(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
s(a) → a
s(s(x)) → x
s(f(x, y)) → f(s(y), s(x))
s(g(x, y)) → g(s(x), s(y))
f(x, a) → x
f(a, y) → y
f(g(x, y), g(u, v)) → g(f(x, u), f(y, v))
g(a, a) → a
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:
S(f(x, y)) → F(s(y), s(x))
S(f(x, y)) → S(y)
S(f(x, y)) → S(x)
S(g(x, y)) → G(s(x), s(y))
S(g(x, y)) → S(x)
S(g(x, y)) → S(y)
F(g(x, y), g(u, v)) → G(f(x, u), f(y, v))
F(g(x, y), g(u, v)) → F(x, u)
F(g(x, y), g(u, v)) → F(y, v)
The TRS R consists of the following rules:
s(a) → a
s(s(x)) → x
s(f(x, y)) → f(s(y), s(x))
s(g(x, y)) → g(s(x), s(y))
f(x, a) → x
f(a, y) → y
f(g(x, y), g(u, v)) → g(f(x, u), f(y, v))
g(a, a) → a
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 with 3 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(x, y), g(u, v)) → F(y, v)
F(g(x, y), g(u, v)) → F(x, u)
The TRS R consists of the following rules:
s(a) → a
s(s(x)) → x
s(f(x, y)) → f(s(y), s(x))
s(g(x, y)) → g(s(x), s(y))
f(x, a) → x
f(a, y) → y
f(g(x, y), g(u, v)) → g(f(x, u), f(y, v))
g(a, a) → a
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
g(x1, x2) = g(x1, x2)
From the DPs we obtained the following set of size-change graphs:
- F(g(x, y), g(u, v)) → F(y, v) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 > 2
- F(g(x, y), g(u, v)) → F(x, u) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 > 2
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(7) TRUE
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
S(f(x, y)) → S(x)
S(f(x, y)) → S(y)
S(g(x, y)) → S(x)
S(g(x, y)) → S(y)
The TRS R consists of the following rules:
s(a) → a
s(s(x)) → x
s(f(x, y)) → f(s(y), s(x))
s(g(x, y)) → g(s(x), s(y))
f(x, a) → x
f(a, y) → y
f(g(x, y), g(u, v)) → g(f(x, u), f(y, v))
g(a, a) → a
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
f(x1, x2) = f(x1, x2)
g(x1, x2) = g(x1, x2)
From the DPs we obtained the following set of size-change graphs:
- S(f(x, y)) → S(x) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
- S(f(x, y)) → S(y) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
- S(g(x, y)) → S(x) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
- S(g(x, y)) → S(y) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(10) TRUE