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:

minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))

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:

minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))

The set Q consists of the following terms:

minus(x0, x1)
cond(x0, x1, x0)
min(0, x0)
min(x0, 0)
min(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:

MINUS(x, y) → COND(min(x, y), x, y)
MINUS(x, y) → MIN(x, y)
COND(y, x, y) → MINUS(x, s(y))
MIN(s(u), s(v)) → MIN(u, v)

The TRS R consists of the following rules:

minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))

The set Q consists of the following terms:

minus(x0, x1)
cond(x0, x1, x0)
min(0, x0)
min(x0, 0)
min(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 1 less node.

(6) Complex Obligation (AND)

(7) Obligation:

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

MIN(s(u), s(v)) → MIN(u, v)

The TRS R consists of the following rules:

minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))

The set Q consists of the following terms:

minus(x0, x1)
cond(x0, x1, x0)
min(0, x0)
min(x0, 0)
min(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.

MIN(s(u), s(v)) → MIN(u, v)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MIN(x1, x2)  =  MIN(x1)
s(x1)  =  s(x1)

Lexicographic path order with status [LPO].
Precedence:

s₁ > MIN₁

Status:

MIN₁: [1]
s₁: [1]

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))

The set Q consists of the following terms:

minus(x0, x1)
cond(x0, x1, x0)
min(0, x0)
min(x0, 0)
min(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:

COND(y, x, y) → MINUS(x, s(y))
MINUS(x, y) → COND(min(x, y), x, y)

The TRS R consists of the following rules:

minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))

The set Q consists of the following terms:

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

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