(0) Obligation:

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

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

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) 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:

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

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

(3) 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)  =  x3
0  =  0
1  =  1
s(x1)  =  s(x1)
g(x1, x2)  =  g(x1, x2)
f(x1, x2, x3)  =  x3

Lexicographic path order with status [LPO].
Precedence:
0 > s1 > 1

Status:
0: []
1: []
s1: [1]
g2: [1,2]

The following usable rules [FROCOS05] were oriented:

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

(4) 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:

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

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) TRUE