(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, g(y)) → g(g(y))
f(g(x), a) → f(x, g(a))
f(g(x), g(y)) → h(g(y), x, g(y))
h(g(x), y, z) → f(y, h(x, y, z))
h(a, y, z) → 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(g(x), a) → F(x, g(a))
F(g(x), g(y)) → H(g(y), x, g(y))
H(g(x), y, z) → F(y, h(x, y, z))
H(g(x), y, z) → H(x, y, z)
The TRS R consists of the following rules:
f(a, g(y)) → g(g(y))
f(g(x), a) → f(x, g(a))
f(g(x), g(y)) → h(g(y), x, g(y))
h(g(x), y, z) → f(y, h(x, y, z))
h(a, y, z) → 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(g(x), a) → F(x, g(a))
F(g(x), g(y)) → H(g(y), x, g(y))
H(g(x), y, z) → F(y, h(x, y, z))
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) =
F(
x0,
x1)
H(
x0,
x1,
x2,
x3) =
H(
x0,
x2)
Tags:
F has argument tags [5,0,0] and root tag 0
H has argument tags [5,4,0,4] and root tag 1
Comparison: MS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
F(
x1,
x2) =
F
g(
x1) =
g(
x1)
a =
a
H(
x1,
x2,
x3) =
H
h(
x1,
x2,
x3) =
h
f(
x1,
x2) =
x2
Lexicographic path order with status [LPO].
Quasi-Precedence:
[F, H] > [g1, a]
h > [g1, a]
Status:
F: []
g1: [1]
a: []
H: []
h: []
The following usable rules [FROCOS05] were oriented:
none
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
H(g(x), y, z) → H(x, y, z)
The TRS R consists of the following rules:
f(a, g(y)) → g(g(y))
f(g(x), a) → f(x, g(a))
f(g(x), g(y)) → h(g(y), x, g(y))
h(g(x), y, z) → f(y, h(x, y, z))
h(a, y, z) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
H(g(x), y, z) → H(x, y, z)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
H(
x0,
x1,
x2,
x3) =
H(
x1,
x3)
Tags:
H has argument tags [1,1,3,3] and root tag 0
Comparison: MS
Underlying order for the size change arcs and the rules of R:
Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial
Status:
H3: [2,3,1]
g1: [1]
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(a, g(y)) → g(g(y))
f(g(x), a) → f(x, g(a))
f(g(x), g(y)) → h(g(y), x, g(y))
h(g(x), y, z) → f(y, h(x, y, z))
h(a, y, z) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE