Term Rewriting System R:
[X]
f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(X), c) -> f(X, c)
f(c, c) -> f(a, a)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(a, a) -> F(a, b)
F(a, b) -> F(s(a), c)
F(s(X), c) -> F(X, c)
F(c, c) -> F(a, a)

Furthermore, R contains one SCC.

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

Dependency Pairs:

F(c, c) -> F(a, a)
F(s(X), c) -> F(X, c)
F(a, b) -> F(s(a), c)
F(a, a) -> F(a, b)

Rules:

f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(X), c) -> f(X, c)
f(c, c) -> f(a, a)

Strategy:

innermost

As we are in the innermost case, we can delete all 4 non-usable-rules.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`           →DP Problem 2`
`             ↳Negative Polynomial Order`

Dependency Pairs:

F(c, c) -> F(a, a)
F(s(X), c) -> F(X, c)
F(a, b) -> F(s(a), c)
F(a, a) -> F(a, b)

Rule:

none

Strategy:

innermost

The following Dependency Pair can be strictly oriented using the given order.

F(c, c) -> F(a, a)

There are no usable rules (regarding the implicit AFS).
Used ordering:
Polynomial Order with Interpretation:

POL( F(x1, x2) ) = x1

POL( c ) = 1

POL( a ) = 0

POL( s(x1) ) = x1

This results in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`           →DP Problem 2`
`             ↳Neg POLO`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pairs:

F(s(X), c) -> F(X, c)
F(a, b) -> F(s(a), c)
F(a, a) -> F(a, b)

Rule:

none

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`           →DP Problem 2`
`             ↳Neg POLO`
`             ...`
`               →DP Problem 4`
`                 ↳Size-Change Principle`

Dependency Pair:

F(s(X), c) -> F(X, c)

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. F(s(X), c) -> F(X, c)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2=2

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

DP: empty set
Oriented Rules: none

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

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

We obtain no new DP problems.

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