(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) 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(g(x)) → F(f(x))
F(g(x)) → F(x)
F'(s(x), y, y) → F'(y, x, s(x))
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.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F'(s(x), y, y) → F'(y, x, s(x))
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.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F'(s(x), y, y) → F'(y, x, s(x))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F'(
x0,
x1,
x2,
x3) =
F'(
x1,
x2)
Tags:
F' has argument tags [0,0,2,3] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
F'(
x1,
x2,
x3) =
x3
s(
x1) =
s(
x1)
Homeomorphic Embedding Order
The following usable rules [FROCOS05] were oriented:
none
(7) 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))
f'(s(x), y, y) → f'(y, x, s(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(9) TRUE
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(x)) → F(x)
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))
f'(s(x), y, y) → f'(y, x, s(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) 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)
F(g(x)) → F(f(x))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(
x0,
x1) =
F(
x1)
Tags:
F has argument tags [0,1] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
F(
x1) =
x1
g(
x1) =
g(
x1)
f(
x1) =
x1
h(
x1) =
h
Homeomorphic Embedding Order
The following usable rules [FROCOS05] were oriented:
f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
(12) 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))
f'(s(x), y, y) → f'(y, x, s(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(14) TRUE