(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))
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,
x2)
H(
x0,
x1,
x2,
x3) =
H(
x0,
x1,
x2,
x3)
Tags:
F has argument tags [1,0,0] and root tag 0
H has argument tags [1,1,1,0] and root tag 0
Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:
POL(F(x1, x2)) = 0
POL(H(x1, x2, x3)) = 0
POL(a) = 1
POL(f(x1, x2)) = 0
POL(g(x1)) = 0
POL(h(x1, x2, x3)) = x3
The following usable rules [FROCOS05] were oriented:
none
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
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.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
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)
H(
x0,
x1,
x2,
x3) =
H(
x0,
x2)
Tags:
F has argument tags [4,0,0] and root tag 1
H has argument tags [4,7,5,2] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:
POL(F(x1, x2)) = x1
POL(H(x1, x2, x3)) = 1
POL(a) = 0
POL(f(x1, x2)) = 0
POL(g(x1)) = 1 + x1
POL(h(x1, x2, x3)) = x1 + x2 + x3
The following usable rules [FROCOS05] were oriented:
none
(6) 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.
(7) 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 [1,2,1,0] and root tag 0
Comparison: MS
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:
POL(H(x1, x2, x3)) = x2 + x3
POL(g(x1)) = 1 + x1
The following usable rules [FROCOS05] were oriented:
none
(8) 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.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) TRUE