Termination Proof using AProVE

Term Rewriting System R:
[x, y]
h(f(x), y) -> f(g(x, y))
g(x, y) -> h(x, y)

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

H(f(x), y) -> G(x, y)
G(x, y) -> H(x, y)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

Dependency Pairs:

G(x, y) -> H(x, y)
H(f(x), y) -> G(x, y)

Rules:

h(f(x), y) -> f(g(x, y))
g(x, y) -> h(x, y)

The following dependency pair can be strictly oriented:

H(f(x), y) -> G(x, y)

The following rules can be oriented:

h(f(x), y) -> f(g(x, y))
g(x, y) -> h(x, y)

Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

G(x₁, x₂) -> x₁
H(x₁, x₂) -> x₁
f(x₁) -> f(x₁)
h(x₁, x₂) -> x₁
g(x₁, x₂) -> x₁

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 2

             ↳Dependency Graph

Dependency Pair:

G(x, y) -> H(x, y)

Rules:

h(f(x), y) -> f(g(x, y))
g(x, y) -> h(x, y)

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes