(0) Obligation:

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

f(0, y) → 0
f(s(x), y) → f(f(x, y), 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:

f(0, y) → 0
f(s(x), y) → f(f(x, y), y)

The set Q consists of the following terms:

f(0, x0)
f(s(x0), 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(s(x), y) → F(f(x, y), y)
F(s(x), y) → F(x, y)

The TRS R consists of the following rules:

f(0, y) → 0
f(s(x), y) → f(f(x, y), y)

The set Q consists of the following terms:

f(0, x0)
f(s(x0), 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 1 SCC with 1 less node.

(6) Obligation:

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

F(s(x), y) → F(x, y)

The TRS R consists of the following rules:

f(0, y) → 0
f(s(x), y) → f(f(x, y), y)

The set Q consists of the following terms:

f(0, x0)
f(s(x0), x1)

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(8) Obligation:

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

f(0, y) → 0
f(s(x), y) → f(f(x, y), y)

The set Q consists of the following terms:

f(0, x0)
f(s(x0), x1)

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

(9) PisEmptyProof (EQUIVALENT transformation)

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

(10) TRUE