Termination w.r.t. Q of the following Term Rewriting System could be proven:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(0, y) → y
f(x, 0) → x
f(i(x), y) → i(x)
f(f(x, y), z) → f(x, f(y, z))
f(g(x, y), z) → g(f(x, z), f(y, z))
f(1, g(x, y)) → x
f(2, g(x, y)) → y
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f(0, y) → y
f(x, 0) → x
f(i(x), y) → i(x)
f(f(x, y), z) → f(x, f(y, z))
f(g(x, y), z) → g(f(x, z), f(y, z))
f(1, g(x, y)) → x
f(2, g(x, y)) → y
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
F(f(x, y), z) → F(y, z)
F(g(x, y), z) → F(y, z)
F(g(x, y), z) → F(x, z)
F(f(x, y), z) → F(x, f(y, z))
The TRS R consists of the following rules:
f(0, y) → y
f(x, 0) → x
f(i(x), y) → i(x)
f(f(x, y), z) → f(x, f(y, z))
f(g(x, y), z) → g(f(x, z), f(y, z))
f(1, g(x, y)) → x
f(2, g(x, y)) → y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
F(f(x, y), z) → F(y, z)
F(g(x, y), z) → F(y, z)
F(g(x, y), z) → F(x, z)
F(f(x, y), z) → F(x, f(y, z))
The TRS R consists of the following rules:
f(0, y) → y
f(x, 0) → x
f(i(x), y) → i(x)
f(f(x, y), z) → f(x, f(y, z))
f(g(x, y), z) → g(f(x, z), f(y, z))
f(1, g(x, y)) → x
f(2, g(x, y)) → y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- F(f(x, y), z) → F(x, f(y, z))
The graph contains the following edges 1 > 1
- F(f(x, y), z) → F(y, z)
The graph contains the following edges 1 > 1, 2 >= 2
- F(g(x, y), z) → F(x, z)
The graph contains the following edges 1 > 1, 2 >= 2
- F(g(x, y), z) → F(y, z)
The graph contains the following edges 1 > 1, 2 >= 2