YES Termination w.r.t. Q proof of AG01_3.54.ari

(0) Obligation:

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

f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(x))

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(f(x1)) = x1   
POL(f'(x1, x2, x3)) = 2·x1 + x2 + x3   
POL(g(x1)) = x1   
POL(h(x1)) = 2·x1   
POL(s(x1)) = 2 + 2·x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

f'(s(x), y, y) → f'(y, x, s(x))


(2) Obligation:

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

f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))

Q is empty.

(3) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(4) Obligation:

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

f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))

The set Q consists of the following terms:

f(g(x0))
f(h(x0))

(5) DependencyPairsProof (EQUIVALENT transformation)

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

(6) Obligation:

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

F(g(x)) → F(f(x))
F(g(x)) → F(x)

The TRS R consists of the following rules:

f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))

The set Q consists of the following terms:

f(g(x0))
f(h(x0))

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


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

Recursive path order with status [RPO].
Quasi-Precedence:
h > g1

Status:
g1: [1]
h: multiset


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))

(8) Obligation:

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

f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))

The set Q consists of the following terms:

f(g(x0))
f(h(x0))

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) YES