(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 + 2·x2 + x3
POL(g(x1)) = x1
POL(h(x1)) = 1 + 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].
The following pairs can be oriented strictly and are deleted.
F(g(x)) → F(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
The following usable rules [FROCOS05] were oriented:
f(h(x)) → h(g(x))
f(g(x)) → g(f(f(x)))
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(x)) → F(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.
(9) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(g(x)) → F(f(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
The following usable rules [FROCOS05] were oriented:
f(h(x)) → h(g(x))
f(g(x)) → g(f(f(x)))
(10) 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.
(11) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(12) TRUE
(13) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(g(x)) → F(f(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(F(x1)) = x1
POL(f(x1)) = x1
POL(g(x1)) = 1 + x1
POL(h(x1)) = 0
The following usable rules [FROCOS05] were oriented:
f(h(x)) → h(g(x))
f(g(x)) → g(f(f(x)))
(14) 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.