), s(v)) → GT(u, v)

The TRS R consists of the following rules:

f(true, x, y) → f(gt(x, y), x, round(s(y)))
round(0) → 0
round(s(0)) → s(s(0))
round(s(s(x))) → s(s(round(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

The set Q consists of the following terms:

f(true, x0, x1)
round(0)
round(s(0))
round(s(s(x0)))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


GT(s(u), s(v)) → GT(u, v)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
GT(x1, x2)  =  x2
s(x1)  =  s(x1)
f(x1, x2, x3)  =  f
true  =  true
gt(x1, x2)  =  gt
round(x1)  =  round(x1)
0  =  0
false  =  false

Recursive Path Order [RPO].
Precedence:
round1 > 0 > [s1, f, true, gt, false]


The following usable rules [FROCOS05] were oriented:

f(true, x, y) → f(gt(x, y), x, round(s(y)))
round(0) → 0
round(s(0)) → s(s(0))
round(s(s(x))) → s(s(round(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

(9) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

f(true, x, y) → f(gt(x, y), x, round(s(y)))
round(0) → 0
round(s(0)) → s(s(0))
round(s(s(x))) → s(s(round(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

The set Q consists of the following terms:

f(true, x0, x1)
round(0)
round(s(0))
round(s(s(x0)))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

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

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

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

ROUND(s(s(x))) → ROUND(x)

The TRS R consists of the following rules:

f(true, x, y) → f(gt(x, y), x, round(s(y)))
round(0) → 0
round(s(0)) → s(s(0))
round(s(s(x))) → s(s(round(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

The set Q consists of the following terms:

f(true, x0, x1)
round(0)
round(s(0))
round(s(s(x0)))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ROUND(s(s(x))) → ROUND(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ROUND(x1)  =  x1
s(x1)  =  s(x1)
f(x1, x2, x3)  =  f
true  =  true
gt(x1, x2)  =  gt
round(x1)  =  round(x1)
0  =  0
false  =  false

Recursive Path Order [RPO].
Precedence:
round1 > 0 > [s1, f, true, gt, false]


The following usable rules [FROCOS05] were oriented:

f(true, x, y) → f(gt(x, y), x, round(s(y)))
round(0) → 0
round(s(0)) → s(s(0))
round(s(s(x))) → s(s(round(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

(14) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

f(true, x, y) → f(gt(x, y), x, round(s(y)))
round(0) → 0
round(s(0)) → s(s(0))
round(s(s(x))) → s(s(round(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

The set Q consists of the following terms:

f(true, x0, x1)
round(0)
round(s(0))
round(s(s(x0)))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

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

(15) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(16) TRUE

(17) Obligation:

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

F(true, x, y) → F(gt(x, y), x, round(s(y)))

The TRS R consists of the following rules:

f(true, x, y) → f(gt(x, y), x, round(s(y)))
round(0) → 0
round(s(0)) → s(s(0))
round(s(s(x))) → s(s(round(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

The set Q consists of the following terms:

f(true, x0, x1)
round(0)
round(s(0))
round(s(s(x0)))
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

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