Term Rewriting System R:
[x, y]
f(x, c(y)) -> f(x, s(f(y, y)))
f(s(x), s(y)) -> f(x, s(c(s(y))))

Termination of R to be shown.

`   R`
`     ↳Overlay and local confluence Check`

The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(x, c(y)) -> F(x, s(f(y, y)))
F(x, c(y)) -> F(y, y)
F(s(x), s(y)) -> F(x, s(c(s(y))))

Furthermore, R contains two SCCs.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳Usable Rules (Innermost)`
`           →DP Problem 2`
`             ↳UsableRules`

Dependency Pair:

F(s(x), s(y)) -> F(x, s(c(s(y))))

Rules:

f(x, c(y)) -> f(x, s(f(y, y)))
f(s(x), s(y)) -> f(x, s(c(s(y))))

Strategy:

innermost

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

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`             ...`
`               →DP Problem 3`
`                 ↳Size-Change Principle`
`           →DP Problem 2`
`             ↳UsableRules`

Dependency Pair:

F(s(x), s(y)) -> F(x, s(c(s(y))))

Rule:

none

Strategy:

innermost

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

which lead(s) to this/these maximal multigraph(s):
{1} , {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:
c(x1) -> c(x1)
s(x1) -> s(x1)

We obtain no new DP problems.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳Usable Rules (Innermost)`

Dependency Pair:

F(x, c(y)) -> F(y, y)

Rules:

f(x, c(y)) -> f(x, s(f(y, y)))
f(s(x), s(y)) -> f(x, s(c(s(y))))

Strategy:

innermost

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

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`             ...`
`               →DP Problem 4`
`                 ↳Size-Change Principle`

Dependency Pair:

F(x, c(y)) -> F(y, y)

Rule:

none

Strategy:

innermost

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

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
2>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:
c(x1) -> c(x1)

We obtain no new DP problems.

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