(0) Obligation:

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

f(a, y) → f(y, g(y))
g(a) → b
g(b) → b

Q is empty.

(1) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is

g(a) → b
g(b) → b

The TRS R 2 is

f(a, y) → f(y, g(y))

The signature Sigma is {f}

(2) Obligation:

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

f(a, y) → f(y, g(y))
g(a) → b
g(b) → b

The set Q consists of the following terms:

f(a, x0)
g(a)
g(b)

(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(a, y) → F(y, g(y))
F(a, y) → G(y)

The TRS R consists of the following rules:

f(a, y) → f(y, g(y))
g(a) → b
g(b) → b

The set Q consists of the following terms:

f(a, x0)
g(a)
g(b)

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(a, y) → F(y, g(y))

The TRS R consists of the following rules:

f(a, y) → f(y, g(y))
g(a) → b
g(b) → b

The set Q consists of the following terms:

f(a, x0)
g(a)
g(b)

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

Recursive Path Order [RPO].
Precedence:
a > F2
a > g > b
f > g > b

The following usable rules [FROCOS05] were oriented:

f(a, y) → f(y, g(y))
g(a) → b
g(b) → b

(8) Obligation:

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

f(a, y) → f(y, g(y))
g(a) → b
g(b) → b

The set Q consists of the following terms:

f(a, x0)
g(a)
g(b)

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