Termination Proof using AProVE

Term Rewriting System R:
[x, y, z]
f(s(a), s(b), x) -> f(x, x, x)
g(f(s(x), s(y), z)) -> g(f(x, y, z))
cons(x, y) -> x
cons(x, y) -> y

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

F(s(a), s(b), x) -> F(x, x, x)
G(f(s(x), s(y), z)) -> G(f(x, y, z))
G(f(s(x), s(y), z)) -> F(x, y, z)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
F(s(a), s(b), x) -> F(x, x, x)

Rules:

f(s(a), s(b), x) -> f(x, x, x)
g(f(s(x), s(y), z)) -> g(f(x, y, z))
cons(x, y) -> x
cons(x, y) -> y
Dependency Pair:
G(f(s(x), s(y), z)) -> G(f(x, y, z))

Rules:

f(s(a), s(b), x) -> f(x, x, x)
g(f(s(x), s(y), z)) -> g(f(x, y, z))
cons(x, y) -> x
cons(x, y) -> y

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
F(s(a), s(b), x) -> F(x, x, x)

Rules:

f(s(a), s(b), x) -> f(x, x, x)
g(f(s(x), s(y), z)) -> g(f(x, y, z))
cons(x, y) -> x
cons(x, y) -> y
Dependency Pair:
G(f(s(x), s(y), z)) -> G(f(x, y, z))

Rules:

f(s(a), s(b), x) -> f(x, x, x)
g(f(s(x), s(y), z)) -> g(f(x, y, z))
cons(x, y) -> x
cons(x, y) -> y

Termination of R could not be shown.
Duration:
0:00 minutes