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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Usable Rules (Innermost)`

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

As we are in the innermost case, we can delete all 2 non-usable-rules.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`           →DP Problem 2`
`             ↳Size-Change Principle`

Dependency Pairs:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. H(x, y) -> G(x, f(y))
2. G(f(x), y) -> H(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1=1
{2} , {2}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{1} , {2}
1>1
{2} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
f(x1) -> f(x1)

We obtain no new DP problems.

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