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

(0) Obligation:

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

digits → d(0)
d(x) → if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x)
if(true, x) → cons(x, d(s(x)))
if(false, x) → nil
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)

Q is empty.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

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

(2) Obligation:

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

digits → d(0)
d(x) → if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x)
if(true, x) → cons(x, d(s(x)))
if(false, x) → nil
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:

digits
d(x0)
if(true, x0)
if(false, x0)
le(0, x0)
le(s(x0), 0)
le(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:

DIGITS → D(0)
D(x) → IF(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x)
D(x) → LE(x, s(s(s(s(s(s(s(s(s(0))))))))))
IF(true, x) → D(s(x))
LE(s(x), s(y)) → LE(x, y)

The TRS R consists of the following rules:

digits → d(0)
d(x) → if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x)
if(true, x) → cons(x, d(s(x)))
if(false, x) → nil
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:

digits
d(x0)
if(true, x0)
if(false, x0)
le(0, x0)
le(s(x0), 0)
le(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 2 SCCs with 2 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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:

digits → d(0)
d(x) → if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x)
if(true, x) → cons(x, d(s(x)))
if(false, x) → nil
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:

digits
d(x0)
if(true, x0)
if(false, x0)
le(0, x0)
le(s(x0), 0)
le(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.

LE(s(x), s(y)) → LE(x, y)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LE(x1, x2)  =  x2
s(x1)  =  s(x1)
digits  =  digits
d(x1)  =  d(x1)
0  =  0
if(x1, x2)  =  if
le(x1, x2)  =  le(x2)
true  =  true
cons(x1, x2)  =  cons
false  =  false
nil  =  nil

Lexicographic path order with status [LPO].
Quasi-Precedence:

[s₁, le₁] > true > [0, cons, false, nil]
digits > d₁ > if > [0, cons, false, nil]

Status:

s₁: [1]
digits: []
d₁: [1]
0: []
if: []
le₁: [1]
true: []
cons: []
false: []
nil: []

The following usable rules [FROCOS05] were oriented:

digits → d(0)
d(x) → if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x)
if(true, x) → cons(x, d(s(x)))
if(false, x) → nil
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)

(9) Obligation:

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

digits → d(0)
d(x) → if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x)
if(true, x) → cons(x, d(s(x)))
if(false, x) → nil
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:

digits
d(x0)
if(true, x0)
if(false, x0)
le(0, x0)
le(s(x0), 0)
le(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:

IF(true, x) → D(s(x))
D(x) → IF(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x)

The TRS R consists of the following rules:

digits → d(0)
d(x) → if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x)
if(true, x) → cons(x, d(s(x)))
if(false, x) → nil
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:

digits
d(x0)
if(true, x0)
if(false, x0)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

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