(0) Obligation:

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

f(0, 1, x) → f(s(x), x, x)
f(x, y, s(z)) → s(f(0, 1, z))

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, 1, x) → f(s(x), x, x)
f(x, y, s(z)) → s(f(0, 1, z))

The set Q consists of the following terms:

f(0, 1, x0)
f(x0, x1, s(x2))

(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(0, 1, x) → F(s(x), x, x)
F(x, y, s(z)) → F(0, 1, z)

The TRS R consists of the following rules:

f(0, 1, x) → f(s(x), x, x)
f(x, y, s(z)) → s(f(0, 1, z))

The set Q consists of the following terms:

f(0, 1, x0)
f(x0, x1, s(x2))

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


F(x, y, s(z)) → F(0, 1, z)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(x1, x2, x3)  =  F(x3)
0  =  0
1  =  1
s(x1)  =  s(x1)
f(x1, x2, x3)  =  f(x3)

Lexicographic Path Order [LPO].
Precedence:
F1 > 0 > s1 > 1
f1 > s1 > 1

The following usable rules [FROCOS05] were oriented:

f(0, 1, x) → f(s(x), x, x)
f(x, y, s(z)) → s(f(0, 1, z))

(6) Obligation:

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

F(0, 1, x) → F(s(x), x, x)

The TRS R consists of the following rules:

f(0, 1, x) → f(s(x), x, x)
f(x, y, s(z)) → s(f(0, 1, z))

The set Q consists of the following terms:

f(0, 1, x0)
f(x0, x1, s(x2))

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(8) TRUE